Neural Network Toolbox For Use with MATLAB ® Howard Demuth Mark Beale Computation Visualization Programming User’s Guide Version 3.0 How to Contact The MathWorks: ☎ 508-647-7000 Phone 508-647-7001 Fax The MathWorks, Inc. 24 Prime Park Way Natick, MA 01760-1500 Mail http://www.mathworks.com Web Anonymous FTP server Newsgroup PHONE FAX ✉ MAIL INTERNET ftp.mathworks.com comp.soft-sys.matlab @ [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] Technical support Product enhancement suggestions Bug reports Documentation error reports Subscribing user registration Order status, license renewals, passcodes Sales, pricing, and general information Neural Network Toolbox User’s Guide COPYRIGHT 1992 - 1997 by The MathWorks, Inc. All Rights Reserved. The software described in this document is furnished under a license agreement. The software may be used or copied only under the terms of the license agreement. No part of this manual may be photocopied or reproduced in any form without prior written consent from The MathWorks, Inc. U.S. GOVERNMENT: If Licensee is acquiring the Programs on behalf of any unit or agency of the U.S. Government, the following shall apply: (a) For units of the Department of Defense: the Government shall have only the rights specified in the license under which the commercial computer software or commercial software documentation was obtained, as set forth in subparagraph (a) of the Rights in Commercial Computer Software or Commercial Software Documentation Clause at DFARS 227.7202-3, therefore the rights set forth herein shall apply; and (b) For any other unit or agency: NOTICE: Notwithstanding any other lease or license agreement that may pertain to, or accompany the delivery of, the computer software and accompanying documentation, the rights of the Government regarding its use, reproduction, and disclosure are as set forth in Clause 52.227-19 (c)(2) of the FAR. 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Printing History: June 1992 April 1993 January 1997 July 1997 First printing Second printing Third printing Fourth Printing Contents Introduction 1 Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-2 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-4 Basic Chapters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-4 Help and Installation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-4 Whats New in 3.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-5 Other New Algorithms and Functions . . . . . . . . . . . . . . . . . . . . . 1-5 Modular Network Representation . . . . . . . . . . . . . . . . . . . . . . . . 1-5 Simulink Simulation Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-5 General Toolbox Improvements . . . . . . . . . . . . . . . . . . . . . . . . . . 1-5 Reduced Memory Levenberg-Marquardt Algorithm . . . . . . . 1-6 Other New Networks, Algorithms and Improvements . . . . . 1-7 Resilient Backpropagation (Rprop) . . . . . . . . . . . . . . . . . . . . . . . 1-7 Conjugate Gradient Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . 1-7 Quasi-Newton Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-7 BFGS Quasi Newton Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 1-7 A One Step Secant Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 1-7 Speed Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-8 Improving Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-9 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-9 Early Stopping With Validation . . . . . . . . . . . . . . . . . . . . . . . . 1-9 Pre and Post Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-9 Scale Minimum and Maximum . . . . . . . . . . . . . . . . . . . . . . . 1-10 Scale Mean and Standard Deviation . . . . . . . . . . . . . . . . . . . 1-10 Principal Component Analysis . . . . . . . . . . . . . . . . . . . . . . . . 1-10 Post-training Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-10 New Training Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-10 Probabilistic Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . . . 1-11 Generalized Regression Networks . . . . . . . . . . . . . . . . . . . . . . . 1-11 i Modular Network Representation . . . . . . . . . . . . . . . . . . . . . 1-12 Better Simulink Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-13 General Toolbox Improvements . . . . . . . . . . . . . . . . . . . . . . . 1-14 Simpler and More Extensible Toolbox . . . . . . . . . . . . . . . . . . . 1-14 Custom Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-15 Neural Network Applications . . . . . . . . . . . . . . . . . . . . . . . . . Aerospace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Automotive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Banking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Defense . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Entertainment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Financial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Insurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Manufacturing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Medical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Oil and Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Robotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Speech . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Securities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Telecommunications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transportation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-16 1-16 1-16 1-16 1-16 1-16 1-17 1-17 1-17 1-17 1-17 1-17 1-17 1-18 1-18 1-18 1-18 1-18 Neural Network Design Book . . . . . . . . . . . . . . . . . . . . . . . . . 1-19 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-20 Neuron Model and Network Architectures 2 ii Contents Basic Chapters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mathematical Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mathematical and Code Equivalents . . . . . . . . . . . . . . . . . . . 2-2 2-2 2-2 2-2 Neuron Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simple Neuron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Neuron With Vector Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-4 2-4 2-5 2-7 Network Architectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Layer of Neurons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inputs and Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiple Layers of Neurons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-10 2-10 2-11 2-13 Data Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation With Concurrent Inputs in a Static Network . . . . Simulation With Sequential Inputs in a Dynamic Network . . Simulation With Concurrent Inputs in a Dynamic Network . 2-15 2-15 2-16 2-18 Training Styles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Incremental Training (of Adaptive and Other Networks) . . . . Incremental Training with Static Networks . . . . . . . . . . . . Incremental Training With Dynamic Networks . . . . . . . . . Batch Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Batch Training With Static Networks . . . . . . . . . . . . . . . . . Batch Training With Dynamic Networks . . . . . . . . . . . . . . . 2-20 2-20 2-20 2-22 2-22 2-22 2-24 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figures and Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simple Neuron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hard Limit Transfer Function . . . . . . . . . . . . . . . . . . . . . . . Purelin Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . . . Log Sigmoid Transfer Function . . . . . . . . . . . . . . . . . . . . . . Neuron With Vector Input . . . . . . . . . . . . . . . . . . . . . . . . . . Net Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Single Neuron Using Abbreviated Notation . . . . . . . . . . . . Icons for Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . . Layer of Neurons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Three Layers of Neurons . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weight Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Layer of Neurons, Abbreviated Notation . . . . . . . . . . . . . . . Layer of Neurons Showing Indices . . . . . . . . . . . . . . . . . . . . Three Layers, Abbreviated Notation . . . . . . . . . . . . . . . . . . Linear Neuron With Two Element Vector Input. . . . . . . . . Dynamic Network With One Delay. . . . . . . . . . . . . . . . . . . . 2-26 2-28 2-28 2-28 2-28 2-29 2-29 2-29 2-30 2-30 2-30 2-31 2-31 2-32 2-32 2-33 2-34 2-34 iii Perceptrons 3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-2 Important Perceptron Functions . . . . . . . . . . . . . . . . . . . . . . . . . 3-3 Neuron Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-4 Perceptron Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-6 Creating a Perceptron (NEWP) . . . . . . . . . . . . . . . . . . . . . . . . . 3-7 Simulation (SIM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-8 Initialization (INIT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-9 Learning Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-11 Perceptron Learning Rule (LEARNP) . . . . . . . . . . . . . . . . . . 3-12 Adaptive Training (ADAPT) . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-15 Limitations and Cautions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-21 Outliers and the Normalized Perceptron Rule . . . . . . . . . . . . . 3-21 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figures and Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Perceptron Neuron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Perceptron Transfer Function, hardlim . . . . . . . . . . . . . . Decision Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Perceptron Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Perceptron Learning Rule . . . . . . . . . . . . . . . . . . . . . . . One Perceptron Neuron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . New Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-23 3-23 3-23 3-24 3-24 3-25 3-25 3-26 3-26 Adaptive Linear Filters 4 iv Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-2 Important Linear Network Functions . . . . . . . . . . . . . . . . . . . . 4-3 Neuron Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-4 Network Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-5 Single ADALINE (NEWLIN) . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-6 Mean Square Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-9 Linear System Design (NEWLIND) . . . . . . . . . . . . . . . . . . . . . 4-10 LMS Algorithm (LEARNWH) . . . . . . . . . . . . . . . . . . . . . . . . . . 4-11 Linear Classification (TRAIN) . . . . . . . . . . . . . . . . . . . . . . . . . 4-13 Adaptive Filtering (ADAPT) . . . . . . . . . . . . . . . . . . . . . . . . . . . Tapped Delay Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adaptive Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adaptive Filter Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Prediction Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Noise Cancellation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiple Neuron Adaptive Filters . . . . . . . . . . . . . . . . . . . . . . . 4-16 4-16 4-17 4-18 4-20 4-21 4-23 Limitations and Cautions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Overdetermined Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Underdetermined Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linearly Dependent Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . Too Large a Learning Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-25 4-25 4-25 4-25 4-26 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figures and Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linear Neuron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Purelin Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . . . MADALINE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ADALINE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Decision Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mean Square Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LMS (Widrow-Hoff) Algorithm . . . . . . . . . . . . . . . . . . . . . . . Tapped Delay Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adaptive Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adaptive Filter Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-27 4-28 4-28 4-28 4-29 4-29 4-30 4-30 4-30 4-31 4-32 4-32 v Prediction Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Noise Cancellation Example . . . . . . . . . . . . . . . . . . . . . . . . . Multiple Neuron Adaptive Filter . . . . . . . . . . . . . . . . . . . . . Abbreviated Form of Adaptive Filter . . . . . . . . . . . . . . . . . . Specific Small Adaptive Filter . . . . . . . . . . . . . . . . . . . . . . . New Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-33 4-34 4-35 4-35 4-36 4-36 Backpropagation 5 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-2 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Neuron Model (TANSIG, LOGSIG, PURELIN) . . . . . . . . . . . Feedforward Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation (SIM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Backpropagation Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . Faster Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variable Learning Rate (TRAINGDA, TRAINGDX) . . . . . . . . Resilient Backpropagation (TRAINRP) . . . . . . . . . . . . . . . . . . Conjugate Gradient Algorithms . . . . . . . . . . . . . . . . . . . . . . . . Fletcher-Reeves Update (TRAINCGF) . . . . . . . . . . . . . . . . . Polak-Ribiére Update (TRAINCGP) . . . . . . . . . . . . . . . . . . Powell-Beale Restarts (TRAINCGB) . . . . . . . . . . . . . . . . . . Scaled Conjugate Gradient (TRAINSCG) . . . . . . . . . . . . . . Line Search Routines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Golden Section Search (SRCHGOL) . . . . . . . . . . . . . . . . . . . Brent’s Search (SRCHBRE) . . . . . . . . . . . . . . . . . . . . . . . . . Hybrid Bisection-Cubic Search (SRCHHYB) . . . . . . . . . . . Charalambous’ Search (SRCHCHA) . . . . . . . . . . . . . . . . . . . Backtracking (SRCHBAC) . . . . . . . . . . . . . . . . . . . . . . . . . . Quasi-Newton Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BFGS Algorithm (TRAINBFG) . . . . . . . . . . . . . . . . . . . . . . . One Step Secant Algorithm (TRAINOSS) . . . . . . . . . . . . . . Levenberg-Marquardt (TRAINLM) . . . . . . . . . . . . . . . . . . . . . Reduced Memory Levenberg-Marquardt (TRAINLM) . . . . . . . vi Contents 5-3 5-3 5-3 5-5 5-8 5-8 5-9 5-16 5-16 5-18 5-20 5-20 5-22 5-24 5-25 5-26 5-26 5-27 5-27 5-28 5-28 5-29 5-29 5-30 5-31 5-33 Speed and Memory Comparison . . . . . . . . . . . . . . . . . . . . . . . 5-35 Improving Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modified Performance Function . . . . . . . . . . . . . . . . . . . . . . Automated Regularization (TRAINBR) . . . . . . . . . . . . . . . . Early Stopping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-37 5-38 5-38 5-39 5-41 Preprocessing and Postprocessing . . . . . . . . . . . . . . . . . . . . . Min and Max (PREMNMX, POSTMNMX, TRAMNMX) . . . . . Mean and Stand. Dev. (PRESTD, POSTSTD, TRASTD) . . . . . Principal Component Analysis (PREPCA, TRAPCA) . . . . . . . Post-training Analysis (POSTREG) . . . . . . . . . . . . . . . . . . . . . 5-44 5-44 5-45 5-46 5-47 Sample Training Session . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-49 Limitations and Cautions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-54 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-56 Radial Basis Networks 6 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-2 Important Radial Basis Functions . . . . . . . . . . . . . . . . . . . . . . . 6-2 Radial Basis Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Neuron Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Network Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exact Design (NEWRBE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . More Efficient Design (NEWRB) . . . . . . . . . . . . . . . . . . . . . . . . . Demonstrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-3 6-3 6-4 6-5 6-7 6-8 Generalized Regression Networks . . . . . . . . . . . . . . . . . . . . . . 6-9 Network Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-9 Design (NEWGRNN) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-11 vii Probabilistic Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . 6-12 Network Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-12 Design (NEWPNN) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-13 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Radial Basis Neuron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Radbas Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . . . . Radial Basis Network Architecture . . . . . . . . . . . . . . . . . . . Generalized Regression Neural Network Architecture . . . . Probabilistic Neural Network Architecture . . . . . . . . . . . . . New Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-15 6-16 6-16 6-16 6-17 6-17 6-18 6-19 Self-Organizing Networks 7 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-2 Important Self-Organizing Functions . . . . . . . . . . . . . . . . . . . . . 7-2 Competitive Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Creating a Competitive Neural Network (NEWC) . . . . . . . . . . Kohonen Learning Rule (LEARNK) . . . . . . . . . . . . . . . . . . . . . . Bias Learning Rule (LEARNCON) . . . . . . . . . . . . . . . . . . . . . . . Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graphical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Self-Organizing Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Topologies (GRIDTOP, HEXTOP, RANDTOP) . . . . . . . . . . . . Distance Functions (DIST, LINKDIST, MANDIST, BOXDIST) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Creating a Self Organizing MAP Neural Network (NEWSOM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Training (LEARNSOM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Phase 1: Ordering Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . Phase 2: Tuning Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Contents 7-3 7-3 7-4 7-5 7-5 7-6 7-8 7-10 7-12 7-16 7-19 7-20 7-22 7-22 7-22 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-25 One-Dimensional Self-Organizing Map . . . . . . . . . . . . . . . . 7-25 Two-Dimensional Self-Organizing Map . . . . . . . . . . . . . . . . 7-27 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Competitive Network Architecture . . . . . . . . . . . . . . . . . . . . Self Organizing Feature Map Architecture . . . . . . . . . . . . . New Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-32 7-32 7-32 7-33 7-33 Learning Vector Quantization 8 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-2 Important LVQ Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-2 Network Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-3 Creating an LVQ Network (NEWLVQ) . . . . . . . . . . . . . . . . . . . 8-5 LVQ Learning Rule(LEARNLV2) . . . . . . . . . . . . . . . . . . . . . . . 8-9 Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LVQ Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . New Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-14 8-14 8-14 8-14 ix 9 Recurrent Networks Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-2 Important Recurrent Network Functions . . . . . . . . . . . . . . . . . . 9-2 Elman Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Creating an Elman Network (NEWELM) . . . . . . . . . . . . . . . . . Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Training an Elman Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-3 9-3 9-4 9-4 9-6 Hopfield Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-9 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-9 Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-9 Design(NEWHOP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-11 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-13 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elman Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hopfield Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . New Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-16 9-17 9-17 9-17 9-18 Applications 10 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-2 Application Scripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-2 Applin1: Linear Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Network Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Network Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thoughts and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x Contents 10-3 10-3 10-4 10-4 10-6 Applin2: Adaptive Prediction . . . . . . . . . . . . . . . . . . . . . . . . . 10-7 Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-7 Network Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-8 Network Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-8 Network Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-8 Thoughts and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-10 Applin3: Linear System Identification . . . . . . . . . . . . . . . . Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Network Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Network Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thoughts and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-11 10-11 10-12 10-12 10-14 Applin4: Adaptive System Identification . . . . . . . . . . . . . . Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Network Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Network Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Network Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thoughts and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-15 10-15 10-16 10-17 10-17 10-17 Appelm1: Amplitude Detection . . . . . . . . . . . . . . . . . . . . . . . Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Network Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Network Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Network Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Network Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Improving Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-19 10-19 10-20 10-20 10-21 10-22 10-23 Appcs1: Nonlinear System Identification . . . . . . . . . . . . . . Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Network Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Network Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Network Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thoughts and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-24 10-25 10-26 10-26 10-26 10-29 xi Appcs2: Model Reference Control . . . . . . . . . . . . . . . . . . . . . Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Neural Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Network Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Network Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Network Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thoughts and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-30 10-30 10-31 10-32 10-32 10-33 10-36 10-37 Appcr1: Character Recognition . . . . . . . . . . . . . . . . . . . . . . . Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Neural Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Training Without Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . Training With Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Training Without Noise Again . . . . . . . . . . . . . . . . . . . . . . System Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-38 10-38 10-39 10-40 10-40 10-40 10-41 10-41 10-42 10-42 10-43 Advanced Topics 11 xii Contents Custom Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-2 Custom Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-4 Network Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-5 Architecture Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-5 Number of Outputs and Targets . . . . . . . . . . . . . . . . . . . . . . 11-7 Subobject Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-8 Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-8 Network Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-12 Weight and Bias Values . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-13 Network Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-14 Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-14 Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-14 Additional Toolbox Functions . . . . . . . . . . . . . . . . . . . . . . . . Initialization Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . randnc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . randnr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . satlin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . softmax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . tribas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Learning Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . learnh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . learnhd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . learnis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . learnos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-17 11-17 11-17 11-17 11-17 11-17 11-17 11-17 11-18 11-18 11-18 11-18 11-18 Custom Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Net Input Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weight Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Initialization Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Network Initialization Functions . . . . . . . . . . . . . . . . . . . . Layer Initialization Functions . . . . . . . . . . . . . . . . . . . . . . Weight and Bias Initialization Functions . . . . . . . . . . . . . Learning Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Training Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adapt Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Performance Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weight and Bias Learning Functions . . . . . . . . . . . . . . . . . Self-Organizing Map Functions . . . . . . . . . . . . . . . . . . . . . . . Topology Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Distance Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-19 11-20 11-20 11-22 11-24 11-26 11-26 11-26 11-27 11-29 11-29 11-32 11-34 11-36 11-39 11-39 11-40 xiii Network Object Reference 12 xiv Contents Network Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-2 Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-2 numInputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-2 numLayers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-2 biasConnect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-3 inputConnect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-4 layerConnect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-4 outputConnect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-4 targetConnect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-5 numOutputs (read-only) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-5 numTargets (read-only) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-5 numInputDelays (read-only) . . . . . . . . . . . . . . . . . . . . . . . . . 12-5 numLayerDelays (read-only) . . . . . . . . . . . . . . . . . . . . . . . . . 12-6 Subobject Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-6 inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-7 layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-7 outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-7 targets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-8 biases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-8 inputWeights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-9 layerWeights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-9 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-10 adaptFcn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-10 initFcn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-10 performFcn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-11 trainFcn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-11 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-13 adaptParam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-13 initParam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-13 performParam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-13 trainParam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-14 Weight and Bias Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-14 IW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-14 LW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-15 b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-16 Other . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-16 userdata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-16 Subobject Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . userdata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . distanceFcn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . distances (read-only) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . initFcn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . netInputFcn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . positions (read-only) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . topologyFcn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . transferFcn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . userdata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . size (read-only) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . userdata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Targets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . size (read-only) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . userdata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Biases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . initFcn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . learn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . learnFcn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . learnParam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . size (read-only) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . userdata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Input Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . initFcn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . learn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . learnFcn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . learnParam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . size (read-only) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . userdata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . weightFcn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-17 12-17 12-17 12-17 12-18 12-18 12-18 12-19 12-19 12-20 12-20 12-21 12-22 12-22 12-23 12-24 12-25 12-25 12-25 12-25 12-25 12-25 12-26 12-26 12-26 12-27 12-27 12-28 12-28 12-28 12-28 12-29 12-29 12-30 12-31 12-31 12-31 12-32 xv LayerWeights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . initFcn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . learn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . learnFcn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . learnParam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . size (read-only) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . userdata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . weightFcn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-32 12-32 12-32 12-33 12-33 12-35 12-35 12-35 12-36 Reference 13 Functions Listed by Network Type . . . . . . . . . . . . . . . . . . . . 13-2 Functions by Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13-3 Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13-13 Transfer Function Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . 13-14 Reference Page Headings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13-18 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13-19 Glossary xvi Contents Notation A Mathematical Notation for Equations and Figures . . . . . . . B-2 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-2 Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-2 Weight Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-2 Scalar Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-2 Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-2 Column Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-2 Row Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-2 Bias Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-2 Scalar Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-2 Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-2 Layer Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-2 Input Weight Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-3 Layer Weight Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-3 Figure and Equation Examples . . . . . . . . . . . . . . . . . . . . . . . . . . B-3 Mathematics and Code Equivalents . . . . . . . . . . . . . . . . . . . . . B-4 Bibliography B Demonstrations and Applications C Tables of Demonstrations and Application . . . . . . . . . . . . . . D-2 Chapter 2 Neuron Model & Network Architectures . . . . . . . . . D-2 Chapter 3 Perceptrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-2 Chapter 4 Adaptive Linear Filters . . . . . . . . . . . . . . . . . . . . . . . D-3 Chapter 5 Backpropagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-3 Chapter 6 Radial Basis Networks . . . . . . . . . . . . . . . . . . . . . . . . D-4 Chapter 7 Self-Organizing Networks . . . . . . . . . . . . . . . . . . . . . D-4 Chapter 8 Learning Vector Quantization . . . . . . . . . . . . . . . . . . D-4 Chapter 9 Recurrent Networks . . . . . . . . . . . . . . . . . . . . . . . . . . D-5 Chapter 10 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-5 xvii Simulink D Block Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-2 Transfer Function Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-3 Net Input Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-3 Weight Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-4 Block Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-7 Changing Input Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-7 Discrete Sample Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-7 Index xviii Contents 1 Introduction Neural Networks . . . . . . . . . . . . . . . . . . 1-2 Getting Started . . . . . . . . . . . . . . . . . . 1-4 Basic Chapters . . . . . . . . . . . . . . . . . . . 1-4 Help and Installation . . . . . . . . . . . . . . . . . 1-4 Whats New in 3.0 . . . . . . . . Other New Algorithms and Functions Modular Network Representation . . Simulink Simulation Support . . . General Toolbox Improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-5 1-5 1-5 1-5 1-5 Reduced Memory Levenberg-Marquardt Algorithm . . 1-6 Other New Networks, Algorithms and Improvements Resilient Backpropagation (Rprop) . . . . . . . . . . Conjugate Gradient Algorithms . . . . . . . . . . . . Quasi-Newton Algorithms . . . . . . . . . . . . . . Speed Comparison . . . . . . . . . . . . . . . . . Improving Generalization . . . . . . . . . . . . . . Pre and Post Processing . . . . . . . . . . . . . . . New Training Options . . . . . . . . . . . . . . . Probabilistic Neural Networks . . . . . . . . . . . . Generalized Regression Networks . . . . . . . . . . . . . . . . . . . . . 1-7 1-7 1-7 1-7 1-8 1-9 1-9 1-10 1-11 1-11 Modular Network Representation . . . . . . . . . . 1-12 Better Simulink Support . . . . . . . . . . . . . . 1-13 General Toolbox Improvements . . . . . . . . . . . 1-14 Simpler and More Extensible Toolbox . . . . . . . . . . 1-14 Custom Functions . . . . . . . . . . . . . . . . . . 1-15 Neural Network Applications . . . . . . . . . . . . 1-16 Neural Network Design Book . . . . . . . . . . . . 1-19 Acknowledgments . . . . . . . . . . . . . . . . . 1-20 1 Introduction Neural Networks Neural networks are composed of simple elements operating in parallel. These elements are inspired by biological nervous systems. As in nature, the network function is determined largely by the connections between elements. We can train a neural network to perform a particular function by adjusting the values of the connections (weights) between elements. Commonly neural networks are adjusted, or trained, so that a particular input leads to a specific target output. Such a situation is shown below. There, the network is adjusted, based on a comparison of the output and the target, until the network output matches the target. Typically many such input/target pairs are used, in this supervised learning, to train a network. Target Input Neural Network including connections (called weights) between neurons Compare Output Adjust weights Neural networks have been trained to perform complex functions in various fields of application including pattern recognition, identification, classification, speech, vision and control systems. A list of applications is given later in this chapter. Today neural networks can be trained to solve problems that are difficult for conventional computers or human beings. Throughout the toolbox emphasis is placed on neural network paradigms that build up to or are themselves used in engineering, financial and other practical applications. The supervised training methods are commonly used, but other networks can be obtained from unsupervised training techniques or from direct design methods. Unsupervised networks can be used, for instance, to identify groups of data. Certain kinds of linear networks and Hopfield networks are designed 1-2 Neural Networks directly. In summary, there are a variety of kinds of design and learning techniques that enrich the choices that a user can make. The field of neural networks has a history of some five decades but has found solid application only in the past fifteen years, and the field is still developing rapidly. Thus, it is distinctly different from the fields of control systems or optimization where the terminology, basic mathematics, and design procedures have been firmly established and applied for many years. We do not view the Neural Network Toolbox as simply a summary of established procedures that are known to work well. Rather, we hope that it will be a useful tool for industry, education and research, a tool that will help users find what works and what doesn’t, and a tool that will help develop and extend the field of neural networks. Because the field and the material are so new, this toolbox will explain the procedures, tell how to apply them, and illustrate their successes and failures with examples. We believe that an understanding of the paradigms and their application is essential to the satisfactory and successful use of this toolbox, and that without such understanding user complaints and inquiries would bury us. So please be patient if we include a lot of explanatory material. We hope that such material will be helpful to you. This chapter includes a few comments on getting started with the Neural Network Toolbox. It also describes the new graphical user interface, and new algorithms and architectures; and it explains the increased flexibility of the Toolbox due to its use of modular network object representation. Finally this chapter gives a list of some practical neural network applications and describes a new text, Neural Network Design. This book presents the theory of neural networks as well as their design and application, and makes considerable use of MATLAB® and the Neural Network Toolbox. 1-3 1 Introduction Getting Started Basic Chapters Chapter 2 contains basic material about network architectures and notation specific to this toolbox. Chapter 3 includes the first reference to basic functions such as init and adapt. Chapter 4 describes the use of the functions designd and train, and discusses delays. Chapter 2, 3 and 4 should be read before going to later chapters Help and Installation The Neural Network Toolbox is contained in a directory called nnet. Type help nnet for a listing of help topics. A number of demonstrations are included in the Toolbox. Each example states a problem, shows the network used to solve the problem and presents the final results. Lists of the neural network demonstration and application scripts that are discussed in this guide can be found by typing help nndemos Instructions for installing the Neural Network Toolbox are found in one of two MATLAB documents, the Installation Guide for MS-Windows and Macintosh or the Installation Guide for UNIX. 1-4 Whats New in 3.0 Whats New in 3.0 A few of the new features and improvements introduced with this version of the Neural Network Toolbox are listed below. Reduced Memory Levenberg-Marquardt Algorithm The Neural Network Toolbox version 2.0 introduced the Levenberg-Marquardt (LM) algorithm which is faster than other algorithms by a factor of from 10 to 100. Now version 3.0 introduces the Reduced Memory Levenberg-Marquardt algorithm, which allows for a time/memory trade off. This means that the LM algorithm can now be used in much larger problems, with perhaps only a slight increase in running time. Other New Algorithms and Functions Conjugate gradient and R-Prop algorithms have been added, as have Probabilistic, and Generalized Regression Networks. Automatic regularization, new training options and a method for early stopping of training have also been included here for the first time. New training options, including training on variations of mean square error for better generalization, training against a validation set, and training until the gradient of the error reaches a minimum are now available. Finally, various pre and post processing function have been included. Modular Network Representation The modular representation in the Toolbox version 3.0 allows a great deal of flexibility for the design of one’s own custom networks. Virtually any combination of neurons and layers, with delays if required, can be trained in any one of Other New Networks, Algorithms and Improvements. Simulink® Simulation Support You can now generate network simulation blocks for use with Simulink. General Toolbox Improvements This toolbox is simpler but more powerful than ever. It has fewer functions but each of them, including INIT (initialization), SIM (simulation), TRAIN, (training) and ADAPT (adaptive learning) can be applied to a broad variety of networks. 1-5 1 Introduction Reduced Memory Levenberg-Marquardt Algorithm A low-memory-use Levenberg-Marquardt algorithm has been developed by Professor Martin Hagan of Oklahoma State University for the Neural Network Toolbox. This algorithm achieves nearly the same speed of the original very fast Levenberg-Marquardt algorithm, but uses less memory required by the original. (See Neural Network Toolbox Version 2.0 and Hagan, M.T., and M. Menhaj, “Training Feedforward Networks with the Marquardt Algorithm,” IEEE Transactions on Neural Networks, vol. 5, no. 6, 1994.) There is a drawback to using memory reduction. A significant computational overhead is associated with computing the Jacobian in submatrices. If you have enough memory available, then it is better to set mem_reduc to 1 and to compute the full Jacobian. If you have a large training set, and you are running out of memory, then you should set mem_reduc to 2, and try again. If you still run out of memory, continue to increase mem_reduc. Even if you use memory reduction, the Levenberg-Marquardt algorithm will always compute the approximate Hessian matrix, which has dimensions n × n . If your network is very large, then you may run out of memory. If this is the case, then you will want to try trainoss, trainrp, or one of the conjugate gradient algorithms. 1-6 Other New Networks, Algorithms and Improvements Other New Networks, Algorithms and Improvements Resilient Backpropagation (Rprop) The resilient backpropagation (Rprop) training algorithm eliminates the harmful effect of having a small slope at the extreme ends of sigmoid squashing transfer functions. Only the sign of the derivative of the transfer function is used to determine the direction of the weight update; the magnitude of the derivative has no effect on the weight update. Rprop is generally much faster than the standard steepest descent algorithm. It also has the nice property that it requires only a modest increase in memory requirements. Conjugate Gradient Algorithms Various forms of a conjugate gradient backprop algorithm have been added. In the conjugate gradient algorithms a search is performed along conjugate directions, which produces generally faster convergence than steepest descent directions. This is a well know, highly efficient algorithm that gives good results on a broad spectrum of problems. Quasi-Newton Algorithms Quasi-Newton (or secant) methods are based on Newton’s method but don’t require calculation of second derivatives. They update an approximate Hessian matrix at each iteration of the algorithm. The update is computed as a function of the gradient. Two quasi-newton algorithms are included in the Neural Network Toolbox. BFGS Quasi Newton Algorithm This algorithm requires more computation in each iteration and more storage than the conjugate gradient methods, although it generally converges in fewer iterations. For very large networks it may be better to use Rprop or one of the conjugate gradient algorithms. For smaller networks, however, trainbfg can be an efficient training function. A One Step Secant Algorithm This algorithm requires less storage and computation per epoch than the BFGS algorithm. It requires slightly more storage and computation per epoch than the conjugate gradient algorithms. It can be considered a compromise between full quasi-Newton algorithms and conjugate gradient algorithms. 1-7 1 Introduction Speed Comparison The following table gives some example convergence times for the various algorithms on one particular regression problem. In this problem a 1-10-1 network was trained on a data set with 41 input/output pairs until a mean square error performance of 0.01 was obtained. Twenty different test runs were made for each training algorithm on a Macintosh Powerbook 1400 to obtain the average numbers shown in the table. These numbers should be used with caution, since the performances shown here may not be typical for these algorithms on other types of problems. (You may notice that there is not a clear relationship between the number of floating point operations and the time required to reach convergence. This is because some of the algorithms can take advantage of efficient built-in MATLAB functions. This is especially true for the Levenberg-Marquardt algorithm.) Function Technique Time Epochs Mflops traingdx Variable Learning Rate 57.71 980 2.50 trainrp Rprop 12.95 185 0.56 trainscg Scaled Conj. Grad. 16.06 106 0.70 traincgf Fletcher-Powell CG 16.40 81 0.99 traincgp Polak-Ribiére CG 19.16 89 0.75 traincgb Powell-Beale CG 15.03 74 0.59 trainoss One-Step-Secant 18.46 101 0.75 trainbfg BFGS quasi-Newton 10.86 44 1.02 trainlm Levenberg-Marquardt 1.87 6 0.46 For most situations, we recommend that you try the Levenberg-Marquardt algorithm first. If this algorithm requires too much memory, then try the BFGS algorithm trainbfg, or one of the conjugate gradient methods. The Rprop algorithm trainrp is also very fast, and has relatively small memory requirements. Radial basis networks can be designed very quickly, typically in less time than it takes the Levenberg-Marquardt algorithm to be trained. You might also 1-8 Other New Networks, Algorithms and Improvements consider them. However, they have the disadvantage that, once designed, the computation associated with their use may be greater than that for conventional feedforward networks. Improving Generalization One of the problems that occurs during neural network training is called overfitting. The error on the training set is driven to a very small value, but when new data is presented to the network the error is large. The network has memorized the training examples, but it has not learned to generalize to new situations. Two solutions to the overfitting problem are presented here. Regularization Regularization involves modifying the performance function, which is normally chosen to be the sum of squares of the network errors on the training set. We have included two routines which will automatically set the optimal performance function to achieve the best generalization. Regularization helps take the mystery out of how to pick the number of neurons in a network and consistently leads to good networks that are not overtrained. Early Stopping With Validation Early stopping is a technique based on dividing the data into three subsets. The first subset is the training set used for computing the gradient and updating the network weights and biases. The second subset is the validation set. The error on the validation set is monitored during the training process. The validation error will normally decrease during the initial phase of training, as does the training set error. However, when the network begins to overfit the data, the error on the validation set will typically begin to rise. When the validation error increases for a specified number of iterations, the training is stopped, and the weights and biases at the minimum of the validation error are returned. Pre and Post Processing Neural network training can be made more efficient if certain preprocessing steps are performed on the network inputs and targets. Thus, we have included the following functions. 1-9 1 Introduction Scale Minimum and Maximum The function premnmx can be used to scale inputs and targets so that they fall in the range [-1,1]. Scale Mean and Standard Deviation The function prestd normalizes the mean and standard deviation of the training set. Principal Component Analysis The principle components analysis program prepca can be used to reduce the dimensions of the input vectors. Post-training Analysis We have included a post training function postreg that performs a regression analysis between the network response and the corresponding targets. New Training Options In this toolbox we can not only minimize mean squared error as before, but we can also: • Minimize with variations of mean squared error for better generalization. Such training simplifies the problem of picking the number of hidden neurons and produces good networks that are not overtrained. • Train with validation to achieve appropriately early stopping. Here the training result is checked against a validation set of input output data to make sure that overtraining has not occurred. • Stop training when the error gradient reaches a minimum. This avoids wasting computation time when further training is having little effect. • The low memory use Levenberg Marquardt algorithm has been incorporated into both new and old algorithms. 1-10 Other New Networks, Algorithms and Improvements Probabilistic Neural Networks Probabilistic neural networks can be used for classification problems. Their design is straightforward and does not depend on training. These networks generalize well. Generalized Regression Networks A generalized regression neural network (GRNN) is often used for function approximation. Given a sufficient number of hidden neurons, GRNNs can approximate a continuous function to an arbitrary accuracy. 1-11 1 Introduction Modular Network Representation The modular representation in the Toolbox version 3.0 allows a great deal of flexibility, including the following options: • Networks can have any number of sets of inputs, layers. • Any input or layer can be connected to any layer with a weight. • Each layer can have a bias or not. • Each layer can be a network output or not. • Weights can have tapped delays. • Weights can be partially connected • Each layer can have a target or not 1-12 Better Simulink Support Better Simulink Support The Neural Network Toolbox Version 3.0 can now generate network simulation blocks for use with Simulink. The Neural Network Toolbox version 2.0 provided transfer function blocks but didn't help import entire networks. We can do that now. 1-13 1 Introduction General Toolbox Improvements Simpler and More Extensible Toolbox The new Toolbox has fewer functions, but each of them does more than the old ones. For instance, the following functions can be applied to a broad variety of networks. • init – initialization • sim – simulation • train – training • adapt – adaptive learning Now the Neural Network Toolbox Version 3.0 is more extensible in the following ways: • Network properties can be altered. • Custom properties can be added to a network object. 1-14 General Toolbox Improvements Custom Functions The toolbox allows you to create and use many kinds of functions, giving you a great deal of control over the algorithms used to initialize, simulate, and train, your networks. The following sections indicate the kinds of functions you can create: • Simulation functions - transfer - net input - weight • Initialization functions - network initialization - layer initialization - weight and bias initialization • Learning functions - network training - network adapt - network performance - weight and bias learning • Self-organizing map functions - topology - distance 1-15 1 Introduction Neural Network Applications The 1988 DARPA Neural Network Study [DARP88] lists various neural network application, s beginning in about 1984 with the adaptive channel equalizer. This device, which is an outstanding commercial success, is a singleneuron network used in long distance telephone systems to stabilize voice signals. The DARPA report goes on to list other commercial applications, including a small word recognizer, a process monitor, a sonar classifier, and a risk analysis system. Neural networks have been applied in many other fields since the DARPA report was written. A list of some applications mentioned in the literature follows: Aerospace • High performance aircraft autopilot, flight path simulation, aircraft control systems, autopilot enhancements, aircraft component simulation, aircraft component fault detection Automotive • Automobile automatic guidance system, warranty activity analysis Banking • Check and other document reading, credit application evaluation Defense • Weapon steering, target tracking, object discrimination, facial recognition, new kinds of sensors, sonar, radar and image signal processing including data compression, feature extraction and noise suppression, signal/image identification Electronics • Code sequence prediction, integrated circuit chip layout, process control, chip failure analysis, machine vision, voice synthesis, nonlinear modeling 1-16 Neural Network Applications Entertainment • Animation, special effects, market forecasting Financial • Real estate appraisal, loan advisor, mortgage screening, corporate bond rating, credit line use analysis, portfolio trading program, corporate financial analysis, currency price prediction Insurance • Policy application evaluation, product optimization Manufacturing • Manufacturing process control, product design and analysis, process and machine diagnosis, real-time particle identification, visual quality inspection systems, beer testing, welding quality analysis, paper quality prediction, computer chip quality analysis, analysis of grinding operations, chemical product design analysis, machine maintenance analysis, project bidding, planning and management, dynamic modeling of chemical process system Medical • Breast cancer cell analysis, EEG and ECG analysis, prosthesis design, optimization of transplant times, hospital expense reduction, hospital quality improvement, emergency room test advisement Oil and Gas • Exploration Robotics • Trajectory control, forklift robot, manipulator controllers, vision systems 1-17 1 Introduction Speech • Speech recognition, speech compression, vowel classification, text to speech synthesis Securities • Market analysis, automatic bond rating, stock trading advisory systems Telecommunications • Image and data compression, automated information services, real-time translation of spoken language, customer payment processing systems Transportation • Truck brake diagnosis systems, vehicle scheduling, routing systems Summary The list of additional neural network applications, the money that has been invested in neural network software and hardware, and the depth and breadth of interest in these devices have been growing rapidly. It is hoped that this toolbox will be useful for neural network educational and design purposes within a broad field of neural network applications. A variety of neural network applications are described in Chapter 10. 1-18 Neural Network Design Book Neural Network Design Book Professor Martin Hagan of Oklahoma State University, and Neural Network Toolbox authors Howard Demuth and Mark Beale have written a textbook, Neural Network Design, published by PWS Publishing Company in 1996 (ISBN 0-534-94332-2). The book presents the theory of neural networks as well as their design and application, and makes considerable use of MATLAB and the Neural Network Toolbox. Demonstration programs from the book are used in various chapters of this Guide. The book has a instructor’s manual containing problem solutions (ISBN 0-534-95049-3), and overheads for class use. (The overheads, in hard copy form, come one to a page for instructor use and three to a page for student use.) For information about obtaining this text, please contact International Thomson Publishing Customer Service, phone 1-800-347-7707. 1-19 1 Introduction Acknowledgments The authors would like to thank: Martin Hagan, Oklahoma State University for providing the original Levenberg-Marquardt algorithm in the Neural Network Toolbox version 2.0 and various algorithms found here in version 3.0, including the new reduced memory use version of the Levenberg-Marquardt algorithm, the coujugate gradient algorithm, RPROP, and generalized regression method. Martin also wrote Chapter 5 of this Toolbox, Backpropagation, which contains descriptions of new algorithms and suggestions for pre and post processing of data. Joe Hicklin, of The MathWorks for getting Howard into neural network research years ago at the University of Idaho, for encouraging Howard to write the toolbox, for providing crucial help in getting the first toolbox version 1.0 out the door, and for continuing to be a good friend. Liz Callanan of The MathWorks for getting us off the such a good start with the Neural Network Toolbox version 1.0. Jim Tung of The MathWorks for his consistent long term support for this project. Jeff Faneuff and Roy Lurie of The MathWorks for their vigilant reviews of the developing material in this version of the toolbox. Matthew Simoneau of The MathWorks for his help with demos, test suite routines, and getting user feedback. Kathy Ford of The MathWorks for her careful help with the documentation. Jane Price of The MathWorks for her help in getting constructive comments from users. Margaret Jenks of Moscow Idaho, for her patient, persistent and effective editing. Teri for running the show while Mark worked on this toolbox and finally, PWS Publishing, for their permission to include various problems, demonstrations and other material from Neural Network Design, Jan. 1996. 1-20 2 Neuron Model and Network Architectures Basic Chapters . . . . . . . . . . . . . . . . . . . 2-2 Notation . . . . . . . . . . . . . . . . . . . . . . 2-2 Neuron Model . . . . Simple Neuron . . . . Transfer Functions . . . Neuron With Vector Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-4 2-4 2-5 2-7 Network Architectures . . . . . . . . . . . . . . . 2-10 A Layer of Neurons . . . . . . . . . . . . . . . . . . 2-10 Multiple Layers of Neurons . . . . . . . . . . . . . . 2-13 Data Structures . . . . . . . . . . . . . . . . . Simulation With Concurrent Inputs in a Static Network . Simulation With Sequential Inputs in a Dynamic Network Simulation With Concurrent Inputs in a Dynamic Network . . . . 2-15 2-15 2-16 2-18 Training Styles . . . . . . . . . . . . . . . . . . . 2-20 Incremental Training (of Adaptive and other Networks) . . . 2-20 Batch Training . . . . . . . . . . . . . . . . . . . 2-22 Summary . . . . . . . . . . . . . . . . . . . . . 2-26 Figures and Equations . . . . . . . . . . . . . . . . 2-28 2 Neuron Model and Network Architectures Basic Chapters The Neural Network Toolbox is written so that if you read Chapter 2, 3 and 4 you can proceed to a later chapter, read it and use its functions without difficulty. To make this possible, Chapter 2 presents the fundamentals of the neuron model, the architectures of neural networks. It also will discuss notation used in the architectures. All of this is basic material. It is to your advantage to understand this Chapter 2 material thoroughly. The neuron model and the architecture of a neural network describe how a network transforms its input into an output. This transformation can be viewed as a computation. The model and the architecture each place limitations on what a particular neural network can compute. The way a network computes its output must be understood before training methods for the network can be explained. Notation Mathematical Notation The basic notation used here for equations and figures is given below. • Scalars-small italic letters.....a,b,c • Vectors - small bold non-italic letters.....a,b,c • Matrices - capital BOLD non-italic letters.....A,B,C • Vector means a column of numbers. Mathematical and Code Equivalents The transition from mathematical to code notation or vice versa can be made with the aid of a few rules. They are listed here for future reference. 2-2 Basic Chapters To change from Mathematical notation to MATLAB notation the user needs to: • Change superscripts to cell array indices 1 For example, p → p { 1 } • Change subscripts to parentheses indices 1 For example, p 2 → p ( 2 ) , and p 2 → p { 1 } ( 2 ) • Change parentheses indices to a second cell array index 1 For example, p ( k – 1 ) → p { 1, k – 1 } • Change mathematics operators to MATLAB operators and toolbox functions For example, ab → a*b See Appendix B for additional information on notation in this Toolbox. 2-3 2 Neuron Model and Network Architectures Neuron Model Simple Neuron A neuron with a single scalar input and no bias is shown on the left below. Input p - Title - bias Neuron without w AA AA n a-=Exp f (wp- ) f a - Title Neuron with- bias Input p w AA AA AAAA n f a b 1 - b) a =- fExp (wp + The scalar input p is transmitted through a connection that multiplies its strength by the scalar weight w, to form the product wp, again a scalar. Here the weighted input wp is the only argument of the transfer function f, which produces the scalar output a. The neuron on the right has a scalar bias, b. You may view the bias as simply being added to the product wp as shown by the summing junction or as shifting the function f to the left by an amount b. The bias is much like a weight, except that it has a constant input of 1. The transfer function net input n, again a scalar, is the sum of the weighted input wp and the bias b. This sum is the argument of the transfer function f. (Chapter 6 discusses a different way to form the net input n.) Here f is a transfer function, typically a step function or a sigmoid function, that takes the argument n and produces the output a. Examples of various transfer functions are given in the next section. Note that w and b are both adjustable scalar parameters of the neuron. The central idea of neural networks is that such parameters can be adjusted so that the network exhibits some desired or interesting behavior. Thus, we can train the network to do a particular job by adjusting the weight or bias parameters, or perhaps the network itself will adjust these parameters to achieve some desired end. All of the neurons in this toolbox have provision for a bias, and a bias is used in many of our examples and will be assumed in most of this toolbox. However, you may omit a bias in a neuron if you wish. 2-4 Neuron Model As noted above, the bias b is an adjustable (scalar) parameter of the neuron. It is not an input. However, the constant 1 that drives the bias is an input and must be treated as such when considering the linear dependence of input vectors in Chapter 4. Transfer Functions Many transfer functions have been included in this toolbox. A complete list of them can be found in “Transfer Function Graphs” in Chapter 13. Three of the most commonly used functions are shown below. a +1 0 -1 n AA a = hardlim(n) Hard Limit Transfer Function The hard limit transfer function shown above limits the output of the neuron to either 0, if the net input argument n is less than 0, or 1, if n is greater than or equal to 0. We will use this function in Chapter 3 “Perceptrons” to create neurons that make classification decisions. The Toolbox has a function, hardlim, to realize the mathematical hard limit transfer function shown above. Your might try the code shown below. n = -5:0.1:5; plot(n,hardlim(n),'c+:'); It produces a plot of the function hardlim over the range -5 to +5. All of the mathematical transfer functions in the toolbox can be realized with a function having the same name. 2-5 2 Neuron Model and Network Architectures The linear transfer function is shown below. a AA AA +1 n 0 -1 a = purelin(n) Linear Transfer Function Neurons of this type are used as linear approximators in “Adaptive Linear Filters” in Chapter 4. The sigmoid transfer function shown below takes the input, which may have any value between plus and minus infinity, and squashes the output into the range 0 to 1. a +1 n 0 -1 AA AA a = logsig(n) Log-Sigmoid Transfer Function This transfer function is commonly used in backpropagation networks, in part because it is differentiable. The symbol in the square to the right of each transfer function graph shown above represents the associated transfer function. These icons will replace the general f in the boxes of network diagrams to show the particular transfer function that is being used. For a complete listing of transfer functions and their icons, see the “Transfer Function Graphs” in Chapter 13. You can also specify your own transfer functions. You are not limited to the transfer functions listed in Chapter 13. You can experiment with a simple neuron and various transfer functions by running the demonstration program nnd2n1. 2-6 Neuron Model Neuron With Vector Input A neuron with a single R-element input vector is shown below. Here the individual element inputs p 1 , p 2 ,... p R are multiplied by weights w 1, 1 , w 1, 2 ,...w 1, R and the weighted values are fed to the summing junction. Their sum is simply Wp, the dot product of the (single row) matrix W and the vector p. Input Neuron w Vector Input AA A A p1 p2 p3 w1,1 pR w1, R n f Where... a R = # Elements in input vector b 1 a = f(Wp +b) The neuron has a bias b, which is summed with the weighted inputs to form the net input n. This sum, n, is the argument of the transfer function f. n = w 1, 1 p 1 + w 1, 2 p 2 + ... + w 1, R p R + b This expression can, of course, be written in MATLAB code as: n = W*p + b However, the user will seldom be writing code at this low level, for such code is already built into functions to define and simulate entire networks. The figure of a single neuron shown above contains a lot of detail. When we consider networks with many neurons and perhaps layers of many neurons, there is so much detail that the main thoughts tend to be lost. Thus, the authors have devised an abbreviated notation for an individual neuron. This 2-7 2 Neuron Model and Network Architectures notation, which will be used later in circuits of multiple neurons, is illustrated in the diagram shown below. Input AA AA AA AA p Rx1 W 1xR 1 R b 1x1 Neuron AA AA AA AA a n 1x1 1x1 f Where... R = # of elements in input vector 1 a = f(Wp +b) Here the input vector p is represented by the solid dark vertical bar at the left. The dimensions of p are shown below the symbol p in the figure as Rx1. (Note that we will use a capital letter, such as R in the previous sentence, when referring to the size of a vector.) Thus, p is a vector of R input elements. These inputs post multiply the single row, R column matrix W. As before, a constant 1 enters the neuron as an input and is multiplied by a scalar bias b. The net input to the transfer function f is n, the sum of the bias b and the product Wp. This sum is passed to the transfer function f to get the neuron’s output a, which in this case is a scalar. Note that if we had more than one neuron, the network output would be a vector. A layer of a network is defined in the figure shown above. A layer includes the combination of the weights, the multiplication and summing operation (here realized as a vector product Wp), the bias b, and the transfer function f. The array of inputs, vector p, will not be included in or called a layer. Each time this abbreviated network notation is used, the size of the matrices will be shown just below their matrix variable names. We hope that this notation will allow you to understand the architectures and follow the matrix mathematics associated with them. 2-8 Neuron Model As discussed previously, when a specific transfer function is to be used in a figure, the symbol for that transfer function will replace the f shown above. Here are some examples. A AA AA AA AA AA AA AA hardlim purelin logsig You can experiment with a 2 element neuron by running the demonstration program nnd2n2. 2-9 2 Neuron Model and Network Architectures Network Architectures Two or more of the neurons shown above may be combined in a layer, and a particular network might contain one or more such layers. First consider a single layer of neurons. A Layer of Neurons A one layer network with R input elements and S neurons is shown below. Input AA AA AA AA AA AA AA AA AA AA AAAA Layer of Neurons n1 w1, 1 p1 p2 p3 pR f a1 b1 1 n2 f a2 b2 1 nS wS, R f aS Where... R = # of elements in input vector S = # Neurons in Layer bS 1 a= f (Wp + b) In this network, each element of the input vector p is connected to each neuron input through the weight matrix W. The ith neuron has a summer that gathers its weighted inputs and bias to form its own scalar output n(i). The various n(i) taken together form an S-element net input vector n. Finally, the neuron layer outputs form a column vector a. We show the expression for a at the bottom of the figure. Note that it is common for the number of inputs to a layer to be different from the number of neurons (i.e. R ≠ S). A layer is not constrained to have the number of its inputs equal to the number of its neurons. You can create a single (composite) layer of neurons having different transfer functions simply by putting two of the networks shown above in parallel. Both 2-10 Network Architectures networks would have the same inputs, and each network would create some of the outputs. The input vector elements enter the network through the weight matrix W. w 1, 1 w 1, 2 … w 1, R W = w 2, 1 w 2, 2 … w 2, R w S, 1 w S, 2 … w S, R Note that the row indices on the elements of matrix W indicate the destination neuron of the weight and the column indices indicate which source is the input for that weight. Thus, the indices in w 1, 2 say that the strength of the signal from the second input element to the first (and only) neuron is w 1, 2 . The S neuron R input one layer network also can be drawn in abbreviated notation. Layer of Neurons Input p AA AA AA AA AA AA Rx1 a W n SxR Sx1 1 R Sx1 R = # of elements in input vector f b Sx1 Where... S = # Neurons in Layer S a= f (Wp + b) Here p is an R length input vector, W is an SxR matrix, and a and b are S length vectors. As defined previously, the neuron layer includes the weight matrix, the multiplication operations, the bias vector b, the summer, and the transfer function boxes. Inputs and Layers We are about to discuss networks having multiple layers so we will need to extend our notation to talk about such networks. Specifically, we need to make a distinction between weight matrices that are connected to inputs and weight 2-11 2 Neuron Model and Network Architectures matrices that are connected between layers. We also need to identify the source and destination for the weight matrices. We will call weight matrices connected to inputs, input weights, and we will call weight matrices coming from layer outputs layer weights. Further, we will use superscripts to identify the source (second index) and the destination (first index) for the various weights and other elements of the network. To illustrate, we have re-drawn the one layer multiple input network shown above in abbreviated form below. Input p R x1 AA AA AA AA IW1,1 S1xR 1 R AA AA AA AA Layer 1 b1 S1x1 n1 S1 x1 Where... a1 S1x 1 f1 S1 R = # of elements in input S1 = # Neurons in layer 1 a1 = f1(IW1,1p +b1) As you can see, we have labeled the weight matrix connected to the input vector p as an Input Weight matrix (IW1,1) having a source 1 (second index) and a destination 1 (first index). Also, elements of layer one, such as its bias, net input and output have a superscript 1 to say that they are associated with the first layer. In the next section we will use Layer Weight (LW) matrices as well as Input Weight (IW) matrices. You might recall from the notation section at the beginning of this chapter that conversion of the layer weight matrix from math to code for a particular network called net is: IW 1, 1 → net.IW { 1, 1 } Thus, we could write the code to obtain the net input to the transfer function as: n{1} = net.IW{1,1}*p + net.b{1}; 2-12 Network Architectures Multiple Layers of Neurons A network can have several layers. Each layer has a weight matrix W, a bias vector b, and an output vector a. To distinguish between the weight matrices, output vectors, etc., for each of these layers in our figures, we will append the number of the layer as a superscript to the variable of interest. You can see the use of this layer notation in the three layer network shown below, and in the equations below the figure. iw1,1 1, 1 p AA AA A 1 n1 1 b1 1 1 p 2 p n1 2 b1 3 2 p 1 1 R AA AA A First Layer Input iw1,1 S, R n 1 1 b1 1 1 S S a1 f1 a1 = f1 (IW1,1p +b1) n2 lw2,1 1 f1 f1 AA AA AA AA AA AA AAAA AAAA Second Layer 1 1,1 a2 1 f2 lw3,2 2 n2 2 f2 a2 2 a S lw2,1 2 n a 2 2 S 1 S ,S b2 2 S 1 a2 = f2 (LW2,1a1 +b2) f3 a3 2 b3 2 1 2 2 f2 n3 2 2 1 1 f3 1 1 b2 1 1 a3 1 b3 1 a1 n3 1,1 b2 1 AA AA AA AA AA AA AAAA AAAA Third Layer S n3 lw3,2 S 3 a3 3 f3 3 S 2 b3 S ,S 3 S 1 a3 =f3 (LW3,2 a2 + b3) a3 =f3 (LW3,2 f2 (LW2,1f1 (IW1,1p +b1)+ b2)+ b3) The network shown above has R1 inputs, S1 neurons in the first layer, S2 neurons in the second layer, etc. It is common for different layers to have different numbers of neurons. A constant input 1 is fed to the biases for each neuron. Note that the outputs of each intermediate layer are the inputs to the following layer. Thus layer 2 can be analyzed as a one layer network with S1 inputs, S2 neurons, and an S1xS2 weight matrix W2. The input to layer 2 is a1, the output is a2. Now that we have identified all the vectors and matrices of layer 2 we can treat it as a single layer network on its own. This approach can be taken with any layer of the network. 2-13 2 Neuron Model and Network Architectures The layers of a multilayer network play different roles. A layer that produces the network output is called an output layer. All other layers are called hidden layers. The three layer network shown above has one output layer (layer 3) and two hidden layers (layer 1 and layer 2). Some authors refer to the inputs as a fourth layer. We will not use that designation. The same three layer network discussed previously also can be drawn using our abbreviated notation. Input AA AA AA p Rx1 IW1,1 S1xR 1 R b 1 S1x1 AA AA AA AA AAAA First Layer Second Layer a1 n1 S1x1 S1x1 S2xS1 f1 1 S1 a1 = f1 (IW1,1p +b1) LW2,1 b 2 S2x1 A AAA AA A AA AAA AAAA AA Third Layer a3 = y a2 n2 S2x1 S2x1 LW3,2 S 3x S 2 f2 n3 S3 x1 b 1 S2 3 S3x1 a2 = f2 (LW2,1 a1 +b2) a3 =f3 (LW3,2 f2 (LW2,1f1 (IW1,1p +b1)+ b2)+ b3 S3 x1 f3 S3 a3 =f3 (LW3,2a2 +b3) = y Multiple layer networks are quite powerful. For instance, a network of two layers, where the first layer is sigmoid and the second layer is linear, can be trained to approximate any function (with a finite number of discontinuities) arbitrarily well. This kind of two-layer network is used extensively in Chapter 5, “Backpropagation.” Note that we have labeled the output of the a3 layer as y. We will use this notation to specify the output of such networks. 2-14 Data Structures Data Structures This section will discuss how the format of input data structures effects the simulation of networks. We will begin with static networks and then move to dynamic networks. We will be concerned about two basic types of input vectors: those that occur concurrently (at the same time, or in no particular time sequence) and those that occur sequentially in time. For sequential vectors, the order in which the vectors appear is important. For concurrent vectors, the order is not important, and if we had a number of networks running in parallel we could present one input vector to each of the networks. Simulation With Concurrent Inputs in a Static Network The simplest situation for simulating a network occurs when the network to be simulated is static (has no feedback or delays). In this case we do not have to be concerned about whether or not the input vectors occur in a particular time sequence, so we can treat the inputs as concurrent. In addition, to make the problem even simpler, we will begin by assuming that the network has only one input vector. We will use the following network as an example. Inputs Linear Neuron AA A A p1 w1,1 p2 w1,2 n a b 1 a = purelin (Wp + b) To set up this feedforward network we can use the following command. net = newlin([-1 1;-1 1],1); For simplicity we will assign the weight matrix and bias to be W = 1 2 ,b = 0 . 2-15 2 Neuron Model and Network Architectures The commands for these assignments are net.IW{1,1} = [1 2]; net.b{1} = 0; Suppose that the network simulation data set consists of Q = 4 concurrent vectors: p1 = 1 , p 2 = 2 , p 3 = 2 , p 4 = 3 2 1 3 1 Concurrent vectors are presented to the network as a single matrix: P = [1 2 2 3; 2 1 3 1]; We can now simulate the network: A = sim(net,P) A = 5 4 8 5 A single matrix of concurrent vectors is presented to the network and the network produces a single matrix of concurrent vectors as output. The result would be the same if there were four networks operating in parallel and each network received one of the input vectors and produced one of the outputs. The ordering of the input vectors is not important as they do not interact with each other. Simulation With Sequential Inputs in a Dynamic Network When a network contains delays, the input to the network would normally be a sequence of input vectors which occur in a certain time order. To illustrate this case we will use a simple network which contains one delay. 2-16 Data Structures Inputs p(t) Linear Neuron AA AA AA AAAAAA w1,1 n(t) D a(t) w1,2 a(t) = w1,1 p(t) + w1,2 p(t - 1) The following commands will create this network: net = newlin([-1 1],1,[0 1]); net.biasConnect = 0; Assign the weight matrix to be W = 1 2 . The command is net.IW{1,1} = [1 2]; Suppose that the input sequence is p(1) = 1 , p(2) = 2 , p(3) = 3 , p(4) = 4 Sequential inputs are presented to the network as elements of a cell array: P = {1 2 3 4}; We can now simulate the network: A = sim(net,P) A = [1] [4] [7] [10] We input a cell array containing a sequence of inputs, and the network produced a cell array containing a sequence of outputs. Note that the order of the inputs is important when they are presented as a sequence. In this case the current output is obtained by multiplying the current input by 1 and the 2-17 2 Neuron Model and Network Architectures preceding input by 2 and summing the result. If we were to change the order of the inputs it would change the numbers we would obtain in the output. Simulation With Concurrent Inputs in a Dynamic Network If we were to apply the same inputs from the previous example as a set of concurrent inputs instead of a sequence of inputs we would obtain a completely different response. (Although it is not clear why we would want to do this with a dynamic network.) It would be as if each input were applied concurrently to a separate parallel network. For the previous example, if we use a concurrent set of inputs we have p1 = 1 , p 2 = 2 , p 3 = 3 , p 4 = 4 , which can be created with the following code: P = [1 2 3 4]; When we simulate with concurrent inputs we obtain A = sim(net,P) A = 1 2 3 4 The result is the same as if we had concurrently applied each one of the inputs to a separate network and computed one output. Note that since we did not assign any initial conditions to the network delays they were assumed to be zero. For this case the output will simply be 1 times the input, since the weight which multiplies the current input is 1. In certain special cases we might want to simulate the network response to several different sequences at the same time. In this case we would want to present the network with a concurrent set of sequences. For example, let’s say we wanted to present the following two sequences to the network: p1 ( 1 ) = 1 , p1 ( 2 ) = 2 , p1 ( 3 ) = 3 , p1 ( 4 ) = 4 , p2 ( 1 ) = 4 , p2 ( 2 ) = 3 , p2 ( 3 ) = 2 , p2 ( 4 ) = 1 . The input P should be a cell array, where each element of the array contains the two elements of the two sequences which occur at the same time: P = {[1 4] [2 3] [3 2] [4 1]}; 2-18 Data Structures We can now simulate the network: A = sim(net,P); The resulting network output would be A = {[ 1 4] [4 11] [7 8] [10 5]} As you can see, the first column of each matrix makes up the output sequence produced by the first input sequence, which was the one we used in an earlier example. The second column of each matrix makes up the output sequence produced by the second input sequence. There is no interaction between the two concurrent sequences. It is as if they were each applied to separate networks running in parallel. The following diagram shows the general format for the input P to the sim function when we have Q concurrent sequences of TS time steps. It covers all cases where there is a single input vector. Each element of the cell array is a matrix of concurrent vectors which correspond to the same point in time for each sequence. If there are multiple input vectors there will be multiple rows of matrices in the cell array. Qth Sequence { [ p 1 ( 1 ), p 2 ( 1 ), …, p Q ( 1 ) ], [ p 1 ( 2 ), p 2 ( 2˙), …, p Q ( 2 ) ], …, [ p 1 ( TS ), p 2 ( TS ), …, p Q ( TS ) ] } First Sequence In this section we have applied sequential and concurrent inputs to dynamic networks. In the previous section we applied concurrent inputs to static networks. It is also possible to apply sequential inputs to static networks. It will not change the simulated response of the network, but it can affect the way in which the network is trained. This will become clear in the next section. 2-19 2 Neuron Model and Network Architectures Training Styles In this section we will describe two different styles of training. In incremental training the weights and biases of the network are updated each time an input is presented to the network. In batch training the weights and biases are only updated after all of the inputs have been presented. Incremental Training (of Adaptive and Other Networks) Incremental training can be applied to both static and dynamic networks, although it is more commonly used with dynamic networks, such as adaptive filters. In this section we will demonstrate how incremental training can be performed on both static and dynamic networks. Incremental Training with Static Networks Consider again the static network we used for our first example. We want to train it incrementally, so that the weights and biases will be updated after each input is presented. In this case we use the function adapt, and we present the inputs and targets as sequences. Suppose we want to train the network to create the linear function t = 2 p1 + p2 . Then for the previous inputs we used, p1 = 1 , p 2 = 2 , p 3 = 2 , p 4 = 3 , 2 1 3 1 the targets would be t1 = 4 , t 2 = 5 , t 3 = 7 , t 4 = 7 . We will first set up the network with zero initial weights and biases. We will also set the learning rate to zero initially, in order to show the effect of the incremental training. net = newlin([-1 1;-1 1],1,0,0); net.IW{1,1} = [0 0]; net.b{1} = 0; 2-20 Training Styles For incremental training we want to present the inputs and targets as sequences: P = {[1;2] [2;1] [2;3] [3;1]}; T = {4 5 7 7}; Recall from the earlier discussion that for a static network the simulation of the network will produce the same outputs whether the inputs are presented as a matrix of concurrent vectors or as a cell array of sequential vectors. This is not true when training the network, however. When using the adapt function, if the inputs are presented as a cell array of sequential vectors, then the weights will be updated as each input is presented (incremental mode). As we will see in the next section, if the inputs are presented as a matrix of concurrent vectors, then the weights will be updated only after all inputs have been presented (batch mode). We are now ready to train the network incrementally. [net,a,e,pf] = adapt(net,P,T); The network outputs will remain zero, since the learning rate is zero, and the weights are not updated. The errors will be equal to the targets: a = [0] e = [4] [0] [5] [0] [7] [0] [7] If we now set the learning rate to 0.1 we can see how the network is adjusted as each input is presented: net.inputWeights{1,1}.learnParam.lr=0.1; net.biases{1,1}.learnParam.lr=0.1; [net,a,e,pf] = adapt(net,P,T); a = [0] [2] [6.0] [5.8] e = [4] [3] [1.0] [1.2] The first output is the same as it was with zero learning rate, since no update is made until the first input is presented. The second output is different, since the weights have been updated. The weights continue to be modified as each error is computed. If the network is capable and the learning rate is set correctly, the error will eventually be driven to zero. 2-21 2 Neuron Model and Network Architectures Incremental Training With Dynamic Networks We can also train dynamic networks incrementally. In fact, this would be the most common situation. Let’s take the linear network with one delay at the input that we used in a previous example. We will initialize the weights to zero and set the learning rate to 0.1. net = newlin([-1 1],1,[0 1],0.1); net.IW{1,1} = [0 0]; net.biasConnect = 0; To train this network incrementally we will present the inputs and targets as elements of cell arrays. Pi = {1}; P = {2 3 4}; T = {3 5 7}; Here we are attempting to train the network to sum the current and previous inputs to create the current output. This is the same input sequence we used in the previous example of using sim, except that we are assigning the first term in the sequence as the initial condition for the delay. We are now ready to sequentially train the network using adapt. [net,a,e,pf] = adapt(net,P,T,Pi); a = [0] [2.4] [ 7.98] e = [3] [2.6] [-1.98] The first output is zero, since the weights have not yet been updated. The weights change at each subsequent time step. Batch Training Batch training, in which weights and biases are only updated after all of the inputs and targets have been presented, can be applied to both static and dynamic networks. We will discuss both types of networks in this section. Batch Training With Static Networks Batch training can be done using either adapt or train, although train is generally the best option, since it typically has access to more efficient training algorithms. Incremental training can only be done with adapt; train can only perform batch training. 2-22 Training Styles Let’s begin with the static network we used in previous examples. The learning rate will be set to 0.1. net = newlin([-1 1;-1 1],1,0,0.1); net.IW{1,1} = [0 0]; net.b{1} = 0; For batch training of a static network with adapt, the input vectors must be placed in one matrix of concurrent vectors. P = [1 2 2 3; 2 1 3 1]; T = [4 5 7 7]; When we call adapt it will invoke adaptwb, which is the default adaptation function for the linear network, and learnwh is the default learning function for the weights and biases. Therefore, Widrow-Hoff learning will be used. [net,a,e,pf] = adapt(net,P,T); a = 0 0 0 0 e = 4 5 7 7 Note that the outputs of the network are all zero, because the weights are not updated until all of the training set has been presented. If we display the weights we find: »net.IW{1,1} ans = 4.9000 »net.b{1} ans = 2.3000 4.1000 This is different that the result we had after one pass of adapt with incremental updating. Now let’s perform the same batch training using train. Since the Widrow-Hoff rule can be used in incremental or batch mode, it can be invoked by adapt or train. There are several algorithms which can only be used in batch mode (e.g., Levenberg-Marquardt), and so these algorithms can only be invoked by train. The network will be set up in the same way. net = newlin([-1 1;-1 1],1,0,0.1); net.IW{1,1} = [0 0]; net.b{1} = 0; 2-23 2 Neuron Model and Network Architectures For this case the input vectors can either be placed in a matrix of concurrent vectors or in a cell array of sequential vectors. Within train any cell array of sequential vectors would be converted to a matrix of concurrent vectors. This is because the network is static, and because train always operates in the batch mode. Concurrent mode operation is generally used whenever possible, because it has a more efficient MATLAB implementation. P = [1 2 2 3; 2 1 3 1]; T = [4 5 7 7]; Now we are ready to train the network. We will train it for only one epoch, since we used only one pass of adapt. The default training function for the linear network is trainwb, and the default learning function for the weights and biases is learnwh, so we should get the same results that we obtained using adapt in the previous example, where the default adaptation function was adaptwb. net.inputWeights{1,1}.learnParam.lr = 0.1; net.biases{1}.learnParam.lr = 0.1; net.trainParam.epochs = 1; net = train(net,P,T); If we display the weights after one epoch of training we find: »net.IW{1,1} ans = 4.9000 »net.b{1} ans = 2.3000 4.1000 This is the same result we had with the batch mode training in adapt. With static networks the adapt function can implement incremental or batch training depending on the format of the input data. If the data is presented as a matrix of concurrent vectors batch training will occur. If the data is presented as a sequence, incremental training will occur. This is not true for train, which always performs batch training, regardless of the format of the input. Batch Training With Dynamic Networks Training static networks is relatively straightforward. If we use train the network will be trained in the batch mode and the inputs will be converted to concurrent vectors (columns of a matrix), even if they are originally passed as a sequence (elements of a cell array). If we use adapt, the format of the input 2-24 Training Styles will determine the method of training. If the inputs are passed as a sequence, then the network will be trained in incremental mode. If the inputs are passed as concurrent vectors, then batch mode training will be used. With dynamic networks batch mode training would typically be done with train only, especially if only one training sequence exists. To illustrate this, let’s consider again the linear network with a delay. We will use a learning rate of 0.02 for the training. (When using a gradient descent algorithm, we will typically use a smaller learning rate for batch mode training than incremental training, because all of the individual gradients are summed together before determining the step change to the weights.) net = newlin([-1 1],1,[0 1],0.02); net.IW{1,1}=[0 0]; net.biasConnect=0; net.trainParam.epochs = 1; Pi = {1}; P = {2 3 4}; T = {3 5 6}; We want to train the network with the same sequence we used for the incremental training earlier, but this time we want to update the weights only after all of the inputs have been applied (batch mode). The network will be simulated in sequential mode because the input is a sequence, but the weights will be updated in batch mode. net=train(net,P,T,Pi); The weights after one epoch of training are »net.IW{1,1} ans = 0.9000 0.6200 These are different weights than we would obtain using incremental training, where the weights would have been updated three times during one pass through the training set. For batch training the weights are only updated once in each epoch. 2-25 2 Neuron Model and Network Architectures Summary The inputs to a neuron include its bias and the sum of its weighted inputs (using the inner product). The output of a neuron depends on the neuron’s inputs and on its transfer function. There are many useful transfer functions. A single neuron cannot do very much. However, several neurons can be combined into a layer or multiple layers that have great power. Hopefully this toolbox makes it easy to create and understand such large networks. The architecture of a network consists of a description of how many layers a network has, the number of neurons in each layer, each layer’s transfer function, and how the layers are connected to each other. The best architecture to use depends on the type of problem to be represented by the network. A network effects a computation by mapping input values to output values. The particular mapping problem to be performed fixes the number of inputs as well as the number of outputs for the network. Aside from the number of neurons in a network’s output layer, the number of neurons in each layer is up to the designer. Except for purely linear networks, the more neurons in a hidden layer the more powerful the network. If a linear mapping needs to be represented linear neurons should be used. However, linear networks cannot perform any nonlinear computation. Use of a nonlinear transfer function makes a network capable of storing nonlinear relationships between input and output. A very simple problem may be represented by a single layer of neurons. However, single layer networks cannot solve certain problems. Multiple feed-forward layers give a network greater freedom. For example, any reasonable function can be represented with a two layer network: a sigmoid layer feeding a linear output layer. Networks with biases can represent relationships between inputs and outputs more easily than networks without biases. (For example, a neuron without a bias will always have a net input to the transfer function of zero when all of its inputs are zero. However, a neuron with a bias can learn to have any net transfer function input under the same conditions by learning an appropriate value for the bias.) Feed-forward networks cannot perform temporal computation. More complex networks with internal feedback paths are required for temporal behavior. 2-26 Summary If several input vectors are to be presented to a network, they may be presented sequentially or concurrently. Batching of concurrent inputs is computationally more efficient and may be what is desired in any case. The matrix notation used in MATLAB makes batching simple. 2-27 2 Neuron Model and Network Architectures Figures and Equations Simple Neuron - Title - bias Neuron without Input AA AA w p n f a a-=Exp f (wp- ) Hard Limit Transfer Function a +1 0 n -1 AA a = hardlim(n) Hard Limit Transfer Function Purelin Transfer Function a +1 n 0 -1 a = purelin(n) AA AA Linear Transfer Function 2-28 - Title Neuron with- bias Input p w AA AA AAAA n f b 1 - b) a =- fExp (wp + a Summary Log Sigmoid Transfer Function a +1 n 0 -1 AA AA a = logsig(n) Log-Sigmoid Transfer Function Neuron With Vector Input Input Neuron w Vector Input AA A A p1 p2 p3 w1,1 pR w1, R n f Where... a R = # Elements in input vector b 1 a = f(Wp +b) Net Input n = w 1, 1 p 1 + w 1, 2 p 2 + ... + w 1, R p R + b 2-29 2 Neuron Model and Network Architectures Single Neuron Using Abbreviated Notation AA AA AA AA AA AA AA A AA Input Neuron p Rx1 a W f 1x1 b R 1x1 n 1xR 1 AA AA AA AA AA AA AA Where... R = # of elements in input vector 1 1x1 a = f(Wp +b) Icons for Transfer Functions hardlim purelin logsig Layer of Neurons Layer of Neurons Input p AA AA AA Rx1 n SxR 1 R Sx1 b Sx1 a= f (Wp + b) 2-30 AA AA AA a W Sx1 Where... R = # of elements in input vector f S S = # Neurons in Layer Summary Three Layers of Neurons Input AA AA AAAA AA AA AA AA AA AA AAAA Layer of Neurons n1 w1, 1 p1 p2 p3 pR f a1 b1 1 n2 f a2 b2 1 nS wS, R f aS Where... R = # of elements in input vector S = # Neurons in Layer bS 1 a= f (Wp + b) Weight Matrix w 1, 1 w 1, 2 … w 1, R W = w 2, 1 w 2, 2 … w 2, R w S, 1 w S, 2 … w S, R 2-31 2 Neuron Model and Network Architectures Layer of Neurons, Abbreviated Notation Layer of Neurons Input p AA AA AA Rx1 n SxR 1 Sx1 b Sx1 R AA AA AA a W Sx1 Where... R = # of elements in input vector f S = # Neurons in Layer S a= f (Wp + b) Layer of Neurons Showing Indices Input p R x1 AA AA AA AA IW1,1 S1xR 1 R AA AA AA AA Layer 1 b1 S1x1 n1 S1 x1 a1 = f1(IW1,1p +b1) 2-32 Where... a1 S1x 1 f1 S1 R = # of elements in input S1 = # Neurons in layer 1 Summary Three Layers of Neurons Input AA AA AA AA AA AA AA AA AA AA AAAA First Layer n1 iw1,1 1, 1 p 1 1 p 2 p 3 p b1 1 R f1 n1 S, R 1 S 1 a1 = f1 (IW1,1p +b1) 1 2 f2 a2 lw2,1 S b3 2 2 b2 1 a2 = f2 (LW2,1a1 +b2) n3 3 S 3 1 a3 2 2 1 lw3,2 2 S f3 2 S 1 S ,S n3 b3 a3 f3 1 1 a2 2 f2 1 1,1 2 n2 2 AA A AA A AA A AA A AA A AAA n3 lw3,2 b2 S b1 1 n2 1 a1 1 f1 a2 f2 1 2 1 S 1 Third Layer b2 2 iw1,1 n2 1,1 a1 f1 2 AA AA AA AA AA AA AA AA AA AA AAAA lw2,1 1 1 n1 b1 1 a1 1 Second Layer a3 3 S f3 2 S ,S b3 3 S 1 a3 =f3 (LW3,2 a2 + b3) a3 =f3 (LW3,2 f2 (LW2,1f1 (IW1,1p +b1)+ b2)+ b3) Three Layers, Abbreviated Notation AAA AA AA AAA AA AA AAA AAAA First Layer Input p Rx1 a1 IW1,1 S1xR n1 S1x1 1 R S1x1 1 S1 a1 = f1 (IW1,1p +b1) LW2,1 S2xS1 f1 b1 S1x1 AA AA AA AA AA AA AAAA AA Second Layer b2 S2x1 Third Layer a3 = y a2 n2 S2x1 S2x1 LW3,2 S 3x S 2 f2 n3 S3 x1 b3 1 S2 S3x1 a2 = f2 (LW2,1 a1 +b2) a3 =f3 (LW3,2 f2 (LW2,1f1 (IW1,1p +b1)+ b2)+ b3 S3 x1 f3 S3 a3 =f3 (LW3,2a2 +b3) = y 2-33 2 Neuron Model and Network Architectures Linear Neuron With Two Element Vector Input. Inputs Linear Neuron AA A A p1 w1,1 p2 w1,2 n a b 1 a = purelin (Wp + b) Dynamic Network With One Delay. Inputs p(t) Linear Neuron AA AA AA AAAAAA w1,1 n(t) D a(t) w1,2 a(t) = w1,1 p(t) + w1,2 p(t - 1) 2-34 3 Perceptrons Introduction . . . . . . . . . . . . . . . . . . . . 3-2 Important Perceptron Functions. . . . . . . . . . . . . 3-3 Neuron Model . . . . . . . . . . . . . . . . . . . 3-4 Perceptron Architecture . . . . . . . . . . . . . . 3-6 Creating a Perceptron (NEWP) . . . . . . . . . . . 3-7 Simulation (SIM) . . . . . . . . . . . . . . . . . . . 3-8 Initialization (INIT) . . . . . . . . . . . . . . . . . 3-9 Learning Rules . . . . . . . . . . . . . . . . . . . 3-11 Perceptron Learning Rule (LEARNP) . . . . . . . . 3-12 Adaptive Training (ADAPT) . . . . . . . . . . . . . 3-15 Limitations and Cautions . . . . . . . . . . . . . . 3-21 Outliers and the Normalized Perceptron Rule . . . . . . . 3-21 Summary . . . . . . . . . . . . . . . . . . . . . 3-23 Figures and Equations . . . . . . . . . . . . . . . . 3-23 New Functions . . . . . . . . . . . . . . . . . . . 3-26 3 Perceptrons Introduction This chapter has a number of objectives. First we want to introduce you to learning rules, methods of deriving the next changes that might be made in a network, and training, a procedure whereby a network is actually adjusted to do a particular job. Along the way we will discuss a toolbox function to create a simple perceptron network, and we will also cover functions to initialize and simulate such networks. We will use the perceptron as a vehicle for tying these concepts together. Rosenblatt [Rose61] created many variations of the perceptron. One of the simplest was a single layer network whose weights and biases could be trained to produce a correct target vector when presented with the corresponding input vector. The training technique used is called the perceptron learning rule. The perceptron generated great interest due to its ability to generalize from its training vectors and learn from initially randomly distributed connections. Perceptrons are especially suited for simple problems in pattern classification. They are fast and reliable networks for the problems they can solve. In addition, an understanding of the operations of the perceptron provides a good basis for understanding more complex networks. In this chapter we will define what we mean by a learning rule, explain the perceptron network and its learning rule, and tell you how to initialize and simulate perceptron networks. The discussion of perceptron in this chapter is necessarily brief. You may wish to read a more thorough discussion such as that given in Chapter 4 “Perceptron Learning Rule,” of [HDB1996]. This Chapter discusses the use of multiple layers of perceptrons to solve more difficult problems beyond the capability of one layer. You also may want to refer to the original book on the perceptron, Rosenblatt, F., Principles of Neurodynamics, Washington D.C.: Spartan Press, 1961. [Rose61]. 3-2 Introduction Important Perceptron Functions Entering help percept at the MATLAB command line displays all the functions which are related to perceptrons. Perceptron networks can be created with the function newp. These networks can be initialized and simulated with the init and sim. The following material describes how perceptrons work and introduces these functions. 3-3 3 Perceptrons Neuron Model A perceptron neuron, which uses the hard limit transfer function hardlim, is shown below. Input Perceptron Neuron AA AA AAAA p1 p2 p3 w1,1 pR w1, R n f a Where... R = # Elements in input vector b 1 a = hardlim - Exp -(Wp + b) Each external input is weighted with an appropriate weight w1j, and the sum of the weighted inputs is sent to the hard limit transfer function, which also has an input of 1 transmitted to it through the bias. The hard limit transfer function, which returns a 0 or a 1, is shown below. a +1 0 -1 n A a = hardlim(n) Hard Limit Transfer Function The perceptron neuron produces a 1 if the net input into the transfer function is equal to or greater than 0, otherwise it produces a 0. The hard limit transfer function gives a perceptron the ability to classify input vectors by dividing the input space into two regions. Specifically, outputs will be 0 if the net input n is less than 0, or 1 if the net input n is 0 or greater. The input space of a 2-input hard limit neuron with the weights w 1, 1 = – 1, w 1, 2 = 1 and a bias b = 1 , is shown below. 3-4 Neuron Model L p W 2 Wp+b > 0 a=1 +1 Wp+b = 0 a=0 -b/w 1,1 p 1 -1 +1 Wp+b < 0 a=0 -b/w -1 1,2 Where... w = -1 w = +1 1,1 1,2 and b = +1 Two classification regions are formed by the decision boundary line L at Wp + b = 0 . This line is perpendicular to the weight matrix W and shifted according to the bias b. Input vectors above and to the left of the line L will result in a net input greater than 0, and therefore cause the hard limit neuron to output a 1. Input vectors below and to the right of the line L cause the neuron to output 0. The dividing line can be oriented and moved anywhere to classify the input space as desired by picking the weight and bias values. Hard limit neurons without a bias will always have a classification line going through the origin. Adding a bias allows the neuron to solve problems where the two sets of input vectors are not located on different sides of the origin. The bias allows the decision boundary to be shifted away from the origin as shown in the plot above. You may wish to run the demonstration program nnd4db. With it you can move a decision boundary around, pick new inputs to classify, and see how the repeated application of the learning rule yields a network that does classify the input vectors properly. 3-5 3 Perceptrons Perceptron Architecture The perceptron network consists of a single layer of S perceptron neurons connected to R inputs through a set of weights wi,j as shown below in two forms. As before, the network indices i and j indicate that wi,j is the strength of the connection from the jth input to the ith neuron. Input #1 AAAA AA AAA AAA Input # 1 Perceptron Layer iw1,1 1, 1 p 1 n11 a11 p R x1 b1 1 p 2 1 iw1,1S1,R n1S 1 n1 S1x1 a 1 1 2 b12 3 pR n 1 AA AA AA AA AA AA a1 IW1,1 S1xR 1 p2 Layer 1 R f1 b1 S1x1 S1 a1 = hardlim(IW1,1p1 +b1) a1S1 b1 1 S 1x 1 Where... S 1 a1 = hardlim (IW1,1p1 + b1) R = # of elements in Input S1 = # of Neurons in layer 1 The perceptron learning rule that we will describe shortly is capable of training only a single layer. Thus, here we will consider only one layer networks. This restriction places limitations on the computation a perceptron can perform. The types of problems that perceptrons are capable of solving are discussed later in this chapter in the “Limitations and Cautions” section. 3-6 Creating a Perceptron (NEWP) Creating a Perceptron (NEWP) A perceptron can be created with the function newp. net = newp(PR, S) where the input arguments are: PR is an Rx2 matrix of minimum and maximum values for R input elements. S is the Number of neurons. Commonly the hardlim function is used in perceptrons, so it is the default. The code below creates a peceptron network with a single one-element input vector and one neuron. The range for the single element of the single input vector is [0 2]. net = newp([0 2],1); We can see what network has been created by executing the following code: inputweights = net.inputweights{1,1} which yields: inputweights = delays: initFcn: learn: learnFcn: learnParam: size: userdata: weightFcn: 0 'initzero' 1 'learnp' [] [1 1] [1x1 struct] 'dotprod' Note that the default learning function is learnp, which will be discussed later in this chapter. The net input to the hardlim transfer function is dotprod, which generates the product of the input vector and weight matrix and adds the bias to compute the net input. Also note that the default initialization function, initzero, is used to set the initial values of the weights to zero. Similarly, biases = net.biases{1} 3-7 3 Perceptrons gives biases = initFcn: learn: learnFcn: learnParam: size: userdata: 'initzero' 1 'learnp' [] 1 [1x1 struct]. We can see that the default initialization for the bias is also 0. Simulation (SIM) To show how sim works we will examine a simple problem. Suppose we take a perceptron with a single two element input vector, like that discussed in the decision boundary figure. We define the network with: net = newp([-2 2;-2 +2],1); As noted above, this will give us zero weights and biases, so if we want a particular set other than zeros, we will have to create them. We can set the two weights and the one bias to -1, 1 and 1 as they were in the decision boundary figure with the following two lines of code. net.IW{1,1}= [-1 1]; net.b{1} = [1]; To make sure that these parameters were set correctly, we will check them with: net.IW{1,1} ans = -1 1 net.b{1} ans = 1 3-8 Creating a Perceptron (NEWP) Now let us see if the network responds to two signals, one on each side of the perceptron boundary. p1 = [1;1]; a1 = sim(net,p1) a1 = 1 and for p2 = [1;-1] a2 = sim(net,p2) a2 = 0 Sure enough, the perceptron has classified the two inputs correctly. Note that we could have presented the two inputs in a sequence and gotten the outputs in a sequence as well. p3 = {[1;1] [1;-1]}; a3 = sim(net,p3) a3 = [1] [0] Initialization (INIT) You can use the function init to reset the network weights and biases to their original values. Suppose, for instance that you start with the network: net = newp([-2 2;-2 +2],1); Now check its weights with wts = net.IW{1,1} which gives, as expected, wts = 0 0 In the same way, you can verify that the bias is 0 with bias = net.b{1} 3-9 3 Perceptrons which gives bias = 0. Now set the weights to the values 3 and 4 and the bias to the value 5 with net.IW{1,1} = [3,4]; net.b{1} = 5; Recheck the weights and bias as shown above to verify that the change has been made. Sure enough, wts = 3 4 bias = 5. Now use init to reset the weights and bias to their original values. net = init(net); We can check as shown above to verify that: wts = 0 bias = 0. 0 We can change the as way that a perceptron is initialized with init. For instance, suppose that we define the network input weights and bias initFcns as rands and then apply init as shown below. net.inputweights{1,1}.initFcn = 'rands'; net.biases{1}.initFcn = 'rands'; net = init(net); Now check on the weights and bias. wts = 0.2309 biases = -0.1106 0.5839 We can see that the weights and bias have been given random numbers. 3-10 Learning Rules Learning Rules We will define a learning rule as a procedure for modifying the weights and biases of a network. (This procedure may also be referred to as a training algorithm.) The learning rule is applied to train the network to perform some particular task. Learning rules in this toolbox fall into two broad categories: supervised learning and unsupervised learning. In supervised learning, the learning rule is provided with a set of examples (the training set) of proper network behavior: {p1, t 1} , {p2, t 2} , …, {pQ, tQ} where p q is an input to the network, and t q is the corresponding correct (target) output. As the inputs are applied to the network, the network outputs are compared to the targets. The learning rule is then used to adjust the weights and biases of the network in order to move the network outputs closer to the targets. The perceptron learning rule falls in this supervised learning category. In unsupervised learning, the weights and biases are modified in response to network inputs only. There are no target outputs available. Most of these algorithms perform clustering operations. They categorize the input patterns into a finite number of classes. This is especially useful in such applications as vector quantization. As noted, the perceptron discussed in this chapter is trained with supervised learning. Hopefully, a network that produces the right output for a particular input will be obtained. 3-11 3 Perceptrons Perceptron Learning Rule (LEARNP) Perceptrons are trained on examples of desired behavior. The desired behavior can be summarized by a set of input, output pairs. p 1 t 1 ,p 2 t 1 ,..., p Q t Q where p is an input to the network and t is the corresponding correct (target) output. The objective is to reduce the error, e which is the difference t – a between the neuron response a, and the target vector t. The perceptron learning rule learnp calculates desired changes to the perceptron’s weights and biases given an input vector p, and the associated error e. The target vector t must contain values of either 0 or 1, as perceptrons (with hardlim transfer functions) can only output such values. Each time learnp is executed, the perceptron will have a better chance of producing the correct outputs. The perceptron rule has been proven to converge on a solution in a finite number of iterations if a solution exists. If a bias is not used, learnp works to find a solution by altering only the weight vector w to point toward input vectors to be classified as 1, and away from vectors to be classified as 0. This results in a decision boundary that is perpendicular to w and which properly classifies the input vectors. There are three conditions that can occur for a single neuron once an input vector p is presented and the network’s response a is calculated: CASE 1. If an input vector is presented and the output of the neuron is correct (a = t, and e = t – a = 0), then the weight vector w is not altered. CASE 2. If the neuron output is 0 and should have been 1 (a = 0 and t = 1, and e = t – a = 1) the input vector p is added to the weight vector w. This makes the weight vector point closer to the input vector, increasing the chance that the input vector will be classified as a 1 in the future. CASE 3. If the neuron output is 1 and should have been 0 (a = 0 and t = 1, and e = t – a = –1) the input vector p is subtracted from the weight vector w. This makes the weight vector point farther away from the input vector, increasing the chance that the input vector will be classified as a 0 in the future. The perceptron learning rule can be written more succinctly in terms of the error e = t – a, and the change to be made to the weight vector ∆w: 3-12 Perceptron Learning Rule (LEARNP) CASE 1. If e = 0, then make a change ∆w equal to 0. CASE 2. If e = 1, then make a change ∆w equal to pT. CASE 3. If e = –1, then make a change ∆w equal to –pT. All three cases can then be written with a single expression: ∆w = ( t – a )p T = ep T We can get the expression for changes in a neuron’s bias by noting that the bias is simply a weight which always has an input of 1: ∆b = ( t – a ) ( 1 ) = e For the case of a layer of neurons we have: ∆W = ( t – a ) ( p ) T = e ( p ) T and ∆b = ( t – a ) = E The Perceptron Learning Rule can be summarized as follows: W b new new = W = b old old + ep T and +e where e = t – a . Now let us try a simple example. We will start with a single neuron having a input vector with just two elements. net = newp([-2 2;-2 +2],1); To simplify matters we will set the bias equal to 0 and the weights to 1 and -0.8. net.biases{1}.value = [0]; w = [1 -0.8]; net.IW{1,1}.value = w; The input target pair is given by: p = [1; 2]; t = [1]; 3-13 3 Perceptrons We can compute the output and error with a = sim(net,p) a = 0 e = t-a e = 1 and finally use the function learnp to find the change in the weights. dw = learnp(w,p,[],[],[],[],e,[],[],[]) dw = 1 2. The new weights, then, are obtained as w = w + dw w = 2.0000 1.2000 The process of finding new weights (and biases) can be repeated until there are no errors. Note that the perceptron learning rule is guaranteed to converge in a finite number of steps for all problems that can be solved by a perceptron. These include all classification problems that are “linearly separable.” The objects to be classified in such cases can be separated by a single line. You might want to try demo nnd4pr. It allows you to pick new input vectors and apply the learning rule to classify them. 3-14 Adaptive Training (ADAPT) Adaptive Training (ADAPT) If sim and learnp are used repeatedly to present inputs to a perceptron, and to change the perceptron weights and biases according to the error, the perceptron will eventually find weight and bias values which solve the problem, given that the perceptron can solve it. Each traverse through all of the training input and target vectors is called a pass. The function adapt carries out such a loop of calculation. In each pass the function adapt will proceed through the specified sequence of inputs, calculating the output, error and network adjustment for each input vector in the sequence as the inputs are presented. Note that adapt does not guarantee that the resulting network does its job. The new values of W and b must be checked by computing the network output for each input vector to see if all targets are reached. If a network does not perform successfully it can be trained further by again calling adapt with the new weights and biases for more training passes, or the problem can be analyzed to see if it is a suitable problem for the perceptron. Problems which are not solvable by the perceptron network are discussed in the “Limitations and Cautions” section. To illustrate the adaptation procedure, we will work through a simple problem. Consider a one neuron perceptron with a single vector input having two elements. Perceptron Neuron Input p 1 p 2 AAAA AAAA w 1,1 w 1, 2 n f a b 1 a = hardlim - Exp -(Wp + b) This network, and the problem we are about to consider are simple enough that you can follow through what is done with hand calculations if you wish. The problem discussed below follows that found in [HDB1996]. 3-15 3 Perceptrons Let us suppose we have the following classification problem and would like to solve it with our single vector input, 2 element perceptron network. 2 , t = 0 p = 1 , t = 1 p = –2 , t = 0 p = –1 , t = 1 p1 = 2 3 4 1 2 3 4 2 –2 2 1 Use the initial weights and bias. We denote the variables at each step of this calculation by using a number in parentheses after the variable. Thus, above, we have the initial values, W(0) and b(0). W(0) = 0 0 b(0) = 0 We start by calculating the perceptron’s output a for the first input vector p1, using the initial weights and bias. a = hardlim ( W ( 0 )p 1 + b ( 0 ) ) = hardlim 0 0 2 + 0 = hardlim ( 0 ) = 1 2 The output a does not equal the target value t1, so we use the perceptron rule to find the incremental changes to the weights and biases based on the error. e = t1 – a = 0 – 1 = –1 T ∆W = ep 1 = ( – 1 ) 2 2 = – 2 – 2 ∆b = e = ( – 1 ) = – 1 You can calculate the new weights and bias using the Perceptron update rules shown previously. W new b 3-16 = W new old = b + ep old T = 0 0 + –2 –2 = –2 –2 = W ( 1 ) + e = 0 + ( –1 ) = –1 = b ( 1 ) Adaptive Training (ADAPT) Now present the next input vector, p2. The output is calculated below. a = hardlim ( W ( 1 )p 2 + b ( 1 ) ) = hardlim – 2 – 2 – 2 – 1 = hardlim ( 1 ) = 1 –2 On this occasion, the target is 1, so the error is zero. Thus there are no changes in weights or bias so W ( 2 ) = W ( 1 ) = – 2 – 2 and p ( 2 ) = p ( 1 ) = – 1 We can continue in this fashion, presenting p3 next, calculating an output and the error, and making changes in the weights and bias, etc. After making one pass through all of the four inputs, you will get the values: W ( 4 ) = – 3 – 1 and b ( 4 ) = 0 . To determine if we have obtained a satisfactory solution, we must make one pass through all input vectors to see if they all produce the desired target values. This is not true for the 4th input, but the algorithm does converge on the 6th presentation of an input. The final values are: W ( 6 ) = – 2 – 3 and b ( 6 ) = 1 . This concludes our hand calculation. Now, how can we do this using the adapt function? The following code defines a perceptron like that shown in the previous figure, with initial weights and bias values of 0. net = newp([-2 2;-2 +2],1); Let us first consider the application of a single input. We will define the first input vector and target as sequences (cell arrays in curly brackets). p = {[2; 2]}; t = {0} Now set passes to 1, so that adapt will go through the input vectors (only one here) just one time. net.adaptParam.passes = 1; [net,a,e] = adapt(net,p,t); 3-17 3 Perceptrons The output and error that are returned are: a = [1] e = [-1] The new weights and bias are: twts = net.IW{1,1} twts = -2 -2 tbiase = net.b{1} tbiase = -1 Thus, the initial weights and bias are 0, and after training on just the first vector they have the values [-2 -2] and -1, just as we hand calculated. We now apply the second input vector p 2 . The output is 1, as it will be until the weights and bias are changed, but now the target is 1, the error will be 0 and the change will be zero. We could proceed in this way, starting from the previous result and applying a new input vector time after time. But we can do this job automatically with adapt. Now let’s apply adapt for one pass through the sequence of all four input vectors. Start with the network definition. net = newp([-2 2;-2 +2],1); net.trainParam.passes = 1; The input vectors and targets are: p = {[2;2] [1;-2] [-2;2] [-1;1]} t = {0 1 0 1}. Now train the network with: [net,a,e] = adapt(net,p,t); 3-18 Adaptive Training (ADAPT) The output and error that are returned are: a = [1] [1] [0] [0] e = [-1] [0] [0] [1] Note that these outputs and errors are the values obtained when each input vector is applied to the network as it existed at the time. The new weights and bias are: twts = -3 tbias = 0 -1 Finally simulate the trained network for each of the inputs. a1 = sim(net,p) a1 = [0] [0] [1] [1] The outputs do not yet equal the targets, so we need to train the network for more than one pass. This time let us run the problem again for two passes. We get the weights twts = -2 tbiase = 1 -3 and the simulated output and errors for the various inputs is: a1 = [0] [1] [0] [1] The second pass does the job. The network has converged and produces the correct outputs for the four input vectors. To check we can find the error for each of the inputs. error = {a1{1}-t{1} a1{2}-t{2} a1{3}-t{3} a1{4}-t{4}} error = [0] [0] [0] [0] Sure enough, all of the inputs are correctly classified. 3-19 3 Perceptrons Note that adapt uses the perceptron learning rule in its pure form, in that corrections to the weights and bias are made after each presentation of an input vector. Thus, adapt will converge in a finite number of steps unless the problem presented can not be solved with a simple perceptron. The function adapt can be used in various ways by other networks as well. You might type help adapt to read more about this basic function. You may wish to try various demonstration programs. For instance, demop1 illustrates classification and training of a simple perceptron. 3-20 Limitations and Cautions Limitations and Cautions Perceptron networks should be trained with adapt, which presents the input vectors to the network one at a time and makes corrections to the network based on the results of each presentation. Use of adapt in this way guarantees that any linearly separable problem will be solved in a finite number of training presentations. Perceptrons can also be trained with the function train, which is presented in the next chapter. When train is used for perceptrons, it presents the inputs to the network in batches, and makes corrections to the network based on the sum of all the individual corrections. Unfortunately, there is no proof that such a training algorithm converges for perceptrons. On that account the use of train for perceptrons is not recommended. Perceptron networks have several limitations. First, the output values of a perceptron can take on only one of two values (0 or 1) due to the hard limit transfer function. Second, perceptrons can only classify linearly separable sets of vectors. If a straight line or a plane can be drawn to separate the input vectors into their correct categories, the input vectors are linearly separable. If the vectors are not linearly separable, learning will never reach a point where all vectors are classified properly. Note, however, that it has been proven that if the vectors are linearly separable, perceptrons trained adaptively will always find a solution in finite time. You might want to try demop6. It shows the difficulty of trying to classify input vectors that are not linearly separable. It is only fair, however, to point out that networks with more than one perceptron can be used to solve more difficult problems. For instance, suppose that you have a set of four vectors that you would like to classify into distinct groups, and that in fact, two lines can be drawn to separate them. A two neuron network can be found such that its two decision boundaries classify the inputs into four categories. For additional discussion about perceptrons and to examine more complex perceptron problems, see [HDB1996]. Outliers and the Normalized Perceptron Rule Long training times can be caused by the presence of an outlier input vector whose length is much larger or smaller than the other input vectors. Applying the perceptron learning rule involves adding and subtracting input vectors from the current weights and biases in response to error. Thus, an input vector with large elements can lead to changes in the weights and biases that take a 3-21 3 Perceptrons long time for a much smaller input vector to overcome. You might wish to try demop4 to see how an outlier effects the training. By changing the perceptron learning rule slightly, training times can be made insensitive to extremely large or small outlier input vectors. Here is the original rule for updating weights: ∆w = ( t – a )p T = ep T As shown above, the larger an input vector p, the larger its effect on the weight vector w. Thus, if an input vector is much larger than other input vectors, the smaller input vectors must be presented many times to have an effect. The solution is to normalize the rule so that effect of each input vector on the weights is of the same magnitude: pT pT ∆w = ( t – a ) -------- = e -------p p The normalized perceptron rule is implemented with the function learnpn which is called exactly like learnpn. The normalized perceptron rule function learnpn takes slightly more time to execute, but reduces number of epochs considerably if there are outlier input vectors. You might try demop5 to see how this normalized training rule works. 3-22 Summary Summary Perceptrons are useful as classifiers. They can classify linearly separable input vectors very well. Convergence is guaranteed in a finite number of steps providing the perceptron can solve the problem. The design of a perceptron network is constrained completely by the problem to be solved. Perceptrons have a single layer of hard limit neurons. The number of network inputs and the number of neurons in the layer are constrained by the number of inputs and outputs required by the problem. Training time is sensitive to outliers, but outlier input vectors do not stop the network from finding a solution. Single-layer perceptrons can solve problems only when data is linearly separable. This is seldom the case. One solution to this difficulty is to use a preprocessing method that results in linearly separable vectors. Or you might use multiple perceptrons in multiple layers. Alternatively, you can use other kinds of networks such as linear networks or backpropagation networks, which can classify nonlinearly separable input vectors. Figures and Equations Perceptron Neuron Input Perceptron Neuron AAAA AAAA p1 p2 p3 w1,1 pR w1, R n f a Where... R = # Elements in input vector b 1 a = hardlim - Exp -(Wp + b) 3-23 3 Perceptrons Perceptron Transfer Function, hardlim a +1 n 0 -1 AA a = hardlim(n) Hard Limit Transfer Function Decision Boundary L p W 2 Wp+b > 0 a=1 +1 Wp+b = 0 a=0 -b/w 1,1 p 1 -1 +1 Wp+b < 0 a=0 -b/w -1 1,2 Where... w = -1 w = +1 1,1 1,2 3-24 and b = +1 Summary Perceptron Architecture Input #1 iw1,11, 1 p 1 b1 n11 1 1 p2 p3 n12 b12 1 p R AA AA AA A A A A A A Input # 1 Perceptron Layer iw1,1S1,R n1 S 1 b1S1 1 a11 p AA AA AA R x1 Layer 1 IW1,1 S1xR a12 1 R AA AA AA a1 n1 S1x1 b1 S1x1 S 1x 1 f 1 S1 a1 = hardlim(IW1,1p1 +b1) a1 1 S Where... R = # of elements in Input S1 = # of Neurons in layer 1 a1 = hardlim (IW1,1p1 + b1) The Perceptron Learning Rule W b new new = W = b old old + ep T +e where e = t–a 3-25 3 Perceptrons One Perceptron Neuron Perceptron Neuron Input p 1 p AA AA AAAA w 1,1 w 2 1, 2 n f a b 1 a = hardlim - Exp -(Wp + b) New Functions This chapter introduces the following new functions: 3-26 Function Description hardlim A hard limit transfer function initzero Zero weight/bias initialization function dotprod Dot product weight function newp Creates a new perceptron network. sim Simulates a neural network. init Initializes a neural network learnp Perceptron learning function adapt Trains a network using a sequence of inputs learnpn Normalized perceptron learning function 4 Adaptive Linear Filters Introduction . . . . . . . . . . . . . . . . . . . . 4-2 Important Linear Network Functions . . . . . . . . . . 4-3 Neuron Model . . . . . . . . . . . . . . . . . . . 4-4 Network Architecture . . . . . . . . . . . . . . . 4-5 Single ADALINE (NEWLIN) . . . . . . . . . . . . . . 4-6 Mean Square Error . . . . . . . . . . . . . . . . . 4-9 . . . . . . . . . 4-10 Linear System Design (NEWLIND) LMS Algorithm (LEARNWH) . . . . . . . . . . . . 4-11 Linear Classification (TRAIN) . . . . . . . . . . . . 4-13 Adaptive Filtering (ADAPT) . Tapped Delay Line . . . . . . Adaptive Filter . . . . . . . Adaptive Filter Example . . . Prediction Example . . . . . . Noise Cancellation Example . . Multiple Neuron Adaptive Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-16 4-16 4-17 4-18 4-20 4-21 4-23 Limitations and Cautions Overdetermined Systems . Underdetermined Systems . Linearly Dependent Vectors Too Large a Learning Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-25 4-25 4-25 4-25 4-26 . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . 4-27 Figures and Equations . . . . . . . . . . . . . . . . 4-28 New Functions . . . . . . . . . . . . . . . . . . . 4-36 4 Adaptive Linear Filters Introduction The ADALINE (Adaptive Linear Neuron networks) networks discussed in this chapter are similar to the perceptron, but their transfer function is linear rather than hard-limiting. This allows their outputs to take on any value, whereas the perceptron output is limited to either 0 or 1. Both the ADALINE and the perceptron can only solve linearly separable problems. However, here we will make use of a so called LMS (Least Mean Squares) learning rule which is much more powerful that the perceptron learning rule. The LMS or Widrow-Hoff learning rule minimizes the mean square error and thus moves the decision boundaries as far as it can from the training patterns. First we will design a linear network that, when presented with a set of given input vectors, produces outputs of corresponding target vectors. For each input vector we can calculate the network’s output vector. The difference between an output vector and its target vector is the error. We would like to find values for the network weights and biases such that the sum of the squares of the errors is minimized or below a specific value. This problem is manageable because linear systems have a single error minimum. In most cases we can calculate a linear network directly, such that its error is a minimum for the given input vectors and targets vectors. In other cases numerical problems prohibit direct calculation. Fortunately, we can always train the network to have a minimum error by using the Widrow-Hoff learning rule. Later we will design an adaptive linear system that responds to changes in its environment as it is operating. Linear networks which are adjusted at each time step based on new input and target vectors can find weights and biases which minimize the network’s sum-squared error for recent input and target vectors. Networks of this sort are often used in error cancellation, signal processing and control systems. The pioneering work in this field was done by Widrow and Hoff, who gave the name ADALINE to adaptive linear elements. The basic reference on this subject is: Widrow B. and S. D. Sterns, Adaptive Signal Processing, New York: Prentice-Hall 1985. 4-2 Introduction Important Linear Network Functions This chapter introduces the newlin, a function that creates a linear layer, the function newlind that designs a linear layer, the function learnwh, the Widrow-Hoff weight/bias learning rule, the function train, that trains a neural network and finally, the function adapt, that changes the weights and biases of a network incrementally during training. You might type help linnet to see a list of linear network functions, demonstrations, and applications. 4-3 4 Adaptive Linear Filters Neuron Model A linear neuron with R inputs is shown below. Linear Neuron w Vector Input Input AA A A p1 p2 p w1,1 pR w1, R n 3 f Where... a R = # Elements in input vector b 1 a = purelin (Wp + b) This network has the same basic structure as the perceptron. The only difference is that the linear neuron uses a linear transfer function which we will give the name purelin. a +1 n 0 -1 a = purelin(n) AA AA Linear Transfer Function The linear transfer function calculates the neuron’s output by simply returning the value passed to it. a = purelin ( n ) = purelin ( Wp + b ) = Wp + b This neuron can be trained to learn an affine function of its inputs, or to find a linear approximation to a nonlinear function. A linear network cannot, of course, be made to perform a nonlinear computation. 4-4 Network Architecture Network Architecture The linear network shown below has one layer of S neurons connected to R inputs through a matrix of weights W. Input Layer of Linear Neurons AA AA AA AA AA AA AA AA AA AA AAAA n1 w1, 1 p1 p 2 p 3 Input a1 p Rx1 b 1 2 a 1 2 R b 2 nS n bS a= purelin (Wp + b) b S Sx1 a= purelin (Wp + b) aS wS, R 1 Sx1 Sx1 1 pR a W SxR 1 n AA AA AA AA AA AA Layer of Linear Neurons Where... R = # of elements in input vector S = # Neurons in Layer This network is sometimes called a MADALINE for Many ADALINES. Note that the figure on the right defines an S-length output vector a. The Widrow-Hoff rule can only train single-layer linear networks. This is not much of a disadvantage, however, as single-layer linear networks are just as capable as multi-layer linear networks. For every multi-layer linear network, there is an equivalent single-layer linear network. 4-5 4 Adaptive Linear Filters Single ADALINE (NEWLIN) Consider a single ADALINE with two inputs. The diagram for this network is shown below. Input Simple ADALINE AA AA AAAA p1 w1,1 p2 w1,2 n a b 1 a = purelin(Wp+b) The weight matrix W in this case has only one row. The network output is: a = purelin ( n ) = purelin ( Wp + b ) = Wp + b or a = w 1, 1 p 1 + w 1, 2 p 2 + b Like the perceptron, the ADALINE has a decision boundary which is determined by the input vectors for which the net input n is zero. For n = 0 the equation Wp + b = 0 specifies such a decision boundary as shown below (adapted with thanks from [HDB96]). p 2 a<0 a>0 -b/w 1,2 W Wp+b=0 p -b/w 1 1,1 Input vectors in the upper right gray area will lead to an output greater than 0. Input vectors in the lower left white area will lead to an output less than 0. Thus, the ADALINE can be used to classify objects into two categories. 4-6 Network Architecture However, it can classify in this way only if the objects are linearly separable. Thus, the ADALINE has the same limitation as the perceptron. We can create a network like that shown above with the command: net = newlin( [-1 1; -1 1],1); The first matrix of arguments specify the range of the two scalar inputs. The last argument, 1, says that the network has a single output. The network weights and biases are set to zero by default. You can see the current values with the commands: W = net.IW{1,1} W = 0 0 and b= net.b{1} b = 0 However, you can give the weights any value that you wish, such as 2 and 3 respectively, with: net.IW{1,1} = [2 3]; W = net.IW{1,1} W = 2 3 The bias can be set and checked in the same way. net.b{1} =[-4]; b = net.b{1} b = -4 You can simulate the ADALINE for a particular input vector. Let us try p = [5;6]; 4-7 4 Adaptive Linear Filters Now you can find the network output with the function sim. a = sim(net,p) a = 24 To summarize, you can create an ADALINE network with newlin, adjust its elements as you wish and simulate it with sim. You can find more about newlin by typing help newlin. 4-8 Mean Square Error Mean Square Error Like the perceptron learning rule, the least mean square error (LMS) algorithm is an example of supervised training, in which the learning rule is provided with a set of examples of desired network behavior: {p 1, t 1} , { p 2, t 2} , …, {p Q, tQ} Here p q is an input to the network, and t q is the corresponding target output. As each input is applied to the network, the network output is compared to the target. The error is calculated as the difference between the target output and the network output. We want to minimize the average of the sum of these errors. 1 mse = ---Q Q ∑ k=1 1 e ( k ) = ---Q 2 Q ∑ (t(k) – a(k)) 2 k=1 The LMS algorithm adjusts the weights and biases of the ADALINE so as to minimize this mean square error. Fortunately, the mean square error performance index for the ADALINE network is a quadratic function. Thus, the performance index will either have one global minimum, a weak minimum or no minimum, depending on the characteristics of the input vectors. Specifically, the characteristics of the input vectors determine whether or not a unique solution exists. You can find more about this topic in Ch. 10 of [HDB96]. 4-9 4 Adaptive Linear Filters Linear System Design (NEWLIND) Unlike most other network architectures, linear networks can be designed directly if all input/target vector pairs are known. Specific network values for weights and biases can be obtained to minimize the mean square error by using the function newlind. Suppose that the inputs and targets are: P = [1 2 3]; T= [2.0 4.1 5.9]; Now you can design a network. net = newlind(P,T); You can simulate the network behavior to check that the design was done properly. Y = sim(net,P) Y = 2.0500 4.0000 5.9500 Note that the network outputs are quite close to the desired targets. You might try demolin1. It shows error surfaces for a particular problem, illustrates the design and plots the designed solution. Next we will discuss the LMS algorithm. We can use it to train a network to minimize the mean square error. 4-10 LMS Algorithm (LEARNWH) LMS Algorithm (LEARNWH) The LMS algorithm or Widrow-Hoff learning algorithm, is based on an approximate steepest descent procedure. Here again, linear networks are trained on examples of correct behavior. Widrow and Hoff had the insight that they could estimate the mean square error by using the squared error at each iteration. If we take the partial derivative of the squared error with respect to the weights and biases at the kth iteration we have: 2 ∂e ( k ) ∂e ( k ) ----------------- = 2e ( k ) --------------∂w 1, j ∂w 1, j for j = 1, 2, …, R and 2 ∂e ( k ) ∂e ( k ) ----------------- = 2e ( k ) -------------∂b ∂b Next look at the partial derivative with respect to the error. ∂[ t ( k ) – a ( k ) ] ∂ ∂e ( k ) --------------- = ------------------------------------ = [ t ( k ) – ( Wp ( k ) + b ) ] or ∂w1, j ∂w 1, j ∂ w1, j ∂ ∂e ( k ) --------------- = ∂w 1, j ∂ w 1, j t(k) – R ∑ w1, i pi ( k ) + b i=1 Here pi(k) is the ith element of the input vector at the kth iteration. Similarly, ∂e ( k ) --------------- = – p j ( k ) ∂w 1, j This can be simplified to: ∂e ( k ) --------------- = – p j ( k ) and ∂w 1, j ∂e ( k ) -------------- = – 1 ∂b 4-11 4 Adaptive Linear Filters Finally, the change to the weight matrix and the bias will be: 2αe ( k )p ( k ) and 2αe ( k ) . These two equations form the basis of the Widrow-Hoff (LMS) learning algorithm. These results can be extended to the case of multiple neurons, and written in matrix form as: T W ( k + 1 ) = W ( k ) + 2αe ( k )p ( k ) b ( k + 1 ) = b ( k ) + 2αe ( k ) . Here the error e and the bias b are vectors and α is a learning rate. If α is large, learning occurs quickly, but if it is too large it may lead to instability and errors may even increase. To ensure stable learning, the learning rate must be less than the reciprocal of the largest eigenvector of the correlation matrix p T p of the input vectors. You might want to read some of Chapter 10 of [HDB96] for more information about the LMS algorithm and its convergence. Fortunately we have a toolbox function learnwh that does all of the calculation for us. It calculates the change in weights as dw = lr*e*p' and the bias change as db = lr*e. The constant 2 shown a few lines above has been absorbed into the code learning rate lr. The function maxlinlr calculates this maximum stable learning rate lr as 0.999 * P'*P. Type help learnwh and help maxlinlr for more details about these two functions. 4-12 Linear Classification (TRAIN) Linear Classification (TRAIN) Linear networks can be trained to perform linear classification with the function train. This function applies each vector of a set of input vectors and calculates the network weight and bias increments due to each of the inputs according to learnp. Then the network is adjusted with the sum of all these corrections. We will call each pass through the input vectors an epoch. This contrasts with adapt, which adjusts weights for each input vector as it is presented. Finally, train applies the inputs to the new network, calculates the outputs, compares them to the associated targets, and calculates a mean square error. If the error goal is met, or if the maximum number of epochs is reached, the training is stopped and train returns the new network and a training record. Otherwise train goes through another epoch. Fortunately, the LMS algorithm converges when this procedure is executed. To illustrate this procedure, we will work through a simple problem. Consider the ADALINE network introduced earlier in this chapter. Input Simple ADALINE AA AA AAAA p1 w1,1 p2 w1,2 n a b 1 a = purelin(Wp+b) Next suppose we have the classification problem presented in Chapter 3 on perceptrons. 2 , t = 0 p = 1 , t = 1 p = –2 , t = 0 p = –1 , t = 1 p1 = 2 3 4 1 2 3 4 2 –2 2 1 Here we have four input vectors, and we would like a network that produces the output corresponding to each input vector when that vector is presented. 4-13 4 Adaptive Linear Filters We will use train to get the weights and biases for a network that produces the correct targets for each input vector. The initial weights and bias for the new network will be 0 by default. We will set the error goal to 0.1 rather than accept its default of 0. P = [2 1 -2 -1;2 -2 2 1]; t = [0 1 0 1]; net = newlin( [-2 2; -2 2],1); net.trainParam.goal= 0.1; [net, tr] = train(net,P,t); The problem runs, producing the following training record. TRAINWB, TRAINWB, TRAINWB, TRAINWB, TRAINWB, Epoch 0/100, MSE 0.5/0.1. Epoch 25/100, MSE 0.181122/0.1. Epoch 50/100, MSE 0.111233/0.1. Epoch 64/100, MSE 0.0999066/0.1. Performance goal met. Thus, the performance goal is met in 64 epochs. The new weights and bias are: weights = net.iw{1,1} weights = -0.0615 -0.2194 bias = net.b(1) bias = [0.5899] We can simulate the new network as shown below. A = sim(net, p) A = 0.0282 0.9672 0.2741 0.4320, We also can calculate the error. err = t - sim(net,P) err = -0.0282 0.0328 -0.2741 0.5680 Note that the targets are not realized exactly. The problem would have run longer in an attempt to get perfect results had we chosen a smaller error goal, but in this problem it is not possible to obtain a goal of 0. The network is limited 4-14 Linear Classification (TRAIN) in its capability. See demolin4 in “Limitations and Cautions” at the end of this chapter for an example of this limitation. This demonstration program demolin2 shows the training of a linear neuron, and plots the weight trajectory and error during training You also might try running the demonstration program nnd10lc. It addresses a classic and historically interesting problem, shows how a network can be trained to classify various patterns, and how the trained network responds when noisy patterns are presented. 4-15 4 Adaptive Linear Filters Adaptive Filtering (ADAPT) The ADALINE network, much like the perceptron, can only solve linearly separable problems. Nevertheless, the ADALINE has been and is today one of the most widely used neural networks found in practical applications. Adaptive filtering is one of its major application areas. Tapped Delay Line We need a new component, the tapped delay line, to make full use of the ADALINE network. Such a delay line is shown below. There the input signal enters from the left, and passes through N-1 delays.The output of the tapped delay line (TDL) is an N-dimensional vector, made up of the input signal at the current time, the previous input signal, etc. TDL AA AA AA AA D pd (k) 2 D pdN (k) N 4-16 pd1(k) Adaptive Filtering (ADAPT) Adaptive Filter We can combine a tapped delay line with an ADALINE network to create the adaptive filter shown below. Linear Layer TDL p(k) AA AA AA AA pd1(k) D p(k - 1) D w1,1 pd (k) 2 pd (k) N AAAA n(k) w1,2 a(k) SxR b 1 w1, N N The output of the filter is given by R a ( k ) = purelin ( Wp + b ) = ∑ w1, i a ( k – i + 1 ) + b i=1 The network shown above is referred to in the digital signal processing field as a finite impulse response (FIR) filter [WiSt85]. Let us take a look at the code that we will use to generate and simulate such an adaptive network. 4-17 4 Adaptive Linear Filters Adaptive Filter Example First we will define a new linear network using newlin. Input p1(t) = p(t) AA AAAA AA AAAA D p2(t) = p(t - 1) Linear Digital Filter D w1,1 w1,2 n(t) a(t) b w1,3 1 p3(t) = p(t - 2) a = purelin - Exp(Wp - + b) Assume that the input values have a range from 0 to 10. We can now define our single output network. net = newlin([0,10],1); We can specify the delays in the tapped delay line with net.inputWeights{1,1}.delays = [0 1 2]; This says that the delay line is connected to the network weight matrix through delays of 0, 1 and 2 time units. (You can specify as many delays as you wish, and can omit some values if you like. They must be in ascending order.) We can give the various weights and the bias values with: net.IW{1,1} = [7 8 9]; net.b{1} = [0]; Finally we will define the initial values of the outputs of the delays as: pi ={1 2} Note that these are ordered from left to right to correspond to the delays taken from top to bottom in the figure. This concludes the setup of the network. Now how about the input? 4-18 Adaptive Filtering (ADAPT) We will assume that the input scalars arrive in a sequence, first the value 3, then the value 4, next the value 5 and finally the value 6. We can indicate this sequence by defining the values as elements of a cell array. (Note the curly brackets.) p = {3 4 5 6} Now we have a network and a sequence of inputs. We can simulate the network to see what its output is as a function of time. [a,pf] = sim(net,p,pi); This yields an output sequence a = [46] [70] [94] [118] and final values for the delay outputs of pf = [5] [6]. The example is sufficiently simple that you can check it by hand to make sure that you understand the inputs, initial values of the delays, etc. The network that we have defined can be trained with the function adapt to produce a particular output sequence. Suppose, for instance, we would like the network to produce the sequence of values 10, 20, 30, and 40. T = {10 20 30 40} We can train our defined network to do this, starting from the initial delay conditions that we used above. We will specify ten passes through the input sequence with: net.adaptParam.passes = 10; Then we can do the training with: [net,y,E pf,af] = adapt(net,p,T,pi); 4-19 4 Adaptive Linear Filters This code returns final weights, bias and output sequence shown below. wts = net.IW{1,1} wts = 0.5059 3.1053 5.7046 bias = net.b{1} bias = -1.5993 y = [11.8558] [20.7735] [29.6679] [39.0036] Presumably if we had run for additional passes the output sequence would have been even closer to the desired values of 10, 20, 30 and 40. Thus, adaptive networks can be specified, simulated and finally trained with adapt. However, the outstanding value of adaptive networks lies in their use to perform a particular function, such as or prediction or noise cancellation. Prediction Example Suppose that we would like to use an adaptive filter to predict the next value of a stationary random process, p(t). We will use the network shown below to do this. Input p1(t) = p(t) AA AA AA A AAAA A D p2(t) = p(t - 1) Linear Digital Filter D p3(t) = p(t - 2) w1,2 n(t) + a(t) e(t) - b w1,3 1 Target = p(t) Adjust weights a = purelin (Wp + b) Predictive Filter: a(t) is approximation to p(t) The signal to be predicted, p(t), enters from the left into a tapped delay line. The previous two values of p(t) are available as outputs from the tapped delay 4-20 Adaptive Filtering (ADAPT) line. The network uses adapt to change the weights on each time step so as to minimize the error e(t) on the far right. If this error is zero, then the network output a(t) is exactly equal to p(t), and the network has done its prediction properly. A detailed analysis of this network is not appropriate here, but we can state the main points. Given the autocorrelation function of the stationary random process p(t), the error surface, the maximum learning rate, and the optimum values of the weights can be calculated. Commonly, of course, one does not have detailed information about the random process, so these calculations cannot be performed. But this lack does not matter to the network. The network, once initialized and operating, adapts at each time step to minimize the error and in a relatively short time is able to predict the input p(t). Chapter 10 of [HDB96] presents this problem, goes through the analysis, and shows the weight trajectory during training. The network finds the optimum weights on its own without any difficulty whatsoever. You also might want to try demonstration program nnd10nc to see an adaptive noise cancellation program example in action. This demonstration allows you to pick a learning rate and momentum, (see Chapter 5), and shows the learning trajectory, and the original and cancellation signals verses time. Noise Cancellation Example Consider a pilot in an airplane. When the pilot speaks into as microphone, the engine noise in the cockpit is added to the voice signal, and the resultant signal heard by passengers would be of low quality. We would like to obtain a signal which contains the pilot’s voice but not the engine noise. We can do this with an adaptive filter if we can obtain a sample of the engine noise and apply it as the input to the adaptive filter. 4-21 4 Adaptive Linear Filters Pilot’s Voice Pilot’s Voice Contaminated with Engine Noise v m Restored Signal e + - Contaminating Noise c "Error" Filtered Noise to Cancel Contamination Noise Path Filter n Engine Noise Adaptive Filter a Adaptive Filter Adjusts to Minimize Error. This removes the engine noise from contaminated signal, leaving the pilot’s voice as the “error.” Here we will adaptively train the neural linear network to predict the combined pilot/engine signal m from an engine signal n. Notice that the engine signal n does not tell the adaptive network anything about the pilot’s voice signal contained in m. However, the engine signal n. does give the network information it can use to predict the engine’s contribution to the pilot/engine signal m. The network will do its best to adaptively output m. However, in this case, the network can only predict the engine interference noise in the pilot/ engine signal m. The network error e will be equal to m, the pilot/engine signal, minus the predicted contaminating engine noise signal. Thus e contains only the pilot’s voice! Our linear adaptive network adaptively learns to cancel the engine noise. Note, in closing, that such adaptive noise canceling generally does a better job than a classical filter because the noise here is subtracted from rather than filtered out of the signal m. 4-22 Adaptive Filtering (ADAPT) Multiple Neuron Adaptive Filters We may want to use more than one neuron in an adaptive system, so we need some additional notation. A tapped delay line can be used with an S linear neurons as shown below. TDL p(k) Linear Layer AA AA AA AA D pd1(k) w1,1 A A A A A n1(k) a1(k) n2(k) a2(k) nS (k) aS (k) b1 pd2(k) 1 p(k - 1) D A A A A A b2 1 pdN (k) wS, N bS 1 N Alternatively, we can show this same network in abbreviated form. AAAAA AA AAAAA AAA AA AAA AA Linear Layer of S Neurons p(k) Qx1 a(k) pd(k) TDL (Q*N) x 1 W n(k) S x1 S x (Q*N) N S x1 1 b S x1 S 4-23 4 Adaptive Linear Filters If we want to show more of the detail of the tapped delay line and there are not too many delays we can use the following notation. Abreviated Notation A AA A AA AA A AA pd(k) p(k) TDL 1x1 3x1 0 1 2 2 W 3x2 1 b 3x1 AA AA AA a(k) n(k) 3x1 3x1 Linear Layer Here we have a tapped delay line that sends the current signal, the previous signal and the signal delayed before that to the weight matrix. We could have a longer list, and some delay values could be omitted if desired. The only requirement is that the delays are shown in increasing order as they go from top to bottom. 4-24 Limitations and Cautions Limitations and Cautions ADALINEs may only learn linear relationships between input and output vectors. Thus ADALINEs cannot find solutions to some problems. However, even if a perfect solution does not exist, the ADALINE will minimize the sum of squared errors if the learning rate lr is sufficiently small. The network will find as close a solution as is possible given the linear nature of the network’s architecture. This property holds because the error surface of a linear network is a multi-dimensional parabola. Since parabolas have only one minimum, a gradient descent algorithm (such as the LMS rule) must produce a solution at that minimum. ADALINES have other various limitations. Some of them are discussed below. Overdetermined Systems Linear networks have a number of limitations. For instance, the system may be overdetermined. Suppose that we have a network to be trained with four 1-element input vectors and four targets. A perfect solution to wp + b = t for each of the inputs may not exist, for there are four constraining equations and only one weight and one bias to adjust. However, the LMS rule will still minimize the error. You might try demolin4 to see how this is done. Underdetermined Systems Consider a single linear neuron with one input. This time, in demolin5, we will train it on only one 1-element input vector and its 1-element target vector: P = [+1.0]; T = [+0.5]; Note that while there is only one constraint arising from the single input/target pair, there are two variables, the weight and the bias. Having more variables than constraints results in an underdetermined problem with an infinite number of solutions. You might wish to try demoin5 to explore this topic. Linearly Dependent Vectors Normally it is a straightforward job to determine whether or not a linear network can solve a problem. Commonly, if a linear network has at least as many degrees of freedom (S*R+S = number of weights and biases) as constraints (Q = pairs of input/target vectors), then the network can solve the 4-25 4 Adaptive Linear Filters problem. This is true except when the input vectors are linearly dependent and they are applied to a network without biases. In this case, as shown with demonstration script demolin6, the network cannot solve the problem with zero error. You might want to try demolin6. Too Large a Learning Rate A linear network can always be trained with the Widrow-Hoff rule to find the minimum error solution for its weights and biases, as long as the learning rate is small enough. Demonstration script demolin7 shows what happens when a neuron with one input and a bias is trained with a learning rate larger than that recommended by maxlinlr. The network is trained with two different learning rates to show the results of using too large a learning rate. 4-26 Summary Summary Single-layer linear networks can perform linear function approximation or pattern association. Single-layer linear networks can be designed directly or trained with the Widrow-Hoff rule to find a minimum error solution. In addition, linear networks can be trained adaptively allowing the network to track changes in its environment. The design of a single-layer linear network is constrained completely by the problem to be solved. The number of network inputs and the number of neurons in the layer are determined by the number of inputs and outputs required by the problem. Multiple layers in a linear network do not result in a more powerful network so the single layer is not a limitation. However, linear networks can solve only linear problems. Nonlinear relationships between inputs and targets cannot be represented exactly by a linear network. The networks discussed in this chapter make a linear approximation with the minimum sum-squared error. If the relationship between inputs and targets is linear or a linear approximation is desired then linear networks are made for the job. Otherwise, backpropagation may be a good alternative. Adaptive linear filters have many practical applications such as noise cancellation and prediction in control and communication systems. 4-27 4 Adaptive Linear Filters Figures and Equations Linear Neuron Linear Neuron w Vector Input Input AA A A p1 p p23 w1,1 pR w1, R n f Where... a b 1 a = purelin (Wp + b) Purelin Transfer Function a +1 n 0 -1 a = purelin(n) AA AA Linear Transfer Function 4-28 R = # Elements in input vector Summary MADALINE Input Layer of Linear Neurons AA AA AA AA AA AA AA AA AA AA AAAA n1 w1, 1 p1 Input a1 p Rx1 b 1 p 2 p 1 2 R b 3 2 nS aS wS, R 1 n b S Sx1 a= purelin (Wp + b) 1 pR Sx1 Sx1 a 2 a W SxR 1 n AA AA AA AA AA AA Layer of Linear Neurons Where... bS R = # of elements in input vector S = # Neurons in Layer a= purelin (Wp + b) ADALINE Input Simple ADALINE AAAA AAAA p1 w1,1 p2 w1,2 n a b 1 a = purelin(Wp+b) 4-29 4 Adaptive Linear Filters Decision Boundary p 2 a<0 a>0 -b/w 1,2 W Wp+b=0 p 1 -b/w 1,1 Mean Square Error 1 mse = ---Q Q ∑ k=1 1 2 e ( k ) = ---Q Q ∑ (t(k) – a(k)) k=1 LMS (Widrow-Hoff) Algorithm T W ( k + 1 ) = W ( k ) + 2αe ( k )p ( k ) b ( k + 1 ) = b ( k ) + 2αe ( k ) . 4-30 2 Summary Tapped Delay Line TDL AA AA AA AA pd1(k) D pd2(k) D pdN (k) N 4-31 4 Adaptive Linear Filters Adaptive Filter Linear Layer TDL p(k) AA AA AA AA pd1(k) D p(k - 1) D w1,1 AAAA pd (k) 2 n(k) SxR w1,2 b 1 pd (k) N w1, N N Adaptive Filter Example Input p1(t) = p(t) AA AA AA AA AAAA D p2(t) = p(t - 1) Linear Digital Filter D w1,1 w1,2 n(t) a(t) b w1,3 1 p3(t) = p(t - 2) a = purelin - Exp(Wp - + b) 4-32 a(k) Summary Prediction Example Input p1(t) = p(t) AA AAAA AA AAAA D p2(t) = p(t - 1) Linear Digital Filter D p3(t) = p(t - 2) w1,2 n(t) + a(t) e(t) - b w1,3 1 Target = p(t) Adjust weights a = purelin (Wp + b) Predictive Filter: a(t) is approximation to p(t) 4-33 4 Adaptive Linear Filters Noise Cancellation Example Pilot’s Voice Pilot’s Voice Contaminated with Engine Noise v m Restored Signal e + - Contaminating Noise c Filtered Noise to Cancel Contamination Noise Path Filter n Engine Noise Adaptive Filter a Adaptive Filter Adjusts to Minimize Error. This removes the engine noise from contaminated signal, leaving the pilot’s voice as the “error.” 4-34 "Error" Summary Multiple Neuron Adaptive Filter TDL p(k) Linear Layer AA AA AA AA D pd1(k) w1,1 A A A A A n1(k) a1(k) n2(k) a2(k) nS (k) aS (k) b1 pd2(k) 1 p(k - 1) D A A A A A b2 1 pdN (k) wS, N bS 1 N Abbreviated Form of Adaptive Filter AA AAA AAAAA AA AAA AA AAA AA Linear Layer of S Neurons p(k) Qx1 a(k) pd(k) TDL (Q*N) x 1 W n(k) S x1 S x (Q*N) N S x1 1 b S x1 S 4-35 4 Adaptive Linear Filters Specific Small Adaptive Filter Abreviated Notation AA A AA A AA A AA AA AA AA pd(k) p(k) TDL 1x1 3x1 0 1 2 a(k) W n(k) 3x2 1 3x1 b 3x1 2 3x1 Linear Layer New Functions This chapter introduces the following new functions: 4-36 Function Description newlin Creates a linear layer. newlind Design a linear layer. learnwh Widrow-Hoff weight/bias learning rule. purelin A hard limit transfer function. train Trains a neural network. 5 Backpropagation Overview . . . . . . . . . . . . . . . . . . . . . 5-2 Fundamentals Architecture . . Simulation (SIM) Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-3 5-3 5-8 5-8 Faster Training . . . . . . . . . . . . . . Variable Learning Rate (TRAINGDA, TRAINGDX) . Resilient Backpropagation (TRAINRP) . . . . . . Conjugate Gradient Algorithms . . . . . . . . . Line Search Routines . . . . . . . . . . . . . Quasi-Newton Algorithms . . . . . . . . . . . Levenberg-Marquardt (TRAINLM) . . . . . . . Reduced Memory Levenberg-Marquardt (TRAINLM) . . . . . . . . . . . . . . . . . . . . . . . . . 5-16 . 5-16 . 5-18 . 5-20 . 5-26 . 5-29 . 5-31 . 5-33 Speed and Memory Comparison . . . . . . . . . . . 5-35 Improving Generalization . . . . . . . . . . . . . . 5-37 Regularization . . . . . . . . . . . . . . . . . . . . 5-38 Early Stopping . . . . . . . . . . . . . . . . . . . . 5-41 Preprocessing and Postprocessing . . . . . . . Min and Max (PREMNMX, POSTMNMX TRAMNMX) Mean and Stand. Dev. (PRESTD, POSTSTD, TRASTD) Principal Component Analysis (PREPCA, TRAPCA) . Post-training Analysis (POSTREG) . . . . . . . . Sample Training Session . . . . . . . . . . . 5-44 . 5-44 . 5-45 . 5-46 . 5-47 . . . . . . . . . . . . . . 5-49 Limitations and Cautions . . . . . . . . . . . . . . 5-54 Summary . . . . . . . . . . . . . . . . . . . . . 5-56 5 Backpropagation Overview Backpropagation was created by generalizing the Widrow-Hoff learning rule to multiple-layer networks and nonlinear differentiable transfer functions. Input vectors and the corresponding output vectors are used to train a network until it can approximate a function, associate input vectors with specific output vectors, or classify input vectors in an appropriate way as defined by you. Networks with biases, a sigmoid layer, and a linear output layer are capable of approximating any function with a finite number of discontinuities. Standard backpropagation is a gradient descent algorithm, as is the Widrow-Hoff learning rule. The term backpropagation refers to the manner in which the gradient is computed for nonlinear multilayer networks. There are a number of variations on the basic algorithm which are based on other standard optimization techniques, such as conjugate gradient and Newton methods. The Neural Network Toolbox implements a number of these variations. This chapter will explain how to use each of these routines and will discuss the advantages and disadvantages of each. Properly-trained backpropagation networks tend to give reasonable answers when presented with inputs that they have never seen. Typically, a new input will lead to an output similar to the correct output for input vectors used in training that are similar to the new input being presented. This generalization property makes it possible to train a network on a representative set of input/ target pairs and get good results without training the network on all possible input/output pairs. There are two features of the Neural Network Toolbox which are designed to improve network generalization - regularization and early stopping. These features and their use will be discussed later in this chapter. This chapter will also discuss preprocessing and postprocessing techniques which can improve the efficiency of network training. Before beginning this chapter you may want to read a basic reference on backpropagation, such as D.E Rumelhart, G.E. Hinton, R.J. Williams, “Learning internal representations by error propagation,”, D. Rumelhart and J. McClelland, editors. Parallel Data Processing, Vol.1, Chapter 8, the M.I.T. Press, Cambridge, MA 1986 pp. 318-362. This subject is also covered in detail in Chapters 11 and 12 of M.T. Hagan, H.B. Demuth, M.H. Beale, Neural Network Design, PWS Publishing Company, Boston, MA 1996. 5-2 Fundamentals Fundamentals Architecture In this section we want to present the architecture of the network which is most commonly used with the backpropagation algorithm - the multilayer feedforward network. The routines in the Neural Network Toolbox can be used to train more general networks, some of these will be briefly discussed in later chapters. Neuron Model (TANSIG, LOGSIG, PURELIN) An elementary neuron with R inputs is shown below. Each input is weighted with an appropriate w. The sum of the weighted inputs and the bias forms the input to the transfer function f. Neurons may use any differentiable transfer function f to generate their output. Input General Neuron AA A A p1 p2 p3 w1,1 pR w1, R n f Where... a b R = # Elements in input vector 1 a =- fExp (Wp-+ b) Multilayer networks often use the log-sigmoid transfer function logsig. a +1 n 0 -1 AA AA a = logsig(n) Log-Sigmoid Transfer Function 5-3 5 Backpropagation The function logsig generates outputs between 0 and 1 as the neuron’s net input goes from negative to positive infinity. Alternatively, multilayer networks may use the tan-sigmoid transfer function tansig. a +1 0 n -1 a = tansig(n) Tan-Sigmoid Transfer Function Occasionally, the linear transfer function purelin is used in backpropagation networks. a +1 n 0 -1 a = purelin(n) AA AA Linear Transfer Function If the last layer of a multilayer network has sigmoid neurons, then the outputs of the network are limited to a small range. If linear output neurons are used the network outputs can take on any value. In backpropagation it is important to be able to calculate the derivatives of any transfer functions used. Each of the transfer functions above, tansig, logsig, and purelin, have a corresponding derivative function: dtansig, dlogsig and dpurelin. To get the name of a transfer function’s associated derivative function, call the transfer function with the string 'deriv'. tansig('deriv') ans = dtansig 5-4 Fundamentals The three transfer functions described here are the most commonly used transfer functions for backpropagation, but other differentiable transfer functions can be created and used with backpropagation if desired. See Chapter 11, “Advanced Topics” for more information. Feedforward Network A single-layer network of S logsig neurons having R inputs is shown below in full detail on the left and with a layer diagram on the right. Input p1 p2 p3 pR w1, 1 Layer of Neurons AAAA AA AA AA AA AA AA AAAA n1 Input a1 p Rx1 b1 1 n2 1 wS, R nS bS 1 AA AA AA W n SxR a2 b2 Layer of Neurons 1 R Sx1 b Sx1 A A A a Sx1 S a= f (Wp + b) aS Where... R = # of elements in input vector S = # Neurons in Layer a= f (Wp + b) Feedforward networks often have one or more hidden layers of sigmoid neurons followed by an output layer of linear neurons. Multiple layers of neurons with nonlinear transfer functions allow the network to learn nonlinear and linear relationships between input and output vectors. The linear output layer lets the network produce values outside the range –1 to +1. On the other hand, if it is desirable to constrain the outputs of a network (such as between 0 and 1) then the output layer should use a sigmoid transfer function (such as logsig). As noted in Chapter 2, for multiple-layer networks we use the number of the layers to determine the superscript on the weight matrices. The appropriate notation is used in the two-layer tansig/purelin network shown next. 5-5 5 Backpropagation Input Hidden Layer Output Layer AAAA AA AAA AAA AA AA AA AAA AAAA AA p1 2 x1 IW1,1 4x2 a1 4x1 n1 LW2,1 3 x4 4 x1 1 2 n2 3 x1 1 b 1 4 x1 a2 4 a1 = tansig (IW1,1p1 +b1) 3 x1 f2 b 2 3 x1 3 a2 =purelin (LW2,1a1 +b2) This network can be used as a general function approximator. It can approximate any function with a finite number of discontinuities, arbitrarily well, given sufficient neurons in the hidden layer. Creating a Network (NEWFF). The first step in training a feedforward network is to create the network object. The function newff creates a trainable feedforward network. It requires four inputs and returns the network object. The first input is an R by 2 matrix of minimum and maximum values for each of the R elements of the input vector. The second input is an array containing the sizes of each layer. The third input is a cell array containing the names of the transfer functions to be used in each layer. The final input contains the name of the training function to be used. For example, the following command will create a two-layer network. There will be one input vector with two elements, three neurons in the first layer and one neuron in the second (output) layer. The transfer function in the first layer will be tan-sigmoid, and the output layer transfer function will be linear. The values for the first element of the input vector will range between -1 and 2, the values of the second element of the input vector will range between 0 and 5, and the training function will be traingd (which will be described in a later section). net=newff([-1 2; 0 5],[3,1],{'tansig','purelin'},'traingd'); This command creates the network object and also initializes the weights and biases of the network; therefore the network is ready for training. There are times when you may wish to re-initialize the weights, or to perform a custom initialization. The next section explains the details of the initialization process. 5-6 Fundamentals Initializing Weights (INIT, INITNW, RANDS). Before training a feedforward network, the weights and biases must be initialized. The initial weights and biases are created with the command init. This function takes a network object as input and returns a network object with all weights and biases initialized. Here is how a network is initialized: net = init(net); The specific technique which is used to initialize a given network will depend on how the network parameters net.initFcn and net.layers{i}.initFcn are set. The parameter net.initFcn is used to determine the overall initialization function for the network. The default initialization function for the feedforward network is initlay, which allows each layer to use its own initialization function. With this setting for net.initFcn, the parameters net.layers{i}.initFcn are used to determine the initialization method for each layer. For feedforward networks there are two different layer initialization methods which are normally used: initwb and initnw. The initwb function causes the initialization to revert to the individual initialization parameters for each weight matrix (net.inputWeights{i,j}.initFcn) and bias. For the feedforward networks the weight initialization is usually set to rands, which sets weights to random values between -1 and 1. It is normally used when the layer transfer function is linear. The function initnw is normally used for layers of feedforward networks where the transfer function is sigmoid. It is based on the technique of Nguyen and Widrow [NgWi90] and generates initial weight and bias values for a layer so that the active regions of the layer's neurons will be distributed roughly evenly over the input space. It has several advantages over purely random weights and biases: (1) few neurons are wasted (since the active regions of all the neurons are in the input space), (2) training works faster (since each area of the input space has active neuron regions). The initialization function init is called by newff, therefore the network is automatically initialized with the default parameters when it is created, and init does not have to be called separately. However, the user may want to re-initialize the weights and biases, or to use a specific method of initialization. For example, in the network that we just created, using newff, the default initialization for the first layer would be initnw. If we wanted to re-initialize 5-7 5 Backpropagation the weights and biases in the first layer using the rands function, we would issue the following commands: net.layers{1}.initFcn = 'initwb'; net.inputWeights{1,1}.initFcn = 'rands'; net.biases{1,1}.initFcn = 'rands'; net.biases{2,1}.initFcn = 'rands'; net = init(net); Simulation (SIM) The function sim simulates a network. sim takes the network input p, and the network object net, and returns the network outputs a. Here is how simuff can be used to simulate the network we created above for a single input vector: p = [1;2]; a = sim(net,p) a = -0.1011 (If you try these commands, your output may be different, depending on the state of your random number generator when the network was initialized.) Below, sim is called to calculate the outputs for a concurrent set of three input vectors. p = [1 3 2;2 4 1]; a=sim(net,p) a = -0.1011 -0.2308 0.4955 Training Once the network weights and biases have been initialized, the network is ready for training. The network can be trained for function approximation (nonlinear regression), pattern association, or pattern classification. The training process requires a set of examples of proper network behavior network inputs p and target outputs t. During training the weights and biases of the network are iteratively adjusted to minimize the network performance function net.performFcn. The default performance function for feedforward networks is mean square error mse - the average squared error between the network outputs a and the target outputs t. 5-8 Fundamentals The remainder of this chapter will describe several different training algorithms for feedforward networks. All of these algorithms use the gradient of the performance function to determine how to adjust the weights to minimize performance. The gradient is determined using a technique called backpropagation, which involves performing computations backwards through the network. The backpropagation computation is derived using the chain rule of calculus and is described in Chapter 11 of [HDB96]. The basic backpropagation training algorithm, in which the weights are moved in the direction of the negative gradient, is described in the next section. Later sections will describe more complex algorithms that increase the speed of convergence. Backpropagation Algorithm There are many variations of the backpropagation algorithm, several of which will be discussed in this chapter. The simplest implementation of backpropagation learning updates the network weights and biases in the direction in which the performance function decreases most rapidly - the negative of the gradient. One iteration of this algorithm can be written xk + 1 = xk – αk g k , where x k is a vector of current weights and biases, g k is the current gradient, and α k is the learning rate. There are two different ways in which this gradient descent algorithm can be implemented: incremental mode and batch mode. In the incremental mode, the gradient is computed and the weights are updated after each input is applied to the network. In the batch mode all of the inputs are applied to the network before the weights are updated. The next section will describe the incremental training, and the following section will describe batch training. Incremental Training(ADAPT) The function adapt is used to train networks in the incremental mode. This function takes the network object and the inputs and the targets from the training set, and returns the trained network object and the outputs and errors of the network for the final weights and biases. There are several network parameters which must be set in order guide the incremental training. The first is net.adaptFcn, which determines which incremental mode training function is to be used. The default for feedforward 5-9 5 Backpropagation networks is adaptwb, which allows each weight and bias to assign its own function. These individual learning functions for the weights and biases are set by the parameters net.biases{i,j}.learnFcn, net.inputWeights{i,j}.learnFcn, and net.layerWeights{i,j}.learnFcn. Gradient Descent (LEARDGD). For the basic steepest (gradient) descent algorithm, the weights and biases are moved in the direction of the negative gradient of the performance function. For this algorithm, the individual learning function parameters for the weights and biases are set to 'learngd'. The following commands illustrate how these parameters are set for the feedforward network we created earlier. net.biases{1,1}.learnFcn = 'learngd'; net.biases{2,1}.learnFcn = 'learngd'; net.layerWeights{2,1}.learnFcn = 'learngd'; net.inputWeights{1,1}.learnFcn = 'learngd'; The function learngd has one learning parameter associated with it - the learning rate lr. The changes to the weights and biases of the network are obtained by multiplying the learning rate times the negative of the gradient. The larger the learning rate, the bigger the step. If the learning rate is made too large the algorithm will become unstable. If the learning rate is set too small, the algorithm will take a long time to converge. See page 12-8 of [HDB96] for a discussion of the choice of learning rate. The learning rate parameter is set to the default value for each weight and bias when the learnFcn is set to learngd, as in the code above, although you can change its value if you desire. The following command demonstrates how you can set the learning rate to 0.2 for the layer weights. The learning rate can be set separately for each weight and bias. net.layerWeights{2,1}.learnParam.lr= 0.2; The final parameter to be set for sequential training is net.adaptParam.passes, which determines the number of passes through the training set during training. Here we set the number of passes to 200. net.adaptParam.passes = 200; 5-10 Fundamentals We are now almost ready to train the network. It remains to set up the training set. Here is a simple set of inputs and targets which we will use to illustrate the training procedure: p = [-1 -1 2 2;0 5 0 5]; t = [-1 -1 1 1]; If we want the learning algorithm to update the weights after each input pattern is presented, we need to convert the matrices of inputs and targets into cell arrays, with a cell for each input vector and target: p = num2cell(p,1); t = num2cell(t,1); We are now ready to perform the incremental training using the adapt function: [net,a,e]=adapt(net,p,t); After the training is complete we can simulate the network to test the quality of the training. a = sim(net,p) a = [-0.9995] [-1.0000] [1.0001] [1.0000] Gradient Descent With Momentum (LEARDGDM). In addition to learngd, there is another incremental learning algorithm for feedforward networks that often provides faster convergence - learngdm, steepest descent with momentum. Momentum allows a network to respond not only to the local gradient, but also to recent trends in the error surface. Acting like a low pass filter, momentum allows the network to ignore small features in the error surface. Without momentum a network may get stuck in a shallow local minimum. With momentum a network can slide through such a minimum. See page 12-9 of [HDB96] for a discussion of momentum. Momentum can be added to backpropagation learning by making weight changes equal to the sum of a fraction of the last weight change and the new change suggested by the backpropagation rule. The magnitude of the effect that the last weight change is allowed to have is mediated by a momentum constant, mc, which can be any number between 0 and 1. When the momentum constant is 0 a weight change is based solely on the gradient. When the 5-11 5 Backpropagation momentum constant is 1 the new weight change is set to equal the last weight change and the gradient is simply ignored. The learngdm function is invoked using the same steps shown above for the learngd function, except that both the mc and lr learning parameters can be set. Different parameter values can be used for each weight and bias, since each weight and bias has its own learning parameters. The following commands will cause the previously created network to be incrementally trained using learngdm with the default learning parameters. net.biases{1,1}.learnFcn = 'learngdm'; net.biases{2,1}.learnFcn = 'learngdm'; net.layerWeights{2,1}.learnFcn = 'learngdm'; net.inputWeights{1,1}.learnFcn = 'learngdm'; [net,a,e]=adapt(net,p,t); Batch Training (TRAIN). The alternative to incremental training is batch training, which is invoked using the function train. In batch mode the weights and biases of the network are updated only after the entire training set has been applied to the network. The gradients calculated at each training example are added together to determine the change in the weights and biases. For a discussion of batch training with the backpropagation algorithm see page 12-7 of [HDB96]. Batch Gradient Descent (TRAINGD). The batching equivalent of the incremental function learngd is traingd, which implements the batching form of the standard steepest descent training function. The weights and biases are updated in the direction of the negative gradient of the performance function. If you wish to train a network using batch steepest descent, you should set the network trainFcn to traingd and then call the function train. Unlike the learning functions in the previous section, which were assigned separately to each weight matrix and bias vector in the network, there is only one training function associated with a given network. There are seven training parameters associated with traingd: epochs, show, goal, time, min_grad, max_fail, and lr. The learning rate lr has the same meaning here as it did for learngd - it is multiplied times the negative of the gradient to determine the changes to the weights and biases. The training status will be displayed every show iterations of the algorithm. The other parameters determine when the training is stopped. The training will stop if the number of iterations exceeds epochs, if the performance function drops 5-12 Fundamentals below goal, if the magnitude of the gradient is less than mingrad, or if the training time is longer than time seconds. We will discuss max_fail, which is associated with the early stopping technique, in the section on improving generalization. The following code will recreate our earlier network, and then train it using batch steepest descent. (Note that for batch training all of the inputs in the training set are placed in one matrix.) net=newff([-1 2; 0 5],[3,1],{'tansig','purelin'},'traingd'); net.trainParam.show = 50; net.trainParam.lr = 0.05; net.trainParam.epochs = 300; net.trainParam.goal = 1e-5; p = [-1 -1 2 2;0 5 0 5]; t = [-1 -1 1 1]; net=train(net,p,t); TRAINGD, Epoch 0/300, MSE 1.59423/1e-05, Gradient 2.76799/ 1e-10 TRAINGD, Epoch 50/300, MSE 0.00236382/1e-05, Gradient 0.0495292/1e-10 TRAINGD, Epoch 100/300, MSE 0.000435947/1e-05, Gradient 0.0161202/1e-10 TRAINGD, Epoch 150/300, MSE 8.68462e-05/1e-05, Gradient 0.00769588/1e-10 TRAINGD, Epoch 200/300, MSE 1.45042e-05/1e-05, Gradient 0.00325667/1e-10 TRAINGD, Epoch 211/300, MSE 9.64816e-06/1e-05, Gradient 0.00266775/1e-10 TRAINGD, Performance goal met. a = sim(net,p) a = -1.0010 -0.9989 1.0018 0.9985 Try the Neural Network Design Demonstration nnd12sd1[HDB96] for an illustration of the performance of the batch gradient descent algorithm. Batch Gradient Descent With Momentum (TRAINGDM). The batch form of gradient descent with momentum is invoked using the training function traingdm. This algorithm is equivalent to learngdm, with two exceptions. First, the gradient is computed by summing the gradients calculated at each training example, and 5-13 5 Backpropagation the weights and biases are only updated after all training examples have been presented. Second, if the new performance function on a given iteration exceeds the performance function on a previous iteration by more than a predefined ratio max_perf_inc (typically 1.04), the new weights and biases are discarded, and the momentum coefficient mc is set to zero. In the following code we recreate our previous network and retrain it using gradient descent with momentum. The training parameters for traingdm are the same as those for traingd, with the addition of the momentum factor mc and the maximum performance increase max_perf_inc. (The training parameters are reset to the default values whenever net.trainFcn is set to traingdm.) net=newff([-1 2; 0 5],[3,1],{'tansig','purelin'},'traingdm'); net.trainParam.show = 50; net.trainParam.lr = 0.05; net.trainParam.mc = 0.9; net.trainParam.epochs = 300; net.trainParam.goal = 1e-5; p = [-1 -1 2 2;0 5 0 5]; t = [-1 -1 1 1]; net=train(net,p,t); TRAINGDM, Epoch 0/300, MSE 3.6913/1e-05, Gradient 4.54729/ 1e-10 TRAINGDM, Epoch 50/300, MSE 0.00532188/1e-05, Gradient 0.213222/1e-10 TRAINGDM, Epoch 100/300, MSE 6.34868e-05/1e-05, Gradient 0.0409749/1e-10 TRAINGDM, Epoch 114/300, MSE 9.06235e-06/1e-05, Gradient 0.00908756/1e-10 TRAINGDM, Performance goal met. a = sim(net,p) a = -1.0026 -1.0044 0.9969 0.9992 5-14 Fundamentals Note that since we re-initialized the weights and biases before training, we obtain a different mean square error than we did using traingd. If we were to re-initialize and train again using traingdm, we would get yet a different mean square error. The random choice of initial weights and biases will affect the performance of the algorithm. If you wish to compare the performance of different algorithms, you should test each using several different sets of initial weights and biases. Try the Neural Network Design Demonstration nnd12mo [HDB96] for an illustration of the performance of the batch momentum algorithm. 5-15 5 Backpropagation Faster Training The previous section presented two backpropagation training algorithms: gradient descent and gradient descent with momentum. These two methods are often too slow for practical problems. In this section we will discuss several high performance algorithms which can converge from ten to one hundred times faster than the algorithms discussed previously. All of the algorithms in this section operate in the batch mode and are invoked using train. These faster algorithms fall into two main categories. The first category uses heuristic techniques, which were developed from an analysis of the performance of the standard steepest descent algorithm. One heuristic modification is the momentum technique, which was presented in the previous section. This section will discuss two more heuristic techniques: variable learning rate backpropagation, traingda, and resilient backpropagation trainrp. The second category of fast algorithms uses standard numerical optimization techniques. (See Chapter 9 of [HDB96] for a review of basic numerical optimization.) Later in this section we will present three types of numerical optimization techniques for neural network training: conjugate gradient (traincgf, traincgp, traincgb, trainscg), quasi-Newton (trainbfg, trainoss), and Levenberg-Marquardt (trainlm). Variable Learning Rate (TRAINGDA, TRAINGDX) With standard steepest descent, the learning rate is held constant throughout training. The performance of the algorithm is very sensitive to the proper setting of the learning rate. If the learning rate is set too high, the algorithm may oscillate and become unstable. If the learning rate is too small, the algorithm will take too long to converge. It is not practical to determine the optimal setting for the learning rate before training, and, in fact, the optimal learning rate changes during the training process, as the algorithm moves across the performance surface. The performance of the steepest descent algorithm can be improved if we allow the learning rate to change during the training process. An adaptive learning rate will attempt to keep the learning step size as large as possible while keeping learning stable. The learning rate is made responsive to the complexity of the local error surface. 5-16 Faster Training An adaptive learning rate requires some changes in the training procedure used by traingd. First, the initial network output and error are calculated. At each epoch new weights and biases are calculated using the current learning rate. New outputs and errors are then calculated. As with momentum, if the new error exceeds the old error by more than a predefined ratio max_perf_inc (typically 1.04), the new weights and biases are discarded. In addition, the learning rate is decreased (typically by multiplying by lr_dec = 0.7). Otherwise the new weights, etc., are kept. If the new error is less than the old error, the learning rate is increased (typically by multiplying by lr_inc = 1.05). This procedure increases the learning rate, but only to the extent that the network can learn without large error increases. Thus a near optimal learning rate is obtained for the local terrain. When a larger learning rate could result in stable learning, the learning rate is increased. When the learning rate is too high to guarantee a decrease in error, it gets decreased until stable learning resumes. Try the Neural Network Design Demonstration nnd12vl [HDB96] for an illustration of the performance of the variable learning rate algorithm. 5-17 5 Backpropagation Backpropagation training with an adaptive learning rate is implemented with the function traingda which is called just like traingd except for the additional training parameters max_perf_inc, lr_dec, and lr_inc. Here is how it is called to train our previous two-layer network: net=newff([-1 2; 0 5],[3,1],{'tansig','purelin'},'traingda'); net.trainParam.show = 50; net.trainParam.lr = 0.05; net.trainParam.lr_inc = 1.05; net.trainParam.epochs = 300; net.trainParam.goal = 1e-5; p = [-1 -1 2 2;0 5 0 5]; t = [-1 -1 1 1]; net=train(net,p,t); TRAINGDA, Epoch 0/300, MSE 1.71149/1e-05, Gradient 2.6397/ 1e-06 TRAINGDA, Epoch 44/300, MSE 7.47952e-06/1e-05, Gradient 0.00251265/1e-06 TRAINGDA, Performance goal met. a = sim(net,p) a = -1.0036 -0.9960 1.0008 0.9991 The function traingdx combines adaptive learning rate with momentum training. It is invoked in the same way as traingda, except that it has the momentum coefficient mc as an additional training parameter. Resilient Backpropagation (TRAINRP) Multilayer networks typically use sigmoid transfer functions in the hidden layers. These functions are often called squashing functions, since they compress an infinite input range into a finite output range. Sigmoid functions are characterized by the fact that their slope must approach zero as the input gets large. This causes a problem when using steepest descent to train a multilayer network with sigmoid functions, since the gradient can have a very small magnitude, and therefore cause small changes in the weights and biases, even though the weights and biases are far from their optimal values. The purpose of the resilient backpropagation (Rprop) training algorithm is to eliminate these harmful effects of the magnitudes of the partial derivatives. Only the sign of the derivative is used to determine the direction of the weight 5-18 Faster Training update; the magnitude of the derivative has no effect on the weight update. The size of the weight change is determined by a separate update value. The update value for each weight and bias is increased by a factor delt_inc whenever the derivative of the performance function with respect to that weight has the same sign for two successive iterations. The update value is decreased by a factor delt_dec whenever the derivative with respect that weight changes sign from the previous iteration. If the derivative is zero, then the update value remains the same. Whenever the weights are oscillating the weight change will be reduced. If the weight continues to change in the same direction for several iterations, then the magnitude of the weight change will be increased. A complete description of the Rprop algorithm is given in [ReBr93]. In the following code we recreate our previous network and train it using the Rprop algorithm. The training parameters for trainrp are epochs, show, goal, time, min_grad, max_fail, delt_inc, delt_dec, delta0, deltamax. We have previously discussed the first eight parameters. The last two are the initial step size and the maximum step size, respectively. The performance of Rprop is not very sensitive to the settings of the training parameters. For the example below we leave most of the training parameters at the default values. We do reduce show below our previous value, because Rprop generally converges much faster than the previous algorithms. net=newff([-1 2; 0 5],[3,1],{'tansig','purelin'},'trainrp'); net.trainParam.show = 10; net.trainParam.epochs = 300; net.trainParam.goal = 1e-5; p = [-1 -1 2 2;0 5 0 5]; t = [-1 -1 1 1]; net=train(net,p,t); TRAINRP, Epoch 0/300, MSE 0.469151/1e-05, Gradient 1.4258/ 1e-06 TRAINRP, Epoch 10/300, MSE 0.000789506/1e-05, Gradient 0.0554529/1e-06 TRAINRP, Epoch 20/300, MSE 7.13065e-06/1e-05, Gradient 0.00346986/1e-06 TRAINRP, Performance goal met. a = sim(net,p) a = -1.0026 -0.9963 0.9978 1.0017 5-19 5 Backpropagation Rprop is generally much faster than the standard steepest descent algorithm. It also has the nice property that it requires only a modest increase in memory requirements. We do need to store the update values for each weight and bias, which is equivalent to storage of the gradient. Conjugate Gradient Algorithms The basic backpropagation algorithm adjusts the weights in the steepest descent direction (negative of the gradient). This is the direction in which the performance function is decreasing most rapidly. It turns out that, although the function decreases most rapidly along the negative of the gradient, this does not necessarily produce the fastest convergence. In the conjugate gradient algorithms a search is performed along conjugate directions, which produces generally faster convergence than steepest descent directions. In this section we will present four different variations of conjugate gradient algorithms. See page 12-14 of [HDB96] for a discussion of conjugate gradient algorithms and their application to neural networks. In most of the training algorithms that we have discussed up to this point, a learning rate is used to determine the length of the weight update (step size). In most of the conjugate gradient algorithms the step size is adjusted at each iteration. A search is made along the conjugate gradient direction to determine the step size which will minimize the performance function along that line. There are five different search functions that are included in the toolbox, and these will be discussed at the end of this section. Any of these search functions can be used interchangeably with a variety of the training functions described in the remainder of this chapter. Some search functions are best suited to certain training functions, although the optimum choice can vary according to the specific application. An appropriate default search function is assigned to each training function, but this can be modified by the user. Fletcher-Reeves Update (TRAINCGF) All of the conjugate gradient algorithms start out by searching in the steepest descent direction (negative of the gradient) on the first iteration. p0 = –g0 5-20 Faster Training A line search is then performed to determine the optimal distance to move along the current search direction: xk + 1 = xk + αk pk Then the next search direction is determined so that it is conjugate to previous search directions. The general procedure for determining the new search direction is to combine the new steepest descent direction with the previous search direction: pk = – gk + βk pk – 1 The various versions of conjugate gradient are distinguished by the manner in which the constant β k is computed. For the Fletcher-Reeves update the procedure is T gk gk β k = --------------------------- . T gk – 1 gk – 1 This is the ratio of the norm squared of the current gradient to the norm squared of the previous gradient. See [FlRe64] or [HDB96] for a discussion of the Fletcher-Reeves conjugate gradient algorithm. In the following code we re-initialize our previous network and retrain it using the Fletcher-Reeves version of the conjugate gradient algorithm. The training parameters for traincgf are epochs, show, goal, time, min_grad, max_fail, srchFcn, scal_tol, alpha, beta, delta, gama, low_lim, up_lim, maxstep, minstep, bmax. We have previously discussed the first six parameters. The parameter srchFcn is the name of the line search function. It can be any of the functions described later in this section (or a user-supplied function). The remaining parameters are associated with specific line search routines and are described later in this section. The default line search routine srchcha is used in this example. traincgf generally converges in fewer iterations than trainrp (although there is more computation required in each iteration). 5-21 5 Backpropagation net=newff([-1 2; 0 5],[3,1],{'tansig','purelin'},'traincgf'); net.trainParam.show = 5; net.trainParam.epochs = 300; net.trainParam.goal = 1e-5; p = [-1 -1 2 2;0 5 0 5]; t = [-1 -1 1 1]; net=train(net,p,t); TRAINCGF-srchcha, Epoch 0/300, MSE 2.15911/1e-05, Gradient 3.17681/1e-06 TRAINCGF-srchcha, Epoch 5/300, MSE 0.111081/1e-05, Gradient 0.602109/1e-06 TRAINCGF-srchcha, Epoch 10/300, MSE 0.0095015/1e-05, Gradient 0.197436/1e-06 TRAINCGF-srchcha, Epoch 15/300, MSE 0.000508668/1e-05, Gradient 0.0439273/1e-06 TRAINCGF-srchcha, Epoch 17/300, MSE 1.33611e-06/1e-05, Gradient 0.00562836/1e-06 TRAINCGF, Performance goal met. a = sim(net,p) a = -1.0001 -1.0023 0.9999 1.0002 The conjugate gradient algorithms are usually much faster than variable learning rate backpropagation, and are sometimes faster than trainrp, although the results will vary from one problem to another. The conjugate gradient algorithms require only a little more storage than the simpler algorithms, so they are often a good choice for networks with a large number of weights. Try the Neural Network Design Demonstration nnd12cg [HDB96] for an illustration of the performance of a conjugate gradient algorithm. Polak-Ribiére Update (TRAINCGP) Another version of the conjugate gradient algorithm was proposed by Polak and Ribiére. As with the Fletcher-Reeves algorithm, the search direction at each iteration is determined by pk = – gk + βk pk – 1 . 5-22 Faster Training For the Polak-Ribiére update, the constant β k is computed by T ∆g k – 1 g k β k = --------------------------- . T gk – 1 gk – 1 This is the inner product of the previous change in the gradient with the current gradient divided by the norm squared of the previous gradient. See [FlRe64] or [HDB96] for a discussion of the Polak-Ribiére conjugate gradient algorithm. In the following code we recreate our previous network and train it using the Polak-Ribiére version of the conjugate gradient algorithm. The training parameters for traincgp are the same as those for traincgf. The default line search routine srchcha is used in this example. The parameters show and epoch are set to the same values as they were for traincgf. net=newff([-1 2; 0 5],[3,1],{'tansig','purelin'},'traincgp'); net.trainParam.show = 5; net.trainParam.epochs = 300; net.trainParam.goal = 1e-5; p = [-1 -1 2 2;0 5 0 5]; t = [-1 -1 1 1]; net=train(net,p,t); TRAINCGP-srchcha, Epoch 0/300, MSE 1.21966/1e-05, Gradient 1.77008/1e-06 TRAINCGP-srchcha, Epoch 5/300, MSE 0.227447/1e-05, Gradient 0.86507/1e-06 TRAINCGP-srchcha, Epoch 10/300, MSE 0.000237395/1e-05, Gradient 0.0174276/1e-06 TRAINCGP-srchcha, Epoch 15/300, MSE 9.28243e-05/1e-05, Gradient 0.00485746/1e-06 TRAINCGP-srchcha, Epoch 20/300, MSE 1.46146e-05/1e-05, Gradient 0.000912838/1e-06 TRAINCGP-srchcha, Epoch 25/300, MSE 1.05893e-05/1e-05, Gradient 0.00238173/1e-06 TRAINCGP-srchcha, Epoch 26/300, MSE 9.10561e-06/1e-05, Gradient 0.00197441/1e-06 TRAINCGP, Performance goal met. a = sim(net,p) a = -0.9967 -1.0018 0.9958 1.0022 5-23 5 Backpropagation The traincgp routine has performance similar to traincgf. It is difficult to predict which algorithm will perform best on a given problem. The storage requirements for Polak-Ribiére (four vectors) are slightly larger than for Fletcher-Reeves (three vectors). Powell-Beale Restarts (TRAINCGB) For all conjugate gradient algorithms, the search direction will be periodically reset to the negative of the gradient. The standard reset point occurs when the number of iterations is equal to the number of network parameters (weights and biases), but there are other reset methods which can improve the efficiency of training. One such reset method was proposed by Powell [Powe77], based on an earlier version proposed by Beale [Beal72]. For this technique we will restart if there is very little orthogonality left between the current gradient and the previous gradient. This is tested with the following inequality: T g k – 1 g k ≥ 0.2 g k 2 . If this condition is satisfied, the search direction is reset to the negative of the gradient. In the following code we recreate our previous network and train it using the Powell-Beale version of the conjugate gradient algorithm. The training parameters for traincgb are the same as those for traincgf. The default line search routine srchcha is used in this example. The parameters show and epoch are set to the same values as they were for traincgf. net=newff([-1 2; 0 5],[3,1],{'tansig','purelin'},'traincgb'); net.trainParam.show = 5; net.trainParam.epochs = 300; net.trainParam.goal = 1e-5; p = [-1 -1 2 2;0 5 0 5]; t = [-1 -1 1 1]; net=train(net,p,t); TRAINCGB-srchcha, Epoch 0/300, MSE 2.5245/1e-05, Gradient 3.66882/1e-06 TRAINCGB-srchcha, Epoch 5/300, MSE 4.86255e-07/1e-05, Gradient 0.00145878/1e-06 TRAINCGB, Performance goal met. a = sim(net,p) a = -0.9997 -0.9998 1.0000 1.0014 5-24 Faster Training The traincgb routine has performance which is somewhat better than traincgp for some problems, although performance on any given problem is difficult to predict. The storage requirements for the Powell-Beale algorithm (six vectors) are slightly larger than for Polak-Ribiére (four vectors). Scaled Conjugate Gradient (TRAINSCG) Each of the conjugate gradient algorithms which we have discussed so far require a line search at each iteration. This line search is computationally expensive, since it requires that the network response to all training inputs be computed several times for each search. The scaled conjugate gradient algorithm (SCG), developed by Moller [Moll93], was designed to avoid the time consuming line search.This algorithm is too complex to explain in a few lines, but the basic idea is to combine the model-trust region approach, which is used in the Levenberg-Marquardt algorithm described later, with the conjugate gradient approach. See {Moll93] for a detailed explanation of the algorithm. In the following code we re-initialize our previous network and retrain it using the scaled conjugate gradient algorithm. The training parameters for trainscg are epochs, show, goal, time, min_grad, max_fail, sigma, lambda. We have previously discussed the first six parameters. The parameter sigma determines the change in the weight for the second derivative approximation. The parameter lambda regulates the indefiniteness of the Hessian. The parameters show and epoch are set to 10 and 300, respectively. net=newff([-1 2; 0 5],[3,1],{'tansig','purelin'},'trainscg'); net.trainParam.show = 10; net.trainParam.epochs = 300; net.trainParam.goal = 1e-5; p = [-1 -1 2 2;0 5 0 5]; t = [-1 -1 1 1]; net=train(net,p,t); TRAINSCG, Epoch 0/300, MSE 4.17697/1e-05, Gradient 5.32455/ 1e-06 TRAINSCG, Epoch 10/300, MSE 2.09505e-05/1e-05, Gradient 0.00673703/1e-06 TRAINSCG, Epoch 11/300, MSE 9.38923e-06/1e-05, Gradient 0.0049926/1e-06 TRAINSCG, Performance goal met. a = sim(net,p) a = -1.0057 -1.0008 1.0019 1.0005 5-25 5 Backpropagation The trainscg routine may require more iterations to converge than the other conjugate gradient algorithms, but the number of computations in each iteration is significantly reduced because no line search is performed. The storage requirements for the scaled conjugate gradient algorithm are about the same as those of Fletcher-Reeves. Line Search Routines Several of the conjugate gradient and quasi-Newton algorithms require that a line search be performed. In this section we describe five different line searches which can be used. In order to use any of these search routines you simply set the training parameter srchFcn equal to the name of the desired search function, as has been described in previous sections. It is often difficult to predict which of these routines will provide the best results for any given problem, but we have set the default search function to an appropriate initial choice for each training function, so you may never need to modify this parameter. Golden Section Search (SRCHGOL) The golden section search srchgol is a linear search which does not require the calculation of the slope. This routine begins by locating an interval in which the minimum of the performance occurs. This is accomplished by evaluating the performance at a sequence of points, starting at a distance of delta and doubling in distance each step, along the search direction. When the performance increases between two successive iterations, a minimum has been bracketed. The next step is to reduce the size of the interval containing the minimum. Two new points are located within the initial interval. The values of the performance at these two points determines a section of the interval which can be discarded, and a new interior point is placed within the new interval. This procedure is continued until the interval of uncertainty is reduced to a width of tol, which is equal to delta/scale_tol. See [HDB96], starting on page 12-16, for a complete description of the golden section search. Try the Neural Network Design Demonstration nnd12sd1 [HDB96] for an illustration of the performance of the golden section search in combination with a conjugate gradient algorithm. 5-26 Faster Training Brent’s Search (SRCHBRE) Brent’s search is a linear search which is a hybrid combination of the golden section search and a quadratic interpolation. Function comparison methods, like the golden section search, have a first-order rate of convergence, while polynomial interpolation methods have an asymptotic rate which is faster than superlinear. On the other hand, the rate of convergence for the golden section search starts when the algorithm is initialized, whereas the asymptotic behavior for the polynomial interpolation methods may take many iterations to become apparent. Brent’s search attempts to combine the best features of both approaches. For Brent’s search we begin with the same interval of uncertainty that we used with the golden section search, but some additional points are computed. A quadratic function is then fitted to these points and the minimum of the quadratic function is computed. If this minimum is within the appropriate interval of uncertainty, it is used in the next stage of the search and a new quadratic approximation is performed. If the minimum falls outside the known interval of uncertainty, then a step of the golden section search is performed. See [Bren73] for a complete description of this algorithm. This algorithm has the advantage that it does not require computation of the derivative. The derivative computation requires a backpropagation through the network, which involves more computation than a forward pass. However, the algorithm may require more performance evaluations than algorithms which use derivative information. Hybrid Bisection-Cubic Search (SRCHHYB) Like Brent’s search, srchhyb is a hybrid algorithm. It is a combination of bisection and cubic interpolation. For the bisection algorithm, one point is located in the interval of uncertainty and the performance and its derivative are computed. Based on this information, half of the interval of uncertainty is discarded. In the hybrid algorithm, a cubic interpolation of the function is obtained by using the value of the performance and its derivative at the two end points. If the minimum of the cubic interpolation falls within the known interval of uncertainty, then it is used to reduce the interval of uncertainty. Otherwise, a step of the bisection algorithm is used. See [Scal85] for a complete description of the hybrid bisection-cubic search. This algorithm does require derivative information, so it performs more computations at each step of the algorithm than the golden section search or Brent’s algorithm. 5-27 5 Backpropagation Charalambous’ Search (SRCHCHA) The method of Charalambous srchcha was designed to be used in combination with a conjugate gradient algorithm for neural network training. Like the previous two methods, it is a hybrid search. It uses a cubic interpolation, together with a type of sectioning. See [Char92] for a description of Charalambous’ search. We have used this routine as the default search for most of the conjugate gradient algorithms, since it appears to produce excellent results for many different problems. It does require the computation of the derivatives (backpropagation) in addition to the computation of performance, but it overcomes this limitation by locating the minimum with fewer steps. This is not true for all problems, and you may want to experiment with other line searches. Backtracking (SRCHBAC) The backtracking search routine srchbac is best suited to use with the quasi-Newton optimization algorithms. It begins with a step multiplier of 1 and then backtracks until an acceptable reduction in the performance is obtained. On the first step it uses the value of performance at the current point and at a step multiplier of 1, and also the derivative of performance at the current point, to obtain a quadratic approximation to the performance function along the search direction. The minimum of the quadratic approximation becomes a tentative optimum point (under certain conditions) and the performance at this point is tested. If the performance is not sufficiently reduced, a cubic interpolation is obtained and the minimum of the cubic interpolation becomes the new tentative optimum point. This process is continued until a sufficient reduction in the performance is obtained. The backtracking algorithm is described in [DeSc83]. It was used as the default line search for the quasi-Newton algorithms, although it may not be the best technique for all problems. 5-28 Faster Training Quasi-Newton Algorithms BFGS Algorithm (TRAINBFG) Newton’s method is an alternative to the conjugate gradient methods for fast optimization. The basic step of Newton’s method is –1 xk + 1 = xk – A k g k , where A k is the Hessian matrix (second derivatives) of the performance index at the current values of the weights and biases. Newton’s method often converges faster than conjugate gradient methods. Unfortunately, it is complex and expensive to compute the Hessian matrix for feedforward neural networks. There is a class of algorithms that are based on Newton’s method but which don’t require calculation of second derivatives. These are called quasi-Newton (or secant) methods. They update an approximate Hessian matrix at each iteration of the algorithm. The update is computed as a function of the gradient. The quasi-Newton method which has been most successful in published studies is the Broyden, Fletcher, Goldfarb, and Shanno (BFGS) update. This algorithm has been implemented in the trainbfg routine. In the following code we re-initialize our previous network and retrain it using the BFGS quasi-Newton algorithm. The training parameters for trainbfg are the same as those for traincgf. The default line search routine srchbac is used in this example. The parameters show and epoch are set to 5 and 300, respectively. 5-29 5 Backpropagation net=newff([-1 2; 0 5],[3,1],{'tansig','purelin'},'trainbfg'); net.trainParam.show = 5; net.trainParam.epochs = 300; net.trainParam.goal = 1e-5; p = [-1 -1 2 2;0 5 0 5]; t = [-1 -1 1 1]; net=train(net,p,t); TRAINBFG-srchbac, Epoch 0/300, MSE 0.492231/1e-05, Gradient 2.16307/1e-06 TRAINBFG-srchbac, Epoch 5/300, MSE 0.000744953/1e-05, Gradient 0.0196826/1e-06 TRAINBFG-srchbac, Epoch 8/300, MSE 7.69867e-06/1e-05, Gradient 0.00497404/1e-06 TRAINBFG, Performance goal met. a = sim(net,p) a = -0.9995 -1.0004 1.0008 0.9945 The BFGS algorithm is described in [DeSc83]. This algorithm requires more computation in each iteration and more storage than the conjugate gradient methods, although it generally converges in fewer iterations. The approximate 2 2 Hessian must be stored, and its dimension is n × n , where n is equal to the number of weights and biases in the network. For very large networks it may be better to use Rprop or one of the conjugate gradient algorithms. For smaller networks, however, trainbfg can be an efficient training function. One Step Secant Algorithm (TRAINOSS) Since the BFGS algorithm requires more storage and computation in each iteration than the conjugate gradient algorithms, there is need for a secant approximation with smaller storage and computation requirements. The one step secant (OSS) method is an attempt to bridge the gap between the conjugate gradient algorithms and the quasi-Newton (secant) algorithms. This algorithm does not store the complete Hessian matrix; it assumes that at each iteration the previous Hessian was the identity matrix. This has the additional advantage that the new search direction can be calculated without computing a matrix inverse. In the following code we re-initialize our previous network and retrain it using the one step secant algorithm. The training parameters for trainoss are the same as those for traincgf. The default line search routine srchbac is used in 5-30 Faster Training this example. The parameters show and epoch are set to 10 and 300, respectively. net=newff([-1 2; 0 5],[3,1],{'tansig','purelin'},'trainoss'); net.trainParam.show = 5; net.trainParam.epochs = 300; net.trainParam.goal = 1e-5; p = [-1 -1 2 2;0 5 0 5]; t = [-1 -1 1 1]; net=train(net,p,t); TRAINOSS-srchbac, Epoch 0/300, MSE 0.665136/1e-05, Gradient 1.61966/1e-06 TRAINOSS-srchbac, Epoch 5/300, MSE 0.000321921/1e-05, Gradient 0.0261425/1e-06 TRAINOSS-srchbac, Epoch 7/300, MSE 7.85697e-06/1e-05, Gradient 0.00527342/1e-06 TRAINOSS, Performance goal met. a = sim(net,p) a = -1.0035 -0.9958 1.0014 0.9997 The one step secant method is described in [Batt92]. This algorithm requires less storage and computation per epoch than the BFGS algorithm. It requires slightly more storage and computation per epoch than the conjugate gradient algorithms. It can be considered a compromise between full quasi-Newton algorithms and conjugate gradient algorithms. Levenberg-Marquardt (TRAINLM) Like the quasi-Newton methods, the Levenberg-Marquardt algorithm was designed to approach second-order training speed without having to compute the Hessian matrix. When the performance function has the form of a sum of squares (as is typical in training feedforward networks), then the Hessian matrix can be approximated as T H = J J, and the gradient can be computed as T g = J e 5-31 5 Backpropagation where J is the Jacobian matrix, which contains first derivatives of the network errors with respect to the weights and biases, and e is a vector of network errors. The Jacobian matrix can be computed through a standard backpropagation technique (see [HaMe94]) that is much less complex than computing the Hessian matrix. The Levenberg-Marquardt algorithm uses this approximation to the Hessian matrix in the following Newton-like update: T –1 T x k + 1 = x k – [ J J + µI ] J e . When the scalar µ is zero, this is just Newton’s method, using the approximate Hessian matrix. When µ is large, this becomes gradient descent with a small step size. Newton’s method is faster and more accurate near an error minimum, so the aim is to shift towards Newton’s method as quickly as possible. Thus, µ is decreased after each successful step (reduction in performance function) and is increased only when a tentative step would increase the performance function. In this way, the performance function will always be reduced at each iteration of the algorithm. In the following code we re-initialize our previous network and retrain it using the Levenberg-Marquardt algorithm. The training parameters for trainlm are epochs, show, goal, time, min_grad, max_fail, mu, mu_dec, mu_inc, mu_max, mem_reduc. We have discussed the first six parameters earlier. The parameter mu is the initial value for µ. This value is multiplied by mu_dec whenever the performance function is reduced by a step. It is multiplied by mu_inc whenever a step would increase the performance function. If mu becomes larger than mu_max, the algorithm is stopped. The parameter mem_reduc is used to control 5-32 Faster Training the amount of memory used by the algorithm. It will be discussed in the next section. The parameters show and epoch are set to 5 and 300, respectively. net=newff([-1 2; 0 5],[3,1],{'tansig','purelin'},'trainlm'); net.trainParam.show = 5; net.trainParam.epochs = 300; net.trainParam.goal = 1e-5; p = [-1 -1 2 2;0 5 0 5]; t = [-1 -1 1 1]; net=train(net,p,t); TRAINLM, Epoch 0/300, MSE 2.7808/1e-05, Gradient 7.77931/1e-10 TRAINLM, Epoch 4/300, MSE 3.67935e-08/1e-05, Gradient 0.000808272/1e-10 TRAINLM, Performance goal met. a = sim(net,p) a = -1.0000 -1.0000 1.0000 0.9996 The original description of the Levenberg-Marquardt algorithm is given in [Marq63]. The application of Levenberg-Marquardt to neural network training is described in [HaMe94] and starting on page 12-19 of [HDB96]. This algorithm appears to be the fastest method for training moderate-sized feedforward neural networks (up to several hundred weights). It also has a very efficient MATLAB implementation, since the solution of the matrix equation is a built-in function, so its attributes become even more pronounced in a MATLAB setting. Try the Neural Network Design Demonstration nnd12m [HDB96] for an illustration of the performance of the batch gradient descent algorithm. Reduced Memory Levenberg-Marquardt (TRAINLM) The main drawback of the Levenberg-Marquardt algorithm is that it requires the storage of some matrices which can be quite large for certain problems. The size of the Jacobian matrix is Q × n , where Q is the number of training sets and n is the number of weights and biases in the network. It turns out that this matrix does not have to be computed and stored as a whole. For example, if we were to divide the Jacobian into two equal submatrices we could compute the approximate Hessian matrix as follows: 5-33 5 Backpropagation T T H = J J = J1 T J2 T T J1 J2 = J1 J1 + J2 J2 . Therefore the full Jacobian does not have to exist at one time. The approximate Hessian can be computed by summing a series of subterms. Once one subterm has been computed, the corresponding submatrix of the Jacobian can be cleared. When using the training function trainlm, the parameter mem_reduc is used to determine how many rows of the Jacobian are to be computed in each submatrix. If mem_reduc is set to 1, then the full Jacobian is computed, and no memory reduction is achieved. If mem_reduc is set to 2, then only half of the Jacobian will be computed at one time. This saves half of the memory used by the calculation of the full Jacobian. There is a drawback to using memory reduction. A significant computational overhead is associated with computing the Jacobian in submatrices. If you have enough memory available, then it is better to set mem_reduc to 1 and to compute the full Jacobian. If you have a large training set, and you are running out of memory, then you should set mem_reduc to 2, and try again. If you still run out of memory, continue to increase mem_reduc. Even if you use memory reduction, the Levenberg-Marquardt algorithm will always compute the approximate Hessian matrix, which has dimensions n × n . If your network is very large, then you may run out of memory. If this is the case, then you will want to try trainoss, trainrp, or one of the conjugate gradient algorithms. 5-34 Speed and Memory Comparison Speed and Memory Comparison It is very difficult to know which training algorithm will be the fastest for a given problem. It will depend on many factors, including the complexity of the problem, the number of data points in the training set, the number of weights and biases in the network, and the error goal. In general, on networks which contain up to a few hundred weights the Levenberg-Marquardt algorithm will have the fastest convergence. This advantage is especially noticeable if very accurate training is required. The quasi-Newton methods are often the next fastest algorithms on networks of moderate size. The BFGS algorithm does require storage of the approximate Hessian matrix, but is generally faster than the conjugate gradient algorithms. Of the conjugate gradient algorithms, the Powell-Beale procedure requires the most storage, but usually has the fastest convergence. Rprop and the scaled conjugate gradient algorithm do not require a line search and have small storage requirements. They are reasonably fast, and are very useful for large problems. The variable learning rate algorithm is usually much slower than the other methods, and has about the same storage requirements as Rprop, but it can still be useful for some problems. There are certain situations in which it is better to converge more slowly. For example, when using early stopping you may have inconsistent results if you use an algorithm which converges too quickly. You may overshoot the point at which the error on the validation set is minimized. For most situations, we recommend that you try the Levenberg-Marquardt algorithm first. If this algorithm requires too much memory, then try the BFGS algorithm, or one of the conjugate gradient methods. The Rprop algorithm is also very fast, and has relatively small memory requirements. The following table gives some example convergence times for the various algorithms on one particular regression problem. In this problem a 1-10-1 network was trained on a data set with 41 input/output pairs until a mean square error performance of 0.01 was obtained. Twenty different test runs were made for each training algorithm on a Macintosh Powerbook 1400 to obtain the average numbers shown in the table. These numbers should be used with caution, since the performances shown here may not be typical for these algorithms on other types of problems. 5-35 5 Backpropagation You may notice that there is not a clear relationship between the number of floating point operations and the time required to reach convergence. This is because some of the algorithms can take advantage of efficient built-in MATLAB functions. This is especially true for the Levenberg-Marquardt algorithm. Function 5-36 Technique Time Epochs Mflops traingdx Variable Learning Rate 57.71 980 2.50 trainrp Rprop 12.95 185 0.56 trainscg Scaled Conj. Grad. 16.06 106 0.70 traincgf Fletcher-Powell CG 16.40 81 0.99 traincgp Polak-Ribiére CG 19.16 89 0.75 traincgb Powell-Beale CG 15.03 74 0.59 trainoss One-Step-Secant 18.46 101 0.75 trainbfg BFGS quasi-Newton 10.86 44 1.02 trainlm Levenberg-Marquardt 1.87 6 0.46 Improving Generalization Improving Generalization One of the problems that occurs during neural network training is called overfitting. The error on the training set is driven to a very small value, but when new data is presented to the network the error is large. The network has memorized the training examples, but it has not learned to generalize to new situations. The following figure shows the response of a 1-20-1 neural network which has been trained to approximate a noisy sine function. The underlying sine function is shown by the dotted line, the noisy measurements are given by the ‘+’ symbols, and the neural network response is given by the solid line. Clearly this network has overfit the data and will not generalize well. Function Approximation 1.5 1 Output 0.5 0 -0.5 -1 -1.5 -1 -0.8 -0.6 -0.4 -0.2 0 Input 0.2 0.4 0.6 0.8 1 One method for improving network generalization is to use a network which is just large enough to provide an adequate fit. The larger a network you use, the more complex the functions that the network can create. If we use a small enough network, it will not have enough power to overfit the data. Run the Neural Network Design Demonstration nnd11gn [HDB96] to investigate how reducing the size of a network can prevent overfitting. The problem is that it is difficult to know beforehand how large a network should be for a specific application. There are two other methods for improving 5-37 5 Backpropagation generalization which are implemented in the Neural Network Toolbox: regularization and early stopping. The next few subsections will describe these two techniques, and the routines to implement them. Regularization The first method for improving generalization is called regularization. This involves modifying the performance function, which is normally chosen to be the sum of squares of the network errors on the training set. The next subsection will explain how the performance function can be modified, and the following subsection will describe two routines which will automatically set the optimal performance function to achieve the best generalization. Modified Performance Function The typical performance function that is used for training feedforward neural networks is the mean sum of squares of the network errors: 1 F = mse = ----N N ∑ i=1 1 ( e i ) = ----N N ∑ ( ti – ai ) 2 2 . i=1 It is possible to improve generalization if we modify the performance function by adding a term that consists of the mean of the sum of squares of the network weights and biases: msereg = γmse + ( 1 – γ )msw , where γ is the performance ratio, and 1 msw = --n n ∑ wj . 2 j=1 Using this performance function will cause the network to have smaller weights and biases, and this will force the network response to be smoother and less likely to overfit. In the following code we re-initialize our previous network and retrain it using the BFGS algorithm with the regularized performance function. Here we set 5-38 Improving Generalization the performance ratio to 0.5, which gives equal weight to the mean square errors and the mean square weights. net=newff([-1 2; 0 5],[3,1],{'tansig','purelin'},'trainbfg'); net.performFcn = 'msereg'; net.performParam.ratio = 0.5; net.trainParam.show = 5; net.trainParam.epochs = 300; net.trainParam.goal = 1e-5; p = [-1 -1 2 2;0 5 0 5]; t = [-1 -1 1 1]; net=train(net,p,t); The problem with regularization is that it is difficult to determine the optimum value for the performance ratio parameter. If we make this parameter too large, we may get overfitting. If the ratio is too small, the network will not adequately fit the training data. In the next section we will describe two routines which automatically set the regularization parameters. Automated Regularization (TRAINBR) It is desirable to determine the optimal regularization parameters in an automated fashion. One approach to this process is the Bayesian framework of David MacKay [MacK92]. In this framework the weights and biases of the network are assumed to be random variables with specified distributions. The regularization parameters are related to the unknown variances associated with these distributions. We can then estimate these parameters using statistical techniques. A detailed discussion of Bayesian regularization is beyond the scope of this users guide. A detailed discussion of the use of Bayesian regularization, in combination with Levenberg-Marquardt training, can be found in [FoHa97]. 5-39 5 Backpropagation Bayesian regularization has been implemented in the function trainbr. The following code shows how we can train a 1-20-1 network using this function to approximate the noisy sine wave shown earlier in this section. net=newff([-1 1],[20,1],{'tansig','purelin'},'trainbr'); net.trainParam.show = 10; net.trainParam.epochs = 50; randn('seed',192736547); p = [-1:.05:1]; t = sin(2*pi*p)+0.1*randn(size(p)); net = init(net); net=train(net,p,t); TRAINBR, Epoch 0/50, SSE 107.962/0, Gradient 9.51e+01/1.00e-10, #Par 6.10e+01/61 TRAINBR, Epoch 10/50, SSE 20.5/0, Gradient 1.88e-01/1.00e-10, #Par 1.82e+01/61 TRAINBR, Epoch 20/50, SSE 20.5/0, Gradient 4.07e-02/1.00e-10, #Par 1.65e+01/61 TRAINBR, Epoch 30/50, SSE 20.5/0, Gradient 5.57e-01/1.00e-10, #Par 1.55e+01/61 TRAINBR, Epoch 40/50, SSE 20.5/0, Gradient 2.76e-01/1.00e-10, #Par 1.48e+01/61 TRAINBR, Epoch 50/50, SSE 20.5/0, Gradient 4.05e-01/1.00e-10, #Par 1.42e+01/61 One feature of this algorithm is that it provides a measure of how many network parameters (weights and biases) are being effectively used by the network. In this case, the final trained network uses approximately 14 parameters (indicated by #Par in the printout), out of the 61 total weights and biases in the 1-20-1 network. This effective number of parameters should remain the same, no matter how large the total number of parameters in the network becomes. The following figure shows the response of the network. In contrast to the previous figure, in which a 1-20-1 network overfit the data, here we see that the network response is very close to the underlying sine function (dotted line), and, therefore, the network will generalize well to new inputs. We could have tried an even larger network, but the network response would never overfit the data. This eliminates the guesswork required in determining the optimum network size. 5-40 Improving Generalization Function Approximation 1.5 1 Output 0.5 0 -0.5 -1 -1.5 -1 -0.8 -0.6 -0.4 -0.2 0 Input 0.2 0.4 0.6 0.8 1 Early Stopping Another method for improving generalization is called early stopping. In this technique the available data is divided into three subsets. The first subset is the training set which is used for computing the gradient and updating the network weights and biases. The second subset is the validation set. The error on the validation set is monitored during the training process. The validation error will normally decrease during the initial phase of training, as does the training set error. However, when the network begins to overfit the data, the error on the validation set will typically begin to rise. When the validation error increases for a specified number of iterations, the training is stopped, and the weights and biases at the minimum of the validation error are returned. The test set error is not used during the training, but it is used to compare different models. It is also useful to plot the test set error during the training process. If the error in the test set reaches a minimum at a significantly different iteration number than the validation set error, this may indicate a poor division of the data set. Early stopping can be used with any of the training functions which were described earlier in this chapter. You simply need to pass the validation data to the training function. The following sequence of commands demonstrates how to use the early stopping function. 5-41 5 Backpropagation First we will create a simple test problem. For our training set we will generate a noisy sine wave with input points ranging from -1 to 1 at steps of 0.05. p = [-1:0.05:1]; t = sin(2*pi*p)+0.1*randn(size(p)); Next we will generate the validation set. The inputs will range from -1 to 1, as in the test set, but we will offset them slightly. To make the problem more realistic, we also add a different noise sequence to the underlying sine wave. Notice that the validation set is contained in a structure which contains both the inputs and the targets. v.P = [-0.975:.05:0.975]; v.T = sin(2*pi*v.P)+0.1*randn(size(v.P)); We will now create a 1-20-1 network, as in our previous example with regularization, and train it. (Notice that the validation structure is passed to train after the initial input and layer conditions, which are null vectors in this case since the network contains no delays.) For this example we will use the training function traingdx, although early stopping can be used with any of the other training functions we have discussed in this chapter. net=newff([-1 1],[20,1],{'tansig','purelin'},'traingdx'); net.trainParam.show = 25; net.trainParam.epochs = 300; net = init(net); [net,tr]=train(net,p,t,[],[],v); TRAINGDX, Epoch 0/300, MSE 9.39342/0, Gradient 17.7789/1e-06 TRAINGDX, Epoch 25/300, MSE 0.312465/0, Gradient 0.873551/1e-06 TRAINGDX, Epoch 50/300, MSE 0.102526/0, Gradient 0.206456/1e-06 TRAINGDX, Epoch 75/300, MSE 0.0459503/0, Gradient 0.0954717/1e-06 TRAINGDX, Epoch 100/300, MSE 0.015725/0, Gradient 0.0299898/1e-06 TRAINGDX, Epoch 125/300, MSE 0.00628898/0, Gradient 0.042467/ 1e-06 TRAINGDX, Epoch 131/300, MSE 0.00650734/0, Gradient 0.133314/ 1e-06 TRAINGDX, Validation stop. 5-42 Improving Generalization In the following figure we have a graph of the network response. We can see that the network did not overfit the data, as in the earlier example, although the response is not extremely smooth, as when using regularization. This is characteristic of early stopping. Function Approximation 1.5 1 Output 0.5 0 -0.5 -1 -1.5 -1 -0.8 -0.6 -0.4 -0.2 0 Input 0.2 0.4 0.6 0.8 1 5-43 5 Backpropagation Preprocessing and Postprocessing Neural network training can be made more efficient if certain preprocessing steps are performed on the network inputs and targets. In this section we describe several preprocessing routines which can be used. Min and Max (PREMNMX, POSTMNMX, TRAMNMX) Before training, it is often useful to scale the inputs and targets so that they always fall within a specified range. The function premnmx can be used to scale inputs and targets so that they fall in the range [-1,1]. The following code illustrates the use of this function. [pn,minp,maxp,tn,mint,maxt] = premnmx(p,t); net=train(net,pn,tn); The original network inputs and targets are given in the matrices p and t. The normalized inputs and targets, pn and tn, that are returned will all fall in the interval [-1,1]. The vectors minp and maxp contain the minimum and maximum values of the original inputs, and the vectors mint and maxt contain the minimum and maximum values of the original targets. After the network has been trained, these vectors should be used to transform any future inputs which are applied to the network. They effectively become a part of the network, just like the network weights and biases. If premnmx is used to scale both the inputs and targets, then the output of the network will be trained to produce outputs in the range [-1,1]. If you want to convert these outputs back into the same units which were used for the original targets, then you should use the routine postmnmx. In the following code we simulate the network which was trained in the previous code and then convert the network output back into the original units. an = sim(net,pn); a = postmnmx(an,mint,maxt); The network output an will correspond to the normalized targets tn. The un-normalized network output a is in the same units as the original targets t. If premnmx is used to preprocess the training set data, then whenever the trained network is used with new inputs they should be preprocessed with the minimum and maximums which were computed for the training set. This can 5-44 Preprocessing and Postprocessing be accomplished with the routine tramnmx. In the following code we apply a new set of inputs to the network we have already trained. pnewn = tramnmx(pnew,minp,maxp); anewn = sim(net,pnewn); anew = postmnmx(anewn,mint,maxt); Mean and Stand. Dev. (PRESTD, POSTSTD, TRASTD) Another approach for scaling network inputs and targets is to normalize the mean and standard deviation of the training set. This procedure is implemented in the function prestd. It normalizes the inputs and targets so that they will have zero mean and unity standard deviation. The following code illustrates the use of prestd. [pn,meanp,stdp,tn,meant,stdt] = prestd(p,t); The original network inputs and targets are given in the matrices p and t. The normalized inputs and targets, pn and tn, that are returned will have zero means and unity standard deviation. The vectors meanp and stdp contain the mean and standard deviations of the original inputs, and the vectors meant and stdt contain the means and standard deviations of the original targets. After the network has been trained, these vectors should be used to transform any future inputs which are applied to the network. They effectively become a part of the network, just like the network weights and biases. If prestd is used to scale both the inputs and targets, then the output of the network will be trained to produce outputs with zero mean and unity standard deviation. If you want to convert these outputs back into the same units which were used for the original targets, then you should use the routine poststd. In the following code we simulate the network which was trained in the previous code and then convert the network output back into the original units. an = sim(net,pn); a = poststd(an,meant,stdt); The network output an will correspond to the normalized targets tn. The un-normalized network output a is in the same units as the original targets t. If prestd is used to preprocess the training set data, then whenever the trained network is used with new inputs they should be preprocessed with the means and standard deviations which were computed for the training set. This can be 5-45 5 Backpropagation accomplished with the routine trastd. In the following code we apply a new set of inputs to the network we have already trained. pnewn = trastd(pnew,meanp,stdp); anewn = sim(net,pnewn); anew = poststd(anewn,meant,stdt); Principal Component Analysis (PREPCA, TRAPCA) In some situations the dimension of the input vector is large, but the components of the vectors are highly correlated (redundant). It is useful in this situation to reduce the dimension of the input vectors. An effective procedure for performing this operation is principal component analysis. This technique has three effects: it orthogonalizes the components of the input vectors (so that they are uncorrelated with each other); it orders the resulting orthogonal components (principal components) so that those with the largest variation come first; and it eliminates those components which contribute the least to the variation in the data set. The following code illustrates the use of prepca, which performs a principal component analysis. [pn,meanp,stdp] = prestd(p); [ptrans,transMat] = prepca(pn,0.02); Note that we first normalize the input vectors, using prestd, so that they have zero mean and unity variance. This is a standard procedure when using principal components. In this example, the second argument passed to prepca is 0.02. This means that prepca will eliminate those principal components which contribute less than 2% to the total variation in the data set. The matrix ptrans contains the transformed input vectors. The matrix transMat contains the principal component transformation matrix. After the network has been trained, this matrix should be used to transform any future inputs which are applied to the network. It effectively becomes a part of the network, just like the network weights and biases. If you multiply the normalized input vectors pn by the transformation matrix transMat, you will obtain the transformed input vectors ptrans. If prepca is used to preprocess the training set data, then whenever the trained network is used with new inputs they should be preprocessed with the transformation matrix which was computed for the training set. 5-46 Preprocessing and Postprocessing This can be accomplished with the routine trapca. In the following code we apply a new set of inputs to a network we have already trained. pnewn = trastd(pnew,meanp,stdp); pnewtrans = trapca(pnewn,transMat); a = sim(net,pnewtrans); Post-training Analysis (POSTREG) The performance of a trained network can be measured to some extent by the errors on the training, validation and test sets, but it is often useful to investigate the network response in more detail. One option is to perform a regression analysis between the network response and the corresponding targets. The routine postreg is designed to perform this analysis. The following commands illustrate how we can perform a regression analysis on the network which we previously trained in the early stopping section. a = sim(net,p); [m,b,r] = postreg(a,t) m = 0.9874 b = -0.0067 r = 0.9935 Here we pass the network output and the corresponding targets to postreg. It returns three parameters. The first two, m and b, correspond to the slope and the y-intercept of the best linear regression relating targets to network outputs. If we had a perfect fit (outputs exactly equal to targets), the slope would be 1, and the y-intercept would be 0. In this example we can see that the numbers are very close. The third variable returned by postreg is the correlation coefficient (R-value) between the outputs and targets. It is a measure of how well the variation in the output is explained by the targets. If this number is equal to 1, then there is perfect correlation between targets and outputs. In our example here the number is very close to 1, which indicates a good fit. The following figure illustrates the graphical output provided by postreg. The network outputs are plotted versus the targets as open circles. The best linear fit is indicated by a dashed line. The perfect fit (output equal to targets) is 5-47 5 Backpropagation indicated by the solid line. In this example it is difficult to distinguish the best linear fit line from the perfect fit line, because the fit is so good. Best Linear Fit: A = (0.987) T + (-0.00667) 1.5 Data Points A=T Best Linear Fit 1 A 0.5 0 -0.5 R = 0.994 -1 -1.5 -1.5 5-48 -1 -0.5 0 T 0.5 1 1.5 Sample Training Session Sample Training Session We have covered a number of different concepts in this chapter. At this point it might be useful to put some of these ideas together with an example of how a typical training session might go. For this example we are going to use data from a medical application [PuLu92]. We want to design an instrument which can determine serum cholesterol levels from measurements of spectral content of a blood sample. We have a total of 264 patients for which we have measurements of 21 wavelengths of the spectrum. For the same patients we also have measurements of hdl, ldl and vldl cholesterol levels, based on serum separation. The first step is to load the data into the workspace and perform a principal component analysis. load choles_all [pn,meanp,stdp,tn,meant,stdt] = prestd(p,t); [ptrans,transMat] = prepca(pn,0.001); Here we have conservatively retained those principal components which account for 99.9% of the variation in the data set. Let’s check the size of the transformed data. [R,Q] = size(ptrans) R = 4 Q = 264 There was apparently significant redundancy in the data set, since the principal component analysis has reduced the size of the input vectors from 21 to 4. The next step is to divide the data up into training, validation and test subsets. We will take one fourth of the data for the validation set, one fourth for the test set and one half for the training set. We will pick the sets as equally spaced points throughout the original data. iitst = 2:4:Q; iival = 4:4:Q; iitr = [1:4:Q 3:4:Q]; v.P = ptrans(:,iival); v.T = tn(:,iival); t.P = ptrans(:,iitst); t.V = tn(:,iitst); ptr = ptrans(:,iitr); ttr = tn(:,iitr); 5-49 5 Backpropagation We are now ready to create a network and train it. For this example we will try a two-layer network, with tan-sigmoid transfer function in the hidden layer and a linear transfer function in the output layer. This is a useful structure for function approximation (or regression) problems. As an initial guess, we will use five neurons in the hidden layer. The network should have three output neurons since there are three targets. We will use the Levenberg-Marquardt algorithm for training. net = newff(minmax(ptr),[5 3],{'tansig' 'purelin'},'trainlm'); [net,tr]=train(net,ptr,ttr,[],[],v,t); TRAINLM, Epoch 0/100, MSE 3.11023/0, Gradient 804.959/1e-10 TRAINLM, Epoch 15/100, MSE 0.330295/0, Gradient 104.219/1e-10 TRAINLM, Validation stop. The training stopped after 15 iterations because the validation error increased. It is a useful diagnostic tool to plot the training, validation and test errors to check the progress of training. We can do that with the following commands. plot(tr.epoch,tr.perf,tr.epoch,tr.vperf,tr.epoch,tr.tperf) legend('Training','Validation','Test',-1); ylabel('Squared Error'); xlabel('Epoch') The result is shown in the following figure. The result here is reasonable, since the test set error and the validation set error have similar characteristics, and it doesn’t appear that any significant overfitting has occurred. 5-50 Sample Training Session 3.5 3 Training Validation Test Squared Error 2.5 2 1.5 1 0.5 0 0 5 10 15 Epoch The next step is to perform some analysis of the network response. We will put the entire data set through the network (training, validation and test) and will perform a linear regression between the network outputs and the corresponding targets. First we will need to un-normalize the network outputs. an = sim(net,ptrans); a = poststd(an,meant,stdt); for i=1:3 figure(i) [m(i),b(i),r(i)] = postreg(a(i,:),t(i,:)); end In this case we have three outputs, so we perform three regressions. The results are shown in the following figures. 5-51 5 Backpropagation Best Linear Fit: A = (0.764) T + (14) 400 350 Data Points A=T Best Linear Fit 300 250 A 200 150 100 R = 0.886 50 0 -50 0 50 100 150 200 T 250 300 350 400 Best Linear Fit: A = (0.753) T + (31.7) 350 300 Data Points A=T Best Linear Fit 250 A 200 150 100 R = 0.862 50 0 0 50 100 150 200 T 5-52 250 300 350 Sample Training Session Best Linear Fit: A = (0.346) T + (28.3) 120 Data Points A=T Best Linear Fit 100 A 80 60 40 R = 0.563 20 0 0 20 40 60 T 80 100 120 The first two outputs seem to track the targets reasonably well (this is a difficult problem), and the R-values are almost 0.9. The third output (vldl levels) is not well modeled. We probably need to work more on that problem. We might go on to try other network architectures (more hidden layer neurons), or to try Bayesian regularization instead of early stopping for our training technique. Of course there is also the possibility that vldl levels cannot be accurately computed based on the given spectral components. 5-53 5 Backpropagation Limitations and Cautions The gradient descent algorithm is generally very slow, because it requires small learning rates for stable learning. The momentum variation is usually faster than simple gradient descent, since it allows higher learning rates while maintaining stability, but it is still too slow for many practical applications. These two methods would normally be used only when incremental training is desired. You would normally use Levenberg-Marquardt training if you have enough memory available. If memory is a problem, then there are a variety of other fast algorithms available. Multi-layered networks are capable of performing just about any linear or nonlinear computation, and can approximate any reasonable function arbitrarily well. Such networks overcome the problems associated with the perceptron and linear networks. However, while the network being trained may be capable theoretically of performing correctly, backpropagation and its variations may not always find a solution. See page 12-8 of [HDB96] for a discussion of convergence to local minimum points. Picking the learning rate for a nonlinear network is a challenge. As with linear networks, a learning rate that is too large leads to unstable learning. Conversely, a learning rate that is too small results in incredibly long training times. Unlike linear networks, there is no easy way of picking a good learning rate for nonlinear multilayer networks. See page 12-8 of [HDB96] for examples of choosing the learning rate. With the faster training algorithms, the default parameter values will normally perform adequately. The error surface of a nonlinear network is more complex than the error surface of a linear network. To understand this complexity see the figures on pages 12-5 to 12-7 of [HDB96], which show three different error surfaces for a multilayer network. The problem is that nonlinear transfer functions in multilayer networks introduce many local minima in the error surface. As gradient descent is performed on the error surface it is possible for the network solution to become trapped in one of these local minima. This may happen depending on the initial starting conditions. Settling in a local minimum may be good or bad depending on how close the local minimum is to the global minimum and how low an error is required. In any case, be cautioned that although a multilayer backpropagation network with enough neurons can implement just about any function, backpropagation will not always find the correct weights for the optimum solution. You may wish to re-initialize the network and retrain several times to guarantee that you have the best solution. 5-54 Limitations and Cautions Networks are also sensitive to the number of neurons in their hidden layers. Too few neurons can lead to underfitting. Too many neurons can contribute to overfitting, in which all training points are well fit, but the fitting curve takes wild oscillations between these points. Ways of dealing with various of these issues have been discussed in the section on improving generalization. This topic is also discussed starting on page 11-21 of [HDB96]. 5-55 5 Backpropagation Summary Backpropagation can train multilayer feed-forward networks with differentiable transfer functions to perform function approximation, pattern association, and pattern classification. (Other types of networks can be trained as well, although the multilayer network is most commonly used.) The term backpropagation refers to the process by which derivatives of network error, with respect to network weights and biases, can be computed. This process can be used with a number of different optimization strategies. The architecture of a multilayer network is not completely constrained by the problem to be solved. The number of inputs to the network is constrained by the problem, and the number of neurons in the output layer is constrained by the number of outputs required by the problem. However, the number of layers between network inputs and the output layer and the sizes of the layers are up to the designer. The two-layer sigmoid/linear network can represent any functional relationship between inputs and outputs if the sigmoid layer has enough neurons. There are several different backpropagation training algorithms. They have a variety of different computation and storage requirements, and no one algorithm is best suited to all locations. The following list summarizes the training algorithms included in the toolbox. 5-56 Function Description traingd Basic gradient descent. Slow response, can be used in incremental mode training. traingdm Gradient descent with momentum. Generally faster than traingd. Can be used in incremental mode training. traingdx Adaptive learning rate. Faster training than traingd, but can only be used in batch mode training. trainrp Resilient backpropagation. Simple batch mode training algorithm with fast convergence and minimal storage requirements. Summary Function Description traincgf Fletcher-Reeves conjugate gradient algorithm. Has smallest storage requirements of the conjugate gradient algorithms. traincgp Polak-Ribiére conjugate gradient algorithm. Slightly larger storage requirements than traincgf. Faster convergence on some problems. traincgb Powell-Beale conjugate gradient algorithm. Slightly larger storage requirements than traincgp. Generally faster convergence. trainscg Scaled conjugate gradient algorithm. The only conjugate gradient algorithm that requires no line search. trainbfg BFGS quasi-Newton method. Requires storage of approximate Hessian matrix and has more computation in each iteration than conjugate gradient algorithms, but usually converges in fewer iterations. trainoss One step secant method. Compromise between conjugate gradient methods and quasi-Newton methods. trainlm Levenberg-Marquardt algorithm. Fastest training algorithm for networks of moderate size. Has memory reduction feature for use when the training set is large. trainbr Bayesian regularization. Modification of the Levenberg-Marquardt training algorithm to produce networks which generalize well. Reduces the difficulty of determining the optimum network architecture. One problem which can occur when training neural networks is that the network can overfit on the training set and not generalize well to new data outside the training set. This can be prevented by training with trainbr, but it can also be prevented by using early stopping with any of the other training routines. This requires that the user pass a validation set to the training algorithm, in addition to the standard training set. 5-57 5 Backpropagation In order to produce the most efficient training, it is often helpful to preprocess the data before training. It is also helpful to analyze the network response after training is complete. The toolbox contains a number of routines for pre- and post-processing. They are summarized in the following table. 5-58 Function Description premnmx Normalize data to fall in the range [-1,1]. postmnmx Inverse of premnmx. Used to convert data back to standard units. tramnmx Normalize data using previously computed minimums and maximums. Used to preprocess new inputs to networks which have been trained with data normalized with premnmx. prestd Normalize data to have zero mean and unity standard deviation. poststd Inverse of prestd. Used to convert data back to standard units. trastd Normalize data using previously computed means and standard deviations. Used to preprocess new inputs to networks which have been trained with data normalized with prestd. prepca Principal component analysis. Reduces dimension of input vector and un-correlates components of input vectors. trapca Preprocess data using previously computed principal component transformation matrix. Used to preprocess new inputs to networks which have been trained with data transformed with prepca. postreg Linear regression between network outputs and targets. Used to determine the adequacy of network fit. 6 Radial Basis Networks Introduction . . . . . . . . . . . . . . . . . . . . 6-2 Important Radial Basis Functions . . . . . . . . . . . . 6-2 Radial Basis Functions . . . Neuron Model . . . . . . . . Network Architecture . . . . . Exact Design (NEWRBE) . . . More Efficient Design (NEWRB) Demonstrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-3 6-3 6-4 6-5 6-7 6-8 Generalized Regression Networks . . . . . . . . . . 6-9 Network Architecture . . . . . . . . . . . . . . . . . 6-9 Design (NEWGRNN) . . . . . . . . . . . . . . . . . 6-11 Probabilistic Neural Networks . . . . . . . . . . . 6-12 Network Architecture . . . . . . . . . . . . . . . . . 6-12 Design (NEWPNN) . . . . . . . . . . . . . . . . . . 6-13 Summary . . . . . . . . . . . . . . . . . . . . . 6-16 Figures . . . . . . . . . . . . . . . . . . . . . . . 6-17 New Functions . . . . . . . . . . . . . . . . . . . 6-19 6 Radial Basis Networks Introduction Radial basis networks may require more neurons than standard feed-forward backpropagation networks, but often they can be designed in a fraction of the time it takes to train standard feed-forward networks. They work best when many training vectors are available. You may want to consult the following paper on this subject: Chen, S., C.F.N. Cowan, and P. M. Grant, “Orthogonal Least Squares Learning Algorithm for Radial Basis Function Networks,” IEEE Transactions on Neural Networks, vol. 2, no. 2, March 1991, pp. 302-309. This chapter discusses two variants of radial basis networks, Generalized Regression networks (GRNN) and Probabilistic neural networks (PNN). You may wish to read about them in P.D. Wasserman, Advanced Methods in Neural Computing, New York: Van Nostrand Reinhold, 1993 on pp. 155-61, and pp. 35-55 respectively. Important Radial Basis Functions Radial basis networks can be designed with either newrbe or newrb. GRNN and PNN can be designed with newgrnn and newpnn respectively. Type help radbasis to see a listing of all functions and demonstrations related to radial basis networks. 6-2 Radial Basis Functions Radial Basis Functions Neuron Model Here is a radial basis network with R inputs. Input Radial Basis Neuron 1,1 p p1 p2 3 ... w1, R AA A AA A w + _ n || dist || a b p R 1 a = radbas( || w-p || b) Notice that the expression for the net input of a radbas neuron is different from that of neurons in previous chapters. Here the net input to the radbas transfer function is the vector distance between its weight vector w and the input vector p, multiplied by the bias b. (The dist box in this figure accepts the input vector p and the single row input weight matrix, and produces the dot product of the two.) The transfer function for a radial basis neuron is: radbas ( n ) = e –n 2 Here is a plot of the radbas transfer function. a 1.0 0.5 0.0 AA AA n -0.833 +0.833 a = radbas(n) Radial Basis Function 6-3 6 Radial Basis Networks The radial basis function has a maximum of 1 when its input is 0. As the distance between w and p decreases, the output increases. Thus a radial basis neuron acts as a detector which produces 1 whenever the input p is identical to its weight vector p. The bias b allows the sensitivity of the radbas neuron to be adjusted. For example, if a neuron had a bias of 0.1 it would output 0.5 for any input vector p at vector distance of 8.326 (0.8326/b) from its weight vector w. Network Architecture Radial basis networks consist of two layers: a hidden radial basis layer of S1 neurons and an output linear layer of S2 neurons. AAA AAA AAA AA AA AA AA AA AA AAA AAA AAAA AA Radial Basis Layer Input S1xR p Rx1 1 R Linear Layer IW1,1 || dist || a1 S1x1 .* S1x1 n1 S2xS1 a2 S2x1 n2 S2x1 S1x1 b1 S1x1 LW2,1 1 S1 a 1 = radbas ( || IW1,1 - p || b 1) i i i b2 S2x1 Where... R = # elements in input vector S2 a2 = purelin (LW2,1 a1 +b2) S1 = # Neurons in layer 1 S2 = # Neurons in layer 2 ai1 is i th element of a1 where i IW1,1 is a vector made of the i th row of IW1,1 The dist box in this figure accepts the input vector p and the input weight matrix IW1,1, and produces a vector having S1 elements. The elements are the distances between the input vector and vectors iIW1,1 formed from the rows of the input weight matrix. The bias vector b1 and the output of dist are combined with the MATLAB operation .* , which does element-by-element multiplication. The output of the first layer for a feed forward network net can be obtained with the following code: a{1} = radbas(netprod(dist(net.IW{1,1},p),net.b{1})) 6-4 Radial Basis Functions Fortunately, you won’t have to write such lines of code. All of the details of designing this network are built into design functions newrbe and newrb, and their outputs can be obtained with sim. We can understand how this network behaves by following an input vector p through the network to the output a2. If we present an input vector to such a network, each neuron in the radial basis layer will output a value according to how close the input vector is to each neuron’s weight vector. Thus, radial basis neurons with weight vectors quite different from the input vector p will have outputs near zero. These small outputs will have only a negligible effect on the linear output neurons. In contrast, a radial basis neuron with a weight vector close to the input vector p will produce a value near 1. If a neuron has an output of 1 its output weights in the second layer pass their values to the linear neurons in the second layer. In fact, if only one radial basis neuron had an output of 1, and all others had outputs of 0’s (or very close to 0), the output of the linear layer would be the active neuron’s output weights. This would, however, be an extreme case. Typically several neurons are always firing, to varying degrees. Now let us look in detail at how the first layer operates. Each neuron's weighted input is the distance between the input vector and its weight vector, calculated with dist. Each neuron's net input is the element by element product of its weighted input with its bias, calculated with netprod. Each neurons' output is its net input passed through radbas. If a neuron's weight vector is equal to the input vector (transposed) its weighted input will be 0, its net input will be 0, and its output will be 1. If a neuron's weight vector is a distance of spread from the input vector, its weighted input will be spread, its net input will be sqrt(-log(.5)) (or 0.8326), therefore its output will be 0.5. Exact Design (NEWRBE) Radial basis networks can be designed with the function newrbe. This function can produce a network with zero error on training vectors. It is called in the following way net = newrbe(P,T,SPREAD) The function newrbe takes matrices of input vectors P and target vectors T, and a spread constant SPREAD for the radial basis layer, and returns a network with weights and biases such that the outputs are exactly T when the inputs are P. 6-5 6 Radial Basis Networks This function newrbe creates as many radbas neurons as there are input vectors in P, and sets the first layer weights to P'. Thus, we have a layer of radbas neurons in which each neuron acts as a detector for a different input vector. If there are Q input vectors, then there will be Q neurons. Each bias in the first layer is set to 0.8326/SPREAD. This gives radial basis functions that cross 0.5 at weighted inputs of +/- SPREAD. This determines the width of an area in the input space to which each neuron responds. If SPREAD is 4, then each radbas neuron will respond with 0.5 or more to any input vectors within a vector distance of 4 from their weight vector. As we shall see, SPREAD should be large enough that neurons respond strongly to overlapping regions of the input space. The second layer weights IW 2,1 (or in code, IW{2,1}) and biases b2 (or in code, b{2}) are found by simulating the first layer outputs a1 (A{1}), and then solving the following linear expression: [W{2,1} b{2}] * [A{1}; ones] = T We know the inputs to the second layer (A{1}) and the target (T), and the layer is linear. We can use the following code to calculate the weights and biases of the second layer to minimize the sum-squared error. Wb = T/[P; ones(1,Q)] Here Wb contains both weights and biases, with the biases in the last column. The sum-squared error will always be 0, as explained below. We have a problem with C constraints (input/target pairs) and each neuron has C +1 variables (the C weights from the C radbas neurons, and a bias). A linear problem with C constraints and more than C variables has an infinite number of zero error solutions! Thus newrbe creates a network with zero error on training vectors. The only condition we have to meet is to make sure that SPREAD is large enough so that the active input regions of the radbas neurons overlap enough so that several radbas neurons always have fairly large outputs at any given moment. This makes the network function smoother and results in better generalization for new input vectors occurring between input vectors used in the design. (However, SPREAD should not be so large that each neuron is effectively responding in the same, large, area of the input space.) 6-6 Radial Basis Functions The drawback to newrbe is that it produces a network with as many hidden neurons as there are input vectors. For this reason, newrbe does not return an acceptable solution when many input vectors are needed to properly define a network, as is typically the case. More Efficient Design (NEWRB) The function newrb iteratively creates a radial basis network one neuron at a time. Neurons are added to the network until the sum-squared error falls beneath an error goal or a maximum number of neurons has been reached. The call for this function is: net = newrb(P,T,GOAL,SPREAD) The function newrb takes matrices of input and target vectors, P and T, and design parameters GOAL and, SPREAD, and returns the desired network. The design method of newrb is similar to that of newrbe. The difference is that newrb creates neurons one at a time. At each iteration the input vector which will result in lowering the network error the most, is used to create a radbas neuron. The error of the new network is checked, and if low enough newrb is finished. Otherwise the next neuron is added. This procedure is repeated until the error goal is met, or the maximum number of neurons is reached. As with newrbe, it is important that the spread parameter be large enough that the radbas neurons respond to overlapping regions of the input space, but not so large that all the neurons respond in essentially the same manner. Why not always use a radial basis network instead of a standard feed-forward network? Radial basis networks, even when designed efficiently with newrbe, tend to have many times more neurons than a comparable feed-forward network with tansig or logsig neurons in the hidden layer. This is because sigmoid neurons can have outputs over a large region of the input space, while radbas neurons only respond to relatively small regions of the input space. The result is that the larger the input space (in terms of number of inputs, and the ranges those inputs vary over) the more radbas neurons required. On the other hand, designing a radial basis network often takes much less time than training a sigmoid/linear network, and can sometimes result in fewer neurons being used, as can be seen in the next demonstration. 6-7 6 Radial Basis Networks Demonstrations The demonstration script demorb1 shows how a radial basis network is used to fit a function. Here the problem is solved with only five neurons. Demonstration scripts demorb3 and demorb4 examine how the spread constant affects the design process for radial basis networks. In demorb3, a radial basis network is designed to solve the same problem as in demorb1. However, this time the spread constant used is 0.01. Thus, each radial basis neuron returns 0.5 or lower, for any input vectors with a distance of 0.01 or more from its weight vector. Because the training inputs occur at intervals of 0.1, no two radial basis neurons have a strong output for any given input. In demorb3, it was demonstrated that having too small a spread constant can result in a solution which does not generalize from the input/target vectors used in the design. This demonstration, demorb4, shows the opposite problem. If the spread constant is large enough, the radial basis neurons will output large values (near 1.0) for all the inputs used to design the network. If all the radial basis neurons always output 1, any information presented to the network becomes lost. No matter what the input, the second layer outputs ones. The function newrb will attempt to find a network, but will not be able to do so because to numerical problems that arise in this situation. The moral of the story is, choose a spread constant larger than the distance between adjacent input vectors, so as to get good generalization, but smaller than the distance across the whole input space. For this problem that would mean picking a spread constant greater than 0.1, the interval between inputs, and less than 2, the distance between the left most and right most inputs. 6-8 Generalized Regression Networks Generalized Regression Networks A generalized regression neural network (GRNN) is often used for function approximation. As discussed below, it has a radial basis layer and a special linear layer. Network Architecture The architecture for the GRNN is shown below. It is similar to the radial basis network, but has a slightly different second layer. AAA AAA AA AAA AA AAA A AA AA A AAA AAA AAA AA A Input Radial Basis Layer Q xR Special Linear Layer Rx1 1 R R = # elements in input vector IW1,1 Q xQ p || dist || LW2,1 Q x1 .* n a 1 n 1 Q x1 nprod Q x1 2 Q x1 Where... a2 Q = # Neurons in layer 1 Q = # Neurons in layer 2 Q = # of input/ target pairs Q x1 b1 Q Q x1 ai1 = radbas ( || iIW1,1 - p || bi1) Q a2 = purelin ( n2) ai1 is i th element of a1 where i IW1,1 is a vector made of the i th row of IW1,1 Here the nprod box shown above (code function normprod) produces S2 elements in vector n2. Each element is the dot product of a row of LW2,1 and the input vector a1, all normalized by the sum of the elements of a1. For instance, suppose that: LW{1,2}= [1 -2;3 4;5 6]; a{1} = [7; -8; Then aout = normprod(LW{1,2},a{1}) aout = -23 11 13 6-9 6 Radial Basis Networks The first layer is just like that for newrbe networks. It has as many neurons as there are input/ target vectors in P. Specifically, the first layer weights are set to P'. The bias b1 is set to a column vector of 0.8326/SPREAD. The user chooses SPREAD, the distance an input vector must be from a neuron’s weight vector to be 0.5. Again, the first layer operates just like the newbe radial basis layer described previously. Each neuron's weighted input is the distance between the input vector and its weight vector, calculated with dist. Each neuron's net input is the product of its weighted input with its bias, calculated with netprod. Each neurons' output is its net input passed through radbas. If a neuron's weight vector is equal to the input vector (transposed), its weighted input will be 0, its net input will be 0, and its output will be 1. If a neuron's weight vector is a distance of spread from the input vector, its weighted input will be spread, and its net input will be sqrt(-log(.5)) (or 0.8326). Therefore its output will be 0.5. The second layer also has as many neurons as input/target vectors, but here LW{2,1} is set to T. Suppose we have an input vector p close to pi, one of the input vectors among the input vector/target pairs used in designing layer one weights. This input p produces a layer 1 ai output close to 1. This leads to a layer 2 output close to ti, one of the targets used forming layer 2 weights. A larger spread leads to a large area around the input vector where layer 1 neurons will respond with significant outputs.Therefore if spread is small the radial basis function is very steep so that the neuron with the weight vector closest to the input will have a much larger output than other neurons. The network will tend to respond with the target vector associated with the nearest design input vector. As spread gets larger the radial basis function's slope gets smoother and several neuron's may respond to an input vector. The network then acts like it is taking a weighted average between target vectors whose design input vectors are closest to the new input vector. As spread gets larger more and more neurons contribute to the average with the result that the network function becomes smoother. 6-10 Generalized Regression Networks Design (NEWGRNN) You can use the function newgrnn to create a GRNN. For instance, suppose that three input and three target vectors are defined as: P = [4 5 6]; T = [1.5 3.6 6.7]; We can now obtain a GRNN with: net = newgrnn(P,T); and simulate it with: P = 4.5; v = sim(net,P) You might want to try demogrn1. It shows how to approximate a function with a GRNN. 6-11 6 Radial Basis Networks Probabilistic Neural Networks Probabilistic neural networks can be used for classification problems. When an input is presented, the first layer computes distances from the input vector to the training input vectors, and produces a vector whose elements indicate how close the input is to a training input. The second layer sums these contributions for each class of inputs to produce as its net output a vector of probabilities. Finally, a compete transfer function on the output of the second layer picks the maximum of these probabilities, and produces a one for that class and a 0 for the other classes. The architecture for this system is shown below. Network Architecture AAA AAA AAA AAA AAA Input Radial Basis Layer Q xR p Rx1 1 R IW1,1 || dist || b 1 Q x1 Q x1 .* Competitive Layer AA AA AA AA AA AA AA AA AA AA a2 n1 a1 Q x1 ai1 = radbas ( || iIW1,1 - p || bi1) Q x1 K x1 n2 LW2,1 K x1 Where... C R = # elements in input vector KxQ Q Q a2 = compet ( LW2,1 a1) ai1 is i th element of a1 where i IW1,1 is a vector made of the i th row of IW1,1 Q = # input/target pairs = # of neurons in layer 1 K = # of classes of input data = # of neurons in layer 2 It is assumed that there are Q input vector/target vector pairs. Each target vector has K elements. One of these element is one and the rest is zero. Thus, each input vector is associated with one of K classes. The first layer input weights, IW1,1 (net.IW{1,1}) are set to the transpose of the matrix formed from the Q training pairs, P'. When an input is presented the ||dist|| box produces a vector whose elements indicate how close the input is to the vectors of the training set. These elements are multiplied, element by element, by the bias and sent the radbas transfer function. An input vector close to a training vector will be represented by a number close to one in the output vector a1. If an input is close to several training vectors of a single class, it will be represented by several elements of a1 that are close to one. 6-12 Probabilistic Neural Networks The second layer weights, LW1,2 (net.LW{2,1}), are set to the matrix T of target vectors. Each vector has a one only in the row associated with that particular class of input, and zeros elsewhere. (A function ind2vec is used to create the proper vectors.) The multiplication Ta1 sums the elements of a1 due to each of the K input classes. Finally, the second layer transfer function, compete, produces a one corresponding to the largest element of n2, and zeros elsewhere. Thus, the network has classified the input vector into a specific one of K classes because that class had the maximum probability of being correct. Design (NEWPNN) You can use the function newpnn to create a PNN. For instance, suppose that seven input vectors and their corresponding targets are: P = [0 0;1 1;0 3;1 4;3 1;4 1;4 3]' which yields P = 0 1 0 0 1 3 Tc = [1 1 2 2 3 3 3]; 1 4 3 1 4 1 4 3 2 3 3 3 Which yields Tc = 1 1 2 We need a target matrix with ones in the right place. We can get it with the function ind2vec. It gives a matrix with zeros except at the correct spots. So execute T = ind2vec(Tc) which gives T = (1,1) (1,2) (2,3) (2,4) (3,5) (3,6) (3,7) 1 1 1 1 1 1 1 6-13 6 Radial Basis Networks Now we can create a network and simulate it, using the input P to make sure that it does produce the correct classifications. We will use the function vec2ind to convert the output Y into a row Yc to make the classifications clear. net = newpnn(P,T); Y = sim(net,P) Yc = vec2ind(Y) Finally we get Yc = 1 1 2 2 3 3 3 We might try classifying vectors other than those that were used to design the network. We will try to classify the vectors shown below in P2. P2 = [1 4;0 1;5 2]' P2 = 1 4 0 1 5 2 Can you guess how these vectors will be classified? If we run the simulation and plot the vectors as we did before we get: Yc = 2 1 3 These results look good, for these test vectors were quite close to members of classes 2, 1 and 3 respectively. The network has managed generalize its operation to properly classify vectors other than those used to design the network. You might want to try demopnn1. It shows how to design a PNN, and how the network can successfully classify a vector not used in the design. 6-14 Summary Summary Radial basis networks can be designed very quickly in two different ways. The first design method, newrbe, finds an exact solution. The function newrbe creates radial basis networks with as many radial basis neurons as there are input vectors in the training data. The second method, newrb, finds the smallest network that can solve the problem within a given error goal. Typically, far fewer neurons are required by newrb than are returned newrbe. However, because the number of radial basis neurons is proportional to the size of the input space, and the complexity of the problem, radial basis networks can still be larger than backpropagation networks. A generalized regression neural network (GRNN) is often used for function approximation. It has been shown that, given a sufficient number of hidden neurons, GRNNs can approximate a continuous function to an arbitrary accuracy. Probabilistic neural networks can be used for classification problems. Their design is straightforward and does not depend on training. A PNN is guaranteed to converge to a Bayesian classifier providing it is given enough training data. These networks generalize well. The GRNN and PNN have many advantages, but they both suffer from one major disadvantage. They are slower to operate because they use more computation than other kinds of networks to do their function approximation or classification. 6-15 6 Radial Basis Networks Figures Radial Basis Neuron Input Radial Basis Neuron 1,1 p p1 p2 3 ... w1, R AA AA w + _ p R n || dist || b 1 A A a = radbas( || w-p || b) Radbas Transfer Function a 1.0 0.5 0.0 AA AA n -0.833 +0.833 a = radbas(n) Radial Basis Function 6-16 a Summary Radial Basis Network Architecture AAA AAA AAA AA AA AAA AA AA AAA AAAA Radial Basis Layer Input IW1,1 S1xR p Rx1 .* S1x1 n1 a2 LW2,1 b1 1 S1 a 1 = radbas ( || IW1,1 - p || b 1) i i i S2x1 n2 S2xS1 S1 = # Neurons in layer 1 S2x1 S1x1 S1x1 Where... R = # elements in input vector AA AA AA a1 S1x1 || dist || 1 R Linear Layer b2 S2x1 S2 S2 = # Neurons in layer 2 a2 = purelin (LW2,1 a1 +b2) ai1 is i th element of a1 where i IW1,1 is a vector made of the i th row of IW1,1 Generalized Regression Neural Network Architecture AAA AA AAA AAA AAAAA AA A AAA AA AAA A AAA AA A Input Radial Basis Layer Q xR Special Linear Layer Rx1 1 R R = # elements in input vector IW1,1 Q xQ p || dist || LW2,1 Q x1 .* n1 a1 Q x1 Q x1 nprod n2 Q x1 b1 Q x1 Q ai1 = radbas ( || iIW1,1 - p || bi1) Q Where... a2 Q = # Neurons in layer 1 Q = # Neurons in layer 2 Q x1 Q = # of input/ target pairs a2 = purelin ( n2) ai1 is i th element of a1 where i IW1,1 is a vector made of the i th row of IW1,1 6-17 6 Radial Basis Networks Probabilistic Neural Network Architecture AAA AAA AAA AA AA A AAA AA AA A AAA AA A Input Radial Basis Layer Q xR p Rx1 1 R Competitive Layer IW1,1 || dist || a2 Q x1 .* n1 a1 Q x1 Q x1 b1 Q x1 ai1 = radbas ( || iIW1,1 - p || bi1) LW2,1 n2 K x1 Where... K x1 C R = # elements in input vector KxQ Q Q a2 = compet ( LW2,1 a1) ai1 is i th element of a1 where i IW1,1 is a vector made of the i th row of IW1,1 Q = # input/target pairs = # of neurons in layer 1 K = # of classes of input data = # of neurons in layer 2 6-18 Summary New Functions This chapter introduces the following new functions: Function Description newrb Design a radial basis network. newrbe Design an exact radial basis network. newgrnn Design a generalized regression neural network. newpnn Design a probabilistic neural network. dist Euclidean distance weight function. negdist Negative euclidean distance weight function dotprod Dot product weight function. normprod Normalized dot product weight function. netprod Product net input function. compet Competitive transfer function. radbas Radial basis transfer function. ind2vec Convert indices to vectors. vec2ind Convert vectors to indices. 6-19 6 Radial Basis Networks 6-20 7 Self-Organizing Networks Introduction . . . . . . . . . . . . . . . . . . . . 7-2 Important Self-Organizing Functions . . . . . . . . . . 7-2 Competitive Learning . . . . . . . . . . Architecture . . . . . . . . . . . . . . . . Creating a Competitive Neural Network (NEWC) Kohonen Learning Rule (LEARNK) . . . . . . Bias Learning Rule (LEARNCON) . . . . . . Training . . . . . . . . . . . . . . . . . Graphical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-3 7-3 7-4 7-5 7-5 7-6 7-8 Self-Organizing Maps . . . . . . . . . . . . . . Topologies (GRIDTOP, HEXTOP, RANDTOP) . . . . . Distance Functions(DIST, LINKD., MAND. BOXD.) . . Architecture . . . . . . . . . . . . . . . . . . . Creating a Self Organizing MAP Neural Net. (NEWSOM) Training (LEARNSOM) . . . . . . . . . . . . . . Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-10 7-12 7-16 7-19 7-20 7-22 7-25 Summary and Conclusions . . . . . . . . . . . . . 7-32 Figures . . . . . . . . . . . . . . . . . . . . . . . 7-32 New Functions . . . . . . . . . . . . . . . . . . . 7-33 7 Self-Organizing Networks Introduction Self-organizing in networks is one of the most fascinating topics in the neural network field. Such networks can learn to detect regularities and correlations in their input and adapt their future responses to that input accordingly. The neurons of competitive networks learn to recognize groups of similar input vectors. Self-organizing maps learn to recognize groups of similar input vectors in such a way that neurons physically close together in the neuron layer respond to similar input vectors. A basic reference is: Kohonen, T. Self-Organization and Associative Memory, 2nd Edition, Berlin: Springer-Verlag, 1987. Important Self-Organizing Functions Competitive layers and self organizing maps can be created with newc and newsom respectively. A listing of all self-organizing functions and demonstrations can be found by typing help selforg. 7-2 Competitive Learning Competitive Learning The neurons in a competitive layer distribute themselves to recognize frequently presented input vectors. Architecture The architecture for a competitive network is shown below. AA AA AAA AAA AA AA Input Competitive Layer S1xR IW1,1 p Rx1 1 R || ndist || b1 S1x1 AA AA AA a1 S1x1 n1 S1x1 S1x1 C S1 The dist box in this figure accepts the input vector p and the input weight matrix IW1,1, and produces a vector having S1 elements. The elements are the negative of the distances between the input vector and vectors iIW1,1 formed from the rows of the input weight matrix. The net input n1 of a competitive layer is computed by finding the negative distance between input vector p and the weight vectors and adding the biases b. If all biases are zero, the maximum net input a neuron can have is 0. This occurs when the input vector p equals that neuron’s weight vector. The competitive transfer function accepts a net input vector for a layer and returns neuron outputs of 0 for all neurons except for the winner, the neuron associated with the most positive element of net input n1. The winner’s output is 1. If all biases are 0 then the neuron whose weight vector is closest to the input vector has the least negative net input, and therefore wins the competition to output a 1. Reasons for using biases with competitive layers are introduced in a later section on training. 7-3 7 Self-Organizing Networks Creating a Competitive Neural Network (NEWC) A competitive neural network can be created with the function newc. We will show how this works with a simple example. Suppose that we wish to divide the following four two element vectors into two classes. p = [.1 .8 p = 0.1000 0.2000 .1 .9; .2 .9 .1 .8] 0.8000 0.9000 0.1000 0.1000 0.9000 0.8000 Thus, we have two vectors near the origin and two vectors near (1,1). First create a two neuron layer is created with two input elements ranging from 0 to 1. The first argument gives the range of the two input vectors and the second argument says that there are to be 2 neurons. net = newc([0 1; 0 1],2); The weights will be initialized to the center of the input ranges with the function midpoint. We can check to see these initial values as follows: wts = net.IW{1,1} wts = 0.5000 0.5000 0.5000 0.5000 These weights are indeed the values at the midpoint of the range (0 to 1) of the inputs, as we would expect when using midpoint for initialization. The biases are computed by initcon, which gives biases = 5.4366 5.4366 Now we have a network, but we need to train it to do the classification job. Recall that each neuron competes to respond to an input vector p. If the biases are all 0, the neuron whose weight vector is closest to p gets the highest net input and therefore wins the competition and outputs 1. All other neurons output 0. We would like to adjust the winning neuron so as to move it closer to the input. A learning rule to do this is discussed in the next section. 7-4 Competitive Learning Kohonen Learning Rule (LEARNK) The weights of the winning neuron (a row of the input weight matrix) are adjusted with the Kohonen learning rule. Supposing that the ith neuron wins, the ith row of the input weight matrix are adjusted as shown below. iIW 1, 1 ( q ) = iIW 1, 1 ( q – 1 ) + α ( p ( q ) – iIW 1, 1 (q – 1)) The Kohonen rule allows the weights of a neuron to learn an input vector, and because of this it is useful in recognition applications. Thus, the neuron whose weight vector was closest to the input vector is updated to be even closer. The result is that the winning neuron is more likely to win the competition the next time a similar vector is presented, and less likely to win when a very different input vector is presented. As more and more inputs are presented, each neuron in the layer closest to a group of input vectors soon adjusts its weight vector toward those input vectors. Eventually, if there are enough neurons, every cluster of similar input vectors will have a neuron which outputs 1 when a vector in the cluster is presented, while outputting a 0 at all other times. Thus the competitive network learns to categorize the input vectors it sees. The function learnk is used to perform the Kohonen learning rule in this toolbox. Bias Learning Rule (LEARNCON) One of the limitations of competitive networks is that some neurons may not always get allocated. In other words, some neuron weight vectors may start out far from any input vectors and never win the competition, no matter how long the training is continued. The result is that their weights do not get to learn and they never win. These unfortunate neurons, referred to as dead neurons, never perform a useful function. To stop this from happening, biases are used to give neurons that only win the competition rarely (if ever) an advantage over neurons which win often. A positive bias, added to the negative distance, makes a distant neuron more likely to win. 7-5 7 Self-Organizing Networks To do this job a running average of neuron outputs is kept. It is equivalent to the percentages of times each output is 1. This average is used to update the biases with the learning function learncon so that the biases of frequently active neurons will get smaller, and biases of infrequently active neurons will get larger. The learning rates for learncon are typically set an order of magnitude or more smaller than for learnk. Doing this helps make sure that the running average is accurate. The result is that biases of neurons which haven’t responded very frequently will increase vs. biases of neurons that have responded frequently. As the biases of infrequently active neurons increase the input space to which that neuron responds increases. As that input space increases the infrequently active neuron responds and moves toward more input vectors. Eventually the neuron will respond to an equal number of vectors as other neurons. This has two good effects. First, if a neuron never wins a competition because its weights are far from any of the input vectors, its bias will eventually get large enough so that it will be able to win. When this happens it will move toward some group of input vectors. Once the neuron’s weights have moved into a group of input vectors and the neuron is winning consistently its bias will decrease to 0. Thus the problem of dead neurons is resolved. The second advantage of biases is that they force each neuron to classify roughly the same percentage of input vectors. Thus, if a region of the input space is associated with a larger number of input vectors than another region, the more densely filled region will attract more neurons and be classified into smaller subsections. Training Now train the network for 500 epochs. Either train or adapt can be used. net.trainParam.epochs = 500 net = train(net,p); Note that train for competitive networks uses the training function trainwb1. You can verify this by executing the following code after creating the network. net.trainFcn 7-6 Competitive Learning This code produces ans = trainwb1 Thus, during each epoch, a single vector is chosen randomly and presented to the network and weight and bias values are updated accordingly. Next supply the original vectors as input to the network, simulate the network and finally convert its output vectors to class indices. a = sim(net,p) ac = vec2ind(a) This yields ac = 1 2 1 2 We see that the network has been trained to classify the input vectors into two groups, those near the origin, class 1, and those near (1,1), class 2. It might be interesting to look at the final weights and biases. They are wts = 0.8208 0.1348 biases = 5.3699 5.5049 0.8263 0.1787 (You may get different answers if you run this problem, as a random seed is used to pick the order of the vectors presented to the network for training.) Note that the first vector (formed from the first row of the weight matrix) is near the input vectors close to (1,1), while the vector formed from the second row of the weight matrix is close to the input vectors near the origin. Thus, the network has been trained, just by exposing it to the inputs, to classify them. During training each neuron in the layer closest to a group of input vectors adjusts its weight vector toward those input vectors. Eventually, if there are enough neurons, every cluster of similar input vectors will have a neuron which outputs 1 when a vector in the cluster is presented, while outputting a 0 at all other times. Thus the competitive network learns to categorize the input. 7-7 7 Self-Organizing Networks Graphical Example Competitive layers can be understood better when their weight vectors and input vectors are shown graphically. The diagram below shows forty-eight two-element input vectors represented as with ‘+’ markers. Input Vectors 1 0.8 0.6 0.4 0.2 0 -0.5 0 0.5 1 The input vectors above appear to fall into clusters. A competitive network of eight neurons will be used to classify the vectors into such clusters. 7-8 Competitive Learning The following plot shows the weights after training. Competitive Learning: 500 epochs 1 0.8 0.6 0.4 0.2 0 -0.5 0 0.5 1 Note that seven of the weight vectors found clusters of input vectors to classify. The eighth neuron’s weight vector is still in the middle of the input vectors. Continued training of the layer would eventually cause this last neuron to move toward some input vectors. You might try democ1 to see a dynamic example of competitive learning. 7-9 7 Self-Organizing Networks Self-Organizing Maps Self-organizing feature maps (SOFM) learn to classify input vectors according to how they are grouped in the input space. They differ from competitive layers in that neighboring neurons in the self-organizing map learn to recognize neighboring sections of the input space. Thus self-organizing maps learn both the distribution (as do competitive layers) and topology of the input vectors they are trained on. The neurons in the layer of an SOFM are arranged originally in physical positions according to a topology function. The functions gridtop, hextop or randtop can arrange the neurons in a grid, hexagonal, or random topology. Distances between neurons are calculated from their positions with a distance function. There are four distance functions, dist, boxdist, linkdist and mandist. Link distance is the most common. These topology and distance functions are described in detail later in this section. Here a self-organizing feature map network identifies a winning neuron i∗ using the same procedure as employed by a competitive layer. However, instead of updating only the winning neuron, all neurons within a certain neighborhood N i∗ ( d ) of the winning neuron are updated using the Kohonen rule. Specifically, we will adjust all such neurons i ∈ N i∗ ( d ) as follows iw ( q ) = iw ( q – 1 ) + α ( p ( q ) – iw ( q – 1 ) ) or iw ( q ) = ( 1 – α ) iw ( q – 1 ) + αp ( q ) Here the neighborhood N i∗ ( d ) contains the indices for all of the neurons that lie within a radius d of the winning neuron i∗ : N i ( d ) = { j, d ij ≤ d } Thus, when a vector p is presented, the weights of the winning neuron and its close neighbors will move towards p . Consequently, after many presentations, neighboring neurons will have learned vectors similar to each other. 7-10 Self-Organizing Maps To illustrate the concept of neighborhoods, consider the figure given below. The left diagram shows a two-dimensional neighborhood of radius d = 1 around neuron 13 . The right diagram shows a neighborhood of radius d = 2 . 1 2 3 4 5 1 2 3 4 5 6 7 8 9 10 6 7 8 9 10 11 12 13 14 15 11 12 13 14 15 16 17 18 19 20 16 17 18 19 20 21 22 23 24 25 21 22 23 24 25 N (1) 13 N (2) 13 These neighborhoods could be written as: N 13 ( 1 ) = { 8, 12, 13, 14, 18 } and N 13 ( 2 ) = { 3, 7, 8, 9, 11, 12, 13, 14, 15, 17, 18, 19, 23 } Note that the neurons in an SOFM do not have to be arranged in a two-dimensional pattern. You use a one-dimensional arrangement, or even three or more dimensions. For a one-dimensional SOFM, a neuron will only have two neighbors within a radius of 1 (or a single neighbor if the neuron is at the end of the line).You can also define distance in different ways, by using rectangular and hexagonal arrangements of neurons and neighborhoods for instance. The performance of the network is not sensitive to the exact shape of the neighborhoods. 7-11 7 Self-Organizing Networks Topologies (GRIDTOP, HEXTOP, RANDTOP) You can specify different topologies for the original neuron locations with the functions gridtop, hextop or randtop. The gridtop topology starts with neurons in a rectangular grid similar to that shown in the previous figure. For example, suppose that you want a 2 by 3 array of six neurons You can get this with: pos = gridtop(2,3) pos = 0 1 0 0 0 1 1 1 0 2 1 2 Here neuron 1 has the position (0,0), neuron 2 has the position (1,0), neuron 3 had the position (0,1), etc. 2 5 6 1 3 4 0 1 2 0 1 gridtop(2,3) Note that had we asked for a gridtop with the arguments reversed we would have gotten a slightly different arrangement. pos = gridtop(3,2) pos = 0 1 2 0 0 0 7-12 0 1 1 1 2 1 Self-Organizing Maps An 8x10 set of neurons in a gridtop topology can be created and plotted with the code shown below pos = gridtop(8,10); plotsom(pos) to give the following graph. Neuron Positions 9 8 7 position(2,i) 6 5 4 3 2 1 0 0 2 4 position(1,i) 6 8 As shown, the neurons in the gridtop topology do indeed lie on a grid. The hextop function creates a similar set of neurons but they are in a hexagonal pattern. A 2 by 3 pattern of hextop neurons is generated as follows: pos = hextop(2,3) pos = 0 1.0000 0 0 0.5000 0.8660 1.5000 0.8660 0 1.7321 1.0000 1.7321 7-13 7 Self-Organizing Networks Note that hextop is the default pattern for SOFM networks generated with newsom. An 8x10 set of neurons in a hextop topology can be created and plotted with the code shown below pos = hextop(8,10); plotsom(pos) to give the following graph. Neuron Positions 7 6 position(2,i) 5 4 3 2 1 0 0 1 2 3 4 position(1,i) 5 6 7 Note the positions of the neurons in a hexagonal arrangement. 7-14 8 Self-Organizing Maps Finally, the randtop function creates neurons in an N dimensional random pattern. The following code generates a random pattern of neurons. pos = randtop(2,3) pos = 0 0.7787 0 0.1925 0.4390 0.6476 1.0657 0.9106 0.1470 1.6490 0.9070 1.4027 An 8x10 set of neurons in a randtop topology can be created and plotted with the code shown below pos = randtop(8,10); plotsom(pos) to give the following graph. Neuron Positions 6 5 position(2,i) 4 3 2 1 0 0 1 2 3 position(1,i) 4 5 6 You might take a look at the help for these topology functions. They contain a number of examples. 7-15 7 Self-Organizing Networks Distance Functions (DIST, LINKDIST, MANDIST, BOXDIST) In this toolbox there are four distinct ways to calculate distances from a particular neuron to its neighbors. Each calculation method is implemented with a special function. The dist function has been discussed before. It calculates the Euclidean distance from a home neuron to any other neuron. Suppose we have three neurons: pos2 = [ 0 1 2; 0 1 2] pos2 = 0 1 2 0 1 2 We will find the distance from each neuron to the other with: D2 = dist(pos2) D2 = 0 1.4142 1.4142 0 2.8284 1.4142 2.8284 1.4142 0 Thus, the distance from neuron 1 to itself is 0, the distance from neuron 1 to neuron 2 is 1.414, etc. These are indeed the Euclidean distances as we know them. The graph below shows a home neuron in a two-dimensional (gridtop) layer of neurons. The home neuron has neighborhoods of increasing diameter surrounding it. A neighborhood of diameter 1 includes the home neuron and its 7-16 Self-Organizing Maps immediate neighbors. The neighborhood of diameter 2 includes the diameter 1 neurons and their immediate neighbors. Columns Home Neuron Rows 2-Dimensional Layer of Neurons Neighborhood 1 Neighborhood 2 Neighborhood 3 As for the dist function, all the neighborhoods for an S neuron layer map are represented by an SxS matrix of distances. The particular distances shown above, 1 in the immediate neighborhood, 2 in neighborhood 2, etc., are generated by the function boxdist. Suppose that we have 6 neurons in a gridtop configuration. pos = gridtop(2,3) pos = 0 1 0 0 0 1 1 1 0 2 1 2 1 1 1 0 1 1 2 2 1 1 0 1 2 2 1 1 1 0 Then the box distances are: d = boxdist(pos) d = 0 1 1 0 1 1 1 1 2 2 2 2 1 1 0 1 1 1 The distance from neuron 1 to 2, 3 and 4 is just 1, for they are in the immediate neighborhood. The distance from neuron 1 to both 5 and 6 is 2. The distance from both 3 and 4 to all other neurons is just 1. 7-17 7 Self-Organizing Networks The link distance from one neuron is just the number of links, or steps, that must be taken to get to the neuron under consideration. Thus if we calculate the distances from the same set of neurons with linkdist we get: dlink = 0 1 1 2 2 3 1 0 2 1 3 2 1 2 0 1 1 2 2 1 1 0 2 1 2 3 1 2 0 1 3 2 2 1 1 0 The Manhattan distance between two vectors x and y is calculated as: D = sum(abs(x-y)) Thus if we have W1 = [ 1 2; 3 4; 5 6] W1 = 1 2 3 4 5 6 and P1= [1;1] P1 = 1 1 then we get for the distances Z1 = mandist(W1,P1) Z1 = 1 5 9 The distances calculated with mandist do indeed follow the mathematical expression given above. 7-18 Self-Organizing Maps Architecture The architecture for this SOFM is shown below. Input AA AAA AAA IW1,1 S1xR p R x1 R A A A A Self Organizing Map Layer || ndist || n1 S1x1 C a1 S1x1 S1 ni1 = - || iIW1,1 - p || a1 = compet (n1) This architecture is like that of a competitive network, except no bias is used here. The competitive transfer function produces a 1 for output element a1i corresponding to i∗ ,the winning neuron. All other output elements in a1 are 0. Now, however, as described above, neurons close to the winning neuron are updated along with the winning neuron. As described previously, one can chose from various topologies of neurons. Similarly, one can choose from various distance expressions to calculate neurons that are close to the winning neuron. 7-19 7 Self-Organizing Networks Creating a Self Organizing MAP Neural Network (NEWSOM) You can create a new SOFM network with the function newsom. This function defines variables used in two phases of learning: • Ordering phase learning rate • Ordering phase steps • Tuning phase learning rate • Tuning phase neighborhood distance These values are used for training and adapting. Consider the following example. Suppose that we wish to create a network having input vectors with two elements that fall in the range 0 to 2 and 0 to 1 respectively. Further suppose that we want to have six neurons in a hexagonal 2 by 3 network. The code to obtain this network is: net = newsom([0 2; 0 1] , [2 3]); Suppose also that the vectors to train on are: P = [.1 .3 1.2 1.1 1.8 1.7 .1 .3 1.2 1.1 1.8 1.7;... 0.2 0.1 0.3 0.1 0.3 0.2 1.8 1.8 1.9 1.9 1.7 1.8] We can plot all of this with plot(P(1,:),P(2,:),'.g','markersize',20) hold on plotsom(net.iw{1,1},net.layers{1}.distances) hold off 7-20 Self-Organizing Maps to give Weight Vectors 2 1.8 1.6 1.4 W(i,2) 1.2 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 W(i,1) The various training vectors are seen as fuzzy gray spots around the perimeter of this figure. The initialization for newsom is midpoint. Thus, the initial network neurons are all concentrated at the black spot at (1, 0.5). When simulating a network, the negative distances between each neuron's weight vector and the input vector are calculated (negdist) to get the weighted inputs. The weighted inputs are also the net inputs (netsum). The net inputs compete (compete) so that only the neuron with the most positive net input will output a 1. 7-21 7 Self-Organizing Networks Training (LEARNSOM) Learning in a self organizing feature map occurs for one vector at a time independent of whether the network is trained directly (trainwb1) or whether it is trained adaptively (adaptwb). In either case, learnsom is the self-organizing map weight learning function. First the network identifies the winning neuron. Then the weights of the winning neuron, and the other neurons in its neighborhood, are moved closer to the input vector at each learning step using the self-organizing map learning function learnsom. The winning neuron's weights are altered proportional to the learning rate. The weights of neurons in its neighborhood are altered proportional to half the learning rate. The learning rate and the neighborhood distance used to determine which neurons are in the winning neuron's neighborhood are altered during training through two phases. Phase 1: Ordering Phase This phase lasts for the given number of steps. The neighborhood distance starts as the maximum distance between two neurons, and decreases to the tuning neighborhood distance. The learning rate starts at the ordering phase learning rate and decreases until it reaches the tuning phase learning rate. As the neighborhood distance and learning rate decrease over this phase the neuron's of the network will typically order themselves in the input space with the same topology which they are ordered physically. Phase 2: Tuning Phase This phase lasts for the rest of training or adaption. The neighborhood distance stays at the tuning neighborhood distance (should include only close neighbors, i.e. typically 1.0). The learning rate continues to decrease from the tuning phase learning rate, but very slowly. The small neighborhood and slowly decreasing learning rate fine tune the network, while keeping the ordering learned in the previous phase stable. The number of epochs for the tuning part of training (or time steps for adaption) should be much larger than the number of steps in the ordering phase, because the tuning phase usually takes much longer. Now let us take a look at some of the specific values commonly used in these networks. 7-22 Self-Organizing Maps Learning occurs according to learnsom's learning parameter, shown here with its default value. LP.order_lr 0.9 Ordering phase learning rate. LP.order_steps 1000 Ordering phase steps. LP.tune_lr 0.02 Tuning phase learning rate. LP.tune_nd 1 Tuning phase neighborhood distance. learnsom calculates the weight change dW for a given neuron from the neuron's input P, activation A2, and learning rate LR: dw = lr*a2*(p'-w) where the activation A2 is found from the layer output A and neuron distances D and the current neighborhood size ND: a2(i,q) = 1, if a(i,q) = 1 = 0.5, if a(j,q) = 1 and D(i,j) <= nd = 0, otherwise The learning rate LR and neighborhood size NS are altered through two phases: an ordering phase and a tuning phase. The ordering phase lasts as many steps as LP.order_steps. During this phase LR is adjusted from LP.order_lr down to LP.tune_lr, and ND is adjusted from the maximum neuron distance down to 1. It is during this phase that neuron weights are expected to order themselves in the input space consistent with the associated neuron positions. During the tuning phase LR decreases slowly from LP.tune_lr and ND is always set to LP.tune_nd. During this phase the weights are expected to spread out relatively evenly over the input space while retaining their topological order found during the ordering phase. Thus, the neuron’s weight vectors initially take large steps all together toward the area of input space where input vectors are occurring. Then as the neighborhood size decreases to 1, the map tends to order itself topologically over the presented input vectors. Once the neighborhood size is 1, the network should be fairly well ordered and the learning rate is slowly decreased over a longer period to give the neurons time to spread out evenly across the input vectors. 7-23 7 Self-Organizing Networks As with competitive layers, the neurons of a self-organizing map will order themselves with approximately equal distances between them if input vectors appear with even probability throughout a section of the input space. Also, if input vectors occur with varying frequency throughout the input space, the feature map layer will tend to allocate neurons to an area in proportion to the frequency of input vectors there. Thus, feature maps, while learning to categorize their input, also learn both the topology and distribution of their input. We can train the network for 1000 epochs with net.trainParam.epochs = 1000; net = train(net,P); This training produces the following plot. Weight Vectors 1.8 1.6 1.4 W(i,2) 1.2 1 0.8 0.6 0.4 0.2 0 7-24 0.5 1 W(i,1) 1.5 2 Self-Organizing Maps We can see that the neurons have started to move toward the various training groups. Additional training will be required to get the neurons closer to the various groups. As noted previously, self-organizing maps differ from conventional competitive learning in terms of which neurons get their weights updated. Instead of updating only the winner, feature maps update the weights of the winner and its neighbors. The result is that neighboring neurons tend to have similar weight vectors and to be responsive to similar input vectors. Examples Two examples are described briefly below. You might try the demonstration scripts demosm1 and demosm2 to see similar examples. One-Dimensional Self-Organizing Map Consider 100 two-element unit input vectors spread evenly between 0˚ and 90˚. angles = 0:0.5∗pi/99:0.5∗pi; Here is a plot of the data P = [sin(angles); cos(angles)]; 1 0.8 0.6 0.4 0.2 0 0 0.5 1 7-25 7 Self-Organizing Networks We will define a a self-organizing map as a one-dimensional layer of 10 neurons. This map is to be trained on these input vectors shown above. Originally these neurons will be at the center of the figure. 1.5 W(i,2) 1 0.5 0 -0.5 -1 0 1 2 W(i,1) Of course, since all the weight vectors start in the middle of the input vector space, all you see now is a single circle. As training starts the weight vectors move together toward the input vectors. They also become ordered as the neighborhood size decreases. Finally the layer has adjusted its weights so that each neuron responds strongly to a region of 7-26 Self-Organizing Maps the input space occupied by input vectors. The placement of neighboring neuron weight vectors also reflects the topology of the input vectors. 1 W(i,2) 0.8 0.6 0.4 0.2 0 0 0.5 W(i,1) 1 Note that self-organizing maps are trained with input vectors in a random order, so starting with the same initial vectors does not guarantee identical training results. Two-Dimensional Self-Organizing Map This example shows how a two-dimensional self-organizing map can be trained. First some random input data is created with the following code. P = rands(2,1000); 7-27 7 Self-Organizing Networks Here is a plot of these 1000 input vectors. 1 0.5 0 -0.5 -1 -1 0 1 A two-dimensional map of 30 neurons is used to classify these input vectors. The two-dimensional map is five neurons by six neurons, with distances calculated according to the Manhattan distance neighborhood function mandist. The map is then trained for 5000 presentation cycles, with displays every 20 cycles. 7-28 Self-Organizing Maps Here is what the self-organizing map looks like after 40 cycles. 1 W(i,2) 0.5 0 -0.5 -1 -0.5 0 0.5 1 W(i,1) The weight vectors, shown with circles, are almost randomly placed. However, even after only 40 presentation cycles, neighboring neurons, connected by lines, have weight vectors close together. Here is the map after 120 cycles. 1 W(i,2) 0.5 0 -0.5 -1 -1 0 W(i,1) 1 7-29 7 Self-Organizing Networks After 120 cycles the map has begun to organize itself according to the topology of the input space which constrains input vectors. The following plot, after 500 cycles, shows the map is more evenly distributed across the input space. 1 W(i,2) 0.5 0 -0.5 -1 -1 0 W(i,1) 1 Finally, after 5000 cycles, the map is rather evenly spread across the input space. In addition, the neurons are very evenly spaced reflecting the even distribution of input vectors in this problem. 1 W(i,2) 0.5 0 -0.5 -1 -1 7-30 0 W(i,1) 1 Self-Organizing Maps Thus a two-dimensional self-organizing map has learned the topology of its inputs’ space. It is important to note that while a self-organizing map does not take long to organize itself so that neighboring neurons recognize similar inputs, it can take a long time for the map to finally arrange itself according to the distribution of input vectors. 7-31 7 Self-Organizing Networks Summary and Conclusions A competitive network learns to categorize the input vectors presented to it. If a neural network only needs to learn to categorize its input vectors, then a competitive network will do. Competitive networks also learn the distribution of inputs by dedicating more neurons to classifying parts of the input space with higher densities of input. A self-organizing map learns to categorize input vectors. It also learns the distribution of input vectors. Feature maps allocate more neurons to recognize parts of the input space where many input vectors occur and allocate fewer neurons to parts of the input space where few input vectors occur. Self-organizing maps also learn the topology of their input vectors. Neurons next to each other in the network learn to respond to similar vectors. The layer of neurons can be imagined to be a rubber net which is stretched over the regions in the input space where input vectors occur. Self-organizing maps allow neurons that are neighbors to the winning neuron to output values. Thus the transition of output vectors is much smoother than that obtained with competitive layers, where only one neuron has an output at a time. Figures Competitive Network Architecture AA AA AAA AAA AA AA Input Competitive Layer S1xR IW1,1 p Rx1 1 R 7-32 || ndist || b1 S1x1 AA AA AA a1 S1x1 n1 S1x1 S1x1 C S1 Summary and Conclusions Self Organizing Feature Map Architecture Input AA AAA AAA IW1,1 S1xR p R x1 R A A A A Self Organizing Map Layer || ndist || n1 S1x1 C a1 S1x1 S1 ni1 = - || iIW1,1 - p || a1 = compet (n1) New Functions This chapter introduces the following new functions: Function Description newc Create a competitive layer. newsom Create a self-organizing map. learncon Conscience bias learning function. boxdist Distance between two position vectors. dist Euclidean distance weight function. linkdist Link distance function mandist Manhattan distance weight function. gridtop Gridtop layer topology function hextop Hexagonal layer topology function. randtop Random layer topology function. 7-33 7 Self-Organizing Networks 7-34 8 Learning Vector Quantization Introduction . . . . . . . . . . . . . . . . . . . . 8-2 Important LVQ Functions . . . . . . . . . . . . . . . 8-2 Network Architecture . . . . . . . . . . . . . . . 8-3 Creating an LVQ Network (NEWLVQ) . . . . . . . . 8-5 LVQ Learning Rule(LEARNLV2) . . . . . . . . . . . 8-9 Training . . . . . . . . . . . . . . . . . . . . . . 8-11 Summary . . . . . . . . . . . . . . . . . . . . . 8-14 Figures . . . . . . . . . . . . . . . . . . . . . . . 8-14 New Functions . . . . . . . . . . . . . . . . . . . 8-14 8 Learning Vector Quantization Introduction Learning vector quantization (LVQ) is a method for training competitive layers in a supervised manner. A competitive layer will automatically learn to classify input vectors. However, the classes that the competitive layer finds are dependent only on the distance between input vectors. If two input vectors are very similar, the competitive layer probably will put them into the same class. There is no mechanism in a strictly competitive layer design to say whether or not any two input vectors are in the same class or different classes. LVQ networks, on the other hand, learn to classify input vectors into target classes chosen by the user. You might consult the following reference: Kohonen, T. Self-Organization and Associative Memory, 2nd Edition, Berlin: Springer-Verlag, 1987. Important LVQ Functions An LVQ network can be created with the function newlvq. For a list of all LVQ functions and demonstrations type help lvq. 8-2 Network Architecture Network Architecture The LVQ network architecture is shown below. A AAAA AA AA AA A AA AA AA A Input Linear Layer Competitive Layer IW1,1 S1xR p R x1 || ndist || n1 a1 S1x1 C S1x1 LW2,1 n2 R n 1 = - || IW1,1 - p || i i a2 S2x1 S2x1 S2xS1 S1 Where... 1 R = # elements in input vector S1= # competitive neurons S2= # linear neurons S2 a2 = purelin(LW2,1 a1) a = compet (n ) 1 1 An LVQ network has a first competitive layer and a second linear layer. The competitive layer learns to classify input vectors in much the same way as the competitive layers of Chapter 7. The linear layer transforms the competitive layer’s classes into target classifications defined by the user. We will refer to the classes learned by the competitive layer as subclasses and the classes of the linear layer as target classes. Both the competitive and linear layers have one neuron per (sub or target) class. Thus, the competitive layer can learn up to S1 subclasses. These, in turn will be combined by the linear layer to form S2 target classes. (S1 will always be larger than S2.) For example, suppose neurons 1, 2, and 3 in the competitive layer all learn subclasses of the input space which belong to the linear layer target class #2. Then competitive neurons 1, 2, and 3, will have LW2,1 weights of 1.0 to neuron n2 in the linear layer, and weights of 0 to all other linear neurons. Thus, the linear neuron produces a 1 if any of the three competitive neurons (1,2, and 3) win the competition and output a 1. This is how the subclasses of the competitive layer are combined into target classes in the linear layer. In short, a 1 in the ith row of a1 (the rest to the elements of a1 will be zero) effectively picks the ith column of LW2,1 as the network output. Each such column contains a single 1, corresponding to a specific class. Thus, subclass 1s out of layer 1 get put into various classes, by the LW2,1a1 multiplication in layer 2. 8-3 8 Learning Vector Quantization We know ahead of time what fraction of the layer one neurons should be classified into the various class outputs of layer 2, so we can specify the elements of LW2,1 at the start. However, we will have to go through a training procedure to get the first layer to produce the correct subclass output for each vector of the training set. We will discuss this training shortly. First consider how to create the original network. 8-4 Creating an LVQ Network (NEWLVQ) Creating an LVQ Network (NEWLVQ) An LVQ network can be created with the function newlvq. net = newlvq(PR,S1,PC,LR,LF) where • PR is an Rx2 matrix of min and max values for R input elements, • S1 is the number of first layer hidden neurons, and • PC is an S2 element vector of typical class percentages. The learning default is 0.01 and the default learning function is 'learnlv2'. Suppose we have ten input vectors. We will create a network that assigns each of these input vectors to one of four subclasses. Thus, we have four neurons in the first competitive layer. These subclasses are then to be assigned to one of two output classes by the two neurons in layer 2. The input vectors and targets are specified by : P = [-3 -2 -2 0 0 0 0 +2 + 2 +3; ... 0 +1 -1 +2 +1 -1 -2 +1 -1 0] and Tc = [1 1 1 2 2 2 2 1 1 1]; It may help to show the details of what we get from these two lines of code. P = -3 0 -2 1 -2 -1 0 2 0 1 0 -1 0 -2 2 1 2 -1 3 0 1 1 2 2 2 2 1 1 1 Tc = 1 8-5 8 Learning Vector Quantization A plot of the input vectors follows. 3 p4 2 p2 1 p1 0 p10 p6 p3 -1 p8 p5 p9 -2 p7 -3 -5 0 5 Input Vectors As you can see, there are four sub-classes of input vectors. We would like a network that will classify p1, p2, p3, p8, p9, and p10 to produce an output of 1 and that will classify vectors p4, p5, p6 and p7 to produce an output of 2. Note that this problem is nonlinearly separable, and so could not be solved by a perceptron, but an LVQ network will have no difficulty. Next we convert the Tc matrix to target vectors. T = ind2vec(Tc) This gives a sparse matrix T that can be displayed in full with: targets = full(T) which gives targets = 1 0 1 0 1 0 0 1 0 1 0 1 0 1 1 0 1 0 1 0 This looks right. It says, for instance, that if we have the first column of P as input, we should get the first column of targets as an output, and that output says the input falls in class 1, which is correct. Now we are ready to call newlvq. 8-6 Creating an LVQ Network (NEWLVQ) We will call newlvq with the proper arguments so that it will create a network with four neurons in the first layer and two neurons in the second layer. The first layer weights will be initialized to the center of the input ranges with the function midpoint. The second layer weights will have 60% (6 of the 10 in Tc above) of its columns with a 1 in the first row, corresponding to class 1, and 40% of its columns will have a 1 in the second row, corresponding to class 2. net = newlvq(range(P),4,[.6 .4]); We can check to see the initial values of the first layer weight matrix: net.IW{1,1} ans = 0 0 0 0 0 0 0 0 These zero weights are indeed the values at the midpoint of the range (-3 to +3) of the inputs, as we would expect when using midpoint for initialization. We can look at the second layer weights with net.LW{2,1} ans = 1 1 0 0 0 1 0 1 This makes sense too. It says that if the competitive layer produces a 1 as the first or second element, the input vector will be classified as class 1. Otherwise it will be in class 2. You may notice that the first two competitive neurons are connected to the first linear neuron (with weights of 1), while the second two competitive neurons are connected to the second linear neuron. All other weights between the competitive neurons and linear neurons have values of 0. Thus, each of the two target classes (the linear neurons) is in fact, the union of two subclasses (the competitive neurons). 8-7 8 Learning Vector Quantization We can simulate the network with sim. We will use the original P matrix as input just to see what we get. Y = sim(net,P); Y = vec2ind(Yb4t) Y = 1 1 1 1 1 1 1 1 1 1 The network classifies all inputs into class 1. This is not what we want of course. We will have to train the network, adjusting the weights of layer 1 only, before we can expect a good result. We will look at the training process shortly, but first will discuss the LVQ learning rule. 8-8 LVQ Learning Rule(LEARNLV2) LVQ Learning Rule(LEARNLV2) LVQ learning in the competitive layer is based on a set of input/target pairs. { p 1, t 1 }, { p 2, t 2 }, …, { p Q, t Q } Each target vector has a single 1. The rest of its elements are 0. The 1 tells the proper classification of the associated input. For instance, consider the following training pair. 0 2 0 p1 = –1 , t1 = 1 0 0 Here we have input vectors of three elements, and each input vector is to be assigned to one of four classes. The network is to be trained so that it will classify the input vector shown above into the third of four classes. To train the network an input vector p is presented, and the distance from p to each row of the input weight matrix IW1,1 is computed with the function ndist. The hidden neurons of layer 1 compete. Let us suppose that the ith element of n1 is most positive, and neuron i* wins the competition. Then the competitive transfer function produces a 1 as the i*th element of a1. All other elements of a1 will be 0. When a1 is multiplied by the layer 2 weights LW2,1, the single 1 in a1 selects the class, k* associated with the input. Thus, the network has assigned the input vector p to class k*. Of course, this assignment may be a good one or a bad one. We will adjust the i*th row of IW1,1 in such a way as to move this row closer to the input vector p if the assignment is correct, and to move the row away from p if the assignment is incorrect. So if p is classified correctly, 2 ( a k∗ = t k∗ = 1 ) we will compute the new value of the i*th row of IW1,1 as: 1, 1 IW i∗ (q) = 1, 1 IW i∗ 1, 1 ( q – 1 ) + α ( p ( q ) – i∗IW (q – 1)) . 8-9 8 Learning Vector Quantization On the other hand, if p is classified incorrectly, 2 ( a k∗ = 1 ≠ t k∗ = 0 ) , we will compute the new value of the i*th row of IW1,1 as: IW i∗ 1, 1 (q) = 1, 1 IW i∗ 1, 1 ( q – 1 ) – α ( p ( q ) – i∗IW (q – 1)) Such corrections move the hidden neuron towards vectors that fall into the class for which it forms a subclass, and away from vectors which fall into other classes. The default learning function that implements these changes in the layer 1 weights in LVQ networks is learnlv2. It will be applied during training. 8-10 Training Training Next we need to train the network to obtain first layer weights that will lead to the correct classification of input vectors. We will do this with train as shown below. First set the training epochs to 1000 and the learning rate to 0.05. Then use train. net.trainParam.epochs = 1000; net.trainParam.lr = 0.05; net = train(net,P,T); Now check on the first layer weights. net.IW{1,1} ans = -2.2506 2.2432 -0.0023 0 -0.0948 -0.0623 1.3048 -1.2692 The following plot shows that these weights are fairly close to their respective classification groups. 3 2 1 0 -1 -2 -3 -5 0 5 Weights (circles) after 1000 epochs 8-11 8 Learning Vector Quantization To check to see that these weights do indeed lead to the correct classification, take the matrix P as input and simulate the network. Then see what classifications are produced by the network. Y = sim(net,P) Yc = vec2ind(Y) This gives Yc = 1 1 1 2 2 2 2 1 1 1 which is what we expected. As a last check, try an input close to a vector that was used in training. pchk1 = [0; 0.5]; Y = sim(net,pchk1); Yc1 = vec2ind(Y) Which gives Yc1 = 2 This looks right, for pchk1 is close to other vectors classified as 2. Similarly, pchk2 = [1; 0]; Y = sim(net,pchk2); Yc2 = vec2ind(Y) gives Yc2 = 1 This looks right too, for pchk2 is close to other vectors classified as 1s. You might try demolvq1 to see an LVQ network problem. Note that you get about the same results by running adapt for 100 epochs through the10 vectors. This will give us 1000 vector presentations just the same as was done when using train. First set the epochs to 100. net.adaptparam.passes = 100; 8-12 Training Recall that the input vectors for adapt are to be in a cell array, so create such an array from the original input signals with the following code. Pseq = con2seq(P) Tseq = con2seq(T) Now apply adapt, and check on the results. net = adapt(net,Pseq,Tseq); wtsl1b = net.IW{1,1} wtsl1b = -2.2157 -0.0032 2.2221 -0.0027 -0.0027 1.2955 -0.0027 -1.2999 Now test the network with our original input vectors as we did before. Y = sim(net,P); Yc = vec2ind(Y); Yc = 1 1 1 2 2 2 2 1 1 1 Here adapt and train give nearly the same results in about the same running time. You may recall that when train is used for competitive networks, it selects the input vectors in random order, and for that reason might give a better result than adapt in some problems. In the problem studied here the results given by train and adapt are about the same. Of course, one could generate 1000 randomly ordered vectors from the 10 input vectors with the following code. TS = 1000; ind = floor(rand(1,TS)*size(P,2))+1; Pseq = con2seq(P(:,ind)); Tseq = con2seq(T(:,ind)); Finally then, we would need to make only one pass. net.adaptparam.passes = 1; net = adapt(net,Pseq,Tseq); This code gives a network much like those from train and adapt that were described previously. 8-13 8 Learning Vector Quantization Summary LVQ networks classify input vectors into target classes by using a competitive layer to find subclasses of input vectors, and then combining them into the target classes. Unlike perceptrons, LVQ networks can classify any set of input vectors, not just linearly separable sets of input vectors. The only requirement is that the competitive layer must have enough neurons, and each class must be assigned enough competitive neurons. To insure that each class is assigned an appropriate amount of competitive neurons, it is important that the target vectors used to initialize the LVQ network have the same distributions of targets as the training data the network will be trained on. If this is done, target classes with more vectors will be the union of more subclasses. Figures LVQ Architecture Input AA AAAA AA IW1,1 S1xR p R x1 R A A A A || ndist || n1 S1x1 C a2 a1 S1x1 S1 ni1 = - || iIW1,1 - p || a1 = compet (n1) AA AA AA AAAA AA Linear Layer Competitive Layer LW2,1 S2x1 n2 S2x1 S2xS1 1 Where... R = # elements in input vector S1= # competitive neurons S2= # linear neurons S2 a2 = purelin(LW2,1 a1) New Functions This chapter introduces the following new functions: 8-14 Function Description newlvq Create a learning vector quantization network. learnlv2 LVQ2 weight learning function. 9‘ Recurrent Networks Introduction . . . . . . . . . . . . . . . . . . . . 9-2 Important Recurrent Network Functions . . . . . . . . . 9-2 Elman Networks . . . . . . . . . Architecture . . . . . . . . . . . . Creating an Elman Network (NEWELM) Training an Elman Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-3 9-3 9-4 9-6 Hopfield Network Fundamentals . . . Architecture . . . . Design(NEWHOP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-9 9-9 9-9 9-11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . 9-16 Figures . . . . . . . . . . . . . . . . . . . . . . . 9-17 New Functions . . . . . . . . . . . . . . . . . . . 9-18 9 Recurrent Networks Introduction Recurrent networks is a topic of considerable interest. This chapter covers two recurrent networks: Elman and Hopfield networks. Elman networks are two-layer backpropagation networks, with the addition of a feedback connection from the output of the hidden layer to its input. This feedback path allows Elman networks to learn to recognize and generate temporal patterns, as well as spatial patterns. The best paper on the Elman network is: Elman, J. L., “Finding structure in time,” Cognitive Science, vol. 14, 1990, pp. 179-211. The Hopfield network is used to store one or more stable target vectors. These stable vectors can be viewed as memories which the network recalls when provided with similar vectors which act as a cue to the network memory. You may want to pursue a basic paper in this field: Li, J., A. N. Michel, and W. Porod, “Analysis and synthesis of a class of neural networks: linear systems operating on a closed hypercube,” IEEE Transactions on Circuits and Systems, vol. 36, no. 11, November 1989, pp. 1405-1422. Important Recurrent Network Functions Elman networks can be created with the function newelm. Hopfield networks can be created with the function newhop. Type help elman or help hopfield to see a list of function and demonstrations related to either of these networks. 9-2 Elman Networks Elman Networks Architecture The Elman network commonly is a two-layer network with feedback from the first layer output to the first layer input. This recurrent connection allows the Elman network to both detect and generate time-varying patterns. A two layer Elman network is shown below. AA AA AA AA AA AA AA AA AA AA AA AA D a (k-1) 1 LW1,1 p R1 x 1 IW1,1 S 1 x R1 a1(k) LW2,1 S1x1 n1 S2xS1 S1x1 1 R1 Input b 1 S1x1 S1 Recurrent tansig layer a1(k) = tansig (IW1,1p +LW1,1a1(k-1) + b1) AA AA AA AA 1 b 2 S1x1 n2 S2x1 A A A A a2(k) S2x1 S2 Output purelin Layer a2(k) = purelin (LW2,1a1(k) + b2) The Elman network has tansig neurons in its hidden (recurrent) layer, and purelin neurons in its output layer. This combination is special in that two-layer networks with these transfer functions can approximate any function (with a finite number of discontinuities) with arbitrary accuracy. The only requirement is that the hidden layer must have enough neurons. More hidden neurons are needed as the function being fit increases in complexity. Note that the Elman network differs from conventional two-layer networks in that the first layer has a recurrent connection. The delay in this connection stores values from the previous time step, which can be used in the current time step. Thus, even if two Elman networks, with the same weights and biases, are given identical inputs at a given time step, their outputs can be different due to different feedback states. 9-3 9 Recurrent Networks Because the network can store information for future reference it is able to learn temporal patterns as well as spatial patterns. The Elman network can be trained to respond to, and to generate, both kinds of patterns. Creating an Elman Network (NEWELM) An Elman network with two or more layers can be created with the function newelm. The hidden layers commonly have tansig transfer functions, so that is the default for newelm. As shown in the architecture diagram, purelin is commonly the output layer transfer function. The default backpropagation training function is trainbfg. One might use trainlm, but it tends to proceed so rapidly that it does not necessarily do well in the Elman network. The backprop weight/bias learning function default is learngdm, and the default performance function is mse. When the network is created, each layers weights and biases are initialized with the Nguyen-Widrow layer initialization method implemented in the function initnw. Now consider an example. Suppose that we have a sequence of single element input vectors in the range from 0 to 1. Suppose further that we want to have five hidden layer tansig neurons and a single logsig output layer. The following code creates the desired network. net = newelm([0 1],[5 1],{'tansig','logsig'}); Simulation Suppose that we would like to find the response of this network to an input sequence of eight digits which are either 0 or 1. P = round(rand(1,8)) P = 0 1 0 1 1 0 0 0 Recall that a sequence to be presented to a network is to be in cell array form. We can convert P to this form with Pseq = con2seq(P) Pseq = [0] [1] [0] 9-4 [1] [1] [0] [0] [0] Elman Networks Now we can find the output of the network with the function sim. Y = sim(net,Pseq) Y = Columns 1 through 5 [1.9875e-04] [0.1146] [5.0677e-05] [0.0017] Columns 6 through 8 [0.0014] [5.7241e-05] [3.6413e-05] [0.9544] We will convert this back to concurrent form with z = seq2con(Y); and can finally display the output in concurrent form with z{1,1} ans = Columns 1 through 7 0.0002 0.1146 0.0001 Column 8 0.0000 0.0017 0.9544 0.0014 0.0001 Thus, once the network is created and the input specified, one need only call sim. 9-5 9 Recurrent Networks Training an Elman Network Elman networks can be trained with either of two functions, train or adapt. When using the function train to train an Elman network the following occurs. At each epoch: 1 The entire input sequence is presented to the network, and its outputs are calculated and compared with the target sequence to generate an error sequence. 2 For each time step the error is backpropagated to find gradients of errors for each weight and bias. This gradient is actually an approximation since the contributions of weights and biases to errors via the delayed recurrent connection are ignored. 3 This gradient is then used to update the weights with the backprop training function chosen by the user. The function traingdx is recommended. When using the function adapt to train an Elman network the following occurs. At each time step: 1 Input vectors are presented to the network, and it generates an error. 2 The error is backpropagated to find gradients of errors for each weight and bias. This gradient is actually an approximation since the contributions of weights and biases to the error, via the delayed recurrent connection are ignored. 3 This approximate gradient is then used to update the weights with the learning function chosen by the user. The function learngdm is recommended. Elman network's are not as reliable as some other kinds of networks because both training and adaption happen using an approximation of the error gradient. For an Elman to have the best chance at learning a problem it needs more hidden neurons in its hidden layer than are actually required for a solution by another method, for while a solution may be available with fewer neurons, the 9-6 Elman Networks Elman network is less able to find the most appropriate weights for hidden neurons since the error gradient is approximated. Therefore having a fair number of neurons to begin with makes it more likely that the hidden neurons will start out dividing up the input space in useful ways. The function train trains an Elman network to generate a sequence of target vectors when it is presented with a given sequence of input vectors. The input vectors and target vectors are passed to train as matrices P and T. Train takes these vectors and the initial weights and biases of the network, trains the network using backpropagation with momentum and an adaptive learning rate, and returns new weights and biases. Let us continue with the example of the previous section, and suppose that we wish to train a network with an input P and targets T as defined below. P = round(rand(1,8)) P = 1 0 1 1 1 0 1 1 T = [0 (P(1:end-1)+P(2:end) == 2)] T = 0 0 0 1 1 0 0 1 and Here T is defined to be 0 except when two ones occur in P, in which case T will be 1. As noted previously, our network will have five hidden neurons in the first layer. net = newelm([0 1],[5 1],{'tansig','logsig'}); We will use trainbfg as the training function and train for 100 epochs. After training we will simulate the network with the input P and calculate the difference between the target output and the simulated network output. net = train(net,Pseq,Tseq); Y = sim(net,Pseq); z = seq2con(Y); z{1,1}; diff1 = T - z{1,1} 9-7 9 Recurrent Networks Note that the difference between the target and the simulated output of the trained network is very small. Thus, the network have been trained to produce the desired output sequence on presentation of the input vector P. See Chapter 10 for an application of the Elman network to the detection of wave amplitudes. 9-8 Hopfield Network Hopfield Network Fundamentals The goal here is to design a network that stores a specific set of equilibrium points such that, when an initial condition is provided, the network eventually comes to rest at such a design point. The network is recursive in that the output is fed back as the input once the network is in operation. Hopefully the network output will settle on one of the original design points The design method that we present is not perfect in that the designed network may have undesired spurious equilibrium points in addition to the desired ones. However, the number of these undesired points is made as small as possible by the design method. Further, the domain of attraction of the designed equilibrium points is as large as possible. The design method is based on a system of first-order linear ordinary differential equations that are defined on a closed hypercube of the state space. The solutions exist on the boundary of the hypercube. These systems have the basic structure of the Hopfield model but are easier to understand and design than the Hopfield model. The material in this section is based on the following paper: Jian-Hua Li, Anthony N. Michel and Wolfgang Porod, “Analysis and synthesis of a class of neural networks: linear systems operating on a closed hypercube,” IEEE Trans. on Circuits and Systems vol 36, no. 11, pp. 1405-22, November 1989. You may wish to read Chapter 18 of [HDB96] for further information on Hopfield networks. Architecture The architecture of the network that we will use follows. 9-9 9 Recurrent Networks AA AA AA AA AA AA AA AA AA AA a1(k-1) p LW1,1 S 1 x R1 a (0) 1 R1 x 1 D n1 a1(k) S1x1 S1x1 1 b1 R1 S1x1 S1 Sym. Sat. Linear Layer a (0) = p and then for k = 1, 2, ... Initial Conditions 1 a1(k) = satlins (LW1,1a1(k-1)) + b1) As noted, the input p to this network merely supplies the initial conditions. The Hopfield network uses the saturated linear transfer function satlins. a +1 n -1 0 +1 -1 AA AA a = satlins(n) Satlins Transfer Function For inputs less than -1 satlins produces -1. For inputs in the range -1 to +1 it simply returns the input value. For inputs greater than +1 it produces +1. This network can be tested with one or more input vectors which are presented as initial conditions to the network. After the initial conditions are given, the network produces an output which is then fed back to become the input. This process is repeated over and over until the output stabilizes. Hopefully, each output vector eventually converges to one of the design equilibrium point vectors that is closest to the input that provoked it. 9-10 Hopfield Network Design(NEWHOP) Li et. al. [LiMi89] have studied a system that has the basic structure of the Hopfield network but is, in Li’s own words, “easier to analyze, synthesize and implement than the Hopfield model.” The authors are enthusiastic about the reference article, as it has many excellent points and is one of the most readable in the field. However, the design is mathematically complex, and even a short justification of it would burden this guide. Thus, we present the Li design method, with thanks to Li et al., as a recipe that is found in the function newhop. Given a set of target equilibrium points represented as a matrix T of vectors, newhop returns weights and biases for a recursive network. The network is guaranteed to have stable equilibrium points at the target vectors, but it may contain other spurious equilibrium points as well. The number of these undesired points is made as small as possible by the design method. Once the network has been designed, it can be tested with one or more input vectors. Hopefully those input vectors close to target equilibrium points will find their targets. As suggested by the network figure, an array of input vectors may be presented at one time or in a batch. The network proceeds to give output vectors which are fed back as inputs. These output vectors can be can be compared to the target vectors to see how the solution is proceeding. The ability to run batches of trial input vectors quickly allows you to check the design in a relatively short time. First you might check to see that the target equilibrium point vectors are indeed contained in the network. Then you could try other input vectors to determine the domains of attraction of the target equilibrium points and the locations of spurious equilibrium points if they are present. Consider the following design example. Suppose that we wish to design a network with two stable points in a three dimensional space. T = [-1 -1 1; 1 -1 1]' T = -1 1 -1 -1 1 1 We can execute the design with net = newhop(T); 9-11 9 Recurrent Networks Next we might check to make sure that the designed network is at these two points. We can do this as follows. (Since Hopfield networks have no inputs, the second argument to sim below is Q = 2 when using matrix notation). Ai = T; [Y,Pf,Af] = sim(net,2,[],Ai); Y This gives us Y = -1 -1 1 1 -1 1 Thus, the network has indeed been designed to be stable at its design points. Next we might try another input condition that is not a design point, such as: Ai = {[-0.9; -0.8; 0.7]} which gives Ai = -0.9000 -0.8000 0.7000 .This point is reasonably close to the first design point, so one might anticipate that the network would converge to that first point. To see if this happens, we will run the following code. Note, incidentally, that we specified the original point in cell array form. This allows us to run the network for more than one step. [Y,Pf,Af] = sim(net,{1 5},{},Ai); Y{1} We get Y = -1 -1 1 Thus, an original condition close to a design point did converge to that point. 9-12 Hopfield Network This is, of course, our hope for all such inputs. Unfortunately, even the best known Hopfield designs occasionally include undesired spurious stable points that attract the solution. Example Consider a Hopfield network with just two neurons. Each neuron has a bias and weights to accommodate two element input vectors weighted. We define the target equilibrium points to be stored in the network as the two columns of the matrix T. T = [1 -1; -1 1]' T = 1 -1 -1 1 Here is a plot of the Hopfield state space with the two stable points labeled with ‘*’ markers. Hopfield Network State Space 1 a(2 ) 0.5 0 -0.5 -1 -1 0 a(1) 1 These target stable points are given to newhop to obtain weights and biases of a Hopfield network. net = newhop(T); 9-13 9 Recurrent Networks The design returns a set of weights and a bias for each neuron. The results are obtained from: W= net.LW{1,1} which gives W = 0.6925 -0.4694 -0.4694 0.6925 and from b = net.b{1,1} which gives b = 1.0e-16 * 0.6900 0.6900 Next the design is tested with the target vectors T to see if they are stored in the network. The targets are used as inputs for the simulation function sim. Ai = T; [Y,Pf,Af] = sim(net,2,[],Ai); Y = 1 -1 -1 1 As hoped, the new network outputs are the target vectors. The solution stays at its initial conditions after a single update, and therefore will stay there for any number of updates. 9-14 Hopfield Network Now you might wonder how the network performs with various random input vectors. Here is a plot showing the paths that the network took through its state space, to arrive at a target point. Hopfield Network State Space 1 a(2 ) 0.5 0 -0.5 -1 -1 0 a(1) 1 This plot show the trajectories of the solution for various starting points. You might try the demonstration demohop1 to see more of this kind of network behavior. Hopfield networks can be designed for an arbitrary number of dimensions. You might try demohop3 to see a three dimensional design. Unfortunately, Hopfield networks may have both unstable equilibrium points and spurious stable points. You might try demonstration programs demohop2 and demohop4 to investigate these issues. 9-15 9 Recurrent Networks Summary Elman networks, by having an internal feedback loop, are capable of learning to detect and generate temporal patterns. This makes Elman networks useful in such areas as signal processing and prediction where time plays a dominant role. Because Elman networks are an extension of the two-layer sigmoid/linear architecture, they inherit the ability to fit any input/output function with a finite number of discontinuities. They are also able to fit temporal patterns, but may need many neurons in the recurrent layer to fit a complex function. Hopfield networks can act as error correction or vector categorization networks. Input vectors are used as the initial conditions to the network, which recurrently updates until it reaches a stable output vector. Hopfield networks are interesting from a theoretical standpoint, but are seldom used in practice. Even the best Hopfield designs may have spurious stable points that lead to incorrect answers. More efficient and reliable error correction techniques, such as backpropagation, are available. 9-16 Summary Figures Elman Network AA AAA AA AAA AA AA AA AA AA AA AA D a1(k-1) LW1,1 p R1 x 1 a1(k) IW1,1 LW2,1 S1x1 n1 S 1 x R1 S2xS1 S1x1 1 b1 S1x1 R1 Input S1 Recurrent tansig layer AA AA AA AA 1 b2 S1x1 AA AA AA LW1,1 S 1 x R1 1 R1 Initial Conditions b S2 a2(k) = purelin (LW2,1a1(k) + b2) D a1(k-1) R1 x 1 S2x1 AA AA Hopfield Network a1(0) S2x1 a2(k) Output purelin Layer a1(k) = tansig (IW1,1p +LW1,1a1(k-1) + b1) p n2 A A A A 1 S1x1 AA AA AA a1(k) S1x1 n1 S1x1 S1 Sym. Sat. Linear Layer a1(0) = p and then for k = 1, 2, ... a1(k) = satlins (LW1,1a1(k-1)) + b1) 9-17 9 Recurrent Networks New Functions This chapter introduces the following new functions: 9-18 Function Description newelm Create an Elman backpropagation network. newhop Create a Hopfield recurrent network. satlins Symmetric saturating linear transfer function. 10 Applications Introduction . . . . . . . . . . . . . . . . . . . . 10-2 Applin1: Linear Design . . . . . . . . . . . . . . . 10-3 Applin2: Adaptive Prediction . . . . . . . . . . . . 10-7 Applin3: Linear System Identification . . . . . . . 10-11 Applin4: Adaptive System Identification . . . . . . 10-15 Appelm1: Amplitude Detection . . . . . . . . . . 10-19 Appcs1: Nonlinear System Identification . . . . . . 10-24 Appcs2: Model Reference Control Appcr1: Character Recognition . . . . . . . . . 10-30 . . . . . . . . . . 10-38 10 Applications Introduction Today, problems of economic importance that could not be approached previously in any practical way can now be solved with neural networks. Some of the recent neural network applications are discussed in this chapter. See Chapter 1 for a list of many areas where neural networks already have been applied. The rest of this chapter describes applications that are practical and make extensive use of the neural network functions described in this User’s Guide. Application Scripts The linear network applications are contained in scripts applin1 through applin4. The Elman network amplitude detection application is contained in the script appelm1. The control system applications are contained in scripts appcs1 and appcs2. The character recognition application is in appcr1. Type help nndemos to see a listing of all neural network demonstrations or applications. 10-2 Applin1: Linear Design Applin1: Linear Design Problem Definition Here is the definition of a signal T which lasts five seconds, and is defined at a sampling rate of 40 samples per second. time = 0:0.025:5; T = sin(time*4*pi); At any given time step, the network is given the last five values of the signal t, and expected to give the next value. The inputs P are found by delaying the signal T from one to five time steps. P = zeros(5,Q); P(1,2:Q) = T(1,1:(Q-1)); P(2,3:Q) = T(1,1:(Q-2)); P(3,4:Q) = T(1,1:(Q-3)); P(4,5:Q) = T(1,1:(Q-4)); P(5,6:Q) = T(1,1:(Q-5)); Here is a plot of the signal T. Signal to be Predicted 1 0.8 0.6 Target Signal 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0 0.5 1 1.5 2 2.5 Time 3 3.5 4 4.5 5 10-3 10 Applications Network Design Because the relationship between past and future values of the signal is not changing, the network can be designed directly from examples using newlind. The problem as defined above has five inputs, the five delayed signal values, and one output, the next signal value. Thus the network solution must consist of a single neuron with five inputs. Linear Neuron Input p1 p2 p3 p4 p5 AA A A w1,1 n w1, 5 a b 1 a = purelin (Wp +b) Here newlind finds the weights and biases, for the neuron above, that minimize the sum-squared error for this problem. net = newlind(P,T); The resulting network can now be tested. Network Testing To test the network, its output a is computed for the five delayed signals P and compared with the actual signal T. a = sim(net,P); Here is a plot of a compared to T. 10-4 Applin1: Linear Design Output and Target Signals 1 0.8 0.6 Output - Target + 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0 0.5 1 1.5 2 2.5 Time 3 3.5 4 4.5 5 The network’s output a and the actual signal t appear to match up perfectly. Just to be sure, let us plot the error e = T – a. Error Signal 0.35 0.3 0.25 Error 0.2 0.15 0.1 0.05 0 -0.05 0 0.5 1 1.5 2 2.5 Time 3 3.5 4 4.5 5 The network did have some error for the first few time steps. This occurred because the network did not actually have five delayed signal values available until the fifth time step. However, after the fifth time step error was negligible. The linear network did a good job. Run the script applin1 to see these plots. 10-5 10 Applications Thoughts and Conclusions While newlind is not able to return a zero error solution for nonlinear problems, it does minimize the sum-squared error. In many cases the solution, while not perfect, may model a nonlinear relationship well enough to meet the application specifications. Giving the linear network many delayed signal values gives it more information with which to find the lowest error linear fit for a nonlinear problem. Of course, if the problem is very nonlinear and/or the desired error is very low, backpropagation or radial basis networks would be more appropriate. 10-6 Applin2: Adaptive Prediction Applin2: Adaptive Prediction In application script applin2, a linear network is trained incrementally with adapt to predict a time series. Because the linear network is trained incrementally, it can respond to changes in the relationship between past and future values of the signal. Problem Definition The signal T to be predicted lasts six seconds with a sampling rate of 20 samples per second. However, after four seconds the signal’s frequency suddenly doubles. time1 = 0:0.05:4; time2 = 4.05:0.024:6; time = [time1 time2]; T = [sin(time1*4*pi) sin(time2*8*pi)]; Since we will be training the network incrementally, we will change t to a sequence. T = con2seq(T); Here is a plot of this signal: Signal to be Predicted 1 0.8 0.6 Target Signal 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0 1 2 3 Time 4 5 6 10-7 10 Applications The input to the network is the same signal which makes up the target: P = T; Network Initialization The network has only one neuron, as only one output value of the signal T is being generated at each time step. This neuron has five inputs, the five delayed values of the signal T. Linear Layer AA A AA A AA A AA pd(k) p(k) TDL 1x1 1 5x1 W 3 4 a(k) n(k) 1x3 2 1 AA AA AA 1x1 1x1 b 1x1 5 The function newlin creates the network shown above. We will use a learning rate of 0.1 for incremental training. lr = 0.1; delays = [1 2 3 4 5]; net = newlin(minmax(cat(2,P{:})),1,delays,lr); [w,b] = initlin(P,t) Network Training The above neuron will be trained incrementally with adapt. Here is the code to train the network on input/target signals P and T. [net,a,e]=adapt(net,P,T); Network Testing Once the network has been adapted, we can plot its output signal and compare it to the target signal. 10-8 Applin2: Adaptive Prediction Output and Target Signals 1.5 1 Output --- Target - - 0.5 0 -0.5 -1 -1.5 0 1 2 3 Time 4 5 6 Initially, it takes the network a second and a half (30 samples) to track the target signal. Then the predictions are accurate until the fourth second when the target signal suddenly changes frequency. However, the adaptive network learns to track the new signal in an even shorter interval as it has already learned a behavior (a sine wave) similar to the new signal. A plot of the error signal makes these effects easier to see. Error Signal 1 0.8 0.6 0.4 Error 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0 1 2 3 Time 4 5 6 10-9 10 Applications Thoughts and Conclusions The linear network was able to adapt very quickly to the change in the target signal. The 30 samples required to learn the wave form are very impressive when one considers that in a typical signal processing application a signal may be sampled at 20 kHz. At such a sampling frequency 30 samples go by in 1.5 milliseconds. 10-10 Applin3: Linear System Identification Applin3: Linear System Identification Linear networks may be used to model real systems. If the real system is linear or near linear then the linear network can act as a zero, or low, error model. Problem Definition Here is an input signal x which is given to a finite impulse response linear system over a period of five seconds at 25 msec intervals. time = 0:0.025:5; X = sin(sin(time).*time*10); Here is a plot of this input signal. Input Signal to the System 1 0.8 0.6 0.4 Input Signal 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0 0.5 1 1.5 2 2.5 Time 3 3.5 4 4.5 5 Here are the measured outputs T of the system. T = filter([1 0.5 –1.5],1,X); Here is a plot of the systems output T. 10-11 10 Applications Output Signal of the System 2 1.5 1 Output Signal 0.5 0 -0.5 -1 -1.5 -2 0 0.5 1 1.5 2 2.5 Time 3 3.5 4 4.5 5 The network attempts to predict the system’s output T given the current and previous two input signals P from X. P = zeros(3,Q); P(1,1:Q) = X(1,1:Q); P(2,2:Q) = X(1,1:(Q-1)); P(3,3:Q) = X(1,1:(Q-2)); Network Design The network required to solve this problem has only one neuron because the system has only one output. That neuron has to have three inputs to receive the current and two delayed values of the input signal. The function newlind designs such a neuron for us. net = newlind(P,T); Network Testing Once the weights and biases are obtained, the network can be tested. Here the linear network computes its approximation a of the system output T given the current and delayed values of the system input. a = sim(net,P); 10-12 Applin3: Linear System Identification The system output T and the network output a are compared in the following plot. Network and System Output Signals 2 1.5 Network Output - System Output + 1 0.5 0 -0.5 -1 -1.5 -2 0 0.5 1 1.5 2 2.5 Time 3 3.5 4 4.5 5 The network appears to do a perfect job of modeling the system. Here is a plot of the difference between network and system outputs. -15 2 Error Signal x 10 Error 1 0 -1 0 0.5 1 1.5 2 2.5 Time 3 3.5 4 4.5 5 Sure enough, the difference is incredibly small (less than 1e–15). 10-13 10 Applications Thoughts and Conclusions As with linear predictors, linear system models can be used to model linear systems with zero error, and nonlinear systems with minimum sum-squared error. Giving the linear network many delayed values from the system’s input signal will allow newlind to minimize the error associated with modeling a nonlinear system. However, if the system is highly nonlinear, a backpropagation or radial basis network would be more appropriate. 10-14 Applin4: Adaptive System Identification Applin4: Adaptive System Identification Demonstration applin4 shows how to train a linear network to adaptively model a linear system. By training the network incrementally, it can change its behavior as the system it is modeling changes. Problem Definition Here is the input signal X to the system over a period of six seconds with a sampling rate of 200 samples per second. time1 = 0:0.005:4; time2 = 4.005:0.005:6; time = [time1 time2]; X = sin(sin(time*4).*time*8); The input signal X to the system is plotted below. Input Signal to System 1 0.8 0.6 0.4 Input Signal 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0 1 2 3 Time 4 5 6 Here is the output signal T of the system. Note that the system acts differently in the intervals before and after the fourth second. steps1 = length(time1); [T1,state] = filter([1 –0.5],1,X(1:steps1)); steps2 = length(time2); T2 = filter([0.9 –0.6],1,x((1:steps2) + steps1),state); T = [T1 T2]; 10-15 10 Applications Here is a plot of the system’s outputs. Note that due to the complexity of the signal, it is not apparent that the system changed at four seconds. Output Signal of System 0.8 0.6 0.4 Output Signal 0.2 0 -0.2 -0.4 -0.6 -0.8 0 1 2 3 Time 4 5 6 The input to the network P is the system input signal X. The network must estimate the system’s output T from the last two values of the input signal. We need to convert the input and output to sequences, so that the training can be done incrementally. T = con2seq(T); P = con2seq(X); Network Initialization The network is created with newlin, which generates the weights and biases for the two-input linear neuron required for this problem. There will be a tapped delay line with two delays at the input of the network. The last two values of the input signal will be used to predict the system’s output. lr = 0.5; delays = [0 1]; net = newlin(minmax(cat(2,P{:})),1,delays,lr); 10-16 Applin4: Adaptive System Identification Network Training These weights and biases can now be trained incrementally on the signal with adapt using the learning rate of 0.5, which was set by newlin. [net,a,e]=adapt(net,P,T); Network Testing To see how well the network did, we could plot the network’s estimate a of the system’s output against the actual output T. It turns out that the signals are complex enough that this plot is not of much help. (You can run applin4 to see it.) Instead we can take a look at the error between the network output and the system output. This plot is easier to interpret. Error Signal 0.25 0.2 0.15 0.1 Error 0.05 0 -0.05 -0.1 -0.15 -0.2 -0.25 0 1 2 3 Time 4 5 6 The error plot shows that the network took 2.5 seconds to track the model with a very high accuracy. Then at four seconds, when the system changed abruptly, another 0.2 seconds were required for network to properly model it again. Thoughts and Conclusions Besides being interesting, an adaptive linear model of a system can be used to obtain a great deal of information. For example, the adaptive model can be analyzed at any given time to determine characteristics of the actual system. 10-17 10 Applications For example, the adaptive network might be monitored so as to give a warning its constants were nearing values that would result in instability. Another use for an adaptive linear model is suggested by its ability to find a minimum sum-squared error linear estimate of a nonlinear system’s behavior. An adaptive linear model will be highly accurate as long as the nonlinear system stays near a given operating point. If the nonlinear system moves to a different operating point, the adaptive linear network will change to model it at the new point. The sampling rate should be high to obtain the linear model of the nonlinear system at its current operating point in the shortest amount of time. However, there is a minimum amount of time that must occur for the network to see enough of the system’s behavior to properly model it. To minimize this time, a small amount of noise can be added to the input signals of the nonlinear system. This allows the network to adapt faster as more of the operating points dynamics will be expressed in a shorter amount of time. Of course, this noise should be small enough so it does not affect the system’s usefulness. 10-18 Appelm1: Amplitude Detection Appelm1: Amplitude Detection Elman networks can be trained to recognize and produce both spatial and temporal patterns. An example of a problem where temporal patterns are recognized and classified with a spatial pattern is amplitude detection. Amplitude detection requires that a wave form be presented to a network through time, and that the network output the amplitude of the wave form. This is not a difficult problem, but it demonstrates the Elman network design process. The following material describes code which is contained in the demonstration script appelm1. Problem Definition The following code defines two sine wave forms, one with an amplitude of 1.0, the other with an amplitude of 2.0: p1 = sin(1:20); p2 = sin(1:20)*2; The target outputs for these wave forms will be their amplitudes. t1 = ones(1,20); t2 = ones(1,20)*2; These wave forms can be combined into a sequence where each wave form occurs twice. These longer wave forms will be used to train the Elman network. p = [p1 p2 p1 p2]; t = [t1 t2 t1 t2]; We want the inputs and targets to be considered a sequence, so we need to make the conversion from the matrix format. Pseq = con2seq(p); Tseq = con2seq(t); 10-19 10 Applications Network Initialization This problem requires that the Elman network detect a single value (the signal), and output a single value (the amplitude), at each time step. Therefore the network must have one input element, and one output neuron. R = 1;% 1 input eleement S2 = 1;% 1 layer 2 output neuron The recurrent layer can have any number of neurons. However, as the complexity of the problem grows, more neurons are needed in the recurrent layer for the network to do a good job. This problem is fairly simple, so only 10 recurrent neurons will be used in the first layer. S1 = 10;% 10 recurrent neurons in the first layer Now the function newelm can be used to create initial weight matrices and bias vectors for a network with one input which can vary between –2 and +2. We will use variable learning rate (traingdx) for this example. net = newelm([-2 2],[S1 S2],{'tansig','purelin'},'traingdx'); Network Training Now call train. [net,tr] = train(net,Pseq,Tseq); As this function finishes training at 500 epochs, it displays the following plot of errors. 10-20 Appelm1: Amplitude Detection Mean Squared Error of Elman Network 1 10 0 Mean Squared Error 10 -1 10 -2 10 0 50 100 150 Epoch 200 250 300 The final mean-squared error was about 1.8e-2. We can test the network to see what this means. Network Testing To test the network, the original inputs are presented, and its outputs are calculated with simuelm. a = sim(net,Pseq); Here is the plot: 10-21 10 Applications Testing Amplitute Detection 2.2 2 Target - - Output --- 1.8 1.6 1.4 1.2 1 0.8 0 10 20 30 40 Time Step 50 60 70 80 The network does a good job. New wave amplitudes are detected with a few samples. More neurons in the recurrent layer and longer training times would result in even better performance. The network has successfully learned to detect the amplitudes of incoming sine waves. Network Generalization Of course, even if the network detects the amplitudes of the training wave forms, it may not detect the amplitude of a sine wave with an amplitude it has not seen before. The following code defines a new wave form made up of two repetitions of a sine wave with amplitude 1.6 and another with amplitude 1.2. p3 = sin(1:20)*1.6; t3 = ones(1,20)*1.6; p4 = sin(1:20)*1.2; t4 = ones(1,20)*1.2; pg = [p3 p4 p3 p4]; tg = [t3 t4 t3 t4]; pgseq = con2seq(pg); The input sequence pg and target sequence tg will be used to test the ability of our network to generalize to new amplitudes. 10-22 Appelm1: Amplitude Detection Once again the function sim is used to simulate the Elman network and the results are plotted. a = sim(net,pgseq); Testing Generalization 2 1.9 1.8 Target - - Output --- 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 0 10 20 30 40 Time Step 50 60 70 80 This time the network did not do as well. It seems to have a vague idea as to what it should do, but is not very accurate! Improved generalization could be obtained by training the network on more amplitudes than just 1.0 and 2.0. The use of three or four different wave forms with different amplitudes can result in a much better amplitude detector. Improving Performance Run appelm1 to see plots similar to those above. Then make a copy of this file and try improving the network by adding more neurons to the recurrent layer, using longer training times, and giving the network more examples in its training data. 10-23 10 Applications Appcs1: Nonlinear System Identification Application script appcs1 demonstrates how to perform system identification on the following inverted pendulum system: 0º u(t) Ø(t) 180º Here the antenna arm angle Ø is controlled by applying a current to a dc motor which is attached to the pendulum. This system may be represented by the following equations: Nonlinear System Model (NSM) x2 d x ----- 1 = dt x 9.81 sin x 1 – 2x 2 +u 2 where: x1 = ∅ d∅ x 2 = -------dt and u is the force applied to the pendulum by a motor. A positive force tends to turn the pendulum clockwise. The 9.81sin x1 term is the force of gravity on the pendulum and the –2x2 term is viscous friction acting against the velocity. 10-24 Appcs1: Nonlinear System Identification The inverted pendulum system is summarized with the function pmodel which takes the current time t, pendulum angle, pendulum velocity, and the current, and returns the derivatives of angle, velocity, and force. x = [angle; vel; force]; dx = pmodel(t,x) Because the current is an input to the system, its derivative is always returned as 0. We can simulate the pendulum from 0 to 0.05 seconds using ode23. [time,X] = ode23(‘pmodel’,[0 0.05],x) This function returns a row vector of times, and the matrix X of state vectors associated with those times. The problem is, given only the behavior of this pendulum, create a network model that behaves in an identical manner. Problem Definition Examples of pendulum behavior must be created so that the network can be trained. The following lines of code define several different pendulum angles, pendulum velocities, and forces. deg2rad = pi/180; angle = [–20:40:200]*deg2rad; vel = [–90:36:90]*deg2rad; force = –30:6:30; By taking all possible combinations of these values, and also a set of steady state conditions at various angles, we get a matrix Pm of 749 pendulum state/ input conditions, where the pendulum’s state is its angle and velocity, and its input is the current. angle2 = [-20:10:200]*deg2rad; Pm = [combvec(angle,vel,force); [angle2; zeros(2,length(angle2))]]; Next each of these 749 initial state/input conditions are applied to the pendulum, and its next state, 0.05 seconds later is measured. The next state consists of the pendulum angle and velocity after 0.05 seconds. The result is a matrix Tm of the 749 2-element next states. (Actually, we will use the difference 10-25 10 Applications between the next state and the current state as the target. The neural network model will learn to predict the change in state over 0.05 seconds. This is done because the state does not change by a large amount in this period of time, and we can improve the performance of the model if we predict only the change in state. If we need to know the actual state we simply need to add the change to the previous state.) Network Initialization The network must transform the pendulum state and input into a next state (or change in state). As there are two states (angle and velocity) and one input, the network requires three inputs and one output. The function initff is used to create a two-layer tansig/purelin network with these number of inputs and outputs and eight hidden neurons. S1 = 8; [S2,Q] = size(Tm); mnet = newff(minmax(Pm),[S1 S2],{'tansig' 'purelin'},'trainlm'); We use this network to model the nonlinear pendulum system. Network Training The Levenberg-Marquardt training function trainlm is used to obtain a solution in the minimum amount of time. The network is trained for up to 500 epochs, displaying progress every epoch, and with a typical error of 0.0037 radians (0.25 degrees) for the 749 training vectors and the network’s two outputs. mnet.trainParam.goal = (0.0037^2); mnet = train(mnet,Pm,Tm); Network Testing The network is tested by simulating its response for the following initial conditions and comparing the results to the pendulum response. angle = 5*deg2rad; vel = 0*deg2rad; force = 0; x = [angle; vel; force]; 10-26 Appcs1: Nonlinear System Identification The following plots may be viewed by running appcs1b. This script loads a previously trained model network. Here is a plot of the pendulum’s response to those initial conditions. Pendulum Response 250 Angle (deg) 200 150 100 50 0 0 0.5 1 1.5 2 Time (sec) 2.5 3 3.5 4 0 0.5 1 1.5 2 Time (sec) 2.5 3 3.5 4 250 Velocity (deg/sec) 200 150 100 50 0 -50 -100 As can be seen, the pendulum starts out with a slightly positive angle (leaning slightly to the right), but quickly falls. After an oscillation it approaches an angle of 180 degrees, pointing straight down. The network is first simulated in an open loop fashion. This means that at each time step the network computes its estimate of the pendulum’s next state. This estimate then becomes the input to the network for the next time step. The network never receives feedback from the real system. The open loop network response is compared to the pendulum in the following plot. 10-27 10 Applications Pendulum and Open Network Response Angle (deg): P + N - 250 200 150 100 50 0 0 0.5 1 1.5 2 Time (sec) 2.5 3 3.5 4 0 0.5 1 1.5 2 Time (sec) 2.5 3 3.5 4 Velocity (deg/sec): P + N - 250 200 150 100 50 0 -50 -100 Note that errors build as the network uses its own estimates as inputs, but the results are still very accurate. To get a better idea of how accurate the model network is, we simulate it in a closed loop manner. At each time step the network is given the actual pendulum state and is required to estimate the pendulum’s next state. Thus, the network’s estimates are not fed back into the network. Here is a plot of the pendulum response and the network closed loop response. Pendulum and Closed Network Response Angle (deg): P + N - 250 200 150 100 50 0 0 0.5 1 1.5 2 Time (sec) 2.5 3 3.5 4 0 0.5 1 1.5 2 Time (sec) 2.5 3 3.5 4 Velocity (deg/sec): P + N - 250 200 150 100 50 0 -50 -100 10-28 Appcs1: Nonlinear System Identification The network response (shown with solid line) is so accurate that it perfectly overlaps the pendulum response (‘+’ markers). Thoughts and Conclusions This example demonstrates how to use a nonlinear network (a two-layer tansig/purelin network) to identify a nonlinear system. Note that the model was obtained in almost the same way as in applin3, where a linear model was created. The only difference is that a linear network was being designed in applin3, so the network could be designed instead of trained. 10-29 10 Applications Appcs2: Model Reference Control The script appcs2 shows how a neural network can be trained to act as a model reference controller, if a neural network model is available for the system. In this script a neural controller is created for the inverted pendulum using the neural model created in appcs1. For model reference control we want to control a system so that its output follows the output of a reference model. In this application we will train a neural network controller which will drive the antenna arm system to follow a linear reference model. Mathematical Model Consider the antenna arm system for which a neural model was created in appcs1: 0º u(t) Ø(t) 180º Here the antenna arm angle Ø is to be controlled by applying a current to a dc motor which is attached to the pendulum. We represent this system by the following equations: Nonlinear System Model (NSM) x2 d x ----- 1 = dt x 9.81 sin x 1 – 2x 2 +u 2 where: x1 = ∅ d∅ x 2 = -------dt 10-30 Appcs2: Model Reference Control and u is the force applied to the pendulum by a motor. A positive force turns the pendulum clockwise. The 9.81sin x1 term is the force of gravity on the pendulum and the –2x2 term is viscous friction acting against the velocity. Suppose that we would like the closed loop system to respond with the dynamics given by the Linear Reference Model (LRM). x2 d x ----- 1 = + 0 dt x – 9x 1 – 6x 2 9r 2 where r is the desired output angle. Now let us train a neural network to help perform this model reference control. Neural Design A neural controller can be created for the case where the mathematical model is not available. All that is required is a neural model of the original system (which is created in appcs1) and a model of our desired system. The desired linear reference model, described mathematically above, is available in the function plinear. It takes the current time t, pendulum angle, pendulum velocity, and the demand angle, and returns the derivatives of angle, velocity, and demand. x = [angle; vel; demand]; dx = pmodel(t,x) Because the demand is an input to the system, its derivative is always returned as 0. We can simulate the desired linear reference model from 0 to 0.05 seconds using ode23. [time,X] = ode23('plinear',0,0.05,x) This function returns a row vector of times, and the matrix X of state vectors associated with those times. We would like to find a controller network which takes the current pendulum angle, velocity, and the demand angle as inputs, and outputs a current which can be applied to the pendulum. This current value should make the pendulum’s next state (in 0.05 seconds) identical to that defined by the desired linear reference model. 10-31 10 Applications Problem Definition Before the controller can be trained, examples of initial states and desired next states must be created. The following lines of code define several different pendulum angles, pendulum velocities, and demand angles. deg2rad = pi/180; angle = [–200:40:200]*deg2rad; vel = [–90:36:90]*deg2rad; demand = –180:40:180*deg2rad; By taking all possible combinations of these values in addition to a set of steady state initial conditions (velocity = 0, demand = angle) we get a matrix P of 381 pendulum state/input conditions, where the pendulum’s state is its angle and velocity, and its input is the current. angle2 = [-10:10:190]*deg2rad; Pc = [combvec(anglevel,demand) [angle2; angle2*0; angle2]]; Next each of these 381 initial state/input conditions are applied to the desired linear model, and its next state, 0.05 seconds later is measured. The next state consists of the desired pendulum angle and velocity after 0.05 seconds. The result is a matrix Tc of the 381 two-element target next states. (As in the identification problem appcs1, we will actually use the difference between the next state and the current state as the target. The neural network model learned to predict the change in state over 0.05 seconds. This was done because the state does not change by a large amount in this period of time, and we can improve the performance of the model if we predict only the change in state. If we need to know the actual state we simply need to add the change to the previous state.) Network Initialization The neural controller, like the neural pendulum model, is a tansig/purelin network. Here newff creates such a network for the controller which will have eight hidden neurons, and one output (the current to the pendulum). S1 = 8; cnet = newff(minmax(Pc),[S1 1],{'tansig' 'purelin'}); 10-32 Appcs2: Model Reference Control Network Training First, take a look at the following diagram of the entire neural controller/ pendulum system. Next State State Demand Controller Network Force Pendulum We would like the pendulum to respond with target states Tc once 0.05 seconds has passed since the pendulum was in the initial state Pc. The problem is that the error between the actual pendulum behavior and the desired linear behavior occurs on the outputs of the pendulum. How can these errors be used to adjust the controller? The trick is to replace the pendulum with its neural model, for purposes of training the controller. Here is a diagram of the neural controller/model system. Linear Reference Model Target State Demand Controller Network Force Model Network Next State Error Now the error occurs at the output of the model network. The derivatives of this error can be backpropagated through the model network to the control network. The derivatives are then backpropagated through the controller and used to adjust its weights and biases. (The neural model’s weights and biases are not changed.) Thus the control network must learn how to control the pendulum (represented temporarily by the model network) so that it behaves like the linear reference model. In the following code we will set up a combination network that includes both the model network and the controller network. There will be two different inputs to this total network, the state and the demand, and there will be four 10-33 10 Applications layers. (The following code uses advanced features of the Neural Network Toolbox. You may want to look through Chapter 11 and Chapter 12 before reviewing the rest of this application.) numInputs = 2; numLayers = 4; tnet = network(numInputs,numLayers); We next need to set up the network connections. tnet.biasConnect = [1 1 1 1]'; tnet.inputConnect = [1 0 1 0; 1 0 0 0]'; tnet.layerConnect = [0 0 0 0; 1 0 0 0; 0 1 0 0; 0 0 1 0]; tnet.outputConnect = [0 0 0 1]; tnet.targetConnect = [0 0 0 1]; Now we define the input and layer parameters. The first input corresponds to the states, and the second input is the demand. tnet.inputs{1}.range = minmax(Pc(1:2,:)); tnet.inputs{2}.range = minmax(Pc(3,:)); Next we define the size of each layer and the transfer functions. tnet.layers{1}.size = S1; tnet.layers{1}.transferFcn tnet.layers{2}.size = 1; tnet.layers{2}.transferFcn tnet.layers{3}.size = 8; tnet.layers{3}.transferFcn tnet.layers{4}.size = 2; tnet.layers{4}.transferFcn = 'tansig'; = 'purelin'; = 'tansig'; = 'purelin'; We will use the quasi-Newton training function trainbfg to train the network to minimize the mean square error. tnet.performFcn = 'mse'; tnet.trainFcn = 'trainbfg'; 10-34 Appcs2: Model Reference Control We now set the initial weights and biases in the total network that correspond to the controller network. These will be adjusted during the training process, so the learn parameter is set to 1. tnet.IW{1,1} = cnet.IW{1,1}(:,1:2); tnet.inputWeights{1,1}.learn = 1; tnet.IW{1,2} = cnet.IW{1,1}(:,3); tnet.inputWeights{1,2}.learn = 1; tnet.b{1} = cnet.b{1}; tnet.biases{1}.learn = 1; tnet.b{2} = cnet.b{2}; tnet.biases{2}.learn = 1; tnet.LW{2,1} = cnet.LW{2,1}; tnet.layerWeights{2,1}.learn = 1; Finally, we set the weights and biases in the total network that correspond to the model network. These will not be adjusted during the training process, so the learn parameter is set to 0. tnet.IW{3,1} = mnet.IW{1,1}(:,1:2); tnet.inputWeights{3,1}.learn = 0; tnet.LW{3,2} = mnet.IW{1,1}(:,3); tnet.layerWeights{3,2}.learn = 0; tnet.b{3} = mnet.b{1}; tnet.biases{3}.learn = 0; tnet.LW{4,3} = mnet.LW{2,1}; tnet.layerWeights{4,3}.learn = 0; tnet.b{4} = mnet.b{2}; tnet.biases{4}.learn = 0; Here train is called to do this training for us. The controller will be trained for up to 600 epochs, displaying progress every 5 epochs, to a typical error of 0.002 radians (0.11 degrees) for each of the 381 training vectors for each of the 2 model network outputs. tnet.trainParam.show = 5; tnet.trainParam.epochs = 600; tnet.trainParam.goal = (0.002^2); [tnet,tr] = train(tnet,{Pc(1:2,:); Pc(3,:)},{Tc}); The network contains the weights and biases of both the controller and model networks, but the model weights do not change. Only the controller weights are 10-35 10 Applications updated. Now we want to place the new controller weights and biases back into the controller network. cnet.IW{1,1}(:,1:2) = tnet.IW{1,1}; cnet.IW{1,1}(:,3) = tnet.IW{1,2}; cnet.b{1} = tnet.b{1}; cnet.b{2} = tnet.b{2}; cnet.LW{2,1} = tnet.LW{2,1}; Network Testing To test the control network, the neural controller/pendulum system is simulated and its response compared to the linear reference model. Run appcs2b to see the following plots. Here are the results of simulating the linear reference model from an initial angle of 10 degrees with a velocity of 0 degrees/second, and a constant demand angle of 90 degrees. Pendulum Response 100 Angle (deg) 80 60 40 20 0 0 0.5 1 1.5 2 Time 2.5 3 3.5 4 0 0.5 1 1.5 2 Time 2.5 3 3.5 4 Velocity (deg/sec) 100 80 60 40 20 0 The linear reference model quickly moves to 90 degrees, and then holds there. 10-36 Appcs2: Model Reference Control Here are the results found by simulating the controlled pendulum: Desired and Actual Response Angle (deg): D + A - 100 80 60 40 20 0 0 0.5 1 1.5 2 Time (sec) 2.5 3 3.5 4 0 0.5 1 1.5 2 Time (sec) 2.5 3 3.5 4 Velocity (deg/sec): D+ A - 100 80 60 40 20 0 The network does a near perfect job of making the nonlinear pendulum system (solid line) act like the linear reference model (‘+’ markers). Thoughts and Conclusions A neural network can be used to control a nonlinear system so that the system output follows the response of a desired reference model. It is first necessary to develop a neural network model of the nonlinear system, which is then used in the training process for the neural network controller. 10-37 10 Applications Appcr1: Character Recognition It is often useful to have a machine perform pattern recognition. In particular, machines which can read symbols are very cost effective. A machine that reads banking checks can process many more checks than a human being in the same time. This kind of application saves time and money, and eliminates the requirement that a human perform such a repetitive task. The script appcr1 demonstrates how character recognition can be done with a backpropagation network. Problem Statement A network is to be designed and trained to recognize the 26 letters of the alphabet. An imaging system that digitizes each letter centered in the system’s field of vision is available. The result is that each letter is represented as a 5 by 7 grid of boolean values. For example, here is the letter A: 10-38 Appcr1: Character Recognition However, the imaging system is not perfect and the letters may suffer from noise: Perfect classification of ideal input vectors is required, and reasonably accurate classification of noisy vectors. The twenty-six 35-element input vectors are defined in the function prprob as a matrix of input vectors called alphabet. The target vectors are also defined in this file with a variable called targets. Each target vector is a 26-element vector with a 1 in the position of the letter it represents, and 0’s everywhere else. For example, the letter A is to be represented by a 1 in the first element (as A is the first letter of the alphabet), and 0’s in elements two through twenty-six. Neural Network The network will receive the 35 boolean values as a 35-element input vector. It will then be required to identify the letter by responding with a 26-element output vector. The 26 elements of the output vector each represent a letter. To operate correctly the network should respond with a 1 in the position of the letter being presented to the network. All other values in the output vector should be 0. In addition, the network should be able to handle noise. In practice the network will not receive a perfect boolean vector as input. Specifically, the network should make as few mistakes as possible when classifying vectors with noise of mean 0 and standard deviation of 0.2 or less. 10-39 10 Applications Architecture The neural network needs 35 inputs and 26 neurons in its output layer to identify the letters. The network is a two-layer log-sigmoid/log-sigmoid network. The log-sigmoid transfer function was picked because its output range (0 to 1) is perfect for learning to output boolean values. Input Hidden Layer AAA AA AA AA AAA AA AAA AAAA p1 35 x 1 IW1,1 10 x 35 1 35 b1 10 x 1 a1 10 x 1 n1 LW2,1 26 x 10 10 x1 1 10 a1 = logsig (IW1,1p1 +b1) b2 26 x 1 Output Layer AA AA AA a2 26 x 1 n2 26 x 1 26 a2 = logsig(LW2,1a1 +b2) The hidden (first) layer has 10 neurons. This number was picked by guesswork and experience. If the network has trouble learning, then neurons can be added to this layer. The network is trained to output a 1 in the correct position of the output vector and to fill the rest of the output vector with 0’s. However, noisy input vectors may result in the network not creating perfect 1’s and 0’s. After the network has been trained the output will be passed through the competitive transfer function compet. This makes sure that the output corresponding to the letter most like the noisy input vector takes on a value of 1 and all others have a value of 0. The result of this post-processing is the output that is actually used. Initialization The two layer network is created with newff. S1 = 10; S2 = 26; net = newff(minmax(P),[S1 S2],{'logsig' 'logsig'},'traingdx'); Training To create a network that can handle noisy input vectors it is best to train the network on both ideal and noisy vectors. To do this the network will first be trained on ideal vectors until it has a low sum-squared error. 10-40 Appcr1: Character Recognition Then the network will be trained on 10 sets of ideal and noisy vectors. The network is trained on two copies of the noise-free alphabet at the same time as it is trained on noisy vectors. The two copies of the noise-free alphabet are used to maintain the network’s ability to classify ideal input vectors. Unfortunately, after the training described above the network may have learned to classify some difficult noisy vectors at the expense of properly classifying a noise free vector. Therefore, the network will again be trained on just ideal vectors. This ensures that the network will respond perfectly when presented with an ideal letter. All training is done using backpropagation with both adaptive learning rate and momentum with the function trainbpx. Training Without Noise The network is initially trained without noise for a maximum of 5000 epochs or until the network sum-squared error falls beneath 0.1. P = alphabet; T = targets; net.performFcn = 'sse'; net.trainParam.goal = 0.1; net.trainParam.show = 20; net.trainParam.epochs = 5000; net.trainParam.mc = 0.95; [net,tr] = train(net,P,T); Training With Noise To obtain a network not sensitive to noise, we trained with two ideal copies and two noisy copies of the vectors in alphabet. The target vectors consist of four copies of the vectors in target. The noisy vectors have noise of mean 0.1 and 0.2 added to them. This forces the neuron to learn how to properly identify noisy letters, while requiring that it can still respond well to ideal vectors. 10-41 10 Applications To train with noise the maximum number of epochs is reduced to 300 and the error goal is increased to 0.6, reflecting that higher error is expected due to more vectors, including some with noise, are being presented. netn = net; netn.trainParam.goal = 0.6; netn.trainParam.epochs = 300; T = [targets targets targets targets]; for pass = 1:10 P = [alphabet, alphabet, ... (alphabet + randn(R,Q)*0.1), ... (alphabet + randn(R,Q)*0.2)]; [netn,tr] = train(netn,P,T); end Training Without Noise Again Once the network has been trained with noise it makes sense to train it without noise once more to ensure that ideal input vectors are always classified correctly. Therefore the network is again trained with code identical to the "Training Without Noise" section. System Performance The reliability of the neural network pattern recognition system is measured by testing the network with hundreds of input vectors with varying quantities of noise. The script file appcr1 tests the network at various noise levels and then graphs the percentage of network errors vs. noise. Noise with mean of 0 and standard deviation from 0 to 0.5 are added to input vectors. At each noise level 100 presentations of different noisy versions of each letter are made and the network’s output is calculated. The output is then passed through the competitive transfer function so that only one of the 26 outputs, representing the letters of the alphabet, has a value of 1. 10-42 Appcr1: Character Recognition The number of erroneous classifications are then added and percentages are obtained: Percentage of Recognition Errors 50 45 40 Network 1 - - Network 2 --- 35 30 25 20 15 10 5 0 0 0.05 0.1 0.15 0.2 0.25 Noise Level 0.3 0.35 0.4 0.45 0.5 The solid line on the graph shows the reliability for the network trained with and without noise. The reliability of the same network when it had only been trained without noise is shown with a dashed line. Thus, training the network on noisy input vectors greatly reduced its errors when it had to classify noisy vectors. The network did not make any errors for vectors with noise of mean 0.00 or 0.05. When noise of mean 0.2 was added to the vectors both networks began to make errors. If a higher accuracy is needed the network could be trained for a longer time or retrained with more neurons in its hidden layer. Also, the resolution of the input vectors could be increased to say, a 10 by 14 grid. Finally, the network could be trained on input vectors with greater amounts of noise if greater reliability were needed for higher levels of noise. 10-43 10 Applications To test the system a letter with noise can be created and presented to the network. noisyJ = alphabet(:,10)+randn(35,1) ∗ 0.2; plotchar(noisyJ); A2 = sim(net,noisyJ); A2 = compet(A2); answer = find(compet(A2) == 1); plotchar(alphabet(:,answer)); Here is the noisy letter and the letter the network picked (correctly): Summary This problem demonstrates how a simple pattern recognition system can be designed. Note that the training process did not consist of a single call to a training function. Instead, the network was trained several times on various input vectors. In this case training a network on different sets of noisy vectors forced the network to learn how to deal with noise, a common problem in the real world. 10-44 12 Network Object Reference Network Properties Architecture . . . . . Subobject Structures . Functions . . . . . . Parameters . . . . . Weight and Bias Values Other . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-2 . 12-2 . 12-6 12-10 12-13 12-14 12-16 Subobject Properties Inputs . . . . . . . Layers . . . . . . . Outputs . . . . . . Targets . . . . . . . Biases . . . . . . . Input Weights . . . . LayerWeights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-17 12-17 12-18 12-25 12-25 12-26 12-28 12-32 12 Network Object Reference Network Properties The properties define the basic features of a network. A later section, “Subobject Properties” describes properties which define network details. Architecture These properties determine the number of network subobjects (which include inputs, layers, outputs, targets, biases, and weights), and how they are connected. numInputs This property defines the number of inputs a network receives. net.numInputs It can be set to 0 or a positive integer. Clarification. The number of network inputs and the size of a network input are not the same thing. The number of inputs defines how many sets of vectors the network receives as input. The size of each input (i.e. the number of elements in each input vector) is determined by the input size (net.inputs{i}.size). Most networks have only one input, whose size is determined by the problem. Side Effects. Any change to this property results in a change in the size of the matrix defining connections to layers from inputs, (net.inputConnect) and the size of the cell array of input subobjects (net.inputs). numLayers This property defines the number of layers a network has. net.numLayers It can be set to 0 or a positive integer. 12-2 Network Properties Side Effects. Any change to this property changes the size of each of these boolean matrices that define connections to and from layers, net.biasConnect net.inputConnect net.layerConnect net.outputConnect net.targetConnect and changes the size each cell array of subobject structures whose size depends on the number of layers, net.biases net.inputWeights net.layerWeights net.outputs net.targets and also changes the size of each of the network’s adjustable parameters properties. net.IW net.LW net.b biasConnect This property defines which layers have biases. net.biasConnect It can be set to any N l × 1 matrix of boolean values, where N l is the number of network layers (net.numLayers). The presence (or absence) of a bias to the ith layer is indicated by a 1 (or 0) at: net.biasConnect(i) Side Effects. Any change to this property will alter the presence or absence of structures in the cell array of biases (net.biases) and in the presence or absence of vectors in the cell array of bias vectors (net.b). 12-3 12 Network Object Reference inputConnect This property defines which layers have weights coming from inputs. net.inputConnect It can be set to any N l × N i matrix of boolean values, where N l is the number of network layers (net.numLayers), and N i is the number of network inputs (net.numInputs). The presence (or absence) of a weight going to the ith layer from the jth input is indicated by a 1 (or 0) at: net.inputConnect(i,j) Side Effects. Any change to this property will alter the presence or absence of structures in the cell array of input weight subobjects (net.inputWeights) and in the presence or absence of matrices in the cell array of input weight matrices (net.IW). layerConnect This property defines which layers have weights coming from other layers. net.layerConnect It can be set to any N l × N l matrix of boolean values, where N l is the number of network layers (net.numLayers). The presence (or absence) of a weight going to the ith layer from the jth layer is indicated by a 1 (or 0) at: net.layerConnect(i,j) Side Effects. Any change to this property will alter the presence or absence of structures in the cell array of layer weight subobjects (net.layerWeights) and in the presence or absence of matrices in the cell array of layer weight matrices (net.LW). outputConnect This property defines which layers generate network outputs. net.outputConnect It can be set to any 1 × N l matrix of boolean values, where N l is the number of network layers (net.numLayers). The presence (or absence) of a network output from the ith layer is indicated by a 1 (or 0) at: net.outputConnect(i) 12-4 Network Properties Side Effects. Any change to this property will alter the number of network outputs (net.numOutputs) and the presence or absence of structures in the cell array of output subobjects (net.outputs). targetConnect This property defines which layers have associated targets. net.targetConnect It can be set to any 1 × N l matrix of boolean values, where N l is the number of network layers (net.numLayers). The presence (or absence) of a target associated with the ith layer is indicated by a 1 (or 0) at: net.targetConnect(i) Side Effects. Any change to this property will alter the number of network targets (net.numTargets) and the presence or absence of structures in the cell array of target subobjects (net.targets). numOutputs (read-only) This property indicates how many outputs the network has. net.numOutputs It is always set to the number 1’s in the matrix of output connections. numOutputs = sum(net.outputConnect) numTargets (read-only) This property indicates how many targets the network has. net.numTargets It is always set to the number of 1’s in the matrix of target connections. numTargets = sum(net.targetConnect) numInputDelays (read-only) This property indicates the number of time steps of past inputs that must be supplied to simulate the network. net.numInputDelays 12-5 12 Network Object Reference It is always set to the maximum delay value associated any of the network’s input weights. numInputDelays = 0; for i=1:net.numLayers for j=1:net.numInputs if net.inputConnect(i,j) numInputDelays = max( ... [numInputDelays net.inputWeights{i,j}.delays]); end end end numLayerDelays (read-only) This property indicates the number of time steps of past layer outputs that must be supplied to simulate the network. net.numLayerDelays It is always set to the maximum delay value associated any of the network’s layer weights. numLayerDelays = 0; for i=1:net.numLayers for j=1:net.numLayers if net.layerConnect(i,j) numLayerDelays = max( ... [numLayerDelays net.layerWeights{i,j}.delays]); end end end Subobject Structures These properties consist of cell arrays of structures which define each of the network’s inputs, layers, outputs, targets, biases, and weights. The properties for each kind of subobject are described in the “Subobject Properties” section which follows this “Network Properties” section. 12-6 Network Properties inputs This property holds structures of properties for each of the network’s inputs. net.inputs It is always an N i × 1 cell array of input structures, where N i is the number of network inputs (net.numInputs). The structure defining the properties of the ith network input is located at: net.inputs{i} Input Properties. See “Inputs” in the “Subobject Properties” section for descriptions of input properties. layers This property holds structures of properties for each of the network’s layers. net.layers It is always an N l × 1 cell array of input structures, where N l is the number of network layers (net.numLayers). The structure defining the properties of the ith layer is located at: net.layers{i} Layer Properties. See “Layers” in the “Subobject Properties” section for descriptions of layer properties. outputs This property holds structures of properties for each of the network’s outputs. net.outputs It is always an N l × 1 cell array, where N l is the number of network layers (net.numLayers). The structure defining the properties of the output from the ith layer (or a null matrix []) is located at: net.outputs{i} 12-7 12 Network Object Reference if the corresponding output connection is 1 (or 0). net.outputConnect(i) Output Properties. See “Outputs” in the “Subobject Properties” section for descriptions of output properties. targets This property holds structures of properties for each of the network’s targets. net.targets It is always an N l × 1 cell array, where N l is the number of network layers (net.numLayers). The structure defining the properties of the target associated with the ith layer (or a null matrix []) is located at: net.targets{i} if the corresponding target connection is 1 (or 0). net.targetConnect(i) Target Properties. See “Targets” in the “Subobject Properties” section for descriptions of target properties. biases This property holds structures of properties for each of the network’s biases. net.biases It is always an N l × 1 cell array, where N l is the number of network layers (net.numLayers). The structure defining the properties of the bias associated with the ith layer (or a null matrix []) is located at: net.biases{i} if the corresponding bias connection is 1 (or 0). net.biasConnect(i) 12-8 Network Properties Bias Properties. See “Biases” in the “Subobject Properties” section for descriptions of bias properties. inputWeights This property holds structures of properties for each of the network’s input weights. net.inputWeights It is always an N l × N i cell array, where N l is the number of network layers (net.numLayers) and N i is the number of network inputs (net.numInputs). The structure defining the properties of the weight going to the ith layer from the jth input (or a null matrix []) is located at: net.inputWeights{i,j} if the corresponding input connection is 1 (or 0). net.inputConnect(i,j) Input Weight Properties. See “Input Weights” in the “Subobject Properties” section for descriptions of input weight properties. layerWeights This property holds structures of properties for each of the network’s layer weights. net.layerWeights It is always an N l × N l cell array, where N l is the number of network layers (net.numLayers). The structure defining the properties of the weight going to the ith layer from the jth layer (or a null matrix []) is located at: net.layerWeights{i,j} if the corresponding layer connection is 1 (or 0). net.layerConnect(i,j) Layer Weight Properties. See “LayerWeights” in the “Subobject Properties” section for descriptions of layer weight properties. 12-9 12 Network Object Reference Functions These properties define the algorithms to use when a network is to adapt, is to be initialized, is to have its performance measured, or is to be trained. adaptFcn This property defines the function to be used when the network adapts. net.adaptFcn It can be set to the name of any network adapt function, including this toolbox function: adaptwb - By-weight-and-bias network adaption function. The network adapt function is used to perform adaption whenever adapt is called. [net,Y,E,Pf,Af] = adapt(NET,P,T,Pi,Ai) Custom Functions. See Chapter 11, “Advanced Topics” for information on creating custom adapt functions. Side Effects. Whenever this property is altered the network’s adaption parameters (net.adaptParam) are set to contain the parameters and default values of the new function. initFcn This property defines the function used to initialize the network’s weight matrices and bias vectors. net.initFcn It can be set to the name of any network initialization function, including this toolbox function. initlay - Layer-by-layer network initialization function. The initialization function is used to initialize the network whenever init is called. net = init(net) 12-10 Network Properties Custom Functions. See Chapter 11, “Advanced Topics” for information on creating custom initialization functions. Side Effects. Whenever this property is altered the network’s initialization parameters (net.initParam) are set to contain the parameters and default values of the new function. performFcn This property defines the function used to measure the network’s performance. net.performFcn It can be set to the name of any performance function, including these toolbox functions: Performance Functions mae Mean absolute error performance function. mse Mean squared error performance function. msereg Mean squared error w/reg performance function. sse Sum squared error performance function. The performance function is used to calculate network performance during training whenever train is called. [net,tr] = train(NET,P,T,Pi,Ai) Custom functions. See Chapter 11, “Advanced Topics” for information on creating custom performance functions. Side Effects. Whenever this property is altered the network’s performance parameters (net.performParam) are set to contain the parameters and default values of the new function. trainFcn This property defines the function used to train the network. net.trainFcn 12-11 12 Network Object Reference It can be set to the name of any training function, including these toolbox functions: Training Functions trainbfg BFGS quasi-Newton backpropagation. trainbr Bayesian regularization. traincgb Powell-Beale conjugate gradient backpropagation. traincgf Fletcher-Powell conjugate gradient backpropagation. traincgp Polak-Ribiere conjugate gradient backpropagation. traingd Gradient descent backpropagation. traingda Gradient descent w/adaptive lr backpropagation. traingdm Gradient descent w/momentum backpropagation. traingdx Gradient descent w/momentum & adaptive lr backprop. trainlm Levenberg-Marquardt backpropagation. trainoss One step secant backpropagation. trainrp Resilient backpropagation (Rprop) trainscg Scaled conjugate gradient backpropagation. trainwb By-weight-and-bias network training function. trainwb1 By-weight-&-bias 1-vector-at-a-time training function. The training function is used to train the network whenever train is called. [net,tr] = train(NET,P,T,Pi,Ai) Custom Functions. See Chapter 11, “Advanced Topics” for information on creating custom training functions. Side Effects. Whenever this property is altered the network’s training parameters (net.trainParam) are set to contain the parameters and default values of the new function. 12-12 Network Properties Parameters adaptParam This property defines the parameters and values of the current adapt function. net.adaptParam The fields of this property depend on the current adapt function (net.adaptFcn). Evaluate the above reference to see the fields of the current adapt function. Call help on the current adapt function to get a description of what each field means. help(net.adaptFcn) initParam This property defines the parameters and values of the current initialization function. net.initParam The fields of this property depend on the current initialization function (net.initFcn). Evaluate the above reference to see the fields of the current initialization function. Call help on the current initialization function to get a description of what each field means. help(net.initFcn) performParam This property defines the parameters and values of the current performance function. net.performParam The fields of this property depend on the current performance function (net.performFcn). Evaluate the above reference to see the fields of the current performance function. 12-13 12 Network Object Reference Call help on the current performance function to get a description of what each field means. help(net.performFcn) trainParam This property defines the parameters and values of the current training function. net.trainParam The fields of this property depend on the current training function (net.trainFcn). Evaluate the above reference to see the fields of the current training function. Call help on the current training function to get a description of what each field means. help(net.trainFcn) Weight and Bias Values These properties define the network’s adjustable parameters: its weight matrices and bias vectors. IW This property defines the weight matrices of weights going to layers from network inputs. net.IW It is always an N l × N i cell array, where N l is the number of network layers (net.numLayers) and N i is the number of network inputs (net.numInputs). The weight matrix for the weight going to the ith layer from the jth input (or a null matrix []) is located at: net.IW{i,j} if the corresponding input connection is 1 (or 0). net.inputConnect(i,j) 12-14 Network Properties The weight matrix will have as many rows as the size of the layer it goes to (net.layers{i}.size). It will have as many columns as the product of the input size with the number of delays associated with the weight. net.inputs{j}.size * length(net.inputWeights{i,j}.delays) These dimensions can also be obtained from the input weight properties. net.inputWeights{i,j}.size LW This property defines the weight matrices of weights going to layers from other layers. net.LW It is always an N l × N l cell array, where N l is the number of network layers (net.numLayers). The weight matrix for the weight going to the ith layer from the jth layer (or a null matrix []) is located at: net.LW{i,j} if the corresponding layer connection is 1 (or 0). net.layerConnect(i,j) The weight matrix will have as many rows as the size of the layer it goes to (net.layers{i}.size). It will have as many columns as the product of the size of the layer it comes from with the number of delays associated with the weight. net.layers{j}.size * length(net.layerWeights{i,j}.delays) These dimensions can also be obtained from the layer weight properties. net.layerWeights{i,j}.size 12-15 12 Network Object Reference b This property defines the bias vectors for each layer with a bias. net.b It is always an N l × 1 cell array, where N l is the number of network layers (net.numLayers). The bias vector for the ith layer (or a null matrix []) is located at: net.b{i} if the corresponding bias connection is 1 (or 0). net.biasConnect(i) The number of elements in the bias vector is always equal to the size of the layer it is associated with (net.layers{i}.size). This dimension can also be obtained from the bias properties. net.biases{i}.size Other The only other property is a user data property. userdata This property provides a place for users to add custom information to a network object. net.userdata Only one field is predefined. It contains a secret message to all Neural Network Toolbox users. net.userdata.note Please keep this information confidential. 12-16 Subobject Properties Subobject Properties These properties define the details of a network’s inputs, layers, outputs, targets, biases, and weights. Inputs These properties define the details of each ith network input. net.inputs{i} range This property defines the ranges of each element of the ith network input. net.inputs{i}.range It can be set to any R i × 2 matrix, where R i is the number of elements in the input (net.inputs{i}.size) and each element in column 1 is less than the element next to it in column 2. Each jth row defines the minimum and maximum values of the jth input element, in that order: net.inputs{i}(j,:) Uses. Some initialization functions use input ranges to find appropriate initial values for input weight matrices. Side Effects. Whenever the number of rows in this property is altered, the layers’s size (net.inputs{i}.size) will change to remain consistent. The size of any weights coming from this input (net.inputWeights{:,i}.size) and the dimensions of their weight matrices (net.IW{:,i}) will also change size. size This property defines the number of elements in the ith network input. net.inputs{i}.size It can be set to 0 or a positive integer. Side Effects. Whenever this property is altered, the input’s ranges (net.inputs{i}.ranges), any input weights (net.inputWeights{:,i}.size) and their weight matrices (net.IW{:,i}) will change size to remain consistent. 12-17 12 Network Object Reference userdata This property provides a place for users to add custom information to the ith network input. net.inputs{i}.userdata Only one field is predefined. It contains a secret message to all Neural Network Toolbox users. net.inputs{i}.userdata.note Layers These properties define the details of each ith network layer. net.layers{i} dimensions This property defines the physical dimensions of the ith layer’s neurons. Being able to arrange a layer’s neurons in a multi-dimensional manner is important for self-organizing maps. net.layers{i}.dimensions It can be set to any row vector of 0 or positive integer elements, where the product of all the elements will become the number of neuron’s in the layer (net.layers{i}.size). Uses. Layer dimensions are used to calculate the neuron positions within the layer (net.layers{i}.positions) using the layer’s topology function (net.layers{i}.topologyFcn). Side Effects. Whenever this property is altered, the layers’s size (net.layers{i}.size) will change to remain consistent. The layer’s neuron positions (net.layers{i}.positions) and the distances between the neurons (net.layers{i}.distances) will also be updated. 12-18 Subobject Properties distanceFcn This property defines the function used to calculate distances between neurons in the ith layer (net.layers{i}.distances) from the neuron positions (net.layers{i}.positions). Neuron distances are used by self-organizing maps. net.layers{i}.distanceFcn It can be set to the name of any distance function, including these toolbox functions: Distance Functions boxdist Distance between two position vectors dist Euclidean distance weight function. linkdist Link distance function. mandist Manhattan distance weight function. Custom Functions. See Chapter 11, “Advanced Topics” for information on creating custom distance functions. Side Effects. Whenever this property is altered, the distance between the layer’s neurons (net.layers{i}.distances) will be updated. distances (read-only) This property defines the distances between neurons in the ith layer. These distances are used by self-organizing maps. net.layers{i}.distances It is always set to the result of applying the layer’s distance function (net.layers{i}.distanceFcn) to the positions of the layers neurons (net.layers{i}.positions). 12-19 12 Network Object Reference initFcn This property defines the initialization function used to initialize the ith layer, if the network initialization function (net.initFcn) is initlay. net.layers{i}.initFcn It can be set to the name of any layer initialization function, including these toolbox functions: Layer Initialization Functions initnw Nguyen-Widrow layer initialization function. initwb By-weight-and-bias layer initialization function If the network initialization is set to initlay, then the function indicated by this property will be used to initialize the layer’s weights and biases when init is called. net = init(net) Custom Functions. See Chapter 11, “Advanced Topics” for information on creating custom initialization functions. netInputFcn This property defines the net input function use to calculate the ith layer’s net input, given the layer’s weighted inputs and bias. net.layers{i}.netInputFcn It can be set to the name of any net input function, including these toolbox functions: Net Input Functions netprod Product net input function. netsum Sum net input function. The net input function is used to simulate the network when sim is called. [Y,Pf,Af] = sim(net,P,Pi,Ai) 12-20 Subobject Properties Custom Functions. See Chapter 11, “Advanced Topics” for information on creating custom net input functions. positions (read-only) This property defines the positions of neurons in the ith layer. These positions are used by self-organizing maps. net.layers{i}.positions It is always set to the result of applying the layer’s topology function (net.layers{i}.topologyFcn) to the positions of the layer’s dimensions (net.layers{i}.dimensions). Plotting. Use plotsom to plot the positions of a layer’s neurons. For instance, if the first layer neurons of a network are arranged with dimensions (net.layers{1}.dimensions) of [4 5] and the topology function (net.layers{1}.topologyFcn) is hextop, the neuron’s positions can be plotted as shown below. plotsom(net.layers{1}.positions) Neuron Positions 3 position(2,i) 2.5 2 1.5 1 0.5 0 0 1 2 position(1,i) 3 12-21 12 Network Object Reference size This property defines the number of neurons in the ith layer. net.layers{i}.size It can be set to 0 or a positive integer. Side Effects. Whenever this property is altered, the sizes of any input weights going to the layer (net.inputWeights{i,:}.size) and any layer weights going to the layer (net.layerWeights{i,:}.size) or coming from the layer (net.inputWeights{i,:}.size), and the layer’s bias (net.biases{i}.size) will change. The dimensions of the corresponding weight matrices (net.IW{i,:}, net.LW{i,:}, net.LW{:,i}) and biases (net.b{i}) will also change. Changing this property also changes the size of the layer’s output (net.outputs{i}.size) and target (net.targets{i}.size) if they exist. Finally, when this property is altered the dimensions of the layer’s neurons (net.layers{i}.dimension) are set to the same value. (This results in a one-dimensional arrangement of neurons. If another arrangement is required set the dimensions property directly instead of using size). topologyFcn This property defines the function used to calculate the ith layer’s neuron positions (net.layers{i}.positions) from the layer’s dimensions (net.layers{i}.dimensions). net.topologyFcn It can be set to the name of any topology function, including these toolbox functions: Topology Functions gridtop Gridtop layer topology function. hextop Hexagonal layer topology function. randtop Random layer topology function. Custom functions. See Chapter 11, “Advanced Topics” for information on creating custom topology functions. 12-22 Subobject Properties Side Effects. Whenever this property is altered, the positions of the layer’s neurons (net.layers{i}.positions) will be updated. Plotting. Use plotsom to plot the positions of a layer’s neurons. For instance, if the first layer neurons of a network are arranged with dimensions (net.layers{1}.dimensions) of [8 10] and the topology function (net.layers{1}.topologyFcn) is randtop, the neuron’s positions will be arranged something like those shown in the plot below. plotsom(net.layers{1}.positions) Neuron Positions 6 5 position(2,i) 4 3 2 1 0 0 1 2 3 position(1,i) 4 5 6 transferFcn This function defines the transfer function use to calculate the ith layer’s output, given the layer’s net input. net.layers{i}.transferFcn 12-23 12 Network Object Reference It can be set to the name of any transfer function, including these toolbox functions: Transfer Functions compet Competitive transfer function. hardlim Hard limit transfer function. hardlims Symmetric hard limit transfer function. logsig Log sigmoid transfer function. poslin Positive linear transfer function. purelin Hard limit transfer function. radbas Radial basis transfer function. satlin Saturating linear transfer function. satlins Symmetric saturating linear transfer function. softmax Soft max transfer function. tansig Hyperbolic tangent sigmoid transfer function. tribas Triangular basis transfer function The transfer function is used to simulate the network when sim is called. [Y,Pf,Af] = sim(net,P,Pi,Ai) Custom functions. See Chapter 11, “Advanced Topics” for information on creating custom transfer functions. userdata This property provides a place for users to add custom information to the ith network layer. net.layers{i}.userdata Only one field is predefined. It contains a secret message to all Neural Network Toolbox users. net.layers{i}.userdata.note 12-24 Subobject Properties Outputs size (read-only) This property defines the number of elements in the ith layer’s output. net.outputs{i}.size It is always set to the size of the ith layer (net.layers{i}.size). userdata This property provides a place for users to add custom information to the ith layer’s output. net.outputs{i}.userdata Only one field is predefined. It contains a secret message to all Neural Network Toolbox users. net.outputs{i}.userdata.note Targets size (read-only) This property defines the number of elements in the ith layer’s target. net.targets{i}.size It is always set to the size of the ith layer (net.layers{i}.size). userdata This property provides a place for users to add custom information to the ith layer’s target. net.targets{i}.userdata Only one field is predefined. It contains a secret message to all Neural Network Toolbox users. net.targets{i}.userdata.note 12-25 12 Network Object Reference Biases initFcn This property defines the function used to initialize the ith layer’s bias vector, if the network initialization function is initlay, and the ith layer’s initialization function is initwb. net.biases{i}.initFcn This function can be set to the name of any bias initialization function, including the toolbox functions: Bias Initialization Functions initcon Conscience bias initialization function. initzero Zero weight/bias initialization function. rands Symmetric random weight/bias initialization function. This function will be used to calculate an initial bias vector for the ith layer (net.b{i}) when init is called, if the network initialization function (net.initFcn) is initlay, and the ith layer’s initialization function (net.layers{i}.initFcn) is initwb. net = init(net) Custom functions. See Chapter 11, “Advanced Topics” for information on creating custom initialization functions. learn This property defines whether the ith bias vector is to be altered during training and adaption. net.biases{i}.learn It can be set to 0 or 1. It enables or disables the bias’ learning during calls to either adapt or train. [net,Y,E,Pf,Af] = adapt(NET,P,T,Pi,Ai) [net,tr] = train(NET,P,T,Pi,Ai) 12-26 Subobject Properties learnFcn This property defines the function used to update the ith layer’s bias vector during training, if the network training function is trainwb or trainwb1, or during adaption, if the network adapt function is adaptwb. net.biases{i}.learnFcn It can be set to the name of any bias learning function, including these toolbox functions: Learning Functions learncon Conscience bias learning function. learngd Gradient descent weight/bias learning function learngdm Grad. descent w/momentum weight/bias learning function learnp Perceptron weight/bias learning function. learnpn Normalized perceptron weight/bias learning function. learnwh Widrow-Hoff weight/bias learning rule The learning function will update the ith bias vector (net.b{i}) during calls to train, if the network training function (net.trainFcn) is trainwb or trainwb1, or during calls to adapt, if the network adapt function (net.adaptFcn) is adaptwb. [net,Y,E,Pf,Af] = adapt(NET,P,T,Pi,Ai) [net,tr] = train(NET,P,T,Pi,Ai) Custom functions. See Chapter 11, “Advanced Topics” for information on creating custom learning functions. Side Effects. Whenever this property is altered the biases’s learning parameters (net.biases{i}.learnParam) are set to contain the fields and default values of the new function. learnParam This property defines the learning parameters and values for the current learning function of the ith layer’s bias. net.biases{i}.learnParam 12-27 12 Network Object Reference The fields of this property depend on the current learning function (net.biases{i}.learnFcn). Evaluate the above reference to see the fields of the current learning function. Call help on the current learning function to get a description of what each field means. help(net.biases{i}.learnFcn) size (read-only) This property defines the size of the ith layer’s bias vector. net.biases{i}.size It is always set to the size of the ith layer (net.layers{i}.size). userdata This property provides a place for users to add custom information to the ith layer’s bias. net.biases{i}.userdata Only one field is predefined. It contains a secret message to all Neural Network Toolbox users. net.biases{i}.userdata.note Input Weights delays This property defines a tapped delay line between the jth input and its weight to the ith layer. net.inputWeights{i,j}.delays It must be set to a row vector of increasing 0 or positive integer values. Side Effects. Whenever this property is altered the weight’s size (net.inputWeights{i,j}.size) and the dimensions of its weight matrix (net.IW{i,j}) are updated. 12-28 Subobject Properties initFcn This property defines the function used to initialize the weight matrix going to the ith layer from the jth input, if the network initialization function is initlay, and the ith layer’s initialization function is initwb. net.inputWeights{i,j}.initFcn This function can be set to the name of any weight initialization function, including these toolbox functions: Weight Initialization Functions initzero Zero weight/bias initialization function. midpoint Midpoint weight initialization function. randnc Normalized column weight initialization function. randnr Normalized row weight initialization function. rands Symmetric random weight/bias initialization function. This function will be used to calculate an initial weight matrix for the weight going to the ith layer from the jth input (net.IW{i,j}) when init is called, if the network initialization function (net.initFcn) is initlay, and the ith layer’s initialization function (net.layers{i}.initFcn) is initwb. net = init(net) Custom Functions. See Chapter 11, “Advanced Topics” for information on creating custom initialization functions. learn This property defines whether the weight matrix to the ith layer from the jth input is to be altered during training and adaption. net.inputWeights{i,j}.learn It can be set to 0 or 1. It enables or disables the weights learning during calls to either adapt or train. [net,Y,E,Pf,Af] = adapt(NET,P,T,Pi,Ai) [net,tr] = train(NET,P,T,Pi,Ai) 12-29 12 Network Object Reference learnFcn This property defines the function used to update the weight matrix going to the ith layer from the jth input during training, if the network training function is trainwb or trainwb1, or during adaption, if the network adapt function is adaptwb. net.inputWeights{i,j}.learnFcn It can be set to the name of any weight learning function, including these toolbox functions: Weight Learning Functions learngd Gradient descent weight/bias learning function learngdm Grad. descent w/momentum weight/bias learning function learnh Hebb weight learning function. learnhd Hebb with decay weight learning function. learnis Instar weight learning function. learnk Kohonen weight learning function. learnlv1 LVQ1 weight learning function. learnlv2 LVQ2 weight learning function. learnos Outstar weight learning function. learnp Perceptron weight/bias learning function. learnpn Normalized perceptron weight/bias learning function. learnsom Self-organizing map weight learning function. learnwh Widrow-Hoff weight/bias learning rule The learning function will update the weight matrix of the ith layer form the jth input (net.IW{i,j}) during calls to train, if the network training function (net.trainFcn) is trainwb or trainwb1, or during calls to adapt, if the network adapt function (net.adaptFcn) is adaptwb. [net,Y,E,Pf,Af] = adapt(NET,P,T,Pi,Ai) [net,tr] = train(NET,P,T,Pi,Ai) 12-30 Subobject Properties Custom Functions. See Chapter 11, “Advanced Topics” for information on creating custom learning functions. learnParam This property defines the learning parameters and values for the current learning function of the ith layer’s weight coming from the jth input. net.inputWeights{i,j}.learnParam The fields of this property depend on the current learning function (net.inputWeights{i,j}.learnFcn). Evaluate the above reference to see the fields of the current learning function. Call help on the current learning function to get a description of what each field means. help(net.inputWeights{i,j}.learnFcn) size (read-only) This property defines the dimensions of the ith layer’s weight matrix from the jth network input. net.inputWeights{i,j}.size It is always set to a two element row vector indicating the number of rows and columns of the associated weight matrix (net.IW{i,j}). The first element is equal to the size of the ith layer (net.layers{i}.size). The second element is equal to the product of the length of the weights delay vectors with the size of the jth input: length(net.inputWeights{i,j}.delays) * net.inputs{j}.size userdata This property provides a place for users to add custom information to the (i,j)th input weight. net.inputWeights{i,j}.userdata Only one field is predefined. It contains a secret message to all Neural Network Toolbox users. net.inputWeights{i,j}.userdata.note 12-31 12 Network Object Reference weightFcn This property defines the function used to apply the ith layer’s weight from the jth input to that input. net.inputWeights{i,j}.weightFcn It can be set to the name of any weight function, including these toolbox functions: Weight Functions dist Conscience bias initialization function. dotprod Zero weight/bias initialization function. mandist Manhattan distance weight function. negdist Normalized column weight initialization function. normprod Normalized row weight initialization function. The weight function is used when sim is called to simulate the network. [Y,Pf,Af] = sim(net,P,Pi,Ai) Custom functions. See Chapter 11, “Advanced Topics” for information on creating custom weight functions. LayerWeights delays This property defines a tapped delay line between the jth layer and its weight to the ith layer. net.layerWeights{i,j}.delays It must be set to a row vector of increasing 0 or positive integer values. initFcn This property defines the function used to initialize the weight matrix going to the ith layer from the jth layer, if the network initialization function is initlay, and the ith layer’s initialization function is initwb. net.layerWeights{i,j}.initFcn 12-32 Subobject Properties This function can be set to the name of any weight initialization function, including the toolbox functions: Weight and Bias Initialization Functions initzero Zero weight/bias initialization function. midpoint Midpoint weight initialization function. randnc Normalized column weight initialization function. randnr Normalized row weight initialization function. rands Symmetric random weight/bias initialization function. This function will be used to calculate an initial weight matrix for the weight going to the ith layer from the jth layer (net.LW{i,j}) when init is called, if the network initialization function (net.initFcn) is initlay, and the ith layer’s initialization function (net.layers{i}.initFcn) is initwb. net = init(net) Custom Functions. See Chapter 11, “Advanced Topics” for information on creating custom initialization functions. learn This property defines whether the weight matrix to the ith layer from the jth layer is to be altered during training and adaption. net.layerWeights{i,j}.learn It can be set to 0 or 1. It enables or disables the weights learning during calls to either adapt or train. [net,Y,E,Pf,Af] = adapt(NET,P,T,Pi,Ai) [net,tr] = train(NET,P,T,Pi,Ai) learnFcn This property defines the function used to update the weight matrix going to the ith layer from the jth layer during training, if the network training function 12-33 12 Network Object Reference is trainwb or trainwb1, or during adaption, if the network adapt function is adaptwb. net.layerWeights{i,j}.learnFcn It can be set to the name of any weight learning function, including these toolbox functions: Learning Functions learngd Gradient descent weight/bias learning function learngdm Grad. descent w/momentum weight/bias learning function learnh Hebb weight learning function. learnhd Hebb with decay weight learning function. learnis Instar weight learning function. learnk Kohonen weight learning function. learnlv1 LVQ1 weight learning function. learnlv2 LVQ2 weight learning function. learnos Outstar weight learning function. learnp Perceptron weight/bias learning function. learnpn Normalized perceptron weight/bias learning function. learnsom Self-organizing map weight learning function. learnwh Widrow-Hoff weight/bias learning rule The learning function will update the weight matrix of the ith layer form the jth layer (net.LW{i,j}) during calls to train, if the network training function (net.trainFcn) is trainwb or trainwb1, or during calls to adapt, if the network adapt function (net.adaptFcn) is adaptwb. [net,Y,E,Pf,Af] = adapt(NET,P,T,Pi,Ai) [net,tr] = train(NET,P,T,Pi,Ai) Custom Functions. See Chapter 11, “Advanced Topics” for information on creating custom learning functions. 12-34 Subobject Properties learnParam This property defines the learning parameters fields and values for the current learning function of the ith layer’s weight coming from the jth layer. net.layerWeights{i,j}.learnParam The sub-fields of this property depend on the current learning function (net.layerWeights{i,j}.learnFcn). Evaluate the above reference to see the fields of the current learning function. Get help on the current learning function to get a description of what each field means. help(net.layerWeights{i,j}.learnFcn) size (read-only) This property defines the dimensions of the ith layer’s weight matrix from the jth layer. net.layerWeights{i,j}.size It is always set to a two element row vector indicating the number of rows and columns of the associated weight matrix (net.LW{i,j}). The first element is equal to the size of the ith layer (net.layers{i}.size). The second element is equal to the product of the length of the weights delay vectors with the size of the jth layer: length(net.layerWeights{i,j}.delays) * net.layers{j}.size userdata This property provides a place for users to add custom information to the (i,j)th layer weight. net.layerWeights{i,j}.userdata Only one field is predefined. It contains a secret message to all Neural Network Toolbox users. net.layerWeights{i,j}.userdata.note 12-35 12 Network Object Reference weightFcn This property defines the function used to apply the ith layer’s weight from the jth layer to that layer’s output. net.layerWeights{i,j}.weightFcn It can be set to the name of any weight function, including these toolbox functions: Weight Functions dist Euclidean distance weight function. dotprod Dot product weight function. mandist Manhattan distance weight function. negdist Dot product weight function. normprod Normalized dot product weight function. The weight function is used when sim is called to simulate the network. [Y,Pf,Af] = sim(net,P,Pi,Ai) Custom Functions. See Chapter 11, “Advanced Topics” for information on creating custom weight functions. 12-36 13 Reference Functions Listed by Network Type . . . . . . . . . . 13-2 Functions by Class . . . . . . . . . . . . . . . . . 13-3 Transfer Functions . . . . . . . . . . . . . . . . 13-13 Transfer Function Graphs . . . . . . . . . . . . 13-14 Reference Page Headings . . . . . . . . . . . . . 13-18 Functions . . . . . . . . . . . . . . . . . . . . 13-19 13 Reference Functions Listed by Network Type Function and type 13-2 assoclr Associative learning rules backprop Backpropagation networks elman Elman recurrent networks hopfield Hopfield recurrent networks linnet Linear networks lvq Learning vector quantization percept Perceptrons radbasis Radial basis networks selforg Self-organizing networks Functions by Class Functions by Class Adapt Function adaptwb By-weight-and-bias network adaption function. Analysis Functions errsurf Error surface of a single input neuron. maxlinr Maximum learning rate for a linear neuron. Distance Functions boxdist Distance between two position vectors dist Euclidean distance weight function. linkdist Link distance function. mandist Manhattan distance weight function. Layer Initialization Functions initnw Nguyen-Widrow layer initialization function. initwb By-weight-and-bias layer initialization function 13-3 13 Reference Learning Functions learncon Conscience bias learning function. learngd Gradient descent weight/bias learning function learngdm Grad. descent w/momentum weight/bias learning function learnh Hebb weight learning function. learnhd Hebb with decay weight learning function. learnis Instar weight learning function. learnk Kohonen weight learning function. learnlv1 LVQ1 weight learning function. learnlv2 LVQ2 weight learning function. learnos Outstar weight learning function. learnp Perceptron weight/bias learning function. learnpn Normalized perceptron weight/bias learning function. learnsom Self-organizing map weight learning function. learnwh Widrow-Hoff weight/bias learning rule Line Search Functions 13-4 srchbac Backtracking search. srchbre Brent's combination golden section/quadratic interpolation. srchcha Charalambous' cubic interpolation. srchgol Golden section search. srchhyb Hybrid bisection/cubic search. Functions by Class Net Input Derivative Functions dnetprod Product net input derivative function. dnetsum Sum net input derivative function. Net Input Functions netprod Product net input function. netsum Sum net input function. Network Initialization Functions initlay Layer-by-layer network initialization function. Network Use sim Simulate a neural network. init Initialize a neural network. adapt Allow a neural network to adapt. train Train a neural network. disp Display a neural network's properties. display Display a neural network variable's name and properties. 13-5 13 Reference New Networks network Create a custom neural network. newc Create a competitive layer. newcf Create a cascade-forward backpropagation network. newelm Create an Elman backpropagation network. newff Create a feed-forward backpropagation network. newfftd Create a feed-forward input-delay backprop network. newgrnn Design a generalized regression neural network. newhop Create a Hopfield recurrent network. newlin Create a linear layer. newlind Design a linear layer. newlvq Create a learning vector quantization network newp Create a perceptron. newpnn Design a probabilistic neural network. newrb Design a radial basis network. newrbe Design an exact radial basis network. newsom Create a self-organizing map. Performance Derivative Functions 13-6 dmae Mean absolute error performance derivatives function. dmse Mean squared error performance derivatives function. dmsereg Mean squared error w/reg performance derivative function. dsse Sum squared error performance derivative function. Functions by Class Performance Functions mae Mean absolute error performance function. mse Mean squared error performance function. msereg Mean squared error w/reg performance function. sse Sum squared error performance function. Plotting Functions hinton Hinton graph of weight matrix. hintonwb Hinton graph of weight matrix and bias vector. plotep Plot weight and bias position on error surface. plotes Plot error surface of single input neuron. plotpc Plot classification line on perceptron vector plot. plotperf Plot network performance. plotpv Plot perceptron input/target vectors. plotsom Plot self-organizing map. plotv Plot vectors as lines from the origin. plotvec Plot vectors with different colors. 13-7 13 Reference Pre and Post Processing postmnmx Unnormalize data which has been norm. by PREMNMX. postreg Post-training regression analysis. poststd Unnormalize data which has been normalized by PRESTD. premnmx Normalize data for maximum of 1 and minimum of -1. prepca Principal component analysis on input data. prestd Normalize data for unity standard deviation and zero mean. tramnmx Transform data with precalculated minimum and max. trapca Transform data with PCA matrix computed by PREPCA. trastd Transform data with precalc. mean & standard deviation. Simulink Support gensim Generate a Simulink block for neural network simulation. Topology Functions 13-8 gridtop Gridtop layer topology function. hextop Hexagonal layer topology function. randtop Random layer topology function. Functions by Class Training Functions trainbfg BFGS quasi-Newton backpropagation. trainbr Bayesian regularization. traincgb Powell-Beale conjugate gradient backpropagation. traincgf Fletcher-Powell conjugate gradient backpropagation. traincgp Polak-Ribiere conjugate gradient backpropagation. traingd Gradient descent backpropagation. traingda Gradient descent w/adaptive lr backpropagation. traingdm Gradient descent w/momentum backpropagation. traingdx Gradient descent w/momentum & adaptive lr backprop. trainlm Levenberg-Marquardt backpropagation. trainoss One step secant backpropagation. trainrp Resilient backpropagation (Rprop) trainscg Scaled conjugate gradient backpropagation. trainwb By-weight-and-bias network training function. trainwb1 By-weight-&-bias 1-vector-at-a-time training function. Transfer Derivative Functions dhardlim Hard limit transfer derivative function. dhardlms Symmetric hard limit transfer derivative function. dlogsig Log sigmoid transfer derivative function. dposlin Positive linear transfer derivative function. 13-9 13 Reference Transfer Derivative Functions dpurelin Hard limit transfer derivative function. dradbas Radial basis transfer derivative function. dsatlin Saturating linear transfer derivative function. dsatlins Symmetric saturating linear transfer derivative function. dtansig Hyperbolic tangent sigmoid transfer derivative function. dtribas Triangular basis transfer derivative function Transfer Functions 13-10 compet Competitive transfer function. hardlim Hard limit transfer function. hardlims Symmetric hard limit transfer function. logsig Log sigmoid transfer function. poslin Positive linear transfer function. purelin Hard limit transfer function. radbas Radial basis transfer function. satlin Saturating linear transfer function. satlins Symmetric saturating linear transfer function. softmax Soft max transfer function. tansig Hyperbolic tangent sigmoid transfer function. tribas Triangular basis transfer function Functions by Class Vectors cell2mat Combines cell array of matrices into one matrix. combvec Create all combinations of vectors. concur Create concurrent bias vectors. con2seq Converts concurrent vectors to sequential vectors. copy Copy matrix or cell array. ind2vec Convert indices to vectors. mat2cell Break matrix up into cell array of matrices. minmax Ranges of matrix rows. normc Normalize columns of matrix. normr Normalize rows of matrix. pnormc Pseudo-normalize columns of matrix. quant Discretize value as multiple of a quantity. seq2con Converts sequential vectors to concurrent vectors. sumsqr Sum squared elements of matrix. vec2ind Convert vectors to indices. Weight and Bias Initialization Functions initcon Conscience bias initialization function. initzero Zero weight/bias initialization function. midpoint Midpoint weight initialization function. randnc Normalized column weight initialization function. randnr Normalized row weight initialization function. rands Symmetric random weight/bias initialization function. 13-11 13 Reference Weight Derivative Functions ddotprod Dot product weight derivative function. Weight Functions 13-12 dist Euclidean distance weight function. dotprod Dot product weight function. mandist Manhattan distance weight function. negdist Dot product weight function. normprod Normalized dot product weight function. Transfer Functions Transfer Functions Transfer Function compet Competitive transfer function. hardlim Hard limit transfer function. hardlims Symmetric hard limit transfer function logsig Log sigmoid transfer function. poslin Positive linear transfer function purelin Linear transfer function. radbas Radial basis transfer function. satlin Saturating linear transfer function. satlins Symmetric saturating linear transfer function softmax Soft max transfer function. tansig Hyperbolic tangent sigmoid transfer function. tribas Triangular basis transfer function. C SM 13-13 13 Reference Transfer Function Graphs a AA +1 n 0 -1 a = hardlim(n) Hard Limit Transfer Function a +1 n 0 -1 a = hardlims(n) AA AA Symmetric Hard Limit Trans. Funct. a +1 n 0 -1 AA AA a = logsig(n) Log-Sigmoid Transfer Function 13-14 Transfer Function Graphs Transfer Function Graphs (continued) a +1 AA AA n 0 1 -1 a = poslin(n) Positive Linear Transfer Funct. a +1 n 0 -1 a = purelin(n) AA AA Linear Transfer Function a +1 0 -1 n AA a = hardlim(n) Hard Limit Transfer Function 13-15 13 Reference Transfer Function Graphs (continued) AA AA a +1 n -1 0 +1 -1 a = satlin(n) Satlin Transfer Function a +1 n -1 0 +1 -1 AA AA a = satlins(n) Satlins Transfer Function a +1 0 n -1 a = tansig(n) Tan-Sigmoid Transfer Function 13-16 Transfer Function Graphs Transfer Function Graphs (continued) a +1 -1 0 +1 n -1 a = tribas(n) Triangular Basis Function 13-17 13 Reference Reference Page Headings Following is a list of headings used in this section. Not every function will have all this material, but the material that is included will be ordered as shown. - Purpose - Graph and Symbol - Syntax - To Get Help - Description - Properties - Examples - Network Use - Algorithm - Limitations - Notes - See Also - References 13-18 adapt Purpose adapt Allow a neural network to adapt Syntax [net,Y,E,Pf,Af] = adapt(net,P,T,Pi,Ai) To Get Help Type help network/adapt Description [net,Y,E,Pf,Af] = adapt(net,P,T,Pi,Ai) takes, net - Network. P - Network inputs. T - Network targets, default = zeros. Pi - Initial input delay conditions, default = zeros. Ai - Initial layer delay conditions, default = zeros. and returns the following after applying the adapt function net.adaptFcn with the adaption parameters net.adaptParam: net - Updated network. Y - Network outputs. E - Network errors. Pf - Final input delay conditions. Af - Final layer delay conditions. Note that T is optional and only needs to be used for networks that require targets. Pi and Pf are also optional and only needs to be used for networks that have input or layer delays. adapt's signal arguments can have two formats: cell array or matrix. 13-19 adapt The cell array format is easiest to describe. It is most convenient to be used for networks with multiple inputs and outputs, and allows sequences of inputs to be presented: P - Ni x TS cell array, each element P{i,ts} is an Ri x Q matrix. T - Nt x TS cell array, each element P{i,ts} is a Vi x Q matrix. Pi - Ni x ID cell array, each element Pi{i,k} is an Ri x Q matrix. Ai - Nl x LD cell array, each element Ai{i,k} is an Si x Q matrix. Y - NO x TS cell array, each element Y{i,ts} is a Ui x Q matrix. Pf - Ni x ID cell array, each element Pf{i,k} is an Ri x Q matrix. Af - Nl x LD cell array, each element Af{i,k} is an Si x Q matrix. where Ni = net.numInputs Nl = net.numLayers No = net.numOutputs Nt = net.numTargets ID = net.numInputDelays LD = net.numLayerDelays TS = Number of time steps Q = Batch size Ri = net.inputs{i}.size Si = net.layers{i}.size Ui = net.outputs{i}.size Vi = net.targets{i}.size The columns of Pi, Pf, Ai, and Af are ordered from oldest delay condition to most recent: Pi{i,k} = input i at time ts = k-ID. Pf{i,k} = input i at time ts = TS+k-ID. Ai{i,k} = layer output i at time ts = k-LD. Af{i,k} = layer output i at time ts = TS+k-LD. The matrix format can be used if only one time step is to be simulated (TS = 1). It is convenient for network's with only one input and output, but can be used with networks that have more. 13-20 adapt Each matrix argument is found by storing the elements of the corresponding cell array argument in a single matrix: P - (sum of Ri) x Q matrix T - (sum of Vi) x Q matrix Pi - (sum of Ri) x (ID*Q) matrix. Ai - (sum of Si) x (LD*Q) matrix. Y - (sum of Ui) x Q matrix. Pf - (sum of Ri) x (ID*Q) matrix. Af - (sum of Si) x (LD*Q) matrix. Examples Here two sequences of 12 steps (where T1 is known to depend on P1) are used to define the operation of a filter. p1 = {-1 0 1 0 1 1 -1 0 -1 1 0 1}; t1 = {-1 -1 1 1 1 2 0 -1 -1 0 1 1}; Here newlin is used to create a layer with an input range of [-1 1]), one neuron, input delays of 0 and 1, and a learning rate of 0.5. The linear layer is then simulated. net = newlin([-1 1],1,[0 1],0.5); Here the network adapts for one pass through the sequence. The network's mean squared error is displayed. (Since this is the first call of adapt the default Pi is used.) [net,y,e,pf] = adapt(net,p1,t1); mse(e) Note the errors are quite large. Here the network adapts to another 12 time steps (using the previous Pf as the new initial delay conditions.) p2 = {1 -1 -1 1 1 -1 0 0 0 1 -1 -1}; t2 = {2 0 -2 0 2 0 -1 0 0 1 0 -1}; [net,y,e,pf] = adapt(net,p2,t2,pf); mse(e) 13-21 adapt Here the network adapts for 100 passes through the entire sequence. p3 = [p1 p2]; t3 = [t1 t2]; net.adaptParam.passes = 100; [net,y,e] = adapt(net,p3,t3); mse(e) The error after 100 passes through the sequence is very small. The network has adapted to the relationship between the input and target signals. Algorithm adapt calls the function indicated by net.adaptFcn, using the adaption parameter values indicated by net.adaptParam. Given an input sequence with TS steps the network is updated as follows. Each step in the sequence of inputs is presented to the network one at a time. The network's weight and bias values are updated after each step, before the next step in the sequence is presented. Thus the network is updated TS times. See Also 13-22 sim, init, train adaptwb Purpose Syntax adaptwb By-weight-and-bias network adaption function [net,Ac,El] = adaptwb(net,Pd,Tl,Ai,Q,TS) info = adaptwb(code) Description adaptwb is a network function which updates each weight and bias according to its learning function. adaptwb takes these inputs, net - Neural network. Pd - Delayed inputs. Tl - Layer targets. Ai - Initial input conditions. Q - Batch size. TS - Time steps. After training the network with its weight and bias the learning functions returns, net - Updated network. Ac - Collective layer outputs. El - Layer errors. Adaption occurs according to the adaptwb's training parameter, shown here with its default value: net.adaptparam.passes 1 Number of passes through sequence Dimensions for these variables are: Pd - No x Ni x TS cell array, each element P{i,j,ts} is a Zij x Q matrix. Tl - Nl x TS cell array, each element P{i,ts} is a Vi x Q matrix or []. Ai - Nl x LD cell array, each element Ai{i,k} is an Si x Q matrix. Ac - Nl x (LD+TS) cell array, each element Ac{i,k} is an Si x Q matrix. El - Nl x TS cell array, each element El{i,k} is an Si x Q matrix or []. 13-23 adaptwb where Ni = net.numInputs Nl = net.numLayers LD = net.numLayerDelays Ri = net.inputs{i}.size Si = net.layers{i}.size Vi = net.targets{i}.size Zij = Ri * length(net.inputWeights{i,j}.delays) adaptwb(code) returns useful information for each code string: 'pnames' - Names of training parameters. 'pdefaults' - Default training parameters. Network Use You can create a standard network that uses adaptwb by calling newp or newlin. To prepare a custom network to adapt with adaptwb: 1 Set net.adaptfcn to 'adaptwb'. (This will set net.adaptparam to adaptwb's default parameters.) 2 Set each net.inputweights{i,j}.learnfcn to a learning function. Set each net.layerweights{i,j}.learnfcn to a learning function. Set each net.biases{i}.learnfcn to a learning function. (Weight and bias learning parameters will automatically be set to default values for the given learning function.) To allow the network to adapt: 1 Set net.adaptparam properties to desired values. 2 Set weight and bias learning parameters to desired values. 3 Call adapt. See newp and newlin for adaption examples. Algorithm Each weight and bias is updated according to its learning function after each step in the input sequence. See Also newp, newlin, train 13-24 boxdist Purpose boxdist Box distance function Syntax d = boxdist(pos); Description boxdist is a layer distance function that is used to find the distances between the layer's neurons, given their positions. boxdist(pos) takes one argument, pos - N x S matrix of neuron positions. and returns the S x S matrix of distances. boxdist is most commonly used in conjunction with layers whose topology function is gridtop. Examples Here we define a random matrix of positions for 10 neurons arranged in three-dimensional space and then find their distances. pos = rand(3,10); d = boxdist(pos) Network Use You can create a standard network that uses boxdist as a distance function by calling newsom. To change a network so that a layer's topology uses boxdist, set net.layers{i}.distanceFcn to 'boxdist'. In either case, call sim to simulate the network with boxdist. See newsom for training and adaption examples. Algorithm The box distance D between two position vectors Pi and Pj from a set of S vectors is: Dij = max(abs(Pi-Pj)) See Also sim, dist, mandist, linkdist 13-25 cell2mat Purpose cell2mat Combines a cell array of matrices into one matrix Syntax m = cell2mat(c) Description m = cell2mat(C) C - Cell array of matrices: {M11 M12... ; M21 M22... ; ...} returns M - Single matrix: [M11 M12 ...; M21 M22... ; ...] Examples Examples c = {[1 2] [3]; [4 5; 6 7] [8; 9]}; m = cell2mat(c) See Also 13-26 mat2cell combvec Purpose combvec Create all combinations of vectors Syntax combvec(a1,a2) Description combvec(A1,A2) takes two inputs, A1 - Matrix of N1 (column) vectors. A2 - Matrix of N2 (column) vectors. and returns a matrix of N1*N2 column vectors, where the columns consist of all possibilities of A2 vectors, appended to A1 vectors. Examples a1 = [1 2 3; 4 5 6]; a2 = [7 8; 9 10]; a3 = combvec(a1,a2) 13-27 compet Purpose Syntax compet Competitive transfer function A = compet(N) info = compet(code) Description compet is a transfer function. Transfer functions calculate a layer's output from its net input. compet(N) takes one input argument, N - S x Q matrix of net input (column) vectors. and returns output vectors with 1 where each net input vector has its maximum value, and 0 elsewhere. compet(code) returns information about this function. These codes are defined: 'deriv' - Name of derivative function. 'name' - Full name. 'output' - Output range. 'active' - Active input range. compet does not have a derivative function. In many network paradigms it is useful to have a layer whose neurons compete for the ability to output a 1. In biology this is done by strong inhibitory connections between each of the neurons in a layer. The result is that the only neuron that can respond with appreciable output is the neuron whose net input is the highest. All other neurons are inhibited so strongly by the winning neuron that their output’s are negligible. To model this type of layer efficiently on a computer, a competitive transfer function is often used. Such a function transforms the net input vector of a layer of neurons so that the neuron receiving the greatest net input has an output of 1 and all other neurons have outputs of 0. 13-28 compet Examples Here we define a net input vector N, calculate the output, and plot both with bar graphs. n = [0; 1; -0.5; 0.5]; a = compet(n); subplot(2,1,1), bar(n), ylabel('n') subplot(2,1,2), bar(a), ylabel('a') Network Use You can create a standard network that uses compet by calling newc or newpnn. To change a network so a layer uses compet, set net.layers{i,j}.transferFcn to 'compet'. In either case, call sim to simulate the network with compet. See newc or newpnn for simulation examples. See Also sim, softmax 13-29 con2seq Purpose con2seq Convert concurrent vectors to sequential vectors Syntax s = con2seq(b) Description The neural network toolbox arranges concurrent vectors with a matrix, and sequential vectors with a cell array (where the second index is the time step). con2seq and seq2con allow concurrent vectors to be converted to sequential vectors, and back again. con2seq(b)takes one input, b - R x TS matrix. and returns one output, S - 1 x TS cell array of R x 1 vectors. con2seq(b,TS) can also convert multiple batches, b - N x 1 cell array of matrices with M*TS columns. TS - Time steps. and will return, S - N x TS cell array of matrices with M columns. Examples Here a batch of three values is converted to a sequence. p1 = [1 4 2] p2 = con2seq(p1) Here two batches of vectors are converted to two sequences with two time steps. p1 = {[1 3 4 5; 1 1 7 4]; [7 3 4 4; 6 9 4 1]} p2 = con2seq(p1,2) See Also 13-30 seq2con, concur concur Purpose concur Create concurrent bias vectors Syntax concur(b,q) Description concur(B,Q) B - S x 1 bias vector (or Nl x 1 cell array of vectors). Q - Concurrent size. Returns an S x B matrix of copies of B (or Nl x 1 cell array of matrices). Examples Here concur creates three copies of a bias vector. b = [1; 3; 2; -1]; concur(b,3) Network Use To calculate a layer's net input, the layer's weighted inputs must be combined with its biases. The following expression calculates the net input for a layer with the netsum net input function, two input weights, and a bias: n = netsum(z1,z2,b) The above expression works if Z1, Z2, and B are all S x 1 vectors. However, if the network is being simulated by sim (or adapt or train) in response to Q concurrent vectors, then Z1 and Z2 will be S x Q matrices. Before B can be combined with Z1 and Z2 we must make Q copies of it. n = netsum(z1,z2,concur(b,q)) See Also netsum, netprod, sim, seq2con, con2seq 13-31 ddotprod Purpose Syntax ddotprod Dot product weight derivative function dZ_dP = ddotprod('p',W,P,Z) dZ_dW = ddotprod('w',W,P,Z) Description ddotprod is a weight derivative function. ddotprod('p',W,P,Z) takes three arguments, W - S x R weight matrix. P - R x Q Inputs. Z - S x Q weighted input. and returns the S x R derivative dZ/dP. ddotprod('w',W,P,Z) returns the R x Q derivative dZ/dW. Examples Here we define a weight W and input P for an input with 3 elements and a layer with 2 neurons. W = [0 -1 0.2; -1.1 1 0]; P = [0.1; 0.6; -0.2]; Here we calculate the weighted input with dotprod, then calculate each derivative with ddotprod. Z = dotprod(W,P) dZ_dP = ddotprod('p',W,P,Z) dZ_dW = ddotprod('w',W,P,Z) Algorithm The derivative of a product of two elements with respect to one element is the other element. dZ/dP = W dZ/dW = P See Also 13-32 dotprod dhardlim Purpose dhardlim Derivative of hard limit transfer function Syntax dA_dN = dhardlim(N,A) Description dhardlim is the derivative function for hardlim. dhardlim(N,A) takes two arguments, N - S x Q net input. A - S x Q output. and returns the S x Q derivative dA/dN. Examples Here we define the net input N for a layer of 3 hardlim neurons. N = [0.1; 0.8; -0.7]; We calculate the layer's output A with hardlim and then the derivative of A with respect to N. A = hardlim(N) dA_dN = dhardlim(N,A) Algorithm The derivative of hardlim is calculated as follows: d = 0 See Also hardlim 13-33 dhardlms Purpose dhardlms Derivative of symmetric hard limit transfer function Syntax dA_dN = dhardlms(N,A) Description dhardlms is the derivative function for hardlims. dhardlms(N,A) takes two arguments, N - S x Q net input. A - S x Q output. and returns the S x Q derivative dA/dN. Examples Here we define the net input N for a layer of 3 hardlims neurons. N = [0.1; 0.8; -0.7]; We calculate the layer's output A with hardlims and then the derivative of A with respect to N. A = hardlims(N) dA_dN = dhardlms(N,A) Algorithm The derivative of hardlims is calculated as follows: d = 0 See Also 13-34 hardlims disp Purpose disp Display a neural network's properties Syntax disp(net) To Get Help Type help network/disp Description disp(net) displays a network's properties. Examples Here a perceptron is created and displayed. net = newp([-1 1; 0 2],3); disp(net) See Also display, sim, init, train, adapt 13-35 display Purpose display Display the name and properties of a neural network variables Syntax display(net) To Get Help Type help network/disp Description display(net) displays a network variable's name and properties. Examples Here a perceptron variable is defined and displayed. net = newp([-1 1; 0 2],3); display(net) display is automatically called as follows: net See Also 13-36 disp, sim, init, train, adapt dist Purpose Syntax dist Euclidean distance weight function Z = dist(W,P) df = dist('deriv') D = dist(pos) Description dist is the Euclidean distance weight function. Weight functions apply weights to an input to get weighted inputs. dist (W,P) takes these inputs, W - S x R weight matrix. P - R x Q matrix of Q input (column) vectors. and returns the S x Q matrix of vector distances. dist('deriv') returns '' because dist does not have a derivative function. dist is also a layer distance function which can be used to find the distances between neurons in a layer. dist(pos) takes one argument, pos - N x S matrix of neuron positions. and returns the S x S matrix of distances. Examples Here we define a random weight matrix W and input vector P and calculate the corresponding weighted input Z. W = rand(4,3); P = rand(3,1); Z = dist(W,P) Here we define a random matrix of positions for 10 neurons arranged in three dimensional space and find their distances. pos = rand(3,10); D = dist(pos) 13-37 dist Network Use You can create a standard network that uses dist by calling newpnn or newgrnn. To change a network so an input weight uses dist, set net.inputWeight{i,j}.weightFcn to 'dist'. For a layer weight set net.inputWeight{i,j}.weightFcn to 'dist'. To change a network so that a layer's topology uses dist, set net.layers{i}.distanceFcn to 'dist'. In either case, call sim to simulate the network with dist. See newpnn or newgrnn for simulation examples. Algorithm The Euclidean distance d between two vectors X and Y is: d = sum((x-y).^2).^0.5 See Also 13-38 sim, dotprod, negdist, normprod, mandist, linkdist dlogsig Purpose dlogsig Log sigmoid transfer derivative function Syntax dA_dN = dlogsig(N,A) Description dlogsig is the derivative function for logsig. dlogsig(N,A) takes two arguments, N - S x Q net input. A - S x Q output. and returns the S x Q derivative dA/dN. Examples Here we define the net input N for a layer of 3 tansig neurons. N = [0.1; 0.8; -0.7]; We calculate the layer's output A with logsig and then the derivative of A with respect to N. A = logsig(N) dA_dN = dlogsig(N,A) Algorithm The derivative of logsig is calculated as follows: d = a * (1 - a) See Also logsig, tansig, dtansig 13-39 dmae Purpose Syntax dmae Mean absolute error performance derivative function dPerf_dE = dmae('e',E,X,perf,PP) dPerf_dX = dmae('x',E,X,perf,PP) Description dmae is the derivative function for mae. dmae('d',E,X,PERF,PP) takes these arguments, E - Matrix or cell array of error vector(s). X - Vector of all weight and bias values. perf - Network performance (ignored). PP - Performance parameters (ignored). and returns the derivative dPerf/dE. dmae('x',E,X,PERF,PP) returns the derivative dPerf/dX. Examples Here we define E and X for a network with one 3-element output and six weight and bias values. E = {[1; -2; 0.5]}; X = [0; 0.2; -2.2; 4.1; 0.1; -0.2]; Here we calculate the network's mean absolute error performance, and derivatives of performance. perf = mae(E) dPerf_dE = dmae('e',E,X) dPerf_dX = dmae('x',E,X) Note that mae can be called with only one argument and dmae with only three arguments because the other arguments are ignored. The other arguments exist so that mae and dmae conform to standard performance function argument lists. See Also 13-40 mae dmse Purpose Syntax dmse Mean squared error performance derivatives function dPerf_dE = dmse('e',E,X,perf,PP) dPerf_dX = dmse('x',E,X,perf,PP) Description dmse is the derivative function for mse. dmse('d',E,X,PERF,PP) takes these arguments, E - Matrix or cell array of error vector(s). X - Vector of all weight and bias values. perf - Network performance (ignored). PP - Performance parameters (ignored). and returns the derivative dPerf/dE. dmse('x',E,X,PERF,PP) returns the derivative dPerf/dX. Examples Here we define E and X for a network with one 3-element output and six weight and bias values. E = {[1; -2; 0.5]}; X = [0; 0.2; -2.2; 4.1; 0.1; -0.2]; Here we calculate the network's mean squared error performance, and derivatives of performance. perf = mse(E) dPerf_dE = dmse('e',E,X) dPerf_dX = dmse('x',E,X) Note that mse can be called with only one argument and dmse with only three arguments because the other arguments are ignored. The other arguments exist so that mse and dmse conform to standard performance function argument lists. See Also mse 13-41 dmsereg Purpose Syntax dmsereg Mean squared error w/reg performance derivative function dPerf_dE = dmsereg('e',E,X,perf,PP) dPerf_dX = dmsereg('x',E,X,perf,PP) Description dmsereg is the derivative function for msereg. dmsereg('d',E,X,PERF,PP) takes these arguments, E - Matrix or cell array of error vector(s). X - Vector of all weight and bias values. perf - Network performance (ignored). PP - mse performance parameter. where PP defines one performance parameters, PP.ratio - Relative importance of errors vs. weight and bias values. and returns the derivative dPerf/dE. dmsereg('x',E,X,perf) returns the derivative dPerf/dX. mse has only one performance parameter. Examples Here we define an error E and X for a network with one 3-element output and six weight and bias values. E = {[1; -2; 0.5]}; X = [0; 0.2; -2.2; 4.1; 0.1; -0.2]; Here the ratio performance parameter is defined so that squared errors are 5 times as important as squared weight and bias values. pp.ratio = 5/(5+1); Here we calculate the network's performance, and derivatives of performance. perf = msereg(E,X,pp) dPerf_dE = dmsereg('e',E,X,perf,pp) dPerf_dX = dmsereg('x',E,X,perf,pp) See Also 13-42 msereg dnetprod Purpose dnetprod Derivative of net input product function Syntax dN_dZ = dnetprod(Z,N) Description dnetprod is the net input derivative function for netprod. dnetprod takes two arguments, Z - S x Q weighted input. N - S x Q net input. and returns the S x Q derivative dN/dZ. Examples Here we define two weighted inputs for a layer with three neurons. Z1 = [0; 1; -1]; Z2 = [1; 0.5; 1.2]; We calculate the layer's net input N with netprod and then the derivative of N with respect to each weighted input. N = netprod(Z1,Z2) dN_dZ1 = dnetprod(Z1,N) dN_dZ2 = dnetprod(Z2,N) Algorithm The derivative a product with respect to any element of that product is the product of the other elements. See Also netsum, netprod, dnetsum 13-43 dnetsum Purpose dnetsum Sum net input derivative function Syntax dN_dZ = dnetsum(Z,N) Description dnetsum is the net input derivative function for netsum. dnetsum takes two arguments, Z - S x Q weighted input. N - S x Q net input. and returns the S x Q derivative dN/dZ. Examples Here we define two weighted inputs for a layer with three neurons. Z1 = [0; 1; -1]; Z2 = [1; 0.5; 1.2]; We calculate the layer's net input N with netsum and then the derivative of N with respect to each weighted input. N = netsum(Z1,Z2) dN_dZ1 = dnetsum(Z1,N) dN_dZ2 = dnetsum(Z2,N) Algorithm The derivative of a sum with respect to any element of that sum is always a ones matrix that is the same size as the sum. See Also netsum, netprod, dnetprod 13-44 dotprod Purpose Syntax dotprod Dot product weight function Z = dotprod(W,P) df = dotprod('deriv') Description dotprod is the dot product weight function. Weight functions apply weights to an input to get weighted inputs. dotprod(W,P) takes these inputs, W - S x R weight matrix. P - R x Q matrix of Q input (column) vectors. and returns the S x Q dot product of W and P. Examples Here we define a random weight matrix W and input vector P and calculate the corresponding weighted input Z. W = rand(4,3); P = rand(3,1); Z = dotprod(W,P) Network Use You can create a standard network that uses dotprod by calling newp or newlin. To change a network so an input weight uses dotprod, set net.inputWeight{i,j}.weightFcn to 'dotprod’. For a layer weight, set net.inputWeight{i,j}.weightFcn to 'dotprod’. In either case, call sim to simulate the network with dotprod. See newp and newlin for simulation examples. See Also sim, ddotprod, dist, negdist, normprod 13-45 dposlin Purpose dposlin Derivative of positive linear transfer function Syntax dA_dN = dposlin(N,A) Description dposlin is the derivative function for poslin. dposlin(N,A) takes two arguments, and returns the S x Q derivative dA/dN. Examples Here we define the net input N for a layer of 3 poslin neurons. N = [0.1; 0.8; -0.7]; We calculate the layer's output A with poslin and then the derivative of A with respect to N. A = poslin(N) dA_dN = dposlin(N,A) Algorithm The derivative of poslin is calculated as follows: d = 1, if 0 <= n; 0, Otherwise.. See Also 13-46 poslin dpurelin Purpose dpurelin Linear transfer derivative function Syntax dA_dN = dpurelin(N,A) Description dpurelin is the derivative function for logsig. dpurelin(N,A) takes two arguments, N - S x Q net input. A - S x Q output. and returns the S x Q derivative dA_dN. Examples Here we define the net input N for a layer of 3 purelin neurons. N = [0.1; 0.8; -0.7]; We calculate the layer's output A with purelin and then the derivative of A with respect to N. A = purelin(N) dA_dN = dpurelin(N,A) Algorithm The derivative of purelin is calculated as follows: D(i,q) = 1 See Also purelin 13-47 dradbas Purpose dradbas Derivative of radial basis transfer function Syntax dA_dN = dradbas(N,A) Description dradbas is the derivative function for radbas. dradbas(N,A) takes two arguments, N - S x Q net input. A - S x Q output. and returns the S x Q derivative dA/dN. Examples Here we define the net input N for a layer of 3 radbas neurons. N = [0.1; 0.8; -0.7]; We calculate the layer's output A with radbas and then the derivative of A with respect to N. A = radbas(N) Algorithm The derivative of radbas is calculated as follows: d = -2*n*a See Also 13-48 radbas dsatlin Purpose dsatlin Derivative of saturating linear transfer function Syntax dA_dN = dsatlin(N,A) Description dsatlin is the derivative function for satlin. dsatlin(N,A) takes two arguments, N - S x Q net input. A - S x Q output. and returns the S x Q derivative dA/dN. Examples Here we define the net input N for a layer of 3 satlin neurons. N = [0.1; 0.8; -0.7]; We calculate the layer's output A with satlin and then the derivative of A with respect to N. A = satlin(N) dA_dN = dsatlin(N,A) Algorithm The derivative of satlin is calculated as follows: d = 1, if 0 <= n <= 1; 0, Otherwise.. See Also satlin 13-49 dsatlins Purpose dsatlins Derivative of symmetric saturating linear transfer function Syntax dA_dN = dsatlins(N,A) Description dsatlins is the derivative function for satlins. dsatlins(N,A) takes two arguments, N - S x Q net input. A - S x Q output. and returns the S x Q derivative dA/dN. Examples Here we define the net input N for a layer of 3 satlins neurons. N = [0.1; 0.8; -0.7]; We calculate the layer's output A with satlins and then the derivative of A with respect to N. A = satlins(N) dA_dN = dsatlins(N,A) Algorithm The derivative of satlins is calculated as follows: d = 1, if -1 <= n <= 1; 0, Otherwise. See Also 13-50 satlins dsse Purpose Syntax dsse Sum squared error performance derivative function dPerf_dE = dsse('e',E,X,perf,PP) dPerf_dX = dsse('x',E,X,perf,PP) Description dsse is the derivative function for sse. dsse('d',E,X,perf,PP) takes these arguments, E - Matrix or cell array of error vector(s). X - Vector of all weight and bias values. perf - Network performance (ignored). PP - Performance parameters (ignored). and returns the derivative dPerf_dE. dsse('x',E,X,perf,PP)returns the derivative dPerf_dX. Examples Here we define an error E and X for a network with one 3-element output and six weight and bias values. E = {[1; -2; 0.5]}; X = [0; 0.2; -2.2; 4.1; 0.1; -0.2]; Here we calculate the network's sum squared error performance, and derivatives of performance. perf = sse(E) dPerf_dE = dsse('e',E,X) dPerf_dX = dsse('x',E,X) Note that sse can be called with only one argument and dsse with only three arguments because the other arguments are ignored. The other arguments exist so that sse and dsse conform to standard performance function argument lists. See Also sse 13-51 dtansig Purpose dtansig Hyperbolic tangent sigmoid transfer derivative function Syntax dA_dN = dtansig(N,A) Description dtansig is the derivative function for tansig. dtansig(N,A) takes two arguments, N - S x Q net input. A - S x Q output. and returns the S x Q derivative dA/dN. Examples Here we define the net input N for a layer of 3 tansig neurons. N = [0.1; 0.8; -0.7]; We calculate the layer's output A with tansig and then the derivative of A with respect to N. A = tansig(N) dA_dN = dtansig(N,A) Algorithm The derivative of tansig is calculated as follows: d = 1-a^2 See Also 13-52 tansig, logsig, dlogsig dtribas Purpose dtribas Derivative of triangular basis transfer function Syntax dA_dN = dtribas(N,A) Description dtribas is the derivative function for tribas. dtribas(N,A) takes two arguments, N - S x Q net input. A - S x Q output. and returns the S x Q derivative dA/dN. Examples Here we define the net input N for a layer of 3 tribas neurons. N = [0.1; 0.8; -0.7]; We calculate the layer's output A with tribas and then the derivative of A with respect to N. A = tribas(N) dA_dN = dtribas(N,A) Algorithm The derivative of tribas is calculated as follows: d = 1, if -1 <= n < 0; -1, if 0 < n <= 1; 0, Otherwise. See Also tribas 13-53 errsurf Purpose errsurf Error surface of single input neuron Syntax errsurf(p,t,wv,bv,f) Description errsurf(p,t,wv,bv,f) takes these arguments, P - 1 x Q matrix of input vectors. T - 1 x Q matrix of target vectors. WV - Row vector of values of W. BV - Row vector of values of B. F - Transfer function (string). and returns a matrix of error values over WV and BV. Examples See Also 13-54 p = [-6.0 -6.1 -4.1 -4.0 +4.0 +4.1 +6.0 +6.1]; t = [+0.0 +0.0 +.97 +.99 +.01 +.03 +1.0 +1.0]; wv = -1:.1:1; bv = -2.5:.25:2.5; es = errsurf(p,t,wv,bv,'logsig'); plotes(wv,bv,ES,[60 30]) plotes gensim Purpose gensim Generate a simulink block for neural network simulation Syntax gensim(net,st) To Get Help Type help network/gensim Description gensim(net,st) creates a simulink system containing a block which simulates neural network net. gensim(net,st) takes these inputs, net - Neural network. st - Sample time (default = 1). and creates a Simulink system containing a block which simulates neural network net with a sampling time of st. If net has no input or layer delays (net.numInputDelays and net.numLayerDelays are both 0) then you can use -1 for st to get a continuously sampling network. Examples net = newff([0 1],[5 1]); gensim(net) 13-55 gridtop Purpose gridtop Grid layer topology function Syntax pos = gridtop(dim1,dim2,...,dimN) Description gridtop calculates neuron positions for layers whose neurons are arranged in an N dimensional grid. gridtop(dim1,dim2,...,dimN) takes N arguments, dimi - Length of layer in dimension i. and returns an N x S matrix of N coordinate vectors where S is the product of dim1*dim2*...*dimN. Examples This code creates and displays a two-dimensional layer with 40 neurons arranged in a 8x5 grid. pos = gridtop(8,5); plotsom(pos) This code plots the connections between the same neurons, but shows each neuron at the location of its weight vector. The weights are generated randomly so the layer is very disorganized as is evident in the following plot. W = rands(40,2); plotsom(W,dist(pos)) See Also 13-56 hextop, randtop hardlim Purpose Graph and Symbol hardlim Hard limit transfer function a +1 0 n -1 AA a = hardlim(n) Hard Limit Transfer Function Syntax A = hardlim(N) info = hardlim(code) Description The hard limit transfer function forces a neuron to output a 1 if its net input reaches a threshold, otherwise it outputs 0. This allows a neuron to make a decision or classification. It can say yes or no. This kind of neuron is often trained with the perceptron learning rule. hardlim is a transfer function. Transfer functions calculate a layer's output from its net input. hardlim(N) takes one input, N - S x Q matrix of net input (column) vectors. and returns 1 where N is positive, 0 elsewhere. hardlim(code) returns useful information for each code string, 'deriv' - Name of derivative function. 'name' - Full name. 'output' - Output range. 'active' - Active input range. Examples Here is the code to create a plot of the hardlim transfer function. n = -5:0.1:5; a = hardlim(n); plot(n,a) 13-57 hardlim Network Use You can create a standard network that uses hardlim by calling newp. To change a network so that a layer uses hardlim, set net.layers{i}.transferFcn to 'hardlim'. In either case call sim to simulate the network with hardlim. See newp for simulation examples. Algorithm hardlim(n) = 1, if n >= 0; 0 otherwise. See Also sim, hardlims 13-58 hardlims Purpose Graph and Symbol hardlims Symmetric hard limit transfer function a +1 0 n -1 a = hardlims(n) AA AA Symmetric Hard Limit Trans. Funct. Syntax A = hardlims(N) info = hardlims(code) Description The symmetric hard limit transfer function forces a neuron to output a 1 if its net input reaches a threshold. Otherwise it outputs -1. Like the regular hard limit function, this allows a neuron to make a decision or classification. It can say yes or no. hardlims is a transfer function. Transfer functions calculate a layer's output from its net input. hardlims(N) takes one input, N - S x Q matrix of net input (column) vectors. and returns 1 where N is positive, -1 elsewhere. hardlims(code) return useful information for each code string: 'deriv' - Name of derivative function. 'name' - Full name. 'output' - Output range. 'active' - Active input range. Examples Here is the code to create a plot of the hardlims transfer function. n = -5:0.1:5; a = hardlims(n); plot(n,a) 13-59 hardlims Network Use You can create a standard network that uses hardlims by calling newp. To change a network so that a layer uses hardlims, set net.layers{i}.transferFcn to 'hardlims'. In either case call sim to simulate the network with hardlims. See newp for simulation examples. Algorithm hardlim(n) = 1, if n >= 0; -1 otherwise. See Also sim, hardlim 13-60 hextop Purpose hextop Hexagonal layer topology function Syntax pos = hextop(dim1,dim2,...,dimN) Description hextop calculates the neuron positions for layers whose neurons are arranged in a N dimensional hexagonal pattern. hextop(dim1,dim2,...,dimN) takes N arguments, dimi - Length of layer in dimension i. and returns an N x S matrix of N coordinate vectors where S is the product of dim1*dim2*...*dimN. Examples This code creates and displays a two-dimensional layer with 40 neurons arranged in a 8x5 hexagonal pattern. pos = hextop(8,5); plotsom(pos) This code plots the connections between the same neurons, but shows each neuron at the location of its weight vector. The weights are generated randomly so that the layer is very disorganized, as is evident in the following plot. W = rands(40,2); plotsom(W,dist(pos)) See Also gridtop, randtop 13-61 hintonw Purpose hintonw Hinton graph of weight matrix Syntax hintonw(W,maxw,minw) Description hintonw(W,maxw,minw) takes these inputs, W - S x R weight matrix maxw - Maximum weight, default = max(max(abs(W))). minw - Minimum weight, default = M1/100. and displays a weight matrix represented as a grid of squares. Each square's area represents a weight’s magnitude. Each square's projection (color) represents a weight’s sign; inset (red) for negative weights, projecting (green) for positive. Examples W = rands(4,5); The following code displays the matrix graphically. hintonw(W) 1 Neuron 2 3 4 1 See Also 13-62 hintonwb 2 3 Input 4 5 hintonwb Purpose hintonwb Hinton graph of weight matrix and bias vector Syntax hintonwb(W,b,maxw,minw) Description hintonwb(W,B,maxw,minw) takes these inputs, W - S x R weight matrix. B - S x 1 bias vector. maxw - Maximum weight, default = max(max(abs(W))). minw - Minimum weight, default = M1/100. and displays a weight matrix and a bias vector represented as a grid of squares. Each square's area represents a weight’s magnitude. Each square's projection (color) represents a weight’s sign; inset (red) for negative weights, projecting (green) for positive. The weights are shown on the left. Examples The following code produces the result shown below. W = rands(4,5); b = rands(4,1); hintonwb(W,b) Neuron 1 2 3 4 0 1 2 3 4 5 Input See Also hintonw 13-63 ind2vec Purpose ind2vec Convert indices to vectors Syntax vec = ind2vec(ind) Description ind2vec and vec2ind allow indices to either be represented by themselves, or as vectors containing a 1 in the row of the index they represent. ind2vec(ind) takes one argument, ind - Row vector of indices. and returns a sparse matrix of vectors, with one 1 in each column, as indicated by ind. Examples Here four indices are defined and converted to vector representation. ind = [1 3 2 3] vec = ind2vec(ind) See Also 13-64 vec2ind init Purpose init Initialize a neural network Syntax net = init(net) To Get Help Type help network/init Description init(net) returns neural network net with weight and bias values updated according to the network initialization function, indicated by net.initFcn, and the parameter values, indicated by net.initParam. Examples Here a perceptron is created with a 2-element input (with ranges of 0 to 1, and -2 to 2) and 1 neuron. Once it is created we can display the neuron’s weights and bias. net = newp([0 1;-2 2],1); net.iw{1,1} net.b{1} Training the perceptron alters its weight and bias values. P = [0 1 0 1; 0 0 1 1]; T = [0 0 0 1]; net = train(net,P,T); net.iw{1,1} net.b{1} init reinitializes those weight and bias values. net = init(net); net.iw{1,1} net.b{1} The weights and biases are zeros again, which are the initial values used by perceptron networks (see newp). Algorithm init calls net.initFcn to initialize the weight and bias values according to the parameter values net.initParam. Typically, net.initFcn is set to 'initlay' which initializes each layer's weights and biases according to its net.layers{i}.initFcn. 13-65 init Backpropagation networks have net.layers{i}.initFcn set to 'initnw', which calculates the weight and bias values for layer i using the Nguyen-Widrow initialization method. Other networks have net.layers{i}.initFcn set to 'initwb', which initializes each weight and bias with its own initialization function. The most common weight and bias initialization function is rands, which generates random values between -1 and 1. See Also 13-66 sim, adapt, train, initlay, initnw, initwb, rands initcon Purpose initcon Conscience bias initialization function Syntax b = initcon(s,pr) Description initcon is a bias initialization function that initializes biases for learning with the learncon learning function. initcon (S,PR) takes two arguments, S - Number of rows (neurons). PR - R x 2 matrix of R = [Pmin Pmax], default = [1 1]. and returns an S x 1 bias vector. Note that for biases R is always 1. initcon could also be used to initialize weights, but it is not recommended for that purpose. Examples Here initial bias values are calculated for a 5 neuron layer. b = initcon(5) Network Use You can create a standard network that uses initcon to initialize weights by calling newc. To prepare the bias of layer i of a custom network to initialize with initcon: 1 Set net.initFcn to 'initlay'. (net.initParam will automatically become initlay's default parameters.) 2 Set net.layers{i}.initFcn to 'initwb'. 3 Set net.biases{i}.initFcn to 'initcon'. To initialize the network call init. See newc for initialization examples. Algorithm learncon updates biases so that each bias value b(i) is a function of the average output c(i) of the neuron i associated with the bias. initcon gets initial bias values by assuming that each neuron has responded to equal numbers of vectors in the "past". See Also initwb, initlay, init, learncon 13-67 initlay Purpose Syntax initlay Layer-by-layer network initialization function net = initlay(net) info = initlay(code) Description initlay is a network initialization function which initializes each layer i according to its own initialization function net.layers{i}.initFcn. initlay(net) takes, net - Neural network. and returns the network with each layer updated. initlay(code) returns useful information for each code string: 'pnames' - Names of initialization parameters. 'pdefaults' - Default initialization parameters. initlay does not have any initialization parameters. Network Use You can create a standard network that uses initlay by calling newp, newlin, newff, newcf, and many other new network functions. To prepare a custom network to be initialized with initlay: 1 Set net.initFcn to 'initlay'. (This will set net.initParam to the empty matrix [ ] since initlay has no initialization parameters.) 2 Set each net.layers{i}.initFcn to a layer initialization function. (Examples of such functions are initwb and initnw). To initialize the network call init. See newp and newlin for initialization examples. Algorithm The weights and biases of each layer i are initialized according to net.layers{i}.initFcn. See Also initwb, initnw, init 13-68 initnw Purpose initnw Nguyen-Widrow layer initialization function Syntax net = initnw(net,i) Description initnw is a layer initialization function which initializes a layer's weights and biases according to the Nguyen-Widrow initialization algorithm. This algorithm chooses values in order to distribute the active region of each neuron in the layer evenly across the layer's input space. initnw(net,i) takes two arguments, net - Neural network. i - Index of a layer. and returns the network with layer i's weights and biases updated. Network Use You can create a standard network that uses initnw by calling newff or newcf. To prepare a custom network to be initialized with initnw: 1 Set net.initFcn to 'initlay'. (This will set net.initParam to the empty matrix [ ] since initlay has no initialization parameters.) 2 Set net.layers{i}.initFcn to 'initnw'. To initialize the network call init. See newff and newcf for training examples. Algorithm The Nguyen-Widrow method generates initial weight and bias values for a layer, so that the active regions of the layer's neurons will be distributed roughly evenly over the input space. Advantages over purely random weights and biases are: 1 Few neurons are wasted (since all the neurons are in the input space). 2 Training works faster (since each area of the input space has neurons). The Nguyen-Widrow method can only be applied to layers... - with a bias, - with weights whose "weightFcn" is dotprod, - with "netInputFcn" set to netsum. If these conditions are not met then initnw uses rands to initialize the layer's weights and biases. See Also initwb, initlay, init 13-69 initwb Purpose initwb By-weight-and-bias layer initialization function Syntax net = initwb(net,i) Description initwb is a layer initialization function with initializes a layer's weights and biases according to their own initialization functions. initwb(net,i) takes two arguments, net - Neural network. i - Index of a layer. and returns the network with layer i's weights and biases updated. Network Use You can create a standard network that uses initwb by calling newp or newlin. To prepare a custom network to be initialized with initwb: 1 Set net.initFcn to 'initlay'. (This will set net.initParam to the empty matrix [ ] since initlay has no initialization parameters.) 2 Set net.layers{i}.initFcn to 'initwb'. 3 Set each net.inputWeights{i,j}.learnFcn to a weight initialization function. Set each net.layerWeights{i,j}.learnFcn to a weight initialization function. Set each net.biases{i}.learnFcn to a bias initialization function. (Examples of such functions are rands and midpoint.) To initialize the network call init. See newp and newlin for training examples. Algorithm Each weight (bias) in layer i is set to new values calculated according to its weight (bias) initialization function. See Also initnw, initlay, init 13-70 initzero Purpose Syntax initzero Zero weight/bias initialization function W = initzero(S,PR) b = initzero(S,[1 1]) Description initzero(S,PR) takes two arguments, S - Number of rows (neurons). PR - R x 2 matrix of input value ranges = [Pmin Pmax]. and returns an S x R weight matrix of zeros. initzero(S,[1 1]) returns S x 1 bias vector of zeros. Examples Here initial weights and biases are calculated for a layer with two inputs ranging over [0 1] and [-2 2], and 4 neurons. W = initzero(5,[0 1; -2 2]) b = initzero(5,[1 1]) Network Use You can create a standard network that uses initzero to initialize its weights by calling newp or newlin. To prepare the weights and the bias of layer i of a custom network to be initialized with midpoint: 1 Set net.initFcn to 'initlay'. (net.initParam will automatically become initlay's default parameters.) 2 Set net.layers{i}.initFcn to 'initwb'. 3 Set each net.inputWeights{i,j}.initFcn to 'initzero'. Set each net.layerWeights{i,j}.initFcn to 'initzero'. Set each net.biases{i}.initFcn to 'initzero'. To initialize the network call init. See newp or newlin for initialization examples. See Also initwb, initlay, init 13-71 learncon Purpose Syntax learncon Conscience bias learning function [dB,LS] = learncon(B,P,Z,N,A,T,E,gW,gA,D,LP,LS) info = learncon(code) Description learncon is the conscience bias learning function used to increase the net input to neurons which have the lowest average output until each neuron responds roughly an equal percentage of the time. learncon(B,P,Z,N,A,T,E,gW,gA,D,LP,LS) takes several inputs, B - S x 1 bias vector. P - 1x Q ones vector. Z - S x Q weighted input vectors. N - S x Q net input vectors. A - S x Q output vectors. T - S x Q layer target vectors. E - S x Q layer error vectors. gW - S x R gradient with respect to performance. gA - S x Q output gradient with respect to performance. D - S x S neuron distances. LP - Learning parameters, none, LP = []. LS - Learning state, initially should be = []. and returns dB - S x 1 weight (or bias) change matrix. LS - New learning state. Learning occurs according to learncon's learning parameter, shown here with its default value. LP.lr - 0.001 - Learning rate. learncon(code) returns useful information for each code string, 'pnames' - Names of learning parameters. 'pdefaults' - Default learning parameters. 'needg' - Returns 1 if this function uses gW or gA. NNT 2.0 compatibility: The LP.lr described above equals 1 minus the bias time constant used by trainc in NNT 2.0. 13-72 learncon Examples Here we define a random output A, and bias vector W for a layer with 3 neurons. We also define the learning rate LR. a = rand(3,1); b = rand(3,1); lp.lr = 0.5; Since learncon only needs these values to calculate a bias change (see algorithim below), we will use them to do so. dW = learncon(b,[],[],[],a,[],[],[],[],[],lp,[]) Network Use To prepare the bias of layer i of a custom network to learn with learncon: 1 Set net.trainFcn to 'trainwb1'. (net.trainParam will automatically become adaptwb's default parameters.) 2 Set net.adaptFcn to 'adaptwb'. (net.adaptParam will automatically become adaptwb's default parameters.) 3 Set net.inputWeights{i}.learnFcn to 'learncon'. Set each net.layerWeights{i,j}.learnFcn to 'learncon'. (Each weight learning parameter property will automatically be set to learncon's default parameters.) To train the network (or enable it to adapt): 1 Set net.trainParam (or net.adaptParam) properties as desired. 2 Call train (or adapt). Algorithm learncon calculates the bias change db for a given neuron by first updating each neuron's conscience, i.e. the running average of its output: c = (1-lr)*c + lr*a The conscience is then used to compute a bias for the neuron that is greatest for smaller conscience values. b = exp(1-log(c)) - b (Note that learncon is able to recover C each time it is called from the bias values.) See Also learnk, learnos, adaptwb, trainwb, adapt, train 13-73 learngd Purpose Syntax learngd Gradient descent weight/bias learning function [dW,LS] = learngd(W,P,Z,N,A,T,E,gW,gA,D,LP,LS) [db,LS] = learngd(b,ones(1,Q),Z,N,A,T,E,gW,gA,D,LP,LS) info = learngd(code) Description learngd is the gradient descent weight/bias learning function. learngd(W,P,Z,N,A,T,E,gW,gA,D,LP,LS) takes several inputs, W - S x R weight matrix (or S x 1 bias vector). P - R x Q input vectors (or ones(1,Q)). Z - S x Q weighted input vectors. N - S x Q net input vectors. A - S x Q output vectors. T - S x Q layer target vectors. E - S x Q layer error vectors. gW - S x R gradient with respect to performance. gA - S x Q output gradient with respect to performance. D - S x S neuron distances. LP - Learning parameters, none, LP = []. LS - Learning state, initially should be = []. and returns, dW - S x R weight (or bias) change matrix. LS - New learning state. Learning occurs according to learngd's learning parameter shown here with its default value. LP.lr - 0.01 - Learning rate. learngd(code) returns useful information for each code string: 'pnames' - Names of learning parameters. 'pdefaults' - Default learning parameters. 'needg' - Returns 1 if this function uses gW or gA. 13-74 learngd Examples Here we define a random gradient gW for a weight going to a layer with 3 neurons, from an input with 2 elements. We also define a learning rate of 0.5. gW = rand(3,2); lp.lr = 0.5; Since learngd only needs these values to calculate a weight change (see algorithim below), we will use them to do so. dW = learngd([],[],[],[],[],[],[],gW,[],[],lp,[]) Network Use You can create a standard network that uses learngd with newff, newcf, or newelm. To prepare the weights and the bias of layer i of a custom network to adapt with learngd: 1 Set net.adaptFcn to 'adaptwb'. net.adaptParam will automatically become trainwb's default parameters. 2 Set each net.inputWeights{i,j}.learnFcn to 'learngd'. Set each net.layerWeights{i,j}.learnFcn to 'learngd'. Set net.biases{i}.learnFcn to 'learngd'. Each weight and bias learning parameter property will automatically be set to learngd's default parameters. To allow the network to adapt: 1 Set net.adaptParam properties to desired values. 2 Call adapt with the network. See newff or newcf for examples. Algoritm learngd calculates the weight change dW for a given neuron from the neuron's input P and error E, and the weight (or bias) learning rate LR, according to the gradient descent: dw = lr*gW. See Also learngdm, newff, newcf, adaptwb, trainwb, adapt, train 13-75 learngdm Purpose Syntax learngdm Gradient descent w/ momentum weight/bias learning function [dW,LS] = learngdm(W,P,Z,N,A,T,E,gW,gA,D,LP,LS) [db,LS] = learngdm(b,ones(1,Q),Z,N,A,T,E,gW,gA,D,LP,LS) info = learngdm(code) Description learngdm is the gradient descent with momentum weight/bias learning function. learngdm(W,P,Z,N,A,T,E,gW,gA,D,LP,LS) takes several inputs, W - S x R weight matrix (or S x 1 bias vector). P - R x Q input vectors (or ones(1,Q)). Z - S x Q weighted input vectors. N - S x Q net input vectors. A - S x Q output vectors. T - S x Q layer target vectors. E - S x Q layer error vectors. gW - S x R gradient with respect to performance. gA - S x Q output gradient with respect to performance. D - S x S neuron distances. LP - Learning parameters, none, LP = []. LS - Learning state, initially should be = []. and returns, dW - S x R weight (or bias) change matrix. LS - New learning state. Learning occurs according to learngdm's learning parameters, shown here with their default values. LP.lr - 0.01 - Learning rate. LP.mc - 0.9 13-76 - Momentum constant. learngdm learngdm(code) returns useful information for each code string: 'pnames' - Names of learning parameters. 'pdefaults' - Default learning parameters. 'needg' - Returns 1 if this function uses gW or gA. Examples Here we define a random gradient G for a weight going to a layer with 3 neurons, from an input with 2 elements. We also define a learning rate of 0.5 and momentum constant of 0.8; gW = rand(3,2); lp.lr = 0.5; lp.mc = 0.8; Since learngdm only needs these values to calculate a weight change (see algorithim below), we will use them to do so. We will use the default initial learning state. ls = []; [dW,ls] = learngdm([],[],[],[],[],[],[],gW,[],[],lp,ls) learngdm returns the weight change and a new learning state. Network Use You can create a standard network that uses learngdm with newff, newcf, or newelm. To prepare the weights and the bias of layer i of a custom network to adapt with learngdm: 1 Set net.adaptFcn to 'adaptwb'. net.adaptParam will automatically become trainwb's default parameters. 2 Set each net.inputWeights{i,j}.learnFcn to 'learngdm'. Set each net.layerWeights{i,j}.learnFcn to 'learngdm'. Set net.biases{i}.learnFcn to 'learngdm'. Each weight and bias learning parameter property will automatically be set to learngdm's default parameters. To allow the network to adapt: 1 Set net.adaptParam properties to desired values. 2 Call adapt with the network. See newff or newcf for examples. 13-77 learngdm Algorithm learngdm calculates the weight change dW for a given neuron from the neuron's input P and error E, the weight (or bias) W, learning rate LR, and momentum constant MC, according to gradient descent with momentum: dW = mc*dWprev + (1-mc)*lr*gW The previous weight change dWprev is stored and read from the learning state LS. See Also 13-78 learngd, newff, newcf, adaptwb, trainwb, adapt, train learnh Purpose Syntax learnh Hebb weight learning rule [dW,LS] = learnh(W,P,Z,N,A,T,E,gW,gA,D,LP,LS) info = learnh(code) Description learnh is the Hebb weight learning function. learnh(W,P,Z,N,A,T,E,gW,gA,D,LP,LS) takes several inputs, W - S x R weight matrix (or S x 1 bias vector). P - R x Q input vectors (or ones(1,Q)). Z - S x Q weighted input vectors. N - S x Q net input vectors. A - S x Q output vectors. T - S x Q layer target vectors. E - S x Q layer error vectors. gW - S x R gradient with respect to performance. gA - S x Q output gradient with respect to performance. D - S x S neuron distances. LP - Learning parameters, none, LP = []. LS - Learning state, initially should be = []. and returns, dW - S x R weight (or bias) change matrix. LS - New learning state. Learning occurs according to learnh's learning parameter, shown here with its default value. LP.lr - 0.01 - Learning rate. learnh(code) returns useful information for each code string: 'pnames' - Names of learning parameters. 'pdefaults' - Default learning parameters. 'needg' - Returns 1 if this function uses gW or gA. 13-79 learnh Examples Here we define a random input P and output A for a layer with a 2-element input and 3 neurons. We also define the learning rate LR. p = rand(2,1); a = rand(3,1); lp.lr = 0.5; Since learnh only needs these values to calculate a weight change (see algorithim below), we will use them to do so. dW = learnh([],p,[],[],a,[],[],[],[],[],lp,[]) Network Use To prepare the weights and the bias of layer i of a custom network to learn with learnh: 1 Set net.trainFcn to 'trainwb'. (net.trainParam will automatically become trainwb's default parameters.) 2 Set net.adaptFcn to 'adaptwb'. (net.adaptParam will automatically become trainwb's default parameters.) 3 Set each net.inputWeights{i,j}.learnFcn to 'learnh'. Set each net.layerWeights{i,j}.learnFcn to 'learnh'. Each weight learning parameter property will automatically be set to learnh's default parameters.) To train the network (or enable it to adapt): 1 Set net.trainParam (net.adaptParam) properties to desired values. 2 Call train (adapt). Algorithm learnh calculates the weight change dW for a given neuron from the neuron's input P, output A, and learning rate LR according to the Hebb learning rule: dw = lr*a*p' See Also learnhd, adaptwb, trainwb, adapt, train References Hebb, D.O., The Organization of Behavior, New York: Wiley, 1949. 13-80 learnhd Purpose Syntax learnhd Hebb with decay weight learning rule [dW,LS] = learnhd(W,P,Z,N,A,T,E,gW,gA,D,LP,LS) info = learnhd(code) Description learnhd is the Hebb weight learning function. learnhd(W,P,Z,N,A,T,E,gW,gA,D,LP,LS) takes several inputs, W - S x R weight matrix (or S x 1 bias vector). P - R x Q input vectors (or ones(1,Q)). Z - S x Q weighted input vectors. N - S x Q net input vectors. A - S x Q output vectors. T - S x Q layer target vectors. E - S x Q layer error vectors. gW - S x R gradient with respect to performance. gA - S x Q output gradient with respect to performance. D - S x S neuron distances. LP - Learning parameters, none, LP = []. LS - Learning state, initially should be = []. and returns, dW - S x R weight (or bias) change matrix. LS - New learning state. Learning occurs according to learnhd's learning parameters shown here with default values. LP.dr - 0.01 - Decay rate. LP.lr - 0.1 - Learning rate. learnhd(code) returns useful information for each code string: 'pnames' - Names of learning parameters. 'pdefaults' - Default learning parameters. 'needg' - Returns 1 if this function uses gW or gA. 13-81 learnhd Examples Here we define a random input P, output A, and weights W for a layer with a 2-element input and 3 neurons. We also define the decay and learning rates. p = rand(2,1); a = rand(3,1); w = rand(3,2); lp.dr = 0.05; lp.lr = 0.5; Since learnhd only needs these values to calculate a weight change (see algorithm below), we will use them to do so. dW = learnhd(w,p,[],[],a,[],[],[],[],[],lp,[]) Network Use To prepare the weights and the bias of layer i of a custom network to learn with learnhd: 1 Set net.trainFcn to 'trainwb'. (net.trainParam will automatically become trainwb's default parameters.) 2 Set net.adaptFcn to 'adaptwb'. (net.adaptParam will automatically become trainwb's default parameters.) 3 Set each net.inputWeights{i,j}.learnFcn to 'learnhd'. Set each net.layerWeights{i,j}.learnFcn to 'learnhd'. (Each weight learning parameter property will automatically be set to learnhd's default parameters.) To train the network (or enable it to adapt): 1 Set net.trainParam (net.adaptParam) properties to desired values. 2 Call train (adapt). Algorithm learnhd calculates the weight change dW for a given neuron from the neuron's input P, output A, decay rate DR, and learning rate LR according to the Hebb with decay learning rule: dw = See Also 13-82 lr*a*p' - dr*w learnh, adaptwb, trainwb, adapt, train learnis Purpose Syntax learnis Instar weight learning function [dW,LS] = learnis(W,P,Z,N,A,T,E,gW,gA,D,LP,LS) info = learnis(code) Description learnis is the instar weight learning function. learnis(W,P,Z,N,A,T,E,gW,gA,D,LP,LS) takes several inputs, W - S x R weight matrix (or S x 1 bias vector). P - R x Q input vectors (or ones(1,Q)). Z - S x Q weighted input vectors. N - S x Q net input vectors. A - S x Q output vectors. T - S x Q layer target vectors. E - S x Q layer error vectors. gW - S x R gradient with respect to performance. gA - S x Q output gradient with respect to performance. D - S x S neuron distances. LP - Learning parameters, none, LP = []. LS - Learning state, initially should be = []. and returns, dW - S x R weight (or bias) change matrix. LS - New learning state. Learning occurs according to learnis's learning parameter, shown here with its default value. LP.lr - 0.01 - Learning rate. learnis(code) return useful information for each code string: 'pnames' - Names of learning parameters. 'pdefaults' - Default learning parameters. 'needg' - Returns 1 if this function uses gW or gA. 13-83 learnis Examples Here we define a random input P, output A, and weight matrix W for a layer with a 2-element input and 3 neurons. We also define the learning rate LR. p = rand(2,1); a = rand(3,1); w = rand(3,2); lp.lr = 0.5; Since learnis only needs these values to calculate a weight change (see algorithm below), we will use them to do so. dW = learnis(w,p,[],[],a,[],[],[],[],[],lp,[]) Network Use To prepare the weights and the bias of layer i of a custom network so that it can learn with learnis: 1 Set net.trainFcn to 'trainwb'. (net.trainParam will automatically become trainwb's default parameters.) 2 Set net.adaptFcn to 'adaptwb'. (net.adaptParam will automatically become trainwb's default parameters.) 3 Set each net.inputWeights{i,j}.learnFcn to 'learnis'. Set each net.layerWeights{i,j}.learnFcn to 'learnis'. (Each weight learning parameter property will automatically be set to learnis's default parameters.) To train the network (or enable it to adapt): 1 Set net.trainParam (net.adaptParam) properties to desired values. 2 Call train (adapt). Algorithm learnis calculates the weight change dW for a given neuron from the neuron's input P, output A, and learning rate LR according to the instar learning rule: dw = lr*a*(p'-w) See Also learnk, learnos, adaptwb, trainwb, adapt, train References Grossberg, S., Studies of the Mind and Brain, Drodrecht, Holland: Reidel Press, 1982. 13-84 learnk Purpose Syntax learnk Kohonen weight learning function [dW,LS] = learnk(W,P,Z,N,A,T,E,gW,gA,D,LP,LS) info = learnk(code) Description learnk is the Kohonen weight learning function. learnk(W,P,Z,N,A,T,E,gW,gA,D,LP,LS) takes several inputs, W - S x R weight matrix (or S x 1 bias vector). P - R x Q input vectors (or ones(1,Q)). Z - S x Q weighted input vectors. N - S x Q net input vectors. A - S x Q output vectors. T - S x Q layer target vectors. E - S x Q layer error vectors. gW - S x R gradient with respect to performance. gA - S x Q output gradient with respect to performance. D - S x S neuron distances. LP - Learning parameters, none, LP = []. LS - Learning state, initially should be = []. and returns, dW - S x R weight (or bias) change matrix. LS - New learning state. Learning occurs according to learnk's learning parameter, shown here with its default value. LP.lr - 0.01 - Learning rate. learnk(code) returns useful information for each code string: 'pnames' - Names of learning parameters. 'pdefaults' - Default learning parameters. 'needg' - Returns 1 if this function uses gW or gA. 13-85 learnk Examples Here we define a random input P, output A, and weight matrix W for a layer with a 2-element input and 3 neurons. We also define the learning rate LR. p = rand(2,1); a = rand(3,1); w = rand(3,2); lp.lr = 0.5; Since learnk only needs these values to calculate a weight change (see algorithm below), we will use them to do so. dW = learnk(w,p,[],[],a,[],[],[],[],[],lp,[]) Network Use To prepare the weights of layer i of a custom network to learn with learnk: 1 Set net.trainFcn to 'trainwb1'. (net.trainParam will automatically become trainwb1's default parameters.) 2 Set net.adaptFcn to 'adaptwb'. (net.adaptParam will automatically become trainwb1's default parameters.) 3 Set each net.inputWeights{i,j}.learnFcn to 'learnk'. Set each net.layerWeights{i,j}.learnFcn to 'learnk'. (Each weight learning parameter property will automatically be set to learnk's default parameters.) To train the network (or enable it to adapt): 1 Set net.trainParam (or net.adaptParam) properties as desired. 2 Call train (or adapt). Algorithm learnk calculates the weight change dW for a given neuron from the neuron's input P, output A, and learning rate LR according to the Kohonen learning rule: dw = lr*(p'-w), if a ~= 0; = 0, otherwise. See Also learnis, learnos, adaptwb, trainwb, adapt, train References Kohonen, T., Self-Organizing and Associative Memory, New York: Springer-Verlag, 1984. 13-86 learnlv1 Purpose Syntax learnlv1 LVQ1 weight learning function [dW,LS] = learnlv1(W,P,Z,N,A,T,E,gW,gA,D,LP,LS) info = learnlv1(code) Description learnlv1 is the LVQ1 weight learning function. learnlv1(W,P,Z,N,A,T,E,gW,gA,D,LP,LS) takes several inputs, W - S x R weight matrix (or S x 1 bias vector). P - R x Q input vectors (or ones(1,Q)). Z - S x Q weighted input vectors. N - S x Q net input vectors. A - S x Q output vectors. T - S x Q layer target vectors. E - S x Q layer error vectors. gW - S x R weight gradient with respect to performance. gA - S x Q output gradient with respect to performance. D - S x R neuron distances. LP - Learning parameters, none, LP = []. LS - Learning state, initially should be = []. and returns, dW - S x R weight (or bias) change matrix. LS - New learning state. Learning occurs according to learnlv1's learning parameter shown here with its default value. LP.lr - 0.01 - Learning rate. learnlv1(code) returns useful information for each code string: 'pnames' - Names of learning parameters. 'pdefaults' - Default learning parameters. needg' - Returns 1 if this function uses gW or gA. 13-87 learnlv1 Examples Here we define a random input P, output A, weight matrix W, and output gradient gA for a layer with a 2-element input and 3 neurons. We also define the learning rate LR. p = rand(2,1); w = rand(3,2); a = compet(negdist(w,p)); gA = [-1;1; 1]; lp.lr = 0.5; Since learnlv1 only needs these values to calculate a weight change (see algorithm below), we will use them to do so. dW = learnlv1(w,p,[],[],a,[],[],[],gA,[],lp,[]) Network Use You can create a standard network that uses learnlv1 with newlvq. To prepare the weights of layer i of a custom network to learn with learnlv1: 1 Set net.trainFcn to trainwb1'. (net.trainParam will automatically become trainwb1's default parameters.) 2 Set net.adaptFcn to 'adaptwb'. (net.adaptParam will automatically become trainwb1's default parameters.) 3 Set each net.inputWeights{i,j}.learnFcn to 'learnlv1'. Set each net.layerWeights{i,j}.learnFcn to 'learnlv1'. (Each weight learning parameter property will automatically be set to learnlv1's default parameters.) To train the network (or enable it to adapt): 1 Set net.trainParam (or net.adaptParam) properties as desired. 2 Call train (or adapt). Algorithm learnlv1 calculates the weight change dW for a given neuron from the neuron's input P, output A, output gradient gA and learning rate LR, according to the LVQ1 rule, given i the index of the neuron whose output a(i) is 1: dw(i,:) = +lr*(p-w(i,:)) if gA(i) = 0;= -lr*(p-w(i,:)) if gA(i) = -1 See Also 13-88 learnlv2, adaptwb, trainwb, adapt, train learnlv2 Purpose Syntax learnlv2 LVQ2 weight learning function [dW,LS] = learnlv2(W,P,Z,N,A,T,E,gW,gA,D,LP,LS) info = learnlv2(code) Description learnlv2 is the LVQ2 weight learning function. learnlv2(W,P,Z,N,A,T,E,gW,gA,D,LP,LS) takes several inputs, W - S x R weight matrix (or S x 1 bias vector). P - R x Q input vectors (or ones(1,Q)). Z - S x Q weighted input vectors. N - S x Q net input vectors. A - S x Q output vectors. T - S x Q layer target vectors. E - S x Q layer error vectors. gW - S x R weight gradient with respect to performance. gA - S x Q output gradient with respect to performance. D - S x S neuron distances. LP - Learning parameters, none, LP = []. LS - Learning state, initially should be = []. and returns, dW - S x R weight (or bias) change matrix. LS - New learning state. Learning occurs according to learnlv1's learning parameter, shown here with its default value. LP.lr - 0.01 - Learning rate. learnlv2(code) returns useful information for each code string: 'pnames' - Names of learning parameters. 'pdefaults' - Default learning parameters. 'needg' - Returns 1 if this function uses gW or gA. 13-89 learnlv2 Examples Here we define a sample input P, output A, weight matrix W, and output gradient gA for a layer with a 2-element input and 3 neurons. We also define the learning rate LR. p = rand(2,1); w = rand(3,2); n = negdist(w,p); a = compet(n); gA = [-1;1; 1]; lp.lr = 0.5; Since learnlv2 only needs these values to calculate a weight change (see algorithm below), we will use them to do so. dW = learnlv2(w,p,[],n,a,[],[],[],gA,[],lp,[]) Network Use You can create a standard network that uses learnlv2 with newlvq. To prepare the weights of layer i of a custom network to learn with learnlv2: 1 Set net.trainFcn to 'trainwb1'. (net.trainParam will automatically become trainwb1's default parameters.) 2 Set net.adaptFcn to 'adaptwb'. (net.adaptParam will automatically become trainwb1's default parameters.) 3 Set each net.inputWeights{i,j}.learnFcn to 'learnlv2'. Set each net.layerWeights{i,j}.learnFcn to 'learnlv2'. (Each weight learning parameter property will automatically be set to learnlv2's default parameters.) To train the network (or enable it to adapt): 1 Set net.trainParam (or net.adaptParam) properties as desired. 2 Call train (or adapt). Algorithm learnlv2 calculates the weight change dW for a given neuron from the neuron's input P, output A, output gradient gA and learning rate LR according to the LVQ2 rule, given i the index of the neuron whose output a(i) is 1: dw(i,:) = +lr*(p-w(i,:)) if gA(i) = 0; = -lr*(p-w(i,:)) 13-90 learnlv2 If gA(i) = -1; if gA(i) is -1 then the index j is found of the neuron with the greatest net input n(k), from the neurons whose gA(k) is 1. This neuron's weights are updated as follows: dw(j,:) = +lr*(p-w(i,:)) See Also learnlv1, adaptwb, trainwb, adapt, train 13-91 learnos Purpose Syntax learnos Outstar weight learning function [dW,LS] = learnos(W,P,Z,N,A,T,E,gW,gA,D,LP,LS) info = learnos(code) Description learnos is the outstar weight learning function. learnos(W,P,Z,N,A,T,E,G,D,LP,LS) takes several inputs, W - S x R weight matrix (or S x 1 bias vector). P - R x Q input vectors (or ones(1,Q)). Z - S x Q weighted input vectors. N - S x Q net input vectors. A - S x Q output vectors. T - S x Q layer target vectors. E - S x Q layer error vectors. gW - S x R weight gradient with respect to performance. gA - S x Q output gradient with respect to performance. D - S x S neuron distances. LP - Learning parameters, none, LP = []. LS - Learning state, initially should be = []. and returns dW - S x R weight (or bias) change matrix. LS - New learning state. Learning occurs according to learnos's learning parameter, shown here with its default value. LP.lr - 0.01 - Learning rate. learnos(code) returns useful information for each code string: 'pnames' - Names of learning parameters. 'pdefaults' - Default learning parameters. 'needg' - Returns 1 if this function uses gW or gA. 13-92 learnos Examples Here we define a random input P, output A, and weight matrix W for a layer with a 2-element input and 3 neurons. We also define the learning rate LR. p = rand(2,1); a = rand(3,1); w = rand(3,2); lp.lr = 0.5; Since learnos only needs these values to calculate a weight change (see algorithm below), we will use them to do so. dW = learnos(w,p,[],[],a,[],[],[],[],[],lp,[]) Network Use To prepare the weights and the bias of layer i of a custom network to learn with learnos: 1 Set net.trainFcn to 'trainwb'. (net.trainParam will automatically become trainwb's default parameters.) 2 Set net.adaptFcn to 'adaptwb'. (net.adaptParam will automatically become trainwb's default parameters.) 3 Set each net.inputWeights{i,j}.learnFcn to 'learnos'. Set each net.layerWeights{i,j}.learnFcn to 'learnos'. (Each weight learning parameter property will automatically be set to learnos's default parameters.) To train the network (or enable it to adapt): 1 Set net.trainParam (net.adaptParam) properties to desired values. 2 Call train (adapt). Algorithm learnos calculates the weight change dW for a given neuron from the neuron's input P, output A, and learning rate LR according to the outstar learning rule: dw = lr*(a-w)*p' See Also learnis, learnk, adaptwb, trainwb, adapt, train References Grossberg, S., Studies of the Mind and Brain, Drodrecht, Holland: Reidel Press, 1982. 13-93 learnp Purpose Syntax learnp Perceptron weight/bias learning function [dW,LS] = learnp(W,P,Z,N,A,T,E,gW,gA,D,LP,LS) [db,LS] = learnp(b,ones(1,Q),Z,N,A,T,E,gW,gA,D,LP,LS) info = learnp(code) Description learnp is the perceptron weight/bias learning function. learnp(W,P,Z,N,A,T,E,gW,gA,D,LP,LS) takes several inputs, W - S x R weight matrix (or S x 1 bias vector). P - R x Q input vectors (or ones(1,Q)). Z - S x Q weighted input vectors. N - S x Q net input vectors. A - S x Q output vectors. T - S x Q layer target vectors. E - S x Q layer error vectors. gW - S x R weight gradient with respect to performance. gA - S x Q output gradient with respect to performance. D - S x S neuron distances. LP - Learning parameters, none, LP = []. LS - Learning state, initially should be = []. and returns, dW - S x R weight (or bias) change matrix. LS - New learning state. learnp(code) returns useful information for each code string: 'pnames' - Names of learning parameters. 'pdefaults' - Default learning parameters. 'needg' - Returns 1 if this function uses gW or gA. 13-94 learnp Examples Here we define a random input P and error E to a layer with a 2-element input and 3 neurons. p = rand(2,1); e = rand(3,1); Since learnp only needs these values to calculate a weight change (see algorithm below), we will use them to do so. dW = learnp([],p,[],[],[],[],e,[],[],[],[],[]) Network Use You can create a standard network that uses learnp with newp. To prepare the weights and the bias of layer i of a custom network to learn with learnp: 1 Set net.trainFcn to 'trainwb'. (net.trainParam will automatically become trainwb's default parameters.) 2 Set net.adaptFcn to 'adaptwb'. (net.adaptParam will automatically become trainwb's default parameters.) 3 Set each net.inputWeights{i,j}.learnFcn to 'learnp'. Set each net.layerWeights{i,j}.learnFcn to 'learnp'. Set net.biases{i}.learnFcn to 'learnp'. (Each weight and bias learning parameter property will automatically become the empty matrix since learnp has no learning parameters.) To train the network (or enable it to adapt): 1 Set net.trainParam (net.adaptParam) properties to desired values. 2 Call train (adapt). See newp for adaption and training examples. Algorithm learnp calculates the weight change dW for a given neuron from the neuron's input P and error E according to the perceptron learning rule: dw = 0, if e = = p', if e = 1 = -p', if e = -1 0 13-95 learnp This can be summarized as: dw = e*p' See Also learnpn, newp, adaptwb, trainwb, adapt, train References Rosenblatt, F., Principles of Neurodynamics, Washington D.C.: Spartan Press, 1961. 13-96 learnpn Purpose Syntax learnpn Normalized perceptron weight/bias learning function [dW,LS] = learnpn(W,P,Z,N,A,T,E,gW,gA,D,LP,LS) info = learnpn(code) Description learnpn is a weight/bias learning function. It can result in faster learning than learnp when input vectors have widely varying magnitudes. learnpn(W,P,Z,N,A,T,E,gW,gA,D,LP,LS) takes several inputs, W - S x R weight matrix (or S x 1 bias vector). P - R x Q input vectors (or ones(1,Q)). Z - S x Q weighted input vectors. N - S x Q net input vectors. A - S x Q output vectors. T - S x Q layer target vectors. E - S x Q layer error vectors. gW - S x R weight gradient with respect to performance. gA - S x Q output gradient with respect to performance. D - S x S neuron distances. LP - Learning parameters, none, LP = []. LS - Learning state, initially should be = []. and returns, dW - S x R weight (or bias) change matrix. LS - New learning state. learnpn(code) returns useful information for each code string: 'pnames' - Names of learning parameters. 'pdefaults' - Default learning parameters. 'needg' - Returns 1 if this function uses gW or gA. 13-97 learnpn Examples Here we define a random input P and error E to a layer with a 2-element input and 3 neurons. p = rand(2,1); e = rand(3,1); Since learnpn only needs these values to calculate a weight change (see algorithm below), we will use them to do so. dW = learnpn([],p,[],[],[],[],e,[],[],[],[],[]) Network Use You can create a standard network that uses learnpn with newp. To prepare the weights and the bias of layer i of a custom network to learn with learnpn: 1 Set net.trainFcn to 'trainwb'. (net.trainParam will automatically become trainwb's default parameters.) 2 Set net.adaptFcn to 'adaptwb'. (net.adaptParam will automatically become trainwb's default parameters.) 3 Set each net.inputWeights{i,j}.learnFcn to 'learnpn'. Set each net.layerWeights{i,j}.learnFcn to 'learnpn'. Set net.biases{i}.learnFcn to 'learnpn'. (Each weight and bias learning parameter property will automatically become the empty matrix since learnpn has no learning parameters.) To train the network (or enable it to adapt): 1 Set net.trainParam (net.adaptParam) properties to desired values. 2 Call train (adapt). See newp for adaption and training examples. Algorithm learnpn calculates the weight change dW for a given neuron from the neuron's input P and error E according to the normalized perceptron learning rule: pn = p / sqrt(1 + p(1)^2 + p(2)^2) + ... + p(R)^2) dw = 0, if e = 0 = pn', if e = 1 = -pn', if e = -1 13-98 learnpn The expression for dW can be summarized as: dw = e*pn' Limitations Perceptrons do have one real limitation. The set of input vectors must be linearly separable if a solution is to be found. That is, if the input vectors with targets of 1 cannot be separated by a line or hyperplane from the input vectors associated with values of 0, the perceptron will never be able to classify them correctly. See Also learnp, newp, adaptwb, trainwb, adapt, train 13-99 learnsom Purpose Syntax learnsom Self-organizing map weight learning function [dW,LS] = learnsom(W,P,Z,N,A,T,E,gW,gA,D,LP,LS) info = learnsom(code) Description learnsom is the self-organizing map weight learning function. learnsom(W,P,Z,N,A,T,E,gW,gA,D,LP,LS) takes several inputs, W - S x R weight matrix (or S x 1 bias vector). P - R x Q input vectors (or ones(1,Q)). Z - S x Q weighted input vectors. N - S x Q net input vectors. A - S x Q output vectors. T - S x Q layer target vectors. E - S x Q layer error vectors. gW - S x R weight gradient with respect to performance. gA - S x Q output gradient with respect to performance. D - S x S neuron distances. LP - Learning parameters, none, LP = []. LS - Learning state, initially should be = []. and returns, dW - S x R weight (or bias) change matrix. LS - New learning state. Learning occurs according to learnsom's learning parameter, shown here with its default value. LP.order_lr 13-100 0.9 Ordering phase learning rate. LP.order_steps 1000 Ordering phase steps. LP.tune_lr 0.02 Tuning phase learning rate. LP.tune_nd 1 Tuning phase neighborhood distance. learnsom learnpn(code) returns useful information for each code string: 'pnames' - Names of learning parameters. 'pdefaults' - Default learning parameters. 'needg' - Returns 1 if this function uses gW or gA. Examples Here we define a random input P, output A, and weight matrix W, for a layer with a 2-element input and 6 neurons. We also calculate positions and distances for the neurons which are arranged in a 2x3 hexagonal pattern. Then we define the four learning parameters. p = rand(2,1); a = rand(6,1); w = rand(6,2); pos = hextop(2,3); d = linkdist(pos); lp.order_lr = 0.9; lp.order_steps = 1000; lp.tune_lr = 0.02; lp.tune_nd = 1; Since learnsom only needs these values to calculate a weight change (see algorithm below), we will use them to do so. ls = []; [dW,ls] = learnsom(w,p,[],[],a,[],[],[],[],d,lp,ls) Network Use You can create a standard network that uses learnsom with newsom. 1 Set net.trainFcn to 'trainwb1'. (net.trainParam will automatically become trainwb1's default parameters.) 2 Set net.adaptFcn to 'adaptwb'. (net.adaptParam will automatically become trainwb1's default parameters.) 3 Set each net.inputWeights{i,j}.learnFcn to 'learnsom'. Set each net.layerWeights{i,j}.learnFcn to 'learnsom'. Set net.biases{i}.learnFcn to 'learnsom'. (Each weight learning parameter property will automatically be set to learnsom's default parameters.) 13-101 learnsom To train the network (or enable it to adapt): 1 Set net.trainParam (net.adaptParam) properties to desired values. 2 Call train (adapt). Algorithm learnsom calculates the weight change dW for a given neuron from the neuron's input P, activation A2, and learning rate LR: dw = lr*a2*(p'-w) where the activation A2 is found from the layer output A and neuron distances D and the current neighborhood size ND: a2(i,q) = 1, if a(i,q) = 1 = 0.5, if a(j,q) = 1 and D(i,j) <= nd = 0, otherwise The learning rate LR and neighborhood size NS are altered through two phases: an ordering phase and a tuning phase. The ordering phases lasts as many steps as LP.order_steps. During this phase LR is adjusted from LP.order_lr down to LP.tune_lr, and ND is adjusted from the maximum neuron distance down to 1. It is during this phase that neuron weights are expected to order themselves in the input space consistent with the associated neuron positions. During the tuning phase LR decreases slowly from LP.tune_lr and ND is always set to LP.tune_nd. During this phase the weights are expected to spread out relatively evenly over the input space while retaining their topological order found during the ordering phase. See Also 13-102 adaptwb, trainwb, adapt, train learnwh Purpose Syntax Description learnwh Widrow-Hoff weight/bias learning function [dW,LS] = learnwh(W,P,Z,N,A,T,E,gW,gA,D,LP,LS) [db,LS] = learnwh(b,ones(1,Q),Z,N,A,T,E,gW,gA,D,LP,LS) info = learnwh(code) learnwh is the Widrow-Hoff weight/bias learning function, and is also known as the delta or least mean squared (LMS) rule. learnwh(W,P,Z,N,A,T,E,gW,gA,D,LP,LS) takes several inputs, W - S x R weight matrix (or S x 1 bias vector). P - R x Q input vectors (or ones(1,Q)). Z - S x Q weighted input vectors. N - S x Q net input vectors. A - S x Q output vectors. T - S x Q layer target vectors. E - S x Q layer error vectors. gW - S x R weight gradient with respect to performance. gA - S x Q output gradient with respect to performance. D - S x S neuron distances. LP - Learning parameters, none, LP = []. LS - Learning state, initially should be = []. and returns, dW - S x R weight (or bias) change matrix. LS - New learning state. Learning occurs according to learnwh's learning parameter shown here with its default value. LP.lr - 0.01 - Learning rate. learnwh(code) returns useful information for each code string: 'pnames' - Names of learning parameters. 'pdefaults' - Default learning parameters. 'needg' - Returns 1 if this function uses gW or gA. 13-103 learnwh Examples Here we define a random input P and error E to a layer with a 2-element input and 3 neurons. We also define the learning rate LR learning parameter. p = rand(2,1); e = rand(3,1); lp.lr = 0.5; Since learnwh only needs these values to calculate a weight change (see algorithm below), we will use them to do so. dW = learnwh([],p,[],[],[],[],e,[],[],[],lp,[]) Network Use You can create a standard network that uses learnwh with newlin. To prepare the weights and the bias of layer i of a custom network to learn with learnwh: 1 Set net.trainFcn to 'trainwb'. net.trainParam will automatically become trainwb's default parameters. 2 Set net.adaptFcn to 'adaptwb'. net.adaptParam will automatically become trainwb's default parameters. 3 Set each net.inputWeights{i,j}.learnFcn to 'learnwh'. Set each net.layerWeights{i,j}.learnFcn to 'learnwh'. Set net.biases{i}.learnFcn to 'learnwh'. Each weight and bias learning parameter property will automatically be set to learnwh's default parameters. To train the network (or enable it to adapt): 1 Set net.trainParam (net.adaptParam) properties to desired values. 2 Call train(adapt). See newlin for adaption and training examples. Algorithm learnwh calculates the weight change dW for a given neuron from the neuron's input P and error E, and the weight (or bias) learning rate LR, according to the Widrow-Hoff learning rule: dw = lr*e*pn' 13-104 learnwh See Also newlin, adaptwb, trainwb, adapt, train References Widrow, B., and M. E. Hoff, “Adaptive switching circuits,” 1960 IRE WESCON Convention Record, New York IRE, pp. 96-104, 1960. Widrow B. and S. D. Sterns, Adaptive Signal Processing, New York: Prentice-Hall, 1985. 13-105 linkdist Purpose linkdist Link distance function Syntax d = linkdist(pos) Description linkdist is a layer distance function used to find the distances between the layer's neurons given their positions. linkdist(pos) takes one argument, pos - N x S matrix of neuron positions. and returns the S x S matrix of distances. Examples Here we define a random matrix of positions for 10 neurons arranged in three dimensional space and find their distances. pos = rand(3,10); D = linkdist(pos) Network Use You can create a standard network that uses linkdist as a distance function by calling newsom. To change a network so that a layer's topology uses linkdist, set net.layers{i}.distanceFcn to 'linkdist’. In either case, call sim to simulate the network with dist. See newsom for training and adaption examples. Algorithm The link distance D between two position vectors Pi and Pj from a set of S vectors is: Dij = 0, if i==j = 1, if (sum((Pi-Pj).^2)).^0.5 is <= 1 = 2, if k exists, Dik = Dkj = 1 = 3, if k1, k2 exist, Dik1 = Dk1k2 = Dk2j = 1. = N, if k1..kN exist, Dik1 = Dk1k2 = ...= DkNj = 1 = S, if none of the above conditions apply. See Also 13-106 sim, dist, mandist logsig Purpose Graph and Symbol logsig Log sigmoid transfer function a +1 n 0 -1 AA AA a = logsig(n) Log-Sigmoid Transfer Function Syntax A = logsig(N) info = logsig(code) Description logsig is a transfer function. Transfer functions calculate a layer's output from its net input. logsig(N) takes one input, N - S x Q matrix of net input (column) vectors. and returns each element of N squashed between 0 and 1. logsig(code) returns useful information for each code string: 'deriv' - Name of derivative function. 'name' - Full name. 'output' - Output range. 'active' - Active input range. Examples Here is the code to create a plot of the logsig transfer function. n = -5:0.1:5; a = logsig(n); plot(n,a) Network Use You can create a standard network that uses logsig by calling newff or newcf. 13-107 logsig To change a network so a layer uses logsig set net.layers{i}.transferFcn to 'logsig'. In either case, call sim to simulate the network with purelin. See newff or newcf for simulation examples. Algorithm logsig(n) = 1 / (1 + exp(-n)) See Also sim, dlogsig, tansig 13-108 mae Purpose Syntax mae Mean absolute error performance function perf = mae(e,x,pp) perf = mae(e,net,pp) info = mae(code) Description mae is a network performance function. mae(E,X,PP) takes from one to three arguments, E - Matrix or cell array of error vector(s). X - Vector of all weight and bias values (ignored). PP - Performance parameters (ignored). and returns the mean absolute error. The errors E can be given in cell array form, E - Nt x TS cell array, each element E{i,ts} is a Vi x Q matrix or[]. or as a matrix, E - (sum of Vi) x Q matrix where Nt = net.numTargets TS = Number of time steps Q = Batch size Vi = net.targets{i}.size mae(E,net,PP) can take an alternate argument to X, net - Neural network from which X can be obtained (ignored). mae(code) returns useful information for each code string: 'deriv' - Name of derivative function. 'name' - Full name. 'pnames' - Names of training parameters. 'pdefaults' - Default training parameters. 13-109 mae Examples Here a perceptron is created with a 1-element input ranging from -10 to 10, and one neuron. net = newp([-10 10],1); Here the network is given a batch of inputs P. The error is calculated by subtracting the output A from target T. Then the mean absolute error is calculated. p = [-10 -5 0 5 10]; t = [0 0 1 1 1]; y = sim(net,p) e = t-y perf = mae(e) Note that mae can be called with only one argument because the other arguments are ignored. mae supports those arguments to conform to the standard performance function argument list. Network Use You can create a standard network that uses mae with newp. To prepare a custom network to be trained with mae, set net.performFcn to 'mae'. This will automatically set net.performParam to the empty matrix [], as mae has no performance parameters. In either case, calling train or adapt will result in mae being used to calculate performance. See newp for examples. See Also 13-110 mse, msereg, dmae mandist Purpose Syntax mandist Manhattan distance weight function Z = mandist(W,P) df = mandist('deriv') D = mandist(pos); Description mandist is the Manhattan distance weight function. Weight functions apply weights to an input to get weighted inputs. mandist(W,P) takes these inputs, W - S x R weight matrix. P - R x Q matrix of Q input (column) vectors. and returns the S x Q matrix of vector distances. mandist('deriv') returns '' because mandist does not have a derivative function. mandist is also a layer distance function which can be used to find the distances between neurons in a layer. mandist(pos) takes one argument, pos - An S row matrix of neuron positions. and returns the S x S matrix of distances. Examples Here we define a random weight matrix W and input vector P and calculate the corresponding weighted input Z. W = rand(4,3); P = rand(3,1); Z = mandist(W,P) Here we define a random matrix of positions for 10 neurons arranged in three dimensional space and then find their distances. pos = rand(3,10); D = mandist(pos) Network Use You can create a standard network that uses mandist as a distance function by calling newsom. 13-111 mandist To change a network so an input weight uses mandist, set net.inputWeight{i,j}.weightFcn to 'mandist’. For a layer weight set net.inputWeight{i,j}.weightFcn to 'mandist'. To change a network so a layer's topology uses mandist, set net.layers{i}.distanceFcn to 'mandist'. In either case, call sim to simulate the network with dist. See newpnn or newgrnn for simulation examples. Algorithm The Manhattan distance D between two vectors X and Y is: D = sum(abs(x-y)) See Also 13-112 sim, dist, linkdist mat2cell Purpose mat2cell Break matrix up into a cell array of matrices Syntax mat2cell(M,R,C); Description mat2cell(M,R,C) takes three arguments, M - row x col matrix. R - Vector of row sizes (should sum to row). C - Vector of col sizes (should sum to col). and returns a cell array of matrices, found using R and C. Examples See Also M = [1 2 3 4; 5 6 7 8; 9 10 11 12]; C = mat2cell(M,[1 2],[2 2]) cell2mat 13-113 maxlinlr Purpose maxlinlr Maximum learning rate for a linear layer Syntax lr = maxlinlr(P) lr = maxlinlr(P,'bias') Description maxlinlr is used to calculate learning rates for newlin. maxlinlr(P) takes one argument, P - R x Q matrix of input vectors. and returns the maximum learning rate for a linear layer without a bias that is to be trained only on the vectors in P. maxlinlr(P,'bias') returns the maximum learning rate for a linear layer with a bias. Examples Here we define a batch of 4 2-element input vectors and find the maximum learning rate for a linear layer with a bias. P = [1 2 -4 7; 0.1 3 10 6]; lr = maxlinlr(P,'bias') See Also 13-114 linnet, newlin, newlind midpoint Purpose midpoint Midpoint weight initialization function Syntax W = midpoint(S,PR) Description midpoint is a weight initialization function that sets weight (row) vectors to the center of the input ranges. midpoint(S,PR) takes two arguments, S - Number of rows (neurons). PR - R x 2 matrix of input value ranges = [Pmin Pmax]. and returns an S x R matrix with rows set to (Pmin+Pmax)'/2. Examples Here initial weight values are calculated for a 5 neuron layer with input elements ranging over [0 1] and [-2 2]. W = midpoint(5,[0 1; -2 2]) Network Use You can create a standard network that uses midpoint to initialize weights by calling newc. To prepare the weights and the bias of layer i of a custom network to initialize with midpoint: 1 Set net.initFcn to 'initlay'. (net.initParam will automatically become initlay's default parameters.) 2 Set net.layers{i}.initFcn to 'initwb'. 3 Set each net.inputWeights{i,j}.initFcn to 'midpoint'. Set each net.layerWeights{i,j}.initFcn to 'midpoint'; To initialize the network call init. See Also initwb, initlay, init 13-115 minmax Purpose minmax Ranges of matrix rows Syntax pr = minmax(P) Description minmax(P) takes one argument, PR - R x Q matrix. and returns the R x 2 matrix PR of minimum and maximum values for each row of M. Examples 13-116 p = [0 1 2; -1 -2 -0.5] pr = minmax(p) mse Purpose Syntax mse Mean squared error performance function perf = mse(e,x,pp) perf = mse(e,net,pp) info = mse(code) Description mse is a network performance function. It measures the network’s performance according to the mean of squared errors. mse(E,X,PP) takes from one to three arguments, E - Matrix or cell array of error vector(s). X - Vector of all weight and bias values (ignored). PP - Performance parameters (ignored). and returns the mean squared error. mse(E,net,PP) can take an alternate argument to X, net - Neural network from which X can be obtained (ignored). mse(code) returns useful information for each code string: 'deriv' - Name of derivative function. 'name' - Full name. 'pnames' - Names of training parameters. 'pdefaults' - Default training parameters. Examples Here a two layer feed-forward network is created with a 1-element input ranging from -10 to 10, four hidden tansig neurons, and one purelin output neuron. net = newff([-10 10],[4 1],{'tansig','purelin'}); 13-117 mse Here the network is given a batch of inputs P. The error is calculated by subtracting the output A from target T. Then the mean squared error is calculated. p = [-10 -5 0 5 10]; t = [0 0 1 1 1]; y = sim(net,p) e = t-y perf = mse(e) Note that mse can be called with only one argument because the other arguments are ignored. mse supports those ignored arguments to conform to the standard performance function argument list. Network Use You can create a standard network that uses mse with newff, newcf, or newelm. To prepare a custom network to be trained with mse, set net.performFcn to 'mse'. This will automatically set net.performParam to the empty matrix [], as mse has no performance parameters. In either case, calling train or adapt will result in mse being used to calculate performance. See newff or newcf for examples. See Also 13-118 msereg, mae, dmse msereg Purpose Syntax msereg Mean squared error w/reg performance function perf = mse(e,x,pp) perf = mse(e,net,pp) info = mse(code) Description msereg is a network performance function. It measures network performance as the weight sum of two factors: the mean squared error and the mean squared weight and bias values. msereg(E,X,PP) takes from three arguments, E - Matrix or cell array of error vector(s). X - Vector of all weight and bias values. PP - Performance parameter. where PP defines one performance parameters, PP.ratio - Relative importance of errors vs. weight and bias values. and returns the sum of mean squared errors (times PP.ratio) with the mean squared weight and bias values (times 1-PP.ratio). The errors E can be given in cell array form, E - Nt x TS cell array, each element E{i,ts} is an Vi x Q matrix or []. or as a matrix, E - (sum of Vi) x Q matrix where Nt = net.numTargets TS = Number of time steps Q = Batch size Vi = net.targets{i}.size mse(E,net) takes an alternate argument to X and PP, net - Neural network from which X and PP can be obtained. 13-119 msereg mse(code) returns useful information for each code string: 'deriv' - Name of derivative function. 'name' - Full name. 'pnames' - Names of training parameters. 'pdefaults' - Default training parameters. Examples Here a two layer feed-forward is created with a 1-element input ranging from -2 to 2, four hidden tansig neurons, and one purelin output neuron. net = newff([-2 2],[4 1] {'tansig','purelin'},'trainlm','learngdm','msereg'); Here the network is given a batch of inputs P. The error is calculated by subtracting the output A from target T. Then the mean squared error is calculated using a ratio of 20/(20+1). (Errors are 20 times as important as weight and bias values). p = [-2 -1 0 1 2]; t = [0 1 1 1 0]; y = sim(net,p) e = t-y net.performParam.ratio = 20/(20+1); perf = msereg(e,net) Network Use You can create a standard network that uses msereg with newff, newcf, or newelm. To prepare a custom network to be trained with msereg, set net.performFcn to 'msereg'. This will automatically set net.performParam to msereg's default performance parameters. In either case, calling train or adapt will result in msereg being used to calculate performance. See newff or newcf for examples. See Also 13-120 mse, mae, dmsereg negdist Purpose Syntax negdist Negative distance weight function Z = negdist(W,P) df = negdist('deriv') Description negdist is a weight function. Weight functions apply weights to an input to get weighted inputs. negdist(W,P) takes these inputs, W - S x R weight matrix. P - R x Q matrix of Q input (column) vectors. and returns the S x Q matrix of negative vector distances. negdist('deriv') returns '' because negdist does not have a derivative function. Examples Here we define a random weight matrix W and input vector P and calculate the corresponding weighted input Z. W = rand(4,3); P = rand(3,1); Z = negdist(W,P) Network Use You can create a standard network that uses negdist by calling newc or newsom. To change a network so an input weight uses negdist, set net.inputWeight{i,j}.weightFcn to 'negdist’. For a layer weight set net.inputWeight{i,j}.weightFcn to 'negdist’. In either case, call sim to simulate the network with negdist. See newc or newsom for simulation examples. Algorithm negdist returns the negative Euclidean distance: z = -sqrt(sum(w-p)^2) See Also sim, dotprod, dist 13-121 netprod Purpose Syntax netprod Product net input function N = netprod(Z1,Z2,...) df = netprod('deriv') Description netprod is a net input function. Net input functions calculate a layer's net input by combining its weighted inputs and biases. netprod(Z1,Z2,...,Zn)takes, Zi - S x Q matrices. and returns an element-wise sum of Zi's. netprod('deriv') returns netprod's derivative function. Examples Here netprod combines two sets of weighted input vectors (which we have defined ourselves). z1 = [1 2 4;3 4 1]; z2 = [-1 2 2; -5 -6 1]; n = netprod(z1,z2) Here netprod combines the same weighted inputs with a bias vector. Because Z1 and Z2 each contain three concurrent vectors, three concurrent copies of B must be created with concur so that all sizes match up. b = [0; -1]; n = netprod(z1,z2,concur(b,3)) Network Use You can create a standard network that uses netprod by calling newpnn or newgrnn. To change a network so that a layer uses netprod, set net.layers{i}.netInputFcn to 'netprod'. In either case, call sim to simulate the network with netprod. See newpnn or newgrnn for simulation examples. See Also 13-122 sim, dnetprod, netsum, concur. netsum Purpose Syntax netsum Sum net input function N = netsum(Z1,Z2,...) df = netsum('deriv') Description netsum is a net input function. Net input functions calculate a layer's net input by combining its weighted inputs and biases. netsum(Z1,Z2,...,Zn) takes any number of inputs, Zi - S x Q matrices, and returns N, the element-wise sum of Zi's. netsum('deriv') returns netsum's derivative function. Examples Here netsum combines two sets of weighted input vectors (which we have defined ourselves). z1 = [1 2 4;3 4 1]; z2 = [-1 2 2; -5 -6 1]; n = netsum(z1,z2) Here netsum combines the same weighted inputs with a bias vector. Because Z1 and Z2 each contain three concurrent vectors, three concurrent copies of B must be created with concur so that all sizes match up. b = [0; -1]; n = netsum(z1,z2,concur(b,3)) Network Use You can create a standard network that uses netsum by calling newp or newlin. To change a network so a layer uses netsum, set net.layers{i}.netInputFcn to 'netsum'. In either case, call sim to simulate the network with netsum. See newp or newlin for simulation examples. See Also sim, dnetprod, netprod, concur 13-123 network Purpose Syntax network Create a custom neural network net = network net = network(numInputs,numLayers,biasConnect,inputConnect, layerConnect,outputConnect,targetConnect) To Get Help Type help network/network Description network creates new custom networks. It is used to create networks that are then customized by functions such as newp, newlin, newff, etc. network takes these optional arguments (shown with default values): numInputs - Number of inputs, 0. numLayers - Number of layers, 0. biasConnect - numLayers-by-1 Boolean vector, zeros. inputConnect - numLayers-by-numInputs Boolean matrix, zeros. layerConnect - numLayers-by-numLayers Boolean matrix, zeros. outputConnect - 1-by-numLayers Boolean vector, zeros. targetConnect - 1-by-numLayers Boolean vector, zeros. and returns, net - New network with the given property values. 13-124 network Properties Architecture properties: net.numInputs: 0 or a positive integer. Number of inputs. net.numLayers: 0 or a positive integer. Number of layers. net.biasConnect: numLayer-by-1 Boolean vector. If net.biasConnect(i) is 1 then the layer i has a bias and net.biases{i} is a structure describing that bias. net.inputConnect: numLayer-by-numInputs Boolean vector. If net.inputConnect(i,j) is 1 then layer i has a weight coming from input j and net.inputWeights{i,j} is a structure describing that weight. net.layerConnect: numLayer-by-numLayers Boolean vector. If net.layerConnect(i,j) is 1 then layer i has a weight coming from layer j and net.layerWeights{i,j} is a structure describing that weight. net.outputConnect: 1-by-numLayers Boolean vector. If net.outputConnect(i) is 1 then the network has an output from layer i and net.outputs{i} is a structure describing that output. net.targetConnect: 1-by-numLayers Boolean vector. If net.outputConnect(i) is 1 then the network has a target from layer i and net.targets{i} is a structure describing that target. net.numOutputs: 0 or a positive integer. Read only. Number of network outputs according to net.outputConnect. net.numTargets: 0 or a positive integer. Read only. Number of targets according to net.targetConnect. net.numInputDelays: 0 or a positive integer. Read only. Maximum input delay according to all net.inputWeight{i,j}.delays. net.numLayerDelays: 0 or a positive number. Read only. Maximum layer delay according to all net.layerWeight{i,j}.delays. 13-125 network Subobject structure properties: net.inputs: numInputs-by-1 cell array. net.inputs{i} is a structure defining input i: net.layers: numLayers-by-1 cell array. net.layers{i} is a structure defining layer i: net.biases: numLayers-by-1 cell array. If net.biasConnect(i) is 1, then net.biases{i} is a structure defining the bias for layer i. net.inputWeights: numLayers-by-numInputs cell array. If net.inputConnect(i,j) is 1, then net.inputWeights{i,j} is a structure defining the weight to layer i from input j. net.layerWeights: numLayers-by-numLayers cell array. If net.layerConnect(i,j) is 1, then net.layerWeights{i,j} is a structure defining the weight to layer i from layer j. net.outputs: 1-by-numLayers cell array. If net.outputConnect(i) is 1, then net.outputs{i} is a structure defining the network output from layer i. net.targets: 1-by-numLayers cell array. If net.targetConnect(i) is 1, then net.targets{i} is a structure defining the network target to layer i. Function properties: net.adaptFcn: name of a network adaption function or ''. net.initFcn: name of a network initialization function or ''. net.performFcn: name of a network performance function or ''. net.trainFcn: name of a network training function or ''. Parameter properties: net.adaptParam: network adaption parameters. net.initParam: network initialization parameters. net.performParam: network performance parameters. net.trainParam: network training parameters. 13-126 network Weight and bias value properties: net.IW: numLayers-by-numInputs cell array of input weight values. net.LW: numLayers-by-numLayers cell array of layer weight values. net.b: numLayers-by-1 cell array of bias values. Other properties: net.userdata: structure you can use to store useful values. Examples Here is the code to create a network without any inputs and layers, and then set its number of inputs and layer to 1 and 2 respectively. net = network net.numInputs = 1 net.numLayers = 2 Here is the code to create the same network with one line of code. net = network(1,2) Here is the code to create a 1 input, 2 layer, feed-forward network. Only the first layer will have a bias. An input weight will connect to layer 1 from input 1. A layer weight will connect to layer 2 from layer 1. Layer 2 will be a network output, and have a target. net = network(1,2,[1;0],[1; 0],[0 0; 1 0],[0 1],[0 1]) We can then see the properties of subobjects as follows: net.inputs{1} net.layers{1}, net.layers{2} net.biases{1} net.inputWeights{1,1}, net.layerWeights{2,1} net.outputs{2} net.targets{2} We can get the weight matrices and bias vector as follows: net.iw.{1,1}, net.iw{2,1}, net.b{1} 13-127 network We can alter the properties of any of these subobjects. Here we change the transfer functions of both layers: net.layers{1}.transferFcn = 'tansig'; net.layers{2}.transferFcn = 'logsig'; Here we change the number of elements in input 1 to 2, by setting each element’s range: net.inputs{1}.range = [0 1; -1 1]; Next we can simulate the network for a 2-element input vector: p = [0.5; -0.1]; y = sim(net,p) See Also 13-128 sim newc Purpose newc Create a competitive layer Syntax net = newc(PR,S,KLR,CLR) Description Competitive layers are used to solve classification problems. net = newc(PR,S,KLR,CLR) takes these inputs, PR - R x 2 matrix of min and max values for R input elements. S - Number of neurons. KLR - Kohonen learning rate, default = 0.01. CLR - Conscience learning rate, default = 0.001. and returns a new competitive layer. Properties Competitive layers consist of a single layer with the negdist weight function, netsum net input function, and the compet transfer function. The layer has a weight from the input, and a bias. Weights and biases are initialized with midpoint and initcon. Adaption and training are done with adaptwb and trainwb1, which both update weight and bias values with the learnk and learncon learning functions. Examples Here is a set of four two-element vectors P. P = [.1 .8 .1 .9; .2 .9 .1 .8]; To competitive layer can be used to divide these inputs into two classes. First a two neuron layer is created with two input elements ranging from 0 to 1, then it is trained. net = newc([0 1; 0 1],2); net = train(net,P); The resulting network can then be simulated and its output vectors converted to class indices. Y = sim(net,P) Yc = vec2ind(Y) See Also sim, init, adapt, train, adaptwb, trainwb1 13-129 newcf Purpose newcf Trainable cascade-forward backpropagation network Syntax net = newcf(Pr,[S1 S2...SNl],{TF1 TF2...TFNl},BTF,BLF,PF) Description newcf(PR,[S1 S2...SNl],{TF1 TF2...TFNl},BTF,BLF,PF) takes, PR - R x 2 matrix of min and max values for R input elements. Si - Size of ith layer, for Nl layers. TFi - Transfer function of ith layer, default = 'tansig'. BTF - Backprop network training function, default = 'traingd'. BLF - Backprop weight/bias learning function, default = 'learngdm'. PF - Performance function, default = 'mse'. and returns an N layer cascade-forward backprop network. The transfer functions TFi can be any differentiable transfer function such as tansig, logsig, or purelin. The training function BTF can be any of the backprop training functions such as trainlm, trainbfg, trainrp, traingd, etc. WARNING: trainlm is the default training function because it is very fast, but it requires a lot of memory to run. If you get an out-of-memory error when training try doing one of these: 1 Slow trainlm training, but reduce memory requirements by setting net.trainParam.mem_reduc to 2 or more. (See help trainlm.) 2 Use trainbfg, which is slower but more memory-efficient than trainlm. 3 Use trainrp which is slower but more memory-efficient than trainbfg. The learning function BLF can be either of the backpropagation learning functions such as learngd, or learngdm. The performance function can be any of the differentiable performance functions such as mse or msereg. Examples Here is a problem consisting of inputs P and targets T that we would like to solve with a network. P = [0 1 2 3 4 5 6 7 8 9 10]; T = [0 1 2 3 4 3 2 1 2 3 4]; 13-130 newcf Here a two-layer cascade-forward network is created. The network's input ranges from [0 to 10]. The first layer has five tansig neurons, the second layer has one purelin neuron. The trainlm network training function is to be used. net = newcf([0 10],[5 1],{'tansig' 'purelin'}); Here the network is simulated and its output plotted against the targets. Y = sim(net,P); plot(P,T,P,Y,'o') Here the network is trained for 50 epochs. Again the network's output is plotted. net.trainParam.epochs = 50; net = train(net,P,T); Y = sim(net,P); plot(P,T,P,Y,'o') Algorithm Cascade-forward networks consist of Nl layers using the dotprod weight function, netsum net input function, and the specified transfer functions. The first layer has weights coming from the input. Each subsequent layer has weights coming from the input and all previous layers. All layers have biases. The last layer is the network output. Each layer's weights and biases are initialized with initnw. Adaption is done with adaptwb which updates weights with the specified learning function. Training is done with the specified training function. Performance is measured according to the specified performance function. See Also newff, newelm, sim, init, adapt, train 13-131 newelm Purpose newelm Create an Elman backpropagation network Syntax net = newelm(PR,[S1 S2...SNl],{TF1 TF2...TFNl},BTF,BLF,PF) Description newelm(PR,[S1 S2...SNl],{TF1 TF2...TFNl},BTF,BLF,PF) takes several arguments, PR - R x 2 matrix of min and max values for R input elements. Si - Size of ith layer, for Nl layers. TFi - Transfer function of ith layer, default = 'tansig'. BTF - Backprop network training function, default = 'traingdx'. BLF - Backprop weight/bias learning function, default = 'learngdm'. PF - Performance function, default = 'mse'. and returns an Elman network. The training function BTF can be any of the backprop training functions such as trainlm, trainbfg, trainrp, traingd, etc. WARNING: trainlm is the default training function because it is very fast, but it requires a lot of memory to run. If you get an "out-of-memory" error when training try doing one of these: 1 Slow trainlm training, but reduce memory requirements by setting net.trainParam.mem_reduc to 2 or more. (See help trainlm.) 2 Use trainbfg, which is slower but more memory-efficient than trainlm. 3 Use trainrp which is slower but more memory-efficient than trainbfg. The learning function BLF can be either of the backpropagation learning functions such as learngd, or learngdm. The performance function can be any of the differentiable performance functions such as mse or msereg. Examples Here is a series of Boolean inputs P, and another sequence T, which is 1 wherever P has had two 1's in a row. P = round(rand(1,20)); T = [0 (P(1:end-1)+P(2:end) == 2)]; 13-132 newelm We would like the network to recognize whenever two 1's occur in a row. First we arrange these values as sequences. Pseq = con2seq(P); Tseq = con2seq(T); Next we create an Elman network whose input varies from 0 to 1, and has five hidden neurons and 1 output. net = newelm([0 1],[10 1],{'tansig','logsig'}); Then we train the network with a mean squared error goal of 0.1, and simulate it. net = train(net,Pseq,Tseq); Y = sim(net,Pseq) Algorithm Elman networks consist of Nl layers using the dotprod weight function, netsum net input function, and the specified transfer functions. The first layer has weights coming from the input. Each subsequent layer has a weight coming from the previous layer. All layers except the last have a recurrent weight. All layers have biases. The last layer is the network output. Each layer's weights and biases are initialized with initnw. Adaption is done with adaptwb which updates weights with the specified learning function. Training is done with the specified training function. Performance is measured according to the specified performance function. See Also newff, newcf, sim, init, adapt, train 13-133 newff Purpose newff Create a feed-forward backpropagation network Syntax net = newff(PR,[S1 S2...SNl],{TF1 TF2...TFNl},BTF,BLF,PF) Description newff(PR,[S1 S2...SNl],{TF1 TF2...TFNl},BTF,BLF,PF) takes, PR - R x 2 matrix of min and max values for R input elements. Si - Size of ith layer, for Nl layers. TFi - Transfer function of ith layer, default = 'tansig'. BTF - Backprop network training function, default = 'traingdx'. BLF - Backprop weight/bias learning function, default = 'learngdm'. PF - Performance function, default = 'mse'. and returns an N layer feed-forward backprop network. The transfer functions TFi can be any differentiable transfer function such as tansig, logsig, or purelin. The training function BTF can be any of the backprop training functions such as trainlm, trainbfg, trainrp, traingd, etc. WARNING: trainlm is the default training function because it is very fast, but it requires a lot of memory to run. If you get an "out-of-memory" error when training try doing one of these: 1 Slow trainlm training, but reduce memory requirements by setting net.trainParam.mem_reduc to 2 or more. (See help trainlm.) 2 Use trainbfg, which is slower but more memory-efficient than trainlm. 3 Use trainrp which is slower but more memory-efficient than trainbfg. The learning function BLF can be either of the backpropagation learning functions such as learngd, or learngdm. The performance function can be any of the differentiable performance functions such as mse or msereg. Examples Here is a problem consisting of inputs P and targets T that we would like to solve with a network. P = [0 1 2 3 4 5 6 7 8 9 10]; T = [0 1 2 3 4 3 2 1 2 3 4]; 13-134 newff Here a two-layer feed-forward network is created. The network's input ranges from [0 to 10]. The first layer has five tansig neurons, the second layer has one purelin neuron. The trainlm network training function is to be used. net = newff([0 10],[5 1],{'tansig' 'purelin'}); Here the network is simulated and its output plotted against the targets. Y = sim(net,P); plot(P,T,P,Y,'o') Here the network is trained for 50 epochs. Again the network's output is plotted. net.trainParam.epochs = 50; net = train(net,P,T); Y = sim(net,P); plot(P,T,P,Y,'o') Algorithm Feed-forward networks consist of Nl layers using the dotprod weight function, netsum net input function, and the specified transfer functions. The first layer has weights coming from the input. Each subsequent layer has a weight coming from the previous layer. All layers have biases. The last layer is the network output. Each layer's weights and biases are initialized with initnw. Adaption is done with adaptwb which updates weights with the specified learning function. Training is done with the specified training function. Performance is measured according to the specified performance function. See Also newcf, newelm, sim, init, adapt, train 13-135 newfftd Purpose newfftd Create a feed-forward input-delay backprop network Syntax net = newfftd(PR,ID,[S1 S2...SNl],{TF1 TF2...TFNl},BTF,BLF,PF) Description newfftd(PR,ID,[S1 S2...SNl],{TF1 TF2...TFNl},BTF,BLF,PF) takes, PR - R x 2 matrix of min and max values for R input elements. ID - Input delay vector. Si - Size of ith layer, for Nl layers. TFi - Transfer function of ith layer, default = 'tansig'. BTF - Backprop network training function, default = 'traingdx'. BLF - Backprop weight/bias learning function, default = 'learngdm'. PF - Performance function, default = 'mse'. and returns an N layer feed-forward backprop network. The transfer functions TFi can be any differentiable transfer function such as tansig, logsig, or purelin. The training function BTF can be any of the backprop training functions such as trainlm, trainbfg, trainrp, traingd, etc. WARNING: trainlm is the default training function because it is very fast, but it requires a lot of memory to run. If you get an "out-of-memory" error when training try doing one of these: 1 Slow trainlm training, but reduce memory requirements by setting net.trainParam.mem_reduc to 2 or more. (See help trainlm.) 2 Use trainbfg, which is slower but more memory-efficient than trainlm. 3 Use trainrp which is slower but more memory-efficient than trainbfg. The learning function BLF can be either of the backpropagation learning functions such as learngd, or learngdm. The performance function can be any of the differentiable performance functions such as mse or msereg. 13-136 newfftd Examples Here is a problem consisting of an input sequence P and target sequence T that can be solved by a network with one delay. P = {1 0 0 1 1 0 1 0 0 0 0 1 1 0 0 1}; T = {1 -1 0 1 0 -1 1 -1 0 0 0 1 0 -1 0 1}; Here a two-layer feed-forward network is created with input delays of 0 and 1. The network's input ranges from [0 to 1]. The first layer has five tansig neurons, the second layer has one purelin neuron. The trainlm network training function is to be used. net = newfftd([0 1],[0 1],[5 1],{'tansig' 'purelin'}); Here the network is simulated. Y = sim(net,P) Here the network is trained for 50 epochs. Again the network's output is calculated. net.trainParam.epochs = 50; net = train(net,P,T); Y = sim(net,P) Algorithm Feed-forward networks consist of Nl layers using the dotprod weight function, netsum net input function, and the specified transfer functions. The first layer has weights coming from the input with the specified input delays. Each subsequent layer has a weight coming from the previous layer. All layers have biases. The last layer is the network output. Each layer's weights and biases are initialized with initnw. Adaption is done with adaptwb which updates weights with the specified learning function. Training is done with the specified training function. Performance is measured according to the specified performance function. See Also newcf, newelm, sim, init, adapt, train 13-137 newgrnn Purpose newgrnn Design a generalized regression neural network Syntax net = newgrnn(P,T,spread) Description Generalized regression neural networks are a kind of radial basis network that is often used for function approximation. grnn’s can be designed very quickly. newgrnn(P,T,spread) takes three inputs, P - R x Q matrix of Q input vectors. T - S x Q matrix of Q target class vectors. spread - Spread of radial basis functions, default = 1.0. and returns a new generalized regression neural network. The larger the spread, is the smoother the function approximation will be. To fit data very closely, use a spread smaller than the typical distance between input vectors. To fit the data more smoothly, use a larger spread. Properties newgrnn creates a two layer network. The first layer has radbas neurons, calculates weighted inputs with dist and net input with netprod. The second layer has purelin neurons, calculates weighted input with normprod and net inputs with netsum. Only the first layer has biases. newgrnn sets the first layer weights to P', and the first layer biases are all set to 0.8326/spread, resulting in radial basis functions that cross 0.5 at weighted inputs of +/- spread. The second layer weights W2 are set to T. Examples Here we design a radial basis network given inputs P and targets T. P = [1 2 3]; T = [2.0 4.1 5.9]; net = newgrnn(P,T); Here the network is simulated for a new input. P = 1.5; Y = sim(net,P) See Also sim, newrb, newrbe, newpnn References Wasserman, P.D., Advanced Methods in Neural Computing, New York: Van Nostrand Reinhold, pp. 155-61, 1993. 13-138 newhop Purpose newhop Create a Hopfield recurrent network Syntax net = newhop(T) Description Hopfield networks are used for pattern recall. newhop(T) takes one input argument, T - R x Q matrix of Q target vectors. (Values must be +1 or -1.) and returns a new Hopfield recurrent neural network with stable points at the vectors in T. Properties Hopfield networks consist of a single layer with the dotprod weight function, netsum net input function, and the satlins transfer function. The layer has a recurrent weight from itself and a bias. Examples Here we create a Hopfield network with two three-element stable points T. T = [-1 -1 1; 1 -1 1]'; net = newhop(T); Below we check that the network is stable at these points by using them as initial layer delay conditions. If the network is stable we would expect that the outputs Y will be the same. (Since Hopfield networks have no inputs, the second argument to sim is Q = 2 when using matrix notation). Ai = T; [Y,Pf,Af] = sim(net,2,[],Ai); Y To see if the network can correct a corrupted vector, run the following code which simulates the Hopfield network for five timesteps. (Since Hopfield networks have no inputs, the second argument to sim is {Q TS} = [1 5] when using cell array notation.) Ai = {[-0.9; -0.8; 0.7]}; [Y,Pf,Af] = sim(net,{1 5},{},Ai); Y{1} If you run the above code, Y{1} will equal T(:,1) if the network has managed to convert the corrupted vector Ai to the nearest target vector. 13-139 newhop Algorithm Hopfield networks are designed to have stable layer outputs as defined by user supplied targets. The algorithm minimizes the number of unwanted stable points. See Also sim, satlins References Li, J., A. N. Michel, and W. Porod, "Analysis and synthesis of a class of neural networks: linear systems operating on a closed hypercube," IEEE Transactions on Circuits and Systems, vol. 36, no. 11, pp. 1405-1422, November 1989. 13-140 newlin Purpose Syntax newlin Create a linear layer net = newlin(PR,S,ID,LR) new = newlin Description Linear layers are often used as adaptive filters for signal processing and prediction. newlin(PR,S,ID,LR) takes these arguments, PR - R x 2 matrix of min and max values for R input elements. S - Number of elements in the output vector. ID - Input delay vector, default = [0]. LR - Learning rate, default = 0.01. and returns a new linear layer. net = newlin(PR,S,0,P) takes an alternate argument, P - Matrix of input vectors. and returns a linear layer with the maximum stable learning rate for learning with inputs P. Call newlin without input arguments to define the network's attributes in a dialog window. Examples This code creates a single input (range of [-1 1] linear layer with one neuron, input delays of 0 and 1, and a learning rate of 0.01. It is simulated for an input sequence P1. net = newlin([-1 1],1,[0 1],0.01); P1 = {0 -1 1 1 0 -1 1 0 0 1}; Y = sim(net,P1) Here targets T1 are defined and the layer adapts to them. (Since this is the first call to adapt, the default input delay conditions are used.) T1 = {0 -1 0 2 1 -1 0 1 0 1}; [net,Y,E,Pf] = adapt(net,P1,T1); Y 13-141 newlin Here the linear layer continues to adapt for a new sequence using the previous final conditions PF as initial conditions. P2 = {1 0 -1 -1 1 1 1 0 -1}; T2 = {2 1 -1 -2 0 2 2 1 0}; [net,Y,E,Pf] = adapt(net,P2,T2); Y Here we initialize the layer's weights and biases to new values. net = init(net); Here we train the newly initialized layer on the entire sequence for 200 epochs to an error goal of 0.1. P3 = [P1 P2]; T3 = [T1 T2]; net.trainParam.epochs = 200; net.trainParam.goal = 0.1; net = train(net,P3,T3); Y = sim(net,[P1 P2]) Algorithm Linear layers consist of a single layer with the dotprod weight function, netsum net input function, and purelin transfer function. The layer has a weight from the input and a bias. Weights and biases are initialized with initzero. Adaption and training are done with adaptwb and trainwb, which both update weight and bias values with learnwh. Performance is measured with mse. See Also 13-142 newlind, sim, init, adapt, train newlind Purpose Syntax newlind Design a linear layer net = newlind(P,T) new = newlind Description newlind(P,T) takes two input arguments, P - R x Q matrix of Q input vectors. T - S x Q matrix of Q target class vectors. and returns a linear layer designed to output T (with minimum sum square error) given input P. Call newlind without input arguments to define the network's attributes in a dialog window. Examples We would like a linear layer that outputs T given P for the following definitions. P = [1 2 3]; T = [2.0 4.1 5.9]; Here we use newlind to design such a network and check its response. net = newlind(P,T); Y = sim(net,P) Algorithm newlind calculates weight W and bias B values for a linear layer from inputs P and targets T by solving this linear equation in the least squares sense: [W b] * [P; ones] = T See Also sim, newlin 13-143 newlvq Purpose newlvq Create a learning vector quantization network Syntax net = newlvq(PR,S1,PC,LR,LF) Description LVQ networks are used to solve classification problems. net = newlvq(PR,S1,PC,LR,LF) takes these inputs, PR - R x 2 matrix of min and max values for R input elements. S1 - Number of hidden neurons. PC - S2 element vector of typical class percentages. LR - Learning rate, default = 0.01. LF - Learning function, default = 'learnlv2'. returns a new LVQ network. The learning function LF can be learnlv1 or learnlv2. Properties newlvq creates a two layer network. The first layer uses the compet transfer function, calculates weighted inputs with negdist, and net input with netsum. The second layer has purelin neurons, calculates weighted input with dotprod and net inputs with netsum. Neither layer has biases. First layer weights are initialized with midpoint. The second layer weights are set so that each output neuron i has unit weights coming to it from PC(i) percent of the hidden neurons. Adaption and training are done with adaptwb and trainwb1, which both update the first layer weights with the specified learning functions. Examples The input vectors P and target classes Tc below define a classification problem to be solved by an LVQ network. P = [-3 -2 -2 0 0 0 0 +2 +2 +3; ... 0 +1 -1 +2 +1 -1 -2 +1 -1 0]; Tc = [1 1 1 2 2 2 2 1 1 1]; 13-144 newlvq The target classes Tc are converted to target vectors T. Then, an LVQ network is created (with inputs ranges obtained from P, 4 hidden neurons, and class percentages of 0.6 and 0.4) and is trained. T = ind2vec(Tc); net = newlvq(minmax(P),4,[.6 .4]); net = train(net,P,T); The resulting network can be tested. Y = sim(net,P) Yc = vec2ind(Y) See Also sim, init, adapt, train, adaptwb, trainwb1, learnlv1, learnlv2 13-145 newp Purpose newp Create a perceptron Syntax net = newp(pr,s,tf,lf) Description Perceptrons are used to solve simple (i.e. linearly separable) classification problems. net = newp(PR,S,TF,LF) takes these inputs, PR - R x 2 matrix of min and max values for R input elements. S - Number of neurons. TF - Transfer function, default = 'hardlim'. LF - Learning function, default = 'learnp'. and returns a new perceptron. The transfer function TF can be hardlim or hardlims. The learning function LF can be learnp or learnpn. Call newp without input arguments to define the network's attributes in a dialog window. Properties Perceptrons consist of a single layer with the dotprod weight function, the netsum net input function, and the specified transfer function. The layer has a weight from the input and a bias. Weights and biases are initialized with initzero. Adaption and training are done with adaptwb and trainwb, which both update weight and bias values with the specified learning function. Performance is measured with mae. Examples This code creates a perceptron layer with one 2-element input (ranges [0 1] and [-2 2]) and one neuron. (Supplying only two arguments to newp results in the default perceptron learning function learnp being used.) net = newp([0 1; -2 2],1); Here we simulate the network to a sequence of inputs P. P1 = {[0; 0] [0; 1] [1; 0] [1; 1]}; Y = sim(net,P1) 13-146 newp Here we define a sequence of targets T (together P and T define the operation of an AND gate), and then let the network adapt for 10 passes through the sequence. We then simulate the updated network. T1 = {0 0 0 1}; net.adaptParam.passes = 10; net = adapt(net,P1,T1); Y = sim(net,P1) Now we define a new problem, an OR gate, with batch inputs P and targets T. P2 = [0 0 1 1; 0 1 0 1]; T2 = [0 1 1 1]; Here we initialize the perceptron (resulting in new random weight and bias values), simulate its output, train for a maximum of 20 epochs, and then simulate it again. net = init(net); Y = sim(net,P2) net.trainParam.epochs = 20; net = train(net,P2,T2); Y = sim(net,P2) Notes Perceptrons can classify linearly separable classes in a finite amount of time. If input vectors have a large variance in their lengths, the learnpn can be faster than learnp. See Also sim, init, adapt, train, hardlim, hardlims, learnp, learnpn 13-147 newpnn Purpose newpnn Design a probabilistic neural network Syntax net = newpnn(P,T,spread) Description Probabilistic neural networks are a kind of radial basis network suitable for classification problems. net = newpnn(P,T,spread)takes two or three arguments, P - R x Q matrix of Q input vectors. T - S x Q matrix of Q target class vectors. spread - Spread of radial basis functions, default = 0.1. and returns a new probabilistic neural network. If spread is near zero the network will act as a nearest neighbor classifier. As spread becomes larger the designed network will take into account several nearby design vectors. Examples Here a classification problem is defined with a set of inputs P and class indices Tc. P = [1 2 3 4 5 6 7]; Tc = [1 2 3 2 2 3 1]; Here the class indices are converted to target vectors, and a PNN is designed and tested. T = ind2vec(Tc) net = newpnn(P,T); Y = sim(net,P) Yc = vec2ind(Y) Algorithm newpnn creates a two layer network. The first layer has radbas neurons, and calculates its weighted inputs with dist, and its net input with netprod. The second layer has compet neurons, and calculates its weighted input with dotprod and its net inputs with netsum. Only the first layer has biases. newpnn sets the first layer weights to P', and the first layer biases are all set to 0.8326/spread resulting in radial basis functions that cross 0.5 at weighted inputs of +/- spread. The second layer weights W2 are set to T. 13-148 newpnn See Also sim, ind2vec, vec2ind, newrb, newrbe, newgrnn References Wasserman, P.D., Advanced Methods in Neural Computing, New York: Van Nostrand Reinhold, pp. 35-55, 1993. 13-149 newrb Purpose newrb Design a radial basis network Syntax net = newrb(P,T,goal,spread) Description Radial basis networks can be used to approximate functions. newrb adds neurons to the hidden layer of a radial basis network until it meets the specified mean squared error goal. newrb(P,T,goal,spread) takes two to four arguments, P - R x Q matrix of Q input vectors. T - S x Q matrix of Q target class vectors. goal - Mean squared error goal, default = 0.0. spread - Spread of radial basis functions, default = 1.0. and returns a new radial basis network. The larger that spread is, the smoother the function approximation will be. Too large a spread means a lot of neurons will be required to fit a fast changing function. Too small a spread means many neurons will be required to fit a smooth function, and the network may not generalize well. Call newrb with different spreads to find the best value for a given problem. Examples Here we design a radial basis network given inputs P and targets T. P = [1 2 3]; T = [2.0 4.1 5.9]; net = newrb(P,T); Here the network is simulated for a new input. P = 1.5; Y = sim(net,P) Algorithm 13-150 newrb creates a two layer network. The first layer has radbas neurons, and calculates its weighted inputs with dist, and its net input with netprod. The second layer has purelin neurons, and calculates its weighted input with dotprod and its net inputs with netsum. Both layers have biases. newrb Initially the radbas layer has no neurons. The following steps are repeated until the network's mean squared error falls below goal. 1 The network is simulated. 2 The input vector with the greatest error is found. 3 A radbas neuron is added with weights equal to that vector. 4 The purelin layer weights are redesigned to minimize error. See Also sim, newrbe, newgrnn, newpnn 13-151 newrbe Purpose newrbe Design an exact radial basis network Syntax net = newrbe(P,T,spread) Description Radial basis networks can be used to approximate functions. newrbe very quickly designs a radial basis network with zero error on the design vectors. newrbe(P,T,spread) takes two or three arguments, P - R x Q matrix of Q input vectors. T - S x Q matrix of Q target class vectors. spread - Spread of radial basis functions, default = 1.0. and returns a new exact radial basis network. The larger the spread is, the smoother the function approximation will be. Too large a spread can cause numerical problems. Examples Here we design a radial basis network given inputs P and targets T. P = [1 2 3]; T = [2.0 4.1 5.9]; net = newrbe(P,T); Here the network is simulated for a new input. P = 1.5; Y = sim(net,P) Algorithm newrbe creates a two layer network. The first layer has radbas neurons, and calculates its weighted inputs with dist, and its net input with netprod. The second layer has purelin neurons, and calculates its weighted input with dotprod and its net inputs with netsum. Both layers have biases. newrbe sets the first layer weights to P', and the first layer biases are all set to 0.8326/spread, resulting in radial basis functions that cross 0.5 at weighted inputs of +/- spread. The second layer weights IW{2,1} and biases b{2} are found by simulating the first layer outputs A{1}, and then solving the following linear expression: [W{2,1} b{2}] * [A{1}; ones] = T See Also 13-152 sim, newrb, newgrnn, newpnn newsom Purpose newsom Create a self-organizing map Syntax net = newsom(PR,[d1,d2,...],tfcn,dfcn,olr,osteps,tlr,tnd) Description Competitive layers are used to solve classification problems. net = newsom (PR,[D1,D2,...],TFCN,DFCN,OLR,OSTEPS,TLR,TND) takes, PR - R x 2 matrix of min and max values for R input elements. I - Size of ith layer dimension, defaults = [5 8]. TFCN - Topology function, default ='hextop'. DFCN - Distance function, default ='linkdist'. OLR - Ordering phase learning rate, default = 0.9. OSTEPS - Ordering phase steps, default = 1000. TLR - Tuning phase learning rate, default = 0.02; TND - Tuning phase neighborhood distance, default = 1. and returns a new self-organizing map. The topology function TFCN can be hextop, gridtop, or randtop. The distance function can be linkdist, dist, or mandist. Properties Simms consist of a single layer with the negdist weight function, netsum net input function, and the compet transfer function. The layer has a weight from the input, but no bias. The weight is initialized with midpoint. Adaption and training are done with adaptwb and trainwb1, which both update the weight with learnsom. Examples The input vectors defined below are distributed over an 2-dimension input space varying over [0 2] and [0 1]. This data will be used to train a SOM with dimensions [3 5]. P = [rand(1,400)*2; rand(1,400)]; net = newsom([0 2; 0 1],[3 5]); plotsom(net.layers{1}.positions) 13-153 newsom Here the SOM is trained and the input vectors are plotted with the map which the SOM's weights have formed. net = train(net,P); plot(P(1,:),P(2,:),'.g','markersize',20) hold on plotsom(net.iw{1,1},net.layers{1}.distances) hold off See Also 13-154 sim, init, adapt, train, adaptwb, trainwb1 nncopy Purpose nncopy Copy matrix or cell array Syntax nncopy(x,m,n) Description nncopy(X,M,N) takes two arguments, X - R x C matrix (or cell array). M - Number of vertical copies. N - Number of horizontal copies. and returns a new (R*M) x (C*N) matrix (or cell array). Examples x1 y1 x2 y2 = = = = [1 2 3; 4 5 6]; nncopy(x1,3,2) {[1 2]; [3; 4; 5]} nncopy(x2,2,3) 13-155 nnt2c Purpose nnt2c Update NNT 2.0 competitive layer to NNT 3.0 Syntax net = nnt2c(pr,w,klr,clr) Description nnt2c(PR,W,KLR,CLR) takes these arguments, PR - R x 2 matrix of min and max values for R input elements. W - S x R weight matrix. KLR - Kohonen learning rate, default = 0.01. CLR - Conscience learning rate, default = 0.001. and returns a competitive layer. Once a network has been updated it can be simulated, initialized, or trained with sim, init, adapt, and train. See Also 13-156 newc nnt2elm Purpose nnt2elm Update NNT 2.0 Elman backpropagation network to NNT 3.0 Syntax net = nnt2elm(pr,w1,b1,w2,b2,btf,blf,pf) Description nnt2elm(PR,W1,B1,W2,B2,BTF,BLF,PF) takes these arguments, PR - R x 2 matrix of min and max values for R input elements. W1 - S1 x (R+S1) weight matrix. B1 - S1 x 1 bias vector. W2 - S2 x S1 weight matrix. B2 - S2 x 1 bias vector. BTF - Backprop network training function, default = 'traingdx'. BLF - Backprop weight/bias learning function, default = 'learngdm'. PF - Performance function, default = 'mse'. and returns a feed-forward network. The training function BTF can be any of the backprop training functions such as traingd, traingdm, traingda, and traingdx. Large step-size algorithms, such as trainlm, are not recommended for Elman networks. The learning function BLF can be either of the backpropagation learning functions such as learngd or learngdm. The performance function can be any of the differentiable performance functions such as mse or msereg. Once a network has been updated it can be simulated, initialized, adapted, or trained with sim, init, adapt, and train. See Also newelm 13-157 nnt2ff Purpose nnt2ff Update NNT 2.0 feed-forward network to NNT 3.0 Syntax net = nnt2ff(pr,{w1 w2 ...},{b1 b2 ...},{tf1 tf2 ...},btf,blr,pf) Description nnt2ff(PR,{W1 W2 ...},{B1 B2 ...},{TF1 TF2 ...},BTF,BLR,PF) takes these arguments, PR - R x 2 matrix of min and max values for R input elements. Wi - Weight matrix for the ith layer. Bi - Bias vector for the ith layer. TFi - Transfer function of ith layer, default = 'tansig'. BTF - Backprop network training function, default = 'traingdx'. BLF - Backprop weight/bias learning function, default = 'learngdm'. PF - Performance function, default = 'mse'. and returns a feed-forward network. The training function BTF can be any of the backprop training functions such as traingd, traingdm, traingda, traingdx or trainlm. The learning function BLF can be either of the backpropagation learning functions such as learngd or learngdm. The performance function can be any of the differentiable performance functions such as mse or msereg. Once a network has been updated it can be simulated, initialized, adapted, or trained with sim, init, adapt, and train. See Also 13-158 newff, newcf, newfftd, newelm nnt2hop Purpose nnt2hop Update NNT 2.0 Hopfield recurrent network to NNT 3.0 Syntax net = nnt2p(w,b) Description nnt2hop (W,B) takes these arguments, W - S x S weight matrix. B - S x 1 bias vector and returns a perceptron. Once a network has been updated it can be simulated, initialized, adapted, or trained with sim, init, adapt, and train. See Also newhop 13-159 nnt2lin Purpose nnt2lin Update NNT 2.0 linear layer to NNT 3.0 Syntax net = nnt2lin(pr,w,b,lr) Description nnt2lin(PR,W,B) takes these arguments, PR - R x 2 matrix of min and max values for R input elements. W - S x R weight matrix. B - S x 1 bias vector LR - Learning rate, default = 0.01; and returns a linear layer. Once a network has been updated it can be simulated, initialized, adapted, or trained with sim, init, adapt, and train. See Also 13-160 newlin nnt2lvq Purpose nnt2lvq Update NNT 2.0 learning vector quantization network to NNT 3.0 Syntax net = nnt2lvq(pr,w1,w2,lr,lf) Description nnt2lvq(PR,W1,W2,LR,LF) takes these arguments, PR - R x 2 matrix of min and max values for R input elements. W1 - S1 x R weight matrix. W2 - S2 x S1 weight matrix. LR - Learning rate, default = 0.01. LF - Learning function, default = 'learnlv2'. and returns a radial basis network. The learning function LF can be learnlv1 or learnlv2. Once a network has been updated it can be simulated, initialized, adapted, or trained with sim, init, adapt, and train. See Also newlvq 13-161 nnt2p Purpose nnt2p Update NNT 2.0 perceptron to NNT 3.0 Syntax net = nnt2p(pr,w,b,tf,lf) Description nnt2p(PR,W,B,TF,LF) takes these arguments, PR - R x 2 matrix of min and max values for R input elements. W - S x R weight matrix. B - S x 1 bias vector. TF - Transfer function, default = 'hardlim'. LF - Learning function, default = 'learnp'. and returns a perceptron. The transfer function TF can be hardlim or hardlims. The learning function LF can be learnp or learnpn. Once a network has been updated it can be simulated, initialized, adapted, or trained with sim, init, adapt, and train. See Also 13-162 newp nnt2rb Purpose nnt2rb Update NNT 2.0 radial basis network to NNT 3.0 Syntax net = nnt2rb(pr,w1,b1,w2,b2) Description nnt2rb(PR,W1,B1,W2,B2) takes these arguments, PR - R x 2 matrix of min and max values for R input elements. W1 - S1 x R weight matrix. B1 - S1 x 1 bias vector. W2 - S2 x S1 weight matrix. B2 - S2 x 1 bias vector. and returns a radial basis network. Once a network has been updated it can be simulated, initialized, adapted, or trained with sim, init, adapt, and train. See Also newrb, newrbe, newgrnn, newpnn 13-163 nnt2som Purpose Syntax Description nnt2som Update NNT 2.0 self-organizing map to NNT 3.0 net = nnt2som(pr,[d1 d2 ...],w,olr,osteps,tlr,tnd) nnt2som(PR,[D1,D2,...],W,OLR,OSTEPS,TLR,TND) takes these arguments, PR - R x 2 matrix of min and max values for R input elements. Di - Size of ith layer dimension. W - S x R weight matrix. OLR - Ordering phase learning rate, default = 0.9. OSTEPS - Ordering phase steps, default = 1000. TLR - Tuning phase learning rate, default = 0.02; TND - Tuning phase neighborhood distance, default = 1. and returns a self-organizing map. nnt2som assumes that the self-organizing map has a grid topology (gridtop) using link distances (linkdist). This corresponds with the nbman neighborhood function in NNT 2.0. The new network will only output 1 for the neuron with the greatest net input. In NNT 2.0 the network would also output 0.5 for that neuron's neighbors. Once a network has been updated it can be simulated, initialized, adapted, or trained with sim, init, adapt, and train. See Also 13-164 newsom normc Purpose normc Normalize the columns of a matrix Syntax normc(M) Description normc(M) normalizes the columns of M to a length of 1. Examples See Also m = [1 2; 3 4]; normc(m) ans = 0.3162 0.4472 0.9487 0.8944 normr 13-165 normprod Purpose Syntax normprod Normalized dot product weight function Z = normprod(W,P) df = normprod('deriv') Description normprod is a weight function. Weight functions apply weights to an input to get weighted inputs. normprod(W,P) takes these inputs, W - S x R weight matrix. P - R x Q matrix of Q input (column) vectors. and returns the S x Q matrix of normalized dot products. normprod('deriv') returns '' because normprod does not have a derivative function. Examples Here we define a random weight matrix W and input vector P and calculate the corresponding weighted input Z. W = rand(4,3); P = rand(3,1); Z = normprod(W,P) Network Use You can create a standard network that uses normprod by calling newgrnn. To change a network so an input weight uses normprod, set net.inputWeight{i,j}.weightFcn to 'normprod’. For a layer weight set net.inputWeight{i,j}.weightFcn to 'normprod’. In either case call sim to simulate the network with normprod. See newgrnn for simulation examples. Algorithm normprod returns the dot product normalized by the sum of the input vector elements. z = w*p/sum(p) See Also 13-166 sim, dotprod, negdist, dist normr Purpose normr Normalize the rows of a matrix Syntax normr(M) Description normr(M) normalizes the columns of M to a length of 1. Examples See Also m = [1 2; 3 4]; normr(m) ans = 0.4472 0.6000 0.8944 0.8000 normc 13-167 plotep Purpose plotep Plot a weight-bias position on an error surface Syntax h = plotep(w,b,e) h = plotep(w,b,e,h) Description plotep is used to show network learning on a plot already created by plotes. plotep(W,B,E) takes these arguments, W - Current weight value. B - Current bias value. E - Current error. and returns a vector H, containing information for continuing the plot. plotep(W,B,E,H) continues plotting using the vector H returned by the last call to plotep. H contains handles to dots plotted on the error surface, so they can be deleted next time, as well as points on the error contour, so they can be connected. See Also 13-168 errsurf, plotes plotes Purpose plotes Plot the error surface of a single input neuron Syntax plotes(wv,bv,es,v) Description plotes(WV,BV,ES,V) takes these arguments, WV - 1 x N row vector of values of W. BV - 1 x M row vector of values of B. ES - M x N matrix of error vectors. V - View, default = [-37.5, 30]. and plots the error surface with a contour underneath. Calculate the error surface ES with errsurf. Examples See Also p = [3 2]; t = [0.4 0.8]; wv = -4:0.4:4; bv = wv; ES = errsurf(p,t,wv,bv,'logsig'); plotes(wv,bv,ES,[60 30]) errsurf 13-169 plotpc Purpose plotpc Plot a classification line on a perceptron vector plot Syntax plotpc(W,b) Description plotpc(W,B) takes these inputs, W - S x R weight matrix (R must be 3 or less). B - S x 1 bias vector. and returns a handle to a plotted classification line. plotpc(W,B,H) takes these inputs, H - Handle to last plotted line. and deletes the last line before plotting the new one. This function does not change the current axis and is intended to be called after plotpv. Examples The code below defines and plots the inputs and targets for a perceptron: p = [0 0 1 1; 0 1 0 1]; t = [0 0 0 1]; plotpv(p,t) The following code creates a perceptron with inputs ranging over the values in P, assigns values to its weights and biases, and plots the resulting classification line. net = newp(minmax(p),1); net.iw{1,1} = [-1.2 -0.5]; net.b{1} = 1; plotpc(net.iw{1,1},net.b{1}) See Also 13-170 plotpv plotperf Purpose plotperf Plot network performance Syntax plotperf(tr,goal,name,epoch) Description plotperf(TR,goal,name,epoch) takes these inputs, TR - Training record returned by train. goal - Performance goal, default = NaN. name - Training function name, default = ''. epoch - Number of epochs, default = length of training record. and plots the training performance, and if available, the performance goal, validation performance, and test performance. Examples Here are 8 input values P and associated targets T, plus a like number of validation inputs VV.P and targets VV.T. P = 1:8; T = sin(P); VV.P = P; VV.T = T+rand(1,8)*0.1; The code below creates a network and trains it on this problem. net = newff(minmax(P),[4 1],{'tansig','tansig'}); [net,tr] = train(net,P,T,[],[],VV); During training plotperf was called to display the training record. You can also call plotperf directly with the final training record TR, as shown below. plotperf(tr) 13-171 plotpv Purpose plotpv Plot perceptron input/target vectors Syntax plotpv(p,t) Description plotpv(P,T) take these inputs, P - R x Q matrix of input vectors (R must be 3 or less). T - S x Q matrix of binary target vectors (S must be 3 or less). and plots column vectors in P with markers based on T. plotpv(P,T,V) takes an additional input, V - Graph limits = [x_min x_max y_min y_max] and plots the column vectors with limits set by V. Examples The code below defines and plots the inputs and targets for a perceptron: p = [0 0 1 1; 0 1 0 1]; t = [0 0 0 1]; plotpv(p,t) The following code creates a perceptron with inputs ranging over the values in P, assigns values to its weights and biases, and plots the resulting classification line. net = newp(minmax(p),1); net.iw{1,1} = [-1.2 -0.5]; net.b{1} = 1; plotpc(net.iw{1,1},net.b{1}) See Also 13-172 plotpc plotv Purpose plotv Plot vectors as lines from the origin Syntax plotv(m,t) Description plotv(M,T) takes two inputs, M - R x Q matrix of Q column vectors with R elements. T - (optional) the line plotting type, default = '-'. and plots the column vectors of M. R must be 2 or greater. If R is greater than two, only the first two rows of M are used for the plot. Examples plotv([-.4 0.7 .2; -0.5 .1 0.5],'-') 13-173 plotvec Purpose plotvec Plot vectors with different colors Syntax plotvec(x,c,m) Description plotvec(X,C,M) takes these inputs, X - Matrix of (column) vectors. C - Row vector of color coordinate. M - Marker, default = '+'. and plots each ith vector in X with a marker M and using the ith value in C as the color coordinate. plotvec(X) only takes a matrix X and plots each ith vector in X with marker '+' using the index i as the color coordinate. Examples 13-174 x = [0 1 0.5 0.7; -1 2 0.5 0.1]; c = [1 2 3 4]; plotvec(x,c) pnormc Purpose pnormc Pseudo-normalize columns of a matrix Syntax pnormc(x,r) Description pnormc(M,R) takes these arguments, X - M x N matrix. R - (optional) radius to normalize columns to, default = 1. and returns X with an additional row of elements, which results in new column vector lengths of R. WARNING: For this function to work properly, the columns of X must originally have vector lengths less than R. Examples See Also x = [0.1 0.6; 0.3 0.1]; y = pnormc(x) normc, normr 13-175 poslin Purpose Graph and Symbol poslin Positive linear transfer function a +1 AA AA n 0 1 -1 a = poslin(n) Positive Linear Transfer Funct. Syntax A = poslin(N) info = poslin(code) Description poslin is a transfer function. Transfer functions calculate a layer's output from its net input. poslin(N) takes one input, N - S x Q matrix of net input (column) vectors. and returns the maximum of 0 and each element of N. poslin(code) returns useful information for each code string: 'deriv' - Name of derivative function. 'name' - Full name. 'output' - Output range. 'active' - Active input range. Examples Here is the code to create a plot of the poslin transfer function. n = -5:0.1:5; a = poslin(n); plot(n,a) 13-176 poslin Network Use To change a network so that a layer uses poslin, set net.layers{i}.transferFcn to 'poslin'. Call sim to simulate the network with poslin. Algorithm poslin(n) = n, if n >= 0; = 0, if n <= 0. See Also sim, purelin, satlin, satlins 13-177 postmnmx Purpose Syntax postmnmx Postprocess data which has been preprocessed by premnmx [p,t] = postmnmx(pn,minp,maxp,tn,mint,maxt) [p] = postmnmx(pn,minp,maxp) Description postmnmx postprocesses the network training set which was preprocessed by premnmx. It converts the data back into unnormalized units. postmnmx takes these inputs, PN - R x Q matrix of normalized input vectors. minp- R x 1 vector containing minimums for each P. maxp- R x 1 vector containing maximums for each P. TN - S x Q matrix of normalized target vectors. mint- S x 1 vector containing minimums for each T. maxt- S x 1 vector containing maximums for each T. and returns, P - R x Q matrix of input (column) vectors. T - R x Q matrix of target vectors. Examples In this example we normalize a set of training data with premnmx, create and train a network using the normalized data, simulate the network, unnormalize the output of the network using postmnmx, and perform a linear regression between the network outputs (unnormalized) and the targets to check the quality of the network training. p = [-0.92 0.73 -0.47 0.74 0.29; -0.08 0.86 -0.67 -0.52 0.93]; t = [-0.08 3.4 -0.82 0.69 3.1]; [pn,minp,maxp,tn,mint,maxt] = premnmx(p,t); net = newff(minmax(pn),[5 1],{'tansig' 'purelin'},'trainlm'); net = train(net,pn,tn); an = sim(net,pn); [a] = postmnmx(an,mint,maxt); [m,b,r] = postreg(a,t); Algorithm See Also 13-178 p = 0.5(pn+1)*(maxp-minp) + minp; premnmx, prepca, poststd postreg Purpose postreg Postprocess the trained network response with a linear regression Syntax [m,b,r] = postreg(A,T) Description postreg postprocesses the network training set by performing a linear regression between each element of the network response and the corresponding target. postreg(A,T) takes these inputs, A - 1 x Q array of network outputs. One element of the network output. T - 1 x Q array of targets. One element of the target vector. and returns, M - Slope of the linear regression. B - Y intercept of the linear regression. R - Regression R-value. R=1 means perfect correlation. Examples In this example we normalize a set of training data with prestd, perform a principal component transformation on the normalized data, create and train a network using the pca data, simulate the network, unnormalize the output of the network using poststd, and perform a linear regression between the network outputs (unnormalized) and the targets to check the quality of the network training. p = [-0.92 0.73 -0.47 0.74 0.29; -0.08 0.86 -0.67 -0.52 0.93]; t = [-0.08 3.4 -0.82 0.69 3.1]; [pn,meanp,stdp,tn,meant,stdt] = prestd(p,t); [ptrans,transMat] = prepca(pn,0.02); net = newff(minmax(ptrans),[5 1],{'tansig''purelin'},'trainlm'); net = train(net,ptrans,tn); an = sim(net,ptrans); a = poststd(an,meant,stdt); [m,b,r] = postreg(a,t); Algorithm Performs a linear regression between the network response and the target, and then computes the correlation coefficient (R-value) between the network response and the target. See Also premnmx, prepca 13-179 poststd Purpose Syntax poststd Postprocess data which has been preprocessed by prestd [p,t] = poststd(pn,meanp,stdp,tn,meant,stdt) [p] = poststd(pn,meanp,stdp) Description poststd postprocesses the network training set which was preprocessed by prestd. It converts the data back into unnormalized units. poststd takes these inputs, PN - R x Q matrix of normalized input vectors. meanp - R x 1 vector containing standard deviations for each P. stdp - R x 1 vector containing standard deviations for each P. TN - S x Q matrix of normalized target vectors. meant - S x 1 vector containing standard deviations for each T. stdt - S x 1 vector containing standard deviations for each T. and returns, P - R x Q matrix of input (column) vectors. T - S x Q matrix of target vectors. Examples In this example we normalize a set of training data with prestd, create and train a network using the normalized data, simulate the network, unnormalize the output of the network using poststd, and perform a linear regression between the network outputs (unnormalized) and the targets to check the quality of the network training. p = [-0.92 0.73 -0.47 0.74 0.29; -0.08 0.86 -0.67 -0.52 0.93]; t = [-0.08 3.4 -0.82 0.69 3.1]; [pn,meanp,stdp,tn,meant,stdt] = prestd(p,t); net = newff(minmax(pn),[5 1],{'tansig' 'purelin'},'trainlm'); net = train(net,pn,tn); an = sim(net,pn); a = poststd(an,meant,stdt); [m,b,r] = postreg(a,t); Algorithm See Also 13-180 p = stdp*pn + meanp; premnmx, prepca, postmnmx, prestd premnmx Purpose Syntax premnmx Preprocess data so that minimum is -1 and maximum is 1 [pn,minp,maxp,tn,mint,maxt] = premnmx(p,t) [pn,minp,maxp] = premnmx(p) Description premnmx preprocesses the network training set by normalizing the inputs and targets so that they fall in the interval [-1,1]. premnmx(P,T) takes these inputs, P - R x Q matrix of input (column) vectors. T - S x Q matrix of target vectors. and returns, PN - R x Q matrix of normalized input vectors. minp- R x 1 vector containing minimums for each P. maxp- R x 1 vector containing maximums for each P. TN - S x Q matrix of normalized target vectors. mint- S x 1 vector containing minimums for each T. maxt- S x 1 vector containing maximums for each T. Examples Here is the code to normalize a given data set so that the inputs and targets will fall in the range [-1,1]. p = [-10 -7.5 -5 -2.5 0 2.5 5 7.5 10]; t = [0 7.07 -10 -7.07 0 7.07 10 7.07 0]; [pn,minp,maxp,tn,mint,maxt] = premnmx(p,t); If you just want to normalize the input, [pn,minp,maxp] = premnmx(p); Algorithm See Also pn = 2*(p-minp)/(maxp-minp) - 1; prestd, prepca, postmnmx 13-181 prepca Purpose prepca Principal component analysis Syntax [ptrans,transMat] = prepca(P,min_frac) Description prepca preprocesses the network input training set by applying a principal component analysis. This analysis transforms the input data so that the elements of the input vector set will be uncorrelated. In addition, the size of the input vectors may be reduced by retaining only those components which contribute more than a specified fraction (min_frac) of the total variation in the data set. prepca(p,min_frac) takes these inputs P - R x Q matrix of centered input (column) vectors. min_frac - Minimum fraction variance component to keep. and returns ptrans - Transformed data set. transMat - Transformation matrix. Examples Here is the code to perform a principal component analysis and retain only those components which contribute more than 2 percent to the variance in the data set. prestd is called first to create zero mean data, which are needed for prepca. p=[-1.5 -0.58 0.21 -0.96 -0.79; -2.2 -0.87 0.31 -1.4 [pn,meanp,stdp] = prestd(p); [ptrans,transMat] = prepca(pn,0.02); -1.2]; Since the second row of p is almost a multiple of the first row, this example will produce a transformed data set which contains only one row. Algorithm 13-182 This routine uses singular value decomposition to compute the principal components. The input vectors are multiplied by a matrix whose rows consist of the eigenvectors of the input covariance matrix. This produces transformed input vectors whose components are uncorrelated and ordered according to the magnitude of their variance. prepca Those components which contribute only a small amount to the total variance in the data set are eliminated. It is assumed that the input data set has already been normalized so that it has a zero mean. The function prestd can be used to normalize the data. See Also prestd, premnmx References Jolliffe, I.T.,Principal Component Analysis, New York: Springer-Verlag, 1986. 13-183 prestd Purpose Syntax prestd Preprocess data so that its mean is 0 and the standard deviation is 1 [pn,meanp,stdp,tn,meant,stdt] = prestd(p,t) [pn,meanp,stdp] = prestd(p) Description prestd preprocesses the network training set by normalizing the inputs and targets so that they have means of zero and standard deviations of 1. prestd(p,t) takes these inputs, p - R x Q matrix of input (column) vectors. t - S x Q matrix of target vectors. and returns, pn - R x Q matrix of normalized input vectors. meanp - R x 1 vector containing standard deviations for each P. stdp - R x 1 vector containing standard deviations for each P. tn - S x Q matrix of normalized target vectors. meant - S x 1 vector containing standard deviations for each T. stdt Examples - S x 1 vector containing standard deviations for each T. Here is the code to normalize a given data set so that the inputs and targets will have means of zero and standard deviations of 1. p = [-0.92 0.73 -0.47 0.74 0.29; -0.08 0.86 -0.67 -0.52 0.93]; t = [-0.08 3.4 -0.82 0.69 3.1]; [pn,meanp,stdp,tn,meant,stdt] = prestd(p,t); If you just want to normalize the input, [pn,meanp,stdp] = prestd(p); Algorithm See Also 13-184 pn = (p-meanp)/stdp; premnmx, prepca purelin Purpose purelin Linear transfer function Syntax A = purelin(N) info = purelin(code) Description purelin is a transfer function. Transfer functions calculate a layer's output from its net input. purelin(N) takes one input, N - S x Q matrix of net input (column) vectors. and returns N. purelin(code) returns useful information for each code string: 'deriv' - Name of derivative function. 'name' - Full name. 'output' - Output range. 'active' - Active input range. Examples Here is the code to create a plot of the purelin transfer function. n = -5:0.1:5; a = purelin(n); plot(n,a) Network Use You can create a standard network that uses purelin by calling newlin or newlind. To change a network so a layer uses purelin, set net.layers{i}.transferFcn to 'purelin'. In either case, call sim to simulate the network with purelin. See newlin or newlind for simulation examples. Algorithm See Also purelin(n) = n sim, dpurelin, satlin, satlins 13-185 quant Purpose quant Discretize values as multiples of a quantity Syntax quant(X,q) Description quant(X,q) takes two inputs, X - Matrix, vector or scalar. Q - Minimum value. and returns values in X rounded to nearest multiple of Q. Examples 13-186 x = [1.333 4.756 -3.897]; y = quant(x,0.1) radbas Purpose radbas Radial basis transfer function Syntax A = radbas(N) info = radbas(code) Description radbas is a transfer function. Transfer functions calculate a layer's output from its net input. radbas(N) takes one input, N - S x Q matrix of net input (column) vectors. and returns each element of N passed through a radial basis function. radbas(code) returns useful information for each code string: 'deriv' - Name of derivative function. 'name' - Full name. 'output' - Output range. 'active' - Active input range. Examples Here we create a plot of the radbas transfer function. n = -5:0.1:5; a = radbas(n); plot(n,a) Network Use You can create a standard network that uses radbas by calling newpnn or newgrnn. To change a network so that a layer uses radbas, set net.layers{i}.transferFcn to 'radbas'. In either case, call sim to simulate the network with radbas. See newpnn or newgrnn for simulation examples. Algorithm radbas(N) calculates its output with according to: a = exp(-n2) See Also sim, tribas, dradbas 13-187 randnc Purpose randnc Normalized column weight initialization function Syntax W = randnc(S,PR) W = randnc(S,R) Description randnc is a weight initialization function. randnc(S,P) takes two inputs, S - Number of rows (neurons). PR - R x 2 matrix of input value ranges = [Pmin Pmax]. and returns an S x R random matrix with normalized columns. Can also be called as randnc(S,R). Examples A random matrix of four normalized three-element columns is generated: M = randnc(3,4) M = –0.6007 –0.4715 –0.7628 –0.6967 –0.2395 0.5406 See Also 13-188 randnr –0.2724 –0.9172 –0.2907 0.5596 0.7819 0.2747 randnr Purpose randnr Normalized row weight initialization function Syntax W = randnr(S,PR) W = randnr(S,R) Description randnr is a weight initialization function. randnr(S,P) takes two inputs, S - Number of rows (neurons). PR - R x 2 matrix of input value ranges = [Pmin Pmax]. and returns an S x R random matrix with normalized rows. Can also be called as randnr(S,R). Examples A matrix of three normalized four-element rows is generated: M = randnr(3,4) M = 0.9713 0.0800 0.8228 0.0338 –0.3042 –0.5725 See Also –0.1838 0.1797 0.5436 –0.1282 0.5381 0.5331 randnc 13-189 rands Purpose rands Symmetric random weight/bias initialization function Syntax W = rands(S,PR) M = rands(S,R) v = rands(S); Description rands is a weight/bias initialization function. rands(S,PR) takes, S - Number of neurons. PR - R x 2 matrix of R input ranges. and returns an S-by-R weight matrix of random values between -1 and 1. rands(S,R) returns an S-by-R matrix of random values. rands(S) returns an S-by-1 vector of random values. Examples Here three sets of random values are generated with rands. rands(4,[0 1; -2 2]) rands(4) rands(2,3) Network Use To prepare the weights and the bias of layer i of a custom network to be initialized with rands: 1 Set net.initFcn to 'initlay'. (net.initParam will automatically become initlay's default parameters.) 2 Set net.layers{i}.initFcn to 'initwb'. 3 Set each net.inputWeights{i,j}.initFcn to 'rands'. Set each net.layerWeights{i,j}.initFcn to 'rands'. Set each net.biases{i}.initFcn to 'rands'. To initialize the network call init. See Also 13-190 randnr, randnc, initwb, initlay, init randtop Purpose randtop Random layer topology function Syntax pos = randtop(dim1,dim2,...,dimN) Description randtop calculates the neuron positions for layers whose neurons are arranged in an N dimensional random pattern. randtop(dim1,dim2,...,dimN)) takes N arguments, dimi - Length of layer in dimension i. and returns an N x S matrix of N coordinate vectors, where S is the product of dim1*dim2*...*dimN. Examples This code creates and displays a two-dimensional layer with 192 neurons arranged in a 16x12 random pattern. pos = randtop(16,12); plotsom(pos) This code plots the connections between the same neurons, but shows each neuron at the location of its weight vector. The weights are generated randomly so that the layer is very unorganized, as is evident in the plot. W = rands(192,2); plotsom(W,dist(pos)) See Also gridtop, hextop 13-191 satlin Purpose satlin Saturating linear transfer function Syntax A = satlin(N) info = satlin(code) Description satlin is a transfer function. Transfer functions calculate a layer's output from its net input. satlin(N) takes one input, N - S x Q matrix of net input (column) vectors, and returns values of N truncated into the interval [-1, 1]. satlin(code) returns useful information for each code string: 'deriv' - Name of derivative function. 'name' - Full name. 'output' - Output range. 'active' - Active input range. Examples Here is the code to create a plot of the satlin transfer function. n = -5:0.1:5; a = satlin(n); plot(n,a) Network Use To change a network so that a layer uses satlin, set net.layers{i}.transferFcn to 'satlin'. Call sim to simulate the network with satlin. See newhop for simulation examples. Algorithm satlin(n) = 0, if n <= 0; n, if 0 <= n <= 1; 1, if 1 <= n. See Also sim, poslin, satlins, purelin 13-192 satlins Purpose satlins Symmetric saturating linear transfer function Syntax A = satlins(N) info = satlins(code) Description satlins is a transfer function. Transfer functions calculate a layer's output from its net input. satlins(N) takes one input, N - S x Q matrix of net input (column) vectors. and returns values of N truncated into the interval [-1, 1]. satlins(code) returns useful information for each code string: 'deriv' - Name of derivative function. 'name' - Full name. 'output' - Output range. 'active' - Active input range. Examples Here is the code to create a plot of the satlins transfer function. n = -5:0.1:5; a = satlins(n); plot(n,a) Network Use You can create a standard network that uses satlins by calling newhop. To change a network so that a layer uses satlins, set net.layers{i}.transferFcn to 'satlins'. In either case, call sim to simulate the network with satlins. See newhop for simulation examples. Algorithm satlins(n) = -1, if n <= -1; n, if -1 <= n <= 1; 1, if 1 <= n. See Also sim, satlin, poslin, purelin 13-193 seq2con Purpose seq2con Converts sequential vectors to concurrent vectors Syntax b = seq2con(s) Description The neural network toolbox represents batches of vectors with a matrix, and sequences of vectors with multiple columns of a cell array. seq2con and con2seq allow concurrent vectors to be converted to sequential vectors, and back again. seq2con(S) takes one input, S - N x TS cell array of matrices with M columns. and returns, B - N x 1 cell array of matrices with M*TS columns. Examples Here three sequential values are converted to concurrent values. p1 = {1 4 2} p2 = seq2con(p1) Here two sequences of vectors over three time steps are converted to concurrent vectors. p1 = {[1; 1] [5; 4] [1; 2]; [3; 9] [4; 1] [9; 8]} p2 = seq2con(p1) See Also 13-194 con2seq, concur sim Purpose Syntax sim Simulate a neural network [Y,Pf,Af] = sim(net,P,Pi,Ai) [Y,Pf,Af] = sim(net,{Q TS},Pi,Ai) [Y,Pf,Af] = sim(net,Q,Pi,Ai) To Get Help Type help network/sim Description sim simulates neural networks. [Y,Pf,Af] = sim(net,P,Pi,Ai) takes, net - Network. P - Network inputs. Pi - Initial input delay conditions, default = zeros. Ai - Initial layer delay conditions, default = zeros. and returns, Y - Network outputs. Pf - Final input delay conditions. Af - Final layer delay conditions. Note that arguments Pi, Ai, Pf, and Af are optional and need only be used for networks that have input or layer delays. sim's signal arguments can have two formats: cell array or matrix. The cell array format is easiest to describe. It is most convenient for networks with multiple inputs and outputs, and allows sequences of inputs to be presented: P - Ni x TS cell array, each element P{i,ts} is an Ri x Q matrix. Pi - Ni x ID cell array, each element Pi{i,k} is an Ri x Q matrix. Ai - Nl x LD cell array, each element Ai{i,k} is an Si x Q matrix. Y - NO x TS cell array, each element Y{i,ts} is a Ui x Q matrix. Pf - Ni x ID cell array, each element Pf{i,k} is an Ri x Q matrix. Af - Nl x LD cell array, each element Af{i,k} is an Si x Q matrix. 13-195 sim where Ni = net.numInputs Nl = net.numLayers, No = net.numOutputs D = net.numInputDelays LD = net.numLayerDelays TS = Number of time steps Q = Batch size Ri = net.inputs{i}.size Si = net.layers{i}.size Ui = net.outputs{i}.size The columns of Pi, Ai, Pf, and Af are ordered from oldest delay condition to most recent: Pi{i,k} = input i at time ts=k-ID. Pf{i,k} = input i at time ts=TS+k-ID. Ai{i,k} = layer output i at time ts=k-LD. Af{i,k} = layer output i at time ts=TS+k-LD. The matrix format can be used if only one time step is to be simulated (TS = 1). It is convenient for networks with only one input and output, but can also be used with networks that have more. Each matrix argument is found by storing the elements of the corresponding cell array argument into a single matrix: P - (sum of Ri) x Q matrix Pi - (sum of Ri) x (ID*Q) matrix. Ai - (sum of Si) x (LD*Q) matrix. Y - (sum of Ui) x Q matrix. Pf - (sum of Ri) x (ID*Q) matrix. Af - (sum of Si) x (LD*Q) matrix. 13-196 sim [Y,Pf,Af] = sim(net,{Q TS},Pi,Ai) is used for networks which do not have an input, such as Hopfield networks, when cell array notation is used. [Y,Pf,Af] = sim(net,Q,Pi,Ai) is used for networks which do not have an input, such as Hopfield networks, when matrix notation is used. Examples Here newp is used to create a perceptron layer with a 2-element input (with ranges of [0 1]), and a single neuron. net = newp([0 1;0 1],1); Here the perceptron is simulated for an individual vector, a batch of 3 vectors, and a sequence of 3 vectors. p1 = [.2; .9]; a1 = sim(net,p1) p2 = [.2 .5 .1; .9 .3 .7]; a2 = sim(net,p2) p3 = {[.2; .9] [.5; .3] [.1; .7]}; a3 = sim(net,p3) Here newlind is used to create a linear layer with a 3-element input, 2 neurons. net = newlin([0 2;0 2;0 2],2,[0 1]); Here the linear layer is simulated with a sequence of 2 input vectors using the default initial input delay conditions (all zeros). p1 = {[2; 0.5; 1] [1; 1.2; 0.1]}; [y1,pf] = sim(net,p1) Here the layer is simulated for 3 more vectors using the previous final input delay conditions as the new initial delay conditions. p2 = {[0.5; 0.6; 1.8] [1.3; 1.6; 1.1] [0.2; 0.1; 0]}; [y2,pf] = sim(net,p2,pf) Here newelm is used to create an Elman network with a 1-element input, and a layer 1 with 3 tansig neurons followed by a layer 2 with 2 purelin neurons. Because it is an Elman network it has a tap delay line with a delay of 1 going from layer 1 to layer 1. net = newelm([0 1],[3 2],{'tansig','purelin'}); 13-197 sim Here the Elman network is simulated for a sequence of 3 values using default initial delay conditions. p1 = {0.2 0.7 0.1}; [y1,pf,af] = sim(net,p1) Here the network is simulated for 4 more values, using the previous final delay conditions as the new initial delay conditions. p2 = {0.1 0.9 0.8 0.4}; [y2,pf,af] = sim(net,p2,pf,af) Algorithm sim uses these properties to simulate a network net. net.numInputs, net.numLayers net.outputConnect, net.biasConnect net.inputConnect, net.layerConnect These properties determine the network's weight and bias values, and the number of delays associated with each weight: net.inputWeights{i,j}.value net.layerWeights{i,j}.value net.layers{i}.value net.inputWeights{i,j}.delays net.layerWeights{i,j}.delays These function properties indicate how sim applies weight and bias values to inputs to get each layer's output: net.inputWeights{i,j}.weightFcn net.layerWeights{i,j}.weightFcn net.layers{i}.netInputFcn net.layers{i}.transferFcn See Chapter 2 for more information on network simulation. See Also 13-198 init, adapt, train softmax Purpose Syntax softmax Soft max transfer function A = softmax(N) info = softmax(code) Description softmax is a transfer function. Transfer functions calculate a layer's output from its net input. softmax(N) takes one input argument, N - S x Q matrix of net input (column) vectors. and returns output vectors with elements between 0 and 1, but with their size relations intact. softmax('code') returns information about this function. These codes are defined: 'deriv' - Name of derivative function. 'name' - Full name. 'output' - Output range. 'active' - Active input range. compet does not have a derivative function. Examples Here we define a net input vector N, calculate the output, and plot both with bar graphs. n = [0; 1; -0.5; 0.5]; a = softmax(n); subplot(2,1,1), bar(n), ylabel('n') subplot(2,1,2), bar(a), ylabel('a') Network Use To change a network so that a layer uses softmax, set net.layers{i,j}.transferFcn to 'softmax'. Call sim to simulate the network with softmax. See newc or newpnn for simulation examples. See Also sim, compet 13-199 srchbac Purpose srchbac One-dimensional minimization using backtracking Syntax [a,gX,perf,retcode,delta,tol] = srchbac(net,X,Pd,Tl,Ai,Q,TS,dX,gX,perf,dperf,delta,tol,ch_perf) Description srchbac is a linear search routine. It searches in a given direction to locate the minimum of the performance function in that direction. It uses a technique called backtracking. srchbac(net,X,Pd,Tl,Ai,Q,TS,dX,gX,perf,dperf,delta,TOL,ch_perf) takes these inputs, net - Neural network. X - Vector containing current values of weights and biases. Pd - Delayed input vectors. Tl - Layer target vectors. Ai - Initial input delay conditions. Q - Batch size. TS - Time steps. dX - Search direction vector. gX - Gradient vector. perf - Performance value at current X. dperf - Slope of performance value at current X in direction of dX. delta - Initial step size. tol - Tolerance on search. ch_perf - Change in performance on previous step. and returns, A - Step size which minimizes performance. gX - Gradient at new minimum point. perf - Performance value at new minimum point. retcode - Return code which has three elements. The first two elements correspond to the number of function evaluations in the two stages of the search. The third element is a return code. These will have different 13-200 srchbac meanings for different search algorithms. Some may not be used in this function. 0 - normal; 1 - minimum step taken; 2 - maximum step taken; 3 - beta condition not met. delta - New initial step size. Based on the current step size. tol - New tolerance on search. Parameters used for the backstepping algorithm are: alpha - Scale factor which determines sufficient reduction in perf. beta - Scale factor which determines sufficiently large step size. low_lim - Lower limit on change in step size. up_lim - Upper limit on change in step size. maxstep - Maximum step length. minstep - Minimum step length. scale_tol - Parameter which relates the tolerance tol to the initial step size delta. Usually set to 20. The defaults for these parameters are set in the training function which calls it. See traincgf, traincgb, traincgp, trainbfg, trainoss. Dimensions for these variables are: Pd - No x Ni x TS cell array, each element P{i,j,ts} is a Dij x Q matrix. Tl - Nl x TS cell array, each element P{i,ts} is an Vi x Q matrix. Ai - Nl x LD cell array, each element Ai{i,k} is an Si x Q matrix. where Ni = net.numInputs Nl = net.numLayers LD = net.numLayerDelays Ri = net.inputs{i}.size Si = net.layers{i}.size Vi = net.targets{i}.size Dij = Ri * length(net.inputWeights{i,j}.delays) 13-201 srchbac Examples Here is a problem consisting of inputs P and targets T that we would like to solve with a network. P = [0 1 2 3 4 5]; T = [0 0 0 1 1 1]; Here a two-layer feed-forward network is created. The network's input ranges from [0 to 10]. The first layer has two tansig neurons, and the second layer has one logsig neuron. The traincgf network training function and the srchbac search function are to be used. Create and Test a Network net = newff([0 5],[2 1],{'tansig','logsig'},'traincgf'); a = sim(net,p) Train and Retest the Network net.trainParam.searchFcn = 'srchbac'; net.trainParam.epochs = 50; net.trainParam.show = 10; net.trainParam.goal = 0.1; net = train(net,p,t); a = sim(net,p) Network Use You can create a standard network that uses srchbac with newff, newcf, or newelm. To prepare a custom network to be trained with traincgf, using the line search function srchbac: 1 Set net.trainFcn to 'traincgf'. This will set net.trainParam to traincgf's default parameters. 2 Set net.trainParam.searchFcn to 'srchbac'. The srchbac function can be used with any of the following training functions: traincgf, traincgb, traincgp, trainbfg, trainoss. 13-202 srchbac Algorithm srchbac locates the minimum of the performance function in the search direction dX, using the backtracking algorithm described on page 126 and 328 of Dennis and Schnabel. (Numerical Methods for Unconstrained Optimization and Nonlinear Equations 1983). See Also srchbrc, srchcha, srchgol, srchhyb References Dennis, J. E., and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Englewood Cliffs, NJ: Prentice-Hall, 1983. 13-203 srchbre Purpose srchbre One-dimensional interval location using Brent's method Syntax [a,gX,perf,retcode,delta,tol] = srchbre(net,X,Pd,Tl,Ai,Q,TS,dX,gX,perf,dperf,delta,tol,ch_perf) Description srchbre is a linear search routine. It searches in a given direction to locate the minimum of the performance function in that direction. It uses a technique called Brent’s technique. srchbre(net,X,Pd,Tl,Ai,Q,TS,dX,gX,perf,dperf,delta,tol,ch_perf) takes these inputs, net - Neural network. X - Vector containing current values of weights and biases. Pd - Delayed input vectors. Tl - Layer target vectors. Ai - Initial input delay conditions. Q - Batch size. TS - Time steps. dX - Search direction vector. gX - Gradient vector. perf - Performance value at current X. dperf - Slope of performance value at current X in direction of dX. delta - Initial step size. tol - Tolerance on search. ch_perf - Change in performance on previous step. and returns, A - Step size which minimizes performance. gX - Gradient at new minimum point. perf - Performance value at new minimum point. retcode - Return code, which has three elements. The first two elements correspond to the number of function evaluations in the two stages of the search. The third element is a return code. These will have different 13-204 srchbre meanings for different search algorithms. Some may not be used in this function. 0 - normal; 1 - minimum step taken; 2 - maximum step taken; 3 - beta condition not met. delta - New initial step size. Based on the current step size. tol - New tolerance on search. Parameters used for the brent algorithm are: alpha - Scale factor which determines sufficient reduction in perf. beta - Scale factor which determines sufficiently large step size. bmax - Largest step size. scale_tol - Parameter which relates the tolerance tol to the initial step size delta. Usually set to 20. The defaults for these parameters are set in the training function which calls it. See traincgf, traincgb, traincgp, trainbfg, trainoss. Dimensions for these variables are: Pd - No x Ni x TS cell array, each element P{i,j,ts} is a Dij x Q matrix. Tl - Nl x TS cell array, each element P{i,ts} is an Vi x Q matrix. Ai - Nl x LD cell array, each element Ai{i,k} is an Si x Q matrix. where Ni = net.numInputs Nl = net.numLayers LD = net.numLayerDelays Ri = net.inputs{i}.size Si = net.layers{i}.size Vi = net.targets{i}.size Dij = Ri * length(net.inputWeights{i,j}.delays) Examples Here is a problem consisting of inputs P and targets T that we would like to solve with a network. P = [0 1 2 3 4 5]; T = [0 0 0 1 1 1]; 13-205 srchbre Here a two-layer feed-forward network is created. The network's input ranges from [0 to 10]. The first layer has two tansig neurons, and the second layer has one logsig neuron. The traincgf network training function and the srchbac search function are to be used. Create and Test a Network net = newff([0 5],[2 1],{'tansig','logsig'},'traincgf'); a = sim(net,p) Train and Retest the Network net.trainParam.searchFcn = 'srchbre'; net.trainParam.epochs = 50; net.trainParam.show = 10; net.trainParam.goal = 0.1; net = train(net,p,t); a = sim(net,p) Network Use You can create a standard network that uses srchbre with newff, newcf, or newelm. To prepare a custom network to be trained with traincgf, using the line search function srchbre: 1 Set net.trainFcn to 'traincgf'. This will set net.trainParam to traincgf's default parameters. 2 Set net.trainParam.searchFcn to 'srchbre'. The srchbre function can be used with any of the following training functions: traincgf, traincgb, traincgp, trainbfg, trainoss. Algorithm srchbre brackets the minimum of the performance function in the search direction dX, using Brent's algorithm described on page 46 of Scales (Introduction to Non-Linear Estimation 1985). It is a hybrid algorithm based on the golden section search and the quadratic approximation. See Also srchbac, srchcha, srchgol, srchhyb References Scales, L. E., Introduction to Non-Linear Optimization, New York: Springer-Verlag, 1985. 13-206 srchcha Purpose srchcha One-dimensional minimization using the method of Charalambous Syntax [a,gX,perf,retcode,delta,tol] = srchcha(net,X,Pd,Tl,Ai,Q,TS,dX,gX,perf,dperf,delta,tol,ch_perf) Description srchcha is a linear search routine. It searches in a given direction to locate the minimum of the performance function in that direction. It uses a technique based on the method of Charalambous. srchcha(net,X,Pd,Tl,Ai,Q,TS,dX,gX,perf,dperf,delta,tol,ch_perf) takes these inputs, net - Neural network. X - Vector containing current values of weights and biases. Pd - Delayed input vectors. Tl - Layer target vectors. Ai - Initial input delay conditions. Q - Batch size. TS - Time steps. dX - Search direction vector. gX - Gradient vector. perf - Performance value at current X. dperf - Slope of performance value at current X in direction of dX. delta - Initial step size. tol - Tolerance on search. ch_perf - Change in performance on previous step. and returns, A - Step size which minimizes performance. gX - Gradient at new minimum point. perf - Performance value at new minimum point. retcode - Return code, which has three elements. The first two elements correspond to the number of function evaluations in the two stages of the search. The third element is a return code. These will have different 13-207 srchcha meanings for different search algorithms. Some may not be used in this function. 0 - normal; 1 - minimum step taken; 2 - maximum step taken; 3 - beta condition not met. delta - New initial step size. Based on the current step size. TOL - New tolerance on search. Parameters used for the Charalambous algorithm are: alpha - Scale factor which determines sufficient reduction in perf. beta - Scale factor which determines sufficiently large step size. gama - Parameter to avoid small reductions in performance. Usually set to 0.1. scale_tol - Parameter which relates the tolerance tol to the initial step size delta. Usually set to 20. The defaults for these parameters are set in the training function which calls it. See traincgf, traincgb, traincgp, trainbfg, trainoss. Dimensions for these variables are: Pd - No x Ni x TS cell array, each element P{i,j,ts} is a Dij x Q matrix. Tl - Nl x TS cell array, each element P{i,ts} is an Vi x Q matrix. Ai - Nl x LD cell array, each element Ai{i,k} is an Si x Q matrix. where Ni = net.numInputs Nl = net.numLayers LD = net.numLayerDelays Ri = net.inputs{i}.size Si = net.layers{i}.size Vi = net.targets{i}.size Dij = Ri * length(net.inputWeights{i,j}.delays) Examples Here is a problem consisting of inputs P and targets T that we would like to solve with a network. P = [0 1 2 3 4 5]; T = [0 0 0 1 1 1]; 13-208 srchcha Here a two-layer feed-forward network is created. The network's input ranges from [0 to 10]. The first layer has two tansig neurons, and the second layer has one logsig neuron. The traincgf network training function and the srchcha search function are to be used. Create and Test a Network net = newff([0 5],[2 1],{'tansig','logsig'},'traincgf'); a = sim(net,p) Train and Retest the Network net.trainParam.searchFcn = 'srchcha'; net.trainParam.epochs = 50; net.trainParam.show = 10; net.trainParam.goal = 0.1; net = train(net,p,t); a = sim(net,p) Network Use You can create a standard network that uses srchcha with newff, newcf, or newelm. To prepare a custom network to be trained with traincgf, using the line search function srchcha: 1 Set net.trainFcn to 'traincgf'. This will set net.trainParam to traincgf's default parameters. 2 Set net.trainParam.searchFcn to 'srchcha'. The srchcha function can be used with any of the following training functions: traincgf, traincgb, traincgp, trainbfg, trainoss. Algorithm srchcha locates the minimum of the performance function in the search direction dX, using an algorithm based on the method described in Charalambous (IEEE Proc. vol. 139, no. 3, June 1992). See Also srchbac, srchbre, srchgol, srchhyb References Charalambous, C.,“Conjugate gradient algorithm for efficient training of artificial neural networks,” IEEE Proceedings, vol. 139, no. 3, pp. 301–310, 1992. 13-209 srchgol Purpose srchgol One-dimensional minimization using golden section search Syntax [a,gX,perf,retcode,delta,tol] = srchgol(net,X,Pd,Tl,Ai,Q,TS,dX,gX,perf,dperf,delta,tol,ch_perf) Description srchgol is a linear search routine. It searches in a given direction to locate the minimum of the performance function in that direction. It uses a technique called the golden section search. srchgol(net,X,Pd,Tl,Ai,Q,TS,dX,gX,perf,dperf,delta,tol,ch_perf) takes these inputs, net - Neural network. X - Vector containing current values of weights and biases. Pd - Delayed input vectors. Tl - Layer target vectors. Ai - Initial input delay conditions. Q - Batch size. TS - Time steps. dX - Search direction vector. gX - Gradient vector. perf - Performance value at current X. dperf - Slope of performance value at current X in direction of dX. delta - Initial step size. tol - Tolerance on search. ch_perf - Change in performance on previous step. and returns, A - Step size which minimizes performance. gX - Gradient at new minimum point. perf - Performance value at new minimum point. retcode - Return code, which has three elements. The first two elements correspond to the number of function evaluations in the two stages of the search. The third element is a return code. These will have different 13-210 srchgol meanings for different search algorithms. Some may not be used in this function. 0 - normal; 1 - minimum step taken; 2 - maximum step taken; 3 - beta condition not met. delta - New initial step size. Based on the current step size. tol - New tolerance on search. Parameters used for the golden section algorithm are: alpha - Scale factor which determines sufficient reduction in perf. bmax - Largest step size. scale_tol - Parameter which relates the tolerance tol to the initial step size delta. Usually set to 20. The defaults for these parameters are set in the training function which calls it. See traincgf, traincgb, traincgp, trainbfg, trainoss. Dimensions for these variables are: Pd - No x Ni x TS cell array, each element P{i,j,ts} is a Dij x Q matrix. Tl - Nl x TS cell array, each element P{i,ts} is an Vi x Q matrix. Ai - Nl x LD cell array, each element Ai{i,k} is an Si x Q matrix. where Ni = net.numInputs Nl = net.numLayers LD = net.numLayerDelays Ri = net.inputs{i}.size Si = net.layers{i}.size Vi = net.targets{i}.size Dij = Ri * length(net.inputWeights{i,j}.delays) Examples Here is a problem consisting of inputs P and targets T that we would like to solve with a network. P = [0 1 2 3 4 5]; T = [0 0 0 1 1 1]; 13-211 srchgol Here a two-layer feed-forward network is created. The network's input ranges from [0 to 10]. The first layer has two tansig neurons, and the second layer has one logsig neuron. The traincgf network training function and the srchgol search function are to be used. Create and Test a Network net = newff([0 5],[2 1],{'tansig','logsig'},'traincgf'); a = sim(net,p) Train and Retest the Network net.trainParam.searchFcn = 'srchgol'; net.trainParam.epochs = 50; net.trainParam.show = 10; net.trainParam.goal = 0.1; net = train(net,p,t); a = sim(net,p) Network Use You can create a standard network that uses srchgol with newff, newcf, or newelm. To prepare a custom network to be trained with traincgf, using the line search function srchgol: 1 Set net.trainFcn to 'traincgf'. This will set net.trainParam to traincgf's default parameters. 2 Set net.trainParam.searchFcn to 'srchgol'. The srchgol function can be used with any of the following training functions: traincgf, traincgb, traincgp, trainbfg, trainoss. Algorithm srchgol locates the minimum of the performance function in the search direction dX, using the golden section search. It is based on the algorithm as described on page 33 of Scales (Introduction to Non-Linear Estimation 1985). See Also srchbac, srchbre, srchcha, srchhyb References Scales, L. E.,Introduction to Non-Linear Optimization, New York: Springer-Verlag, 1985. 13-212 srchhyb Purpose srchhyb One-dimensional minimization using a hybrid bisection-cubic search Syntax [a,gX,perf,retcode,delta,tol] = srchhyb(net,X,P,T,Q,TS,dX,gX,perf,dperf,delta,tol,ch_perf) Description srchhyb is a linear search routine. It searches in a given direction to locate the minimum of the performance function in that direction. It uses a technique that is a combination of a bisection and a cubic interpolation. srchhyb(net,X,Pd,Tl,Ai,Q,TS,dX,gX,perf,dperf,delta,tol,ch_perf) takes these inputs, net - Neural network. X - Vector containing current values of weights and biases. Pd - Delayed input vectors. Tl - Layer target vectors. Ai - Initial input delay conditions. Q - Batch size. TS - Time steps. dX - Search direction vector. gX - Gradient vector. perf - Performance value at current X. dperf - Slope of performance value at current X in direction of dX. delta - Initial step size. tol - Tolerance on search. ch_perf - Change in performance on previous step. and returns, A - Step size which minimizes performance. gX - Gradient at new minimum point. perf - Performance value at new minimum point. retcode - Return code, which has three elements. The first two elements correspond to the number of function evaluations in the two stages of the search. The third element is a return code. These will have different 13-213 srchhyb meanings for different search algorithms. Some may not be used in this function. 0 - normal; 1 - minimum step taken; 2 - maximum step taken; 3 - beta condition not met. delta - New initial step size. Based on the current step size. tol - New tolerance on search. Parameters used for the hybrid bisection-cubic algorithm are: alpha - Scale factor which determines sufficient reduction in perf. beta - Scale factor which determines sufficiently large step size. bmax - Largest step size. scale_tol - Parameter which relates the tolerance tol to the initial step size delta. Usually set to 20. The defaults for these parameters are set in the training function which calls it. See traincgf, traincgb, traincgp, trainbfg, trainoss. Dimensions for these variables are: Pd - No x Ni x TS cell array, each element P{i,j,ts} is a Dij x Q matrix. Tl - Nl x TS cell array, each element P{i,ts} is an Vi x Q matrix. Ai - Nl x LD cell array, each element Ai{i,k} is an Si x Q matrix. where Ni = net.numInputs Nl = net.numLayers LD = net.numLayerDelays Ri = net.inputs{i}.size Si = net.layers{i}.size Vi = net.targets{i}.size Dij = Ri * length(net.inputWeights{i,j}.delays) Examples Here is a problem consisting of inputs P and targets T that we would like to solve with a network. P = [0 1 2 3 4 5]; T = [0 0 0 1 1 1]; 13-214 srchhyb Here a two-layer feed-forward network is created. The network's input ranges from [0 to 10]. The first layer has two tansig neurons, and the second layer has one logsig neuron. The traincgf network training function and the srchhyb search function are to be used. Create and Test a Network net = newff([0 5],[2 1],{'tansig','logsig'},'traincgf'); a = sim(net,p) Train and Retest the Network net.trainParam.searchFcn = 'srchhyb'; net.trainParam.epochs = 50; net.trainParam.show = 10; net.trainParam.goal = 0.1; net = train(net,p,t); a = sim(net,p) Network Use You can create a standard network that uses srchhyb with newff, newcf, or newelm. To prepare a custom network to be trained with traincgf, using the line search function srchhyb: 1 Set net.trainFcn to 'traincgf'. This will set net.trainParam to traincgf's default parameters. 2 Set net.trainParam.searchFcn to 'srchhyb'. The srchhyb function can be used with any of the following training functions: traincgf, traincgb, traincgp, trainbfg, trainoss. Algorithm srchhyb locates the minimum of the performance function in the search direction dX, using the hybrid bisection-cubic interpolation algorithm described on page 50 of Scales (Introduction to Non-Linear Estimation 1985). See Also srchbac, srchbre, srchcha, srchgol References Scales, L. E., Introduction to Non-Linear Optimization, New York: Springer-Verlag, 1985. 13-215 sse Purpose Syntax sse Sum squared error performance function perf = sse(e,x,pp) perf = sse(e,net,pp) info = sse(code) Description sse is a network performance function. It measures performance according to the sum of squared errors. sse(E,X,PP) takes from one to three arguments, E - Matrix or cell array of error vector(s). X - Vector of all weight and bias values (ignored). PP - Performance parameters (ignored). and returns the sum squared error. sse(E,net,PP) can take an alternate argument to X, net - Neural network from which X can be obtained (ignored). sse(code) returns useful information for each code string: 'deriv' - Name of derivative function. 'name' - Full name. 'pnames' - Names of training parameters. 'pdefaults' - Default training parameters. Examples Here a two layer feed-forward is created with a 1-element input ranging from -10 to 10, four hidden tansig neurons, and one purelin output neuron. net = newff([-10 10],[4 1],{'tansig','purelin'}); Here the network is given a batch of inputs P. The error is calculated by subtracting the output A from target T. Then the sum squared error is calculated. 13-216 sse p = [-10 -5 0 5 10]; t = [0 0 1 1 1]; y = sim(net,p) e = t-y perf = sse(e) Note that sse can be called with only one argument because the other arguments are ignored. sse supports those arguments to conform to the standard performance function argument list. Network Use To prepare a custom network to be trained with sse set net.performFcn to 'sse'. This will automatically set net.performParam to the empty matrix [], as sse has no performance parameters. Calling train or adapt will result in sse being used to calculate performance. See Also dsse 13-217 sumsqr Purpose sumsqr Sum squared elements of a matrix Syntax sumsqr(m) Description sumsqr(M) returns the sum of the squared elements in M. Examples 13-218 s = sumsqr([1 2;3 4]) tansig Purpose Graph and Symbol tansig Hyperbolic tangent sigmoid transfer function a +1 0 n -1 a = tansig(n) Tan-Sigmoid Transfer Function Syntax A = tansig(N) info = tansig(code) Description tansig is a transfer function. Transfer functions calculate a layer's output from its net input. tansig(N) takes one input, N - S x Q matrix of net input (column) vectors. and returns each element of N squashed between -1 and 1. tansig(code) return useful information for each code string: 'deriv' - Name of derivative function. 'name' - Full name. 'output' - Output range. 'active' - Active input range. tansig is named after the hyperbolic tangent which has the same shape. However, tanh may be more accurate and is recommended for applications that require the hyperbolic tangent. Examples Here is the code to create a plot of the tansig transfer function. n = -5:0.1:5; a = tansig(n); plot(n,a) 13-219 tansig Network Use You can create a standard network that uses tansig by calling newff or newcf. To change a network so a layer uses tansig set net.layers{i,j}.transferFcn to 'tansig'. In either case, call sim to simulate the network with tansig. See newff or newcf for simulation examples. Algorithm tansig(N) calculates its output according to: n = 2/(1+exp(-2*n))-1 This is mathematically equivalent to tanh(N). It differs in that it runs faster than the MATLAB implementation of tanh, but the results can have very small numerical differences. This function is a good trade off for neural networks, where speed is important and the exact shape of the transfer function is not. See Also sim, dtansig, logsig References Vogl, T. P., J.K. Mangis, A.K. Rigler, W.T. Zink, and D.L. Alkon, “Accelerating the convergence of the backpropagation method,” Biological Cybernetics, vol. 59, pp. 257-263, 1988. 13-220 train Purpose Syntax train Train a neural network [net,tr] = train(NET,P,T,Pi,Ai) [net,tr] = train(NET,P,T,Pi,Ai,VV,TV) To Get Help Type help network/train Description train trains a network net according to net.trainFcn and net.trainParam. train(net,P,T,Pi,Ai) takes, net - Network. P - Network inputs. T - Network targets, default = zeros. Pi - Initial input delay conditions, default = zeros. Ai - Initial layer delay conditions, default = zeros. and returns, net - New network. TR - Training record (epoch and perf). Note that T is optional and need only be used for networks that require targets. Pi and Pf are also optional and need only be used for networks that have input or layer delays. train's signal arguments can have two formats: cell array or matrix. The cell array format is easiest to describe. It is most convenient for networks with multiple inputs and outputs, and allows sequences of inputs to be presented: P - Ni x TS cell array, each element P{i,ts} is an Ri x Q matrix. T - Nt x TS cell array, each element P{i,ts} is an Vi x Q matrix. Pi - Ni x ID cell array, each element Pi{i,k} is an Ri x Q matrix. Ai - Nl x LD cell array, each element Ai{i,k} is an Si x Q matrix. 13-221 train where Ni = net.numInputs Nl = net.numLayers Nt = net.numTargets ID = net.numInputDelays LD = net.numLayerDelays TS = Number of time steps Q = Batch size Ri = net.inputs{i}.size Si = net.layers{i}.size Vi = net.targets{i}.size The columns of Pi, Pf, Ai, and Af are ordered from the oldest delay condition to the most recent: Pi{i,k} = input i at time ts=k-ID. Pf{i,k} = input i at time ts=TS+k-ID. Ai{i,k} = layer output i at time ts=k-LD. Af{i,k} = layer output i at time ts=TS+k-LD. The matrix format can be used if only one time step is to be simulated (TS = 1). It is convenient for network's with only one input and output, but can be used with networks that have more. Each matrix argument is found by storing the elements of the corresponding cell array argument into a single matrix: P - (sum of Ri) x Q matrix T - (sum of Vi) x Q matrix Pi - (sum of Ri) x (ID*Q) matrix. Ai - (sum of Si) x (LD*Q) matrix. 13-222 train train(net,P,T,Pi,Ai,VV,TV) takes optional structures of validation and test vectors, VV.P, TV.P - Validation/test inputs. VV.T, VV.T - Validation/test targets, default = zeros. VV.Pi, VV.Pi - Validation/test initial input delay conditions, default= zeros. VV.Ai, VV.Ai - Validation/test layer delay conditions, default = zeros. The validation vectors are used to stop training early if further training on the primary vectors will hurt generalization to the validation vectors. Test vector performance can be used to measure how well the network generalizes beyond primary and validation vectors. If VV.T, VV.Pi, or VV.Ai are set to an empty matrix or cell array, default values will be used. The same is true for VT.T, VT.Pi, VT.Ai. Examples Here input P and targets T define a simple function which we can plot: p = [0 1 2 3 4 5 6 7 8]; t = [0 0.84 0.91 0.14 -0.77 -0.96 -0.28 0.66 0.99]; plot(p,t,'o') Here newff is used to create a two layer feed forward network. The network will have an input (ranging from 0 to 8), followed by a layer of 10 tansig neurons, followed by a layer with 1 purelin neuron. trainlm backpropagation is used. The network is also simulated. net = newff([0 8],[10 1],{'tansig' 'purelin'},'trainlm'); y1 = sim(net,p) plot(p,t,'o',p,y1,'x') Here the network is trained for up to 50 epochs to a error goal of 0.01, and then resimulated. net.trainParam.epochs = 50; net.trainParam.goal = 0.01; net = train(net,p,t); y2 = sim(net,p) plot(p,t,'o',p,y1,'x',p,y2,'*') Algorithm train calls the function indicated by net.trainFcn, using the adaption parameter values indicated by net.trainParam. 13-223 train Typically one epoch of training is defined as a single presentation of all input vectors to the network. The network is then updated according to the results of all those presentations. Training occurs until a maximum number of epochs occurs, the performance goal is met, or any other stopping condition of the function net.trainFcn occurs. Some training functions depart from this norm by presenting only one input vector (or sequence) each epoch. An input vector (or sequence) is chosen randomly each epoch from concurrent input vectors (or sequences). newc and newsom return networks that use trainwb1, a training function that presents only one input vector. See Also 13-224 sim, init, adapt trainbfg Purpose Syntax trainbfg BFGS quasi-Newton backpropagation [net,tr] = trainbfg(net,Pd,Tl,Ai,Q,TS,VV) info = trainbfg(code) Description trainbfg is a network training function that updates weight and bias values according to the BFGS quasi-Newton method. trainbfg(net,Pd,Tl,Ai,Q,TS,VV,TV) takes these inputs, net - Neural network. Pd - Delayed input vectors. Tl - Layer target vectors. Ai - Initial input delay conditions. Q - Batch size. TS - Time steps. VV - Either empty matrix [] or structure of validation vectors. TV - Either empty matrix [] or structure of test vectors. and returns, net - Trained network. TR - Training record of various values over each epoch: TR.epoch - Epoch number. TR.perf - Training performance. TR.vperf - Validation performance. TR.tperf - Test performance. 13-225 trainbfg Training occurs according to the trainbfg's training parameters, shown here with their default values: net.trainParam.epochs 100 net.trainParam.show 25 net.trainParam.goal 0 net.trainParam.time inf net.trainParam.min_grad 1e-6 net.trainParam.max_fail 5 net.trainParam.searchFcn 'srchcha' Maximum number of epochs to train Epochs between showing progress Performance goal Maximum time to train in seconds Minimum performance gradient Maximum validation failures Name of line search routine to use. Parameters related to line search methods (not all used for all methods): net.trainParam.scal_tol 20 Divide into delta to determine tolerance for linear search. net.trainParam.alpha 0.001 Scale factor which determines sufficient reduction in perf. net.trainParam.beta 0.1 Scale factor which determines sufficiently large step size. net.trainParam.delta 0.01 Initial step size in interval location step. net.trainParam.gama 0.1 Parameter to avoid small reductions in performance. Usually set to 0.1. (See use in srch_cha.) net.trainParam.low_lim 0.1 Lower limit on change in step size. net.trainParam.up_lim 0.5 Upper limit on change in step size. net.trainParam.maxstep 100 Maximum step length. net.trainParam.minstep 1.0e-6 Minimum step length. net.trainParam.bmax 13-226 26 Maximum step size. trainbfg Dimensions for these variables are: Pd - No x Ni x TS cell array, each element P{i,j,ts} is a Dij x Q matrix. Tl - Nl x TS cell array, each element P{i,ts} is a Vi x Q matrix. Ai - Nl x LD cell array, each element Ai{i,k} is an Si x Q matrix. where Ni = net.numInputs Nl = net.numLayers LD = net.numLayerDelays Ri = net.inputs{i}.size Si = net.layers{i}.size Vi = net.targets{i}.size Dij = Ri * length(net.inputWeights{i,j}.delays) If VV is not [], it must be a structure of validation vectors, VV.PD - Validation delayed inputs. VV.Tl - Validation layer targets. VV.Ai - Validation initial input conditions. VV.Q - Validation batch size. VV.TS - Validation time steps. which is used to stop training early if the network performance on the validation vectors fails to improve or remains the same for max_fail epochs in a row. If TV is not [], it must be a structure of validation vectors, TV.PD - Validation delayed inputs. TV.Tl - Validation layer targets. TV.Ai - Validation initial input conditions. TV.Q - Validation batch size. TV.TS - Validation time steps. which is used to test the generalization capability of the trained network. 13-227 trainbfg trainbfg(code) returns useful information for each code string: 'pnames' - Names of training parameters. 'pdefaults' - Default training parameters. Examples Here is a problem consisting of inputs P and targets T that we would like to solve with a network. P = [0 1 2 3 4 5]; T = [0 0 0 1 1 1]; Here a two-layer feed-forward network is created. The network's input ranges from [0 to 10]. The first layer has two tansig neurons, and the second layer has one logsig neuron. The trainbfg network training function is to be used. Create and Test a Network net = newff([0 5],[2 1],{'tansig','logsig'},'trainbfg'); a = sim(net,p) Train and Retest the Network net.trainParam.epochs = 50; net.trainParam.show = 10; net.trainParam.goal = 0.1; net = train(net,p,t); a = sim(net,p) See newff, newcf, and newelm for other examples Network Use You can create a standard network that uses trainbfg with newff, newcf, or newelm. To prepare a custom network to be trained with trainbfg: 1 Set net.trainFcn to 'trainbfg'. This will set net.trainParam to trainbfg's default parameters. 2 Set net.trainParam properties to desired values. In either case, calling train with the resulting network will train the network with trainbfg. 13-228 trainbfg Algorithm trainbfg can train any network as long as its weight, net input, and transfer functions have derivative functions. Backpropagation is used to calculate derivatives of performance perf with respect to the weight and bias variables X. Each variable is adjusted according to the following: X = X + a*dX; where dX is the search direction. The parameter a is selected to minimize the performance along the search direction. The line search function searchFcn is used to locate the minimum point. The first search direction is the negative of the gradient of performance. In succeeding iterations the search direction is computed according to the following formula: dX = -H\gX; where gX is the gradient and H is an approximate Hessian matrix. See page 119 of Gill, Murray & Wright (Practical Optimization 1981) for a more detailed discussion of the BFGS quasi-Newton method. Training stops when any of these conditions occur: 1 The maximum number of epochs (repetitions) is reached. 2 The maximum amount of time has been exceeded. 3 Performance has been minimized to the goal. 4 The performance gradient falls below mingrad. 5 Validation performance has increased more than max_fail times since the last time it decreased (when using validation). See Also newff, newcf, traingdm, traingda, traingdx, trainlm, trainrp, traincgf, traincgb, trainscg, traincgp, trainoss. References Gill, P. E.,W. Murray, and M. H. Wright, Practical Optimization, New York: Academic Press, 1981. 13-229 trainbr Purpose Syntax trainbr Bayesian Regulation backpropagation [net,tr] = trainbr(net,Pd,Tl,Ai,Q,TS,VV) info = trainbr(code) Description trainbr is a network training function that updates the weight and bias values according to Levenberg-Marquardt optimization. It minimizes a combination of squared errors and weights, and then determines the correct combination so as to produce a network which generalizes well. The process is called Bayesian regularization. trainbr(net,Pd,Tl,Ai,Q,TS,VV,TV) takes these inputs, net - Neural network. Pd - Delayed input vectors. Tl - Layer target vectors. Ai - Initial input delay conditions. Q - Batch size. TS - Time steps. VV - Either empty matrix [] or structure of validation vectors. and returns, net - Trained network. TR - Training record of various values over each epoch: TR.epoch - Epoch number. TR.perf - Training performance. TR.vperf - Validation performance. TR.tperf - Test performance. TR.mu 13-230 - Adaptive mu value. trainbr Training occurs according to the trainlm's training parameters, shown here with their default values: 100 Maximum number of epochs to train net.trainParam.epochs 0 Performance goal net.trainParam.goal net.trainParam.mu 0.005 Marquardt adjustment parameter net.trainParam.mu_dec 0.1 Decrease factor for mu net.trainParam.mu_inc 10 Increase factor for mu net.trainParam.mu_max 1e-10 Maximum value for mu net.trainParam.max_fail 5 Maximum validation failures net.trainParam.mem_reduc 1 Factor to use for memory/speed trade-off net.trainParam.min_grad 1e-10 Minimum performance gradient net.trainParam.show 25 Epochs between showing progress net.trainParam.time inf Maximum time to train in seconds Dimensions for these variables are: Pd - No x Ni x TS cell array, each element P{i,j,ts} is a Dij x Q matrix. Tl - Nl x TS cell array, each element P{i,ts} is a Vi x Q matrix. Ai - Nl x LD cell array, each element Ai{i,k} is an Si x Q matrix. where Ni = net.numInputs Nl = net.numLayers LD = net.numLayerDelays Ri = net.inputs{i}.size Si = net.layers{i}.size Vi = net.targets{i}.size Dij = Ri * length(net.inputWeights{i,j}.delays) 13-231 trainbr If VV is not [], it must be a structure of validation vectors, VV.PD - Validation delayed inputs. VV.Tl - Validation layer targets. VV.Ai - Validation initial input conditions. VV.Q - Validation batch size. VV.TS - Validation time steps. which is normally used to stop training early if the network performance on the validation vectors fails to improve or remains the same for max_fail epochs in a row. This early stopping is not used for trainbr, but the validation performance is computed for analysis purposes if VV is not []. trainbr(code) returns useful information for each code string: 'pnames' - Names of training parameters. 'pdefaults' - Default training parameters. Examples Here is a problem consisting of inputs p and targets t that we would like to solve with a network. It involves fitting a noisy sine wave. p = [-1:.05:1]; t = sin(2*pi*p)+0.1*randn(size(p)); Here a two-layer feed-forward network is created. The network's input ranges from [-1 to 1]. The first layer has 20 tansig neurons, the second layer has one purelin neuron. The trainbr network training function is to be used. The plot of the resulting network output should show a smooth response, without overfitting. Create a Network net=newff([-1 1],[20,1],{'tansig','purelin'},'trainbr'); Train and Test the Network net.trainParam.epochs = 50; net.trainParam.show = 10; net = train(net,p,t); a = sim(net,p) plot(p,a,p,t,'+') 13-232 trainbr Network Use You can create a standard network that uses trainbr with newff, newcf, or newelm. To prepare a custom network to be trained with trainbr: 1 Set net.trainFcn to 'trainlm'. This will set net.trainParam to trainbr's default parameters. 2 Set net.trainParam properties to desired values. In either case, calling train with the resulting network will train the network with trainbr. See newff, newcf, and newelm for examples. Algorithm trainbr can train any network as long as its weight, net input, and transfer functions have derivative functions. Bayesian regularization minimizes a linear combination of squared errors and weights. It also modifies the linear combination so that at the end of training the resulting network has good generalization qualities. See MacKay (Neural Computation, vol. 4, no. 3, 1992, pp. 415-447) and Foresee and Hagan (Proceedings of the International Joint Conference on Neural Networks, June, 1997) for more detailed discussions of Bayesian regularization. This Bayesian regularization takes place within the Levenberg-Marquardt algorithm. Backpropagation is used to calculate the Jacobian jX of performance perf with respect to the weight and bias variables X. Each variable is adjusted according to Levenberg-Marquardt, jj = jX * jX je = jX * E dX = -(jj+I*mu) \ je where E is all errors and I is the identity matrix. The adaptive value mu is increased by mu_inc until the change shown above results in a reduced performance value. The change is then made to the network and mu is decreased by mu_dec. The parameter mem_reduc indicates how to use memory and speed to calculate the Jacobian jX. If mem_reduc is 1, then trainlm runs the fastest, but can require a lot of memory. Increase mem_reduc to 2, cuts some of the memory required by a factor of two, but slows trainlm somewhat. Higher values 13-233 trainbr continue to decrease the amount of memory needed and increase the training times. Training stops when any of these conditions occur: 1 The maximum number of epochs (repetitions) is reached. 2 The maximum amount of time has been exceeded. 3 Performance has been minimized to the goal. 4 The performance gradient falls below mingrad. 5 mu exceeds mu_max. 6 Validation performance has increased more than max_fail times since the last time it decreased (when using validation). See Also newff, newcf, traingdm, traingda, traingdx, trainlm, trainrp, traincgf, traincgb, trainscg, traincgp, trainoss References Foresee, F. D., and M. T. Hagan, “Gauss-Newton approximation to Bayesian regularization,” Proceedings of the 1997 International Joint Conference on Neural Networks, 1997. MacKay, D. J. C., “Bayesian interpolation,” Neural Computation, vol. 4, no. 3, pp. 415-447, 1992. 13-234 traincgb Purpose Syntax traincgb Conjugate gradient backpropagation with Powell-Beale restarts [net,tr] = traincgb(net,Pd,Tl,Ai,Q,TS,VV) info = traincgb(code) Description traincgb is a network training function that updates weight and bias values according to the conjugate gradient backpropagation with Powell-Beale restarts. traincgb(net,Pd,Tl,Ai,Q,TS,VV,TV) takes these inputs, net - Neural network. Pd - Delayed input vectors. Tl - Layer target vectors. Ai - Initial input delay conditions. Q - Batch size. TS - Time steps. VV - Either empty matrix [] or structure of validation vectors. TV - Either empty matrix [] or structure of test vectors. and returns, net - Trained network. TR - Training record of various values over each epoch: TR.epoch - Epoch number. TR.perf - Training performance. TR.vperf - Validation performance. TR.tperf - Test performance. 13-235 traincgb Training occurs according to the traincgb's training parameters, shown here with their default values: net.trainParam.epochs 100 net.trainParam.show 25 net.trainParam.goal 0 net.trainParam.time inf net.trainParam.min_grad 1e-6 net.trainParam.max_fail 5 net.trainParam.searchFcn 'srchcha' Maximum number of epochs to train Epochs between showing progress Performance goal Maximum time to train in seconds Minimum performance gradient Maximum validation failures Name of line search routine to use. Parameters related to line search methods (not all used for all methods): net.trainParam.scal_tol 20 Divide into delta to determine tolerance for linear search. net.trainParam.alpha 0.001 Scale factor which determines sufficient reduction in perf. net.trainParam.beta 0.1 Scale factor which determines sufficiently large step size. net.trainParam.delta 0.01 Initial step size in interval location step. net.trainParam.gama 0.1 Parameter to avoid small reductions in performance. Usually set to 0.1. (See use in srch_cha.) net.trainParam.low_lim 0.1 Lower limit on change in step size. net.trainParam.up_lim 0.5 Upper limit on change in step size. net.trainParam.maxstep 100 Maximum step length. net.trainParam.minstep 1.0e-6 Minimum step length. net.trainParam.bmax 13-236 26 Maximum step size. traincgb Dimensions for these variables are: Pd - No x Ni x TS cell array, each element P{i,j,ts} is a Dij x Q matrix. Tl - Nl x TS cell array, each element P{i,ts} is a Vi x Q matrix. Ai - Nl x LD cell array, each element Ai{i,k} is an Si x Q matrix. where Ni = net.numInputs Nl = net.numLayers LD = net.numLayerDelays Ri = net.inputs{i}.size Si = net.layers{i}.size Vi = net.targets{i}.size Dij = Ri * length(net.inputWeights{i,j}.delays) If VV is not [], it must be a structure of validation vectors, VV.PD - Validation delayed inputs. VV.Tl - Validation layer targets. VV.Ai - Validation initial input conditions. VV.Q - Validation batch size. VV.TS - Validation time steps. which is used to stop training early if the network performance on the validation vectors fails to improve or remains the same for max_fail epochs in a row. If TV is not [], it must be a structure of validation vectors, TV.PD - Validation delayed inputs. TV.Tl - Validation layer targets. TV.Ai - Validation initial input conditions. TV.Q - Validation batch size. TV.TS - Validation time steps. which is used to test the generalization capability of the trained network. 13-237 traincgb traincgb(code) returns useful information for each code string: 'pnames' - Names of training parameters. 'pdefaults' - Default training parameters. Examples Here is a problem consisting of inputs P and targets T that we would like to solve with a network. P = [0 1 2 3 4 5]; T = [0 0 0 1 1 1]; Here a two-layer feed-forward network is created. The network's input ranges from [0 to 10]. The first layer has two tansig neurons, and the second layer has one logsig neuron. The traincgb network training function is to be used. Create and Test a Network net = newff([0 5],[2 1],{'tansig','logsig'},'traincgb'); a = sim(net,p) Train and Retest the Network net.trainParam.epochs = 50; net.trainParam.show = 10; net.trainParam.goal = 0.1; net = train(net,p,t); a = sim(net,p) See newff, newcf, and newelm for other examples. Network Use You can create a standard network that uses traincgb with newff, newcf, or newelm. To prepare a custom network to be trained with traincgb: 1 Set net.trainFcn to 'traincgb'. This will set net.trainParam to traincgb's default parameters. 2 Set net.trainParam properties to desired values. In either case, calling train with the resulting network will train the network with traincgb. 13-238 traincgb Algorithm traincgb can train any network as long as its weight, net input, and transfer functions have derivative functions. Backpropagation is used to calculate derivatives of performance perf with respect to the weight and bias variables X. Each variable is adjusted according to the following: X = X + a*dX; where dX is the search direction. The parameter a is selected to minimize the performance along the search direction. The line search function searchFcn is used to locate the minimum point. The first search direction is the negative of the gradient of performance. In succeeding iterations the search direction is computed from the new gradient and the previous search direction according to the formula: dX = -gX + dX_old*Z; where gX is the gradient. The parameter Z can be computed in several different ways. The Powell-Beale variation of conjugate gradient is distinguished by two features. First, the algorithm uses a test to determine when to reset the search direction to the negative of the gradient. Second, the search direction is computed from the negative gradient, the previous search direction, and the last search direction before the previous reset. See Powell, Mathematical Programming, Vol. 12 (1977) pp. 241-254, for a more detailed discussion of the algorithm. Training stops when any of these conditions occur: 1 The maximum number of epochs (repetitions) is reached. 2 The maximum amount of time has been exceeded. 3 Performance has been minimized to the goal. 4 The performance gradient falls below mingrad. 5 Validation performance has increased more than max_fail times since the last time it decreased (when using validation). See Also newff, newcf, traingdm, traingda, traingdx, trainlm, traincgp, traincgf, traincgb, trainscg, trainoss, trainbfg References Powell, M. J. D.,“Restart procedures for the conjugate gradient method,” Mathematical Programming, vol. 12, pp. 241-254, 1977. 13-239 traincgf Purpose Syntax traincgf Conjugate gradient backpropagation with Fletcher-Reeves updates [net,tr] = traincgf(net,Pd,Tl,Ai,Q,TS,VV) info = traincgf(code) Description traincgf is a network training function that updates weight and bias values according to the conjugate gradient backpropagation with Fletcher-Reeves updates. traincgf(net,Pd,Tl,Ai,Q,TS,VV,TV) takes these inputs, net - Neural network. Pd - Delayed input vectors. Tl - Layer target vectors. Ai - Initial input delay conditions. Q - Batch size. TS - Time steps. VV - Either empty matrix [] or structure of validation vectors. TV - Either empty matrix [] or structure of test vectors. and returns, net - Trained network. TR - Training record of various values over each epoch: TR.epoch - Epoch number. TR.perf - Training performance. TR.vperf - Validation performance. TR.tperf - Test performance. 13-240 traincgf Training occurs according to the traincgf's training parameters, shown here with their default values: net.trainParam.epochs 100 net.trainParam.show 25 net.trainParam.goal 0 net.trainParam.time inf net.trainParam.min_grad 1e-6 net.trainParam.max_fail 5 net.trainParam.searchFcn 'srchcha' Maximum number of epochs to train Epochs between showing progress Performance goal Maximum time to train in seconds Minimum performance gradient Maximum validation failures Name of line search routine to use Parameters related to line search methods (not all used for all methods): net.trainParam.scal_tol 20 Divide into delta to determine tolerance for linear search. net.trainParam.alpha 0.001 Scale factor which determines sufficient reduction in perf. net.trainParam.beta 0.1 Scale factor which determines sufficiently large step size. net.trainParam.delta 0.01 Initial step size in interval location step. net.trainParam.gama 0.1 Parameter to avoid small reductions in performance. Usually set to 0.1. (See use in srch_cha.) net.trainParam.low_lim 0.1 Lower limit on change in step size. net.trainParam.up_lim 0.5 Upper limit on change in step size. net.trainParam.maxstep 100 Maximum step length. net.trainParam.minstep 1.0e-6 Minimum step length. net.trainParam.bmax 26 Maximum step size. 13-241 traincgf Dimensions for these variables are: Pd - No x Ni x TS cell array, each element P{i,j,ts} is a Dij x Q matrix. Tl - Nl x TS cell array, each element P{i,ts} is a Vi x Q matrix. Ai - Nl x LD cell array, each element Ai{i,k} is an Si x Q matrix. where Ni = net.numInputs Nl = net.numLayers LD = net.numLayerDelays Ri = net.inputs{i}.size Si = net.layers{i}.size Vi = net.targets{i}.size Dij = Ri * length(net.inputWeights{i,j}.delays) If VV is not [], it must be a structure of validation vectors, VV.PD - Validation delayed inputs. VV.Tl - Validation layer targets. VV.Ai - Validation initial input conditions. VV.Q - Validation batch size. VV.TS - Validation time steps. which is used to stop training early if the network performance on the validation vectors fails to improve or remains the same for max_fail epochs in a row. If TV is not [], it must be a structure of validation vectors, TV.PD - Validation delayed inputs. TV.Tl - Validation layer targets. TV.Ai - Validation initial input conditions. TV.Q - Validation batch size. TV.TS - Validation time steps. which is used to test the generalization capability of the trained network. 13-242 traincgf traincgf(code) returns useful information for each code string: 'pnames' - Names of training parameters. 'pdefaults' - Default training parameters. Examples Here is a problem consisting of inputs P and targets T that we would like to solve with a network. P = [0 1 2 3 4 5]; T = [0 0 0 1 1 1]; Here a two-layer feed-forward network is created. The network's input ranges from [0 to 10]. The first layer has two tansig neurons, and the second layer has one logsig neuron. The traincgf network training function is to be used. Create and Test a Network net = newff([0 5],[2 1],{'tansig','logsig'},'traincgf'); a = sim(net,p) Train and Retest the Network net.trainParam.epochs = 50; net.trainParam.show = 10; net.trainParam.goal = 0.1; net = train(net,p,t); a = sim(net,p) See newff, newcf, and newelm for other examples. Network Use You can create a standard network that uses traincgf with newff, newcf, or newelm. To prepare a custom network to be trained with traincgf: 1 Set net.trainFcn to 'traincgf'. This will set net.trainParam to traincgf's default parameters. 2 Set net.trainParam properties to desired values. In either case, calling train with the resulting network will train the network with traincgf. 13-243 traincgf Algorithm traincgf can train any network as long as its weight, net input, and transfer functions have derivative functions. Backpropagation is used to calculate derivatives of performance perf with respect to the weight and bias variables X. Each variable is adjusted according to the following: X = X + a*dX; where dX is the search direction. The parameter a is selected to minimize the performance along the search direction. The line search function searchFcn is used to locate the minimum point. The first search direction is the negative of the gradient of performance. In succeeding iterations the search direction is computed from the new gradient and the previous search direction, according to the formula: dX = -gX + dX_old*Z; where gX is the gradient. The parameter Z can be computed in several different ways. For the Fletcher-Reeves variation of conjugate gradient it is computed according to Z=normnew_sqr/norm_sqr; where norm_sqr is the norm square of the previous gradient and normnew_sqr is the norm square of the current gradient. See page 78 of Scales (Introduction to Non-Linear Optimization 1985) for a more detailed discussion of the algorithm. Training stops when any of these conditions occur: 1 The maximum number of epochs (repetitions) is reached. 2 The maximum amount of time has been exceeded. 3 Performance has been minimized to the goal. 4 The performance gradient falls below mingrad. 5 Validation performance has increased more than max_fail times since the last time it decreased (when using validation). 13-244 traincgf See Also newff, newcf, traingdm, traingda, traingdx, trainlm, traincgp, traincgb, trainscg, traincgp, trainoss, trainbfg References Scales, L. E.,Introduction to Non-Linear Optimization, New York: Springer-Verlag, 1985. 13-245 traincgp Purpose Syntax traincgp Conjugate gradient backpropagation with Polak-Ribiere updates [net,tr] = traincgp(net,Pd,Tl,Ai,Q,TS,VV) info = traincgp(code) Description traincgp is a network training function that updates weight and bias values according to the conjugate gradient backpropagation with Polak-Ribiere updates. traincgp(net,Pd,Tl,Ai,Q,TS,VV,TV) takes these inputs, net - Neural network. Pd - Delayed input vectors. Tl - Layer target vectors. Ai - Initial input delay conditions. Q - Batch size. TS - Time steps. VV - Either empty matrix [] or structure of validation vectors. TV - Either empty matrix [] or structure of test vectors. and returns, net - Trained network. TR - Training record of various values over each epoch: TR.epoch - Epoch number. TR.perf - Training performance. TR.vperf - Validation performance. TR.tperf - Test performance. 13-246 traincgp Training occurs according to the traincgp's training parameters shown here with their default values: net.trainParam.epochs 100 net.trainParam.show 25 net.trainParam.goal 0 net.trainParam.time inf net.trainParam.min_grad 1e-6 net.trainParam.max_fail 5 net.trainParam.searchFcn 'srchcha' Maximum number of epochs to train Epochs between showing progress Performance goal Maximum time to train in seconds Minimum performance gradient Maximum validation failures Name of line search routine to use Parameters related to line search methods (not all used for all methods): net.trainParam.scal_tol 20 Divide into delta to determine tolerance for linear search. net.trainParam.alpha 0.001 Scale factor which determines sufficient reduction in perf. net.trainParam.beta 0.1 Scale factor which determines sufficiently large step size. net.trainParam.delta 0.01 Initial step size in interval location step. net.trainParam.gama 0.1 Parameter to avoid small reductions in performance. Usually set to 0.1. (See use in srch_cha.) net.trainParam.low_lim 0.1 Lower limit on change in step size. net.trainParam.up_lim 0.5 Upper limit on change in step size. net.trainParam.maxstep 100 Maximum step length. net.trainParam.minstep 1.0e-6 Minimum step length. net.trainParam.bmax 26 Maximum step size. 13-247 traincgp Dimensions for these variables are: Pd - No x Ni x TS cell array, each element P{i,j,ts} is a Dij x Q matrix. Tl - Nl x TS cell array, each element P{i,ts} is a Vi x Q matrix. Ai - Nl x LD cell array, each element Ai{i,k} is an Si x Q matrix. where Ni = net.numInputs Nl = net.numLayers LD = net.numLayerDelays Ri = net.inputs{i}.size Si = net.layers{i}.size Vi = net.targets{i}.size Dij = Ri * length(net.inputWeights{i,j}.delays) If VV is not [], it must be a structure of validation vectors, VV.PD - Validation delayed inputs. VV.Tl - Validation layer targets. VV.Ai - Validation initial input conditions. VV.Q - Validation batch size. VV.TS - Validation time steps. which is used to stop training early if the network performance on the validation vectors fails to improve or remains the same for max_fail epochs in a row. If TV is not [], it must be a structure of validation vectors, TV.PD - Validation delayed inputs. TV.Tl - Validation layer targets. TV.Ai - Validation initial input conditions. TV.Q - Validation batch size. TV.TS - Validation time steps. which is used to test the generalization capability of the trained network. 13-248 traincgp traincgp(code) returns useful information for each code string: 'pnames' - Names of training parameters. 'pdefaults' - Default training parameters. Examples Here is a problem consisting of inputs P and targets T that we would like to solve with a network. P = [0 1 2 3 4 5]; T = [0 0 0 1 1 1]; Here a two-layer feed-forward network is created. The network's input ranges from [0 to 10]. The first layer has two tansig neurons, and the second layer has one logsig neuron. The traincgp network training function is to be used. Create and Test a Network net = newff([0 5],[2 1],{'tansig','logsig'},'traincgp'); a = sim(net,p) Train and Retest the Network net.trainParam.epochs = 50; net.trainParam.show = 10; net.trainParam.goal = 0.1; net = train(net,p,t); a = sim(net,p) See newff, newcf, and newelm for other examples. Network Use You can create a standard network that uses traincgp with newff, newcf, or newelm. To prepare a custom network to be trained with traincgp: 1 Set net.trainFcn to 'traincgp'. This will set net.trainParam to traincgp's default parameters. 2 Set net.trainParam properties to desired values. In either case, calling train with the resulting network will train the network with traincgp. 13-249 traincgp Algorithm traincgp can train any network as long as its weight, net input, and transfer functions have derivative functions. Backpropagation is used to calculate derivatives of performance perf with respect to the weight and bias variables X. Each variable is adjusted according to the following: X = X + a*dX; where dX is the search direction. The parameter a is selected to minimize the performance along the search direction. The line search function searchFcn is used to locate the minimum point. The first search direction is the negative of the gradient of performance. In succeeding iterations the search direction is computed from the new gradient and the previous search direction according to the formula: dX = -gX + dX_old*Z; where gX is the gradient. The parameter Z can be computed in several different ways. For the Polak-Ribiere variation of conjugate gradient it is computed according to: Z = ((gX - gX_old)'*gX)/norm_sqr; where norm_sqr is the norm square of the previous gradient and gX_old is the gradient on the previous iteration. See page 78 of Scales (Introduction to Non-Linear Optimization 1985) for a more detailed discussion of the algorithm. Training stops when any of these conditions occur: 1 The maximum number of epochs (repetitions) is reached. 2 The maximum amount of time has been exceeded. 3 Performance has been minimized to the goal. 4 The performance gradient falls below mingrad. 5 Validation performance has increased more than max_fail times since the last time it decreased (when using validation). 13-250 traincgp See Also newff, newcf, traingdm, traingda, traingdx, trainlm, trainrp, traincgf, traincgb, trainscg, trainoss, trainbfg References Scales, L. E.,Introduction to Non-Linear Optimization, New York: Springer-Verlag, 1985. 13-251 traingd Purpose Syntax traingd Gradient descent backpropagation [net,tr] = traingd(net,Pd,Tl,Ai,Q,TS,VV) info = traingd(code) Description traingd is a network training function that updates weight and bias values according to gradient descent. traingd(net,Pd,Tl,Ai,Q,TS,VV) takes these inputs, net - Neural network. Pd - Delayed input vectors. Tl - Layer target vectors. Ai - Initial input delay conditions. Q - Batch size. TS - Time steps. VV - Either an empty matrix [] or a structure of validation vectors. and returns, net - Trained network. TR - Training record of various values over each epoch: TR.epoch - Epoch number. TR.perf - Training performance. TR.vperf - Validation performance. TR.tperf - Test performance. Training occurs according to the traingd's training parameters shown here with their default values: net.trainParam.epochs net.trainParam.goal net.trainParam.lr net.trainParam.max_fail 13-252 10 0 0.01 5 Maximum number of epochs to train Performance goal Learning rate Maximum validation failures net.trainParam.min_grad 1e-10 Minimum performance gradient net.trainParam.show 25 Epochs between showing progress net.trainParam.time inf Maximum time to train in seconds traingd Dimensions for these variables are: Pd - No x Ni x TS cell array, each element P{i,j,ts} is a Dij x Q matrix. Tl - Nl x TS cell array, each element P{i,ts} is an Vi x Q matrix. Ai - Nl x LD cell array, each element Ai{i,k} is an Si x Q matrix. where Ni = net.numInputs Nl = net.numLayers LD = net.numLayerDelays Ri = net.inputs{i}.size Si = net.layers{i}.size Vi = net.targets{i}.size Dij = Ri * length(net.inputWeights{i,j}.delays) If VV is not [], it must be a structure of validation vectors, VV.PD - Validation delayed inputs. VV.Tl - Validation layer targets. VV.Ai - Validation initial input conditions. VV.Q - Validation batch size. VV.TS - Validation time steps. which are used to stop training early if the network performance on the validation vectors fails to improve or remain the same for max_fail epochs in a row. traingd(code) returns useful information for each code string: 'pnames' - Names of training parameters. 'pdefaults' - Default training parameters. Network Use You can create a standard network that uses traingd with newff, newcf, or newelm. 13-253 traingd To prepare a custom network to be trained with traingd: 1 Set net.trainFcn to 'traingd'. This will set net.trainParam to traingd's default parameters. 2 Set net.trainParam properties to desired values. In either case, calling train with the resulting network will train the network with traingd. See newff, newcf, and newelm for examples. Algorithm traingd can train any network as long as its weight, net input, and transfer functions have derivative functions. Backpropagation is used to calculate derivatives of performance perf with respect to the weight and bias variables X. Each variable is adjusted according to gradient descent: dX = lr * dperf/dX Training stops when any of these conditions occurs: 1 The maximum number of epochs (repetitions) is reached. 2 The maximum amount of time has been exceeded. 3 Performance has been minimized to the goal. 4 The performance gradient falls below mingrad. 5 Validation performance has increased more than max_fail times since the last time it decreased (when using validation). See Also 13-254 newff, newcf, traingdm, traingda, traingdx, trainlm traingda Purpose Syntax traingda Gradient descent with adaptive lr backpropagation [net,tr] = traingda(net,Pd,Tl,Ai,Q,TS,VV) info = traingda(code) Description traingda is a network training function that updates weight and bias values according to gradient descent with adaptive learning rate. traingda(NET,Pd,Tl,Ai,Q,TS,VV) takes these inputs, net - Neural network. Pd - Delayed input vectors. Tl - Layer target vectors. Ai - Initial input delay conditions. Q - Batch size. TS - Time steps. VV - Either empty matrix [] or structure of validation vectors. and returns, net - Trained network. TR - Training record of various values over each epoch: TR.epoch - Epoch number. TR.perf - Training performance. TR.vperf - Validation performance. TR.tperf - Test performance. TR.lr - Adaptive learning rate. 13-255 traingda Training occurs according to the traingda's training parameters, shown here with their default values: net.trainParam.epochs 10 net.trainParam.goal 0 Maximum number of epochs to train Performance goal net.trainParam.lr 0.01 Learning rate net.trainParam.lr_inc 1.05 Ratio to increase learning rate net.trainParam.lr_dec 0.7 Ratio to decrease learning rate net.trainParam.max_fail 5 Maximum validation failures net.trainParam.max_perf_inc 1.04 Maximum performance increase net.trainParam.min_grad Minimum performance gradient 1e-10 net.trainParam.show net.trainParam.time 25 Epochs between showing progress inf Maximum time to train in seconds Dimensions for these variables are: Pd - No x Ni x TS cell array, each element P{i,j,ts} is a Dij x Q matrix. Tl - Nl x TS cell array, each element P{i,ts} is a Vi x Q matrix. Ai - Nl x LD cell array, each element Ai{i,k} is an Si x Q matrix. where Ni = net.numInputs Nl = net.numLayers LD = net.numLayerDelays Ri = net.inputs{i}.size Si = net.layers{i}.size Vi = net.targets{i}.size Dij = Ri * length(net.inputWeights{i,j}.delays) If VV is not [], it must be a structure of validation vectors, VV.PD - Validation delayed inputs. VV.Tl - Validation layer targets. VV.Ai - Validation initial input conditions. VV.Q - Validation batch size. VV.TS - Validation time steps. 13-256 traingda which are used to stop training early if the network performance on the validation vectors fails to improve or remains the same for max_fail epochs in a row. traingda(code) return useful information for each code string: 'pnames' - Names of training parameters. 'pdefaults' - Default training parameters. Network Use You can create a standard network that uses traingda with newff, newcf, or newelm. To prepare a custom network to be trained with traingda: 1 Set net.trainFcn to 'traingda'. This will set net.trainParam to traingda's default parameters. 2 Set net.trainParam properties to desired values. In either case, calling train with the resulting network will train the network with traingda. See newff, newcf, and newelm for examples. Algorithm traingda can train any network as long as its weight, net input, and transfer functions have derivative functions. Backpropagation is used to calculate derivatives of performance dperf with respect to the weight and bias variables X. Each variable is adjusted according to gradient descent: dX = lr*dperf/dX At each epoch, if performance decreases toward the goal, then the learning rate is increased by the factor lr_inc. If performance increases by more than the factor max_perf_inc, the learning rate is adjusted by the factor lr_dec and the change, which increased the performance, is not made. 13-257 traingda Training stops when any of these conditions occurs: 1 The maximum number of epochs (repetitions) is reached. 2 The maximum amount of time has been exceeded. 3 Performance has been minimized to the goal. 4 The performance gradient falls below mingrad. 5 Validation performance has increased more than max_fail times since the last time it decreased (when using validation). See Also 13-258 newff, newcf, traingd, traingdm, traingdx, trainlm traingdm Purpose Syntax traingdm Gradient descent w/momentum backpropagation [net,tr] = traingdm(net,Pd,Tl,Ai,Q,TS,VV) info = traingdm(code) Description traingdm is a network training function that updates weight and bias values according to gradient descent with adaptive learning rate. traingdm(net,Pd,Tl,Ai,Q,TS,VV) takes these inputs, net - Neural network. Pd - Delayed input vectors. Tl - Layer target vectors. Ai - Initial input delay conditions. Q - Batch size. TS - Time steps. VV - Either empty matrix [] or structure of validation vectors. and returns, net - Trained network. TR - Training record of various values over each epoch: TR.epoch - Epoch number. TR.perf - Training performance. TR.vperf - Validation performance. TR.tperf - Test performance. 13-259 traingdm TR.lr - adaptive learning rate. Training occurs according to the traingdm's training parameters shown here with their default values: net.trainParam.epochs 10 net.trainParam.goal net.trainParam.lr 0 0.01 net.trainParam.max_fail 5 net.trainParam.mc net.trainParam.min_grad 0.9 1e-10 net.trainParam.show net.trainParam.time 25 Maximum number of epochs to train Performance goal Learning rate Maximum validation failures Momentum constant. Minimum performance gradient Epochs between showing progress inf Maximum time to train in seconds Dimensions for these variables are: Pd - No x Ni x TS cell array, each element P{i,j,ts} is a Dij x Q matrix. Tl - Nl x TS cell array, each element P{i,ts} is a Vi x Q matrix. Ai - Nl x LD cell array, each element Ai{i,k} is an Si x Q matrix. where Ni = net.numInputs Nl = net.numLayers LD = net.numLayerDelays Ri = net.inputs{i}.size Si = net.layers{i}.size Vi = net.targets{i}.size Dij = Ri * length(net.inputWeights{i,j}.delays) If VV is not [], it must be a structure of validation vectors, VV.PD - Validation delayed inputs. VV.Tl - Validation layer targets. VV.Ai - Validation initial input conditions. VV.Q - Validation batch size. VV.TS - Validation time steps. which is used to stop training early if the network performance on the validation vectors fails to improve or remains the same for max_fail epochs in a row. 13-260 traingdm traingdm(code) returns useful information for each code string: 'pnames' - Names of training parameters. 'pdefaults' - Default training parameters. Network Use You can create a standard network that uses traingdm with newff, newcf, or newelm. To prepare a custom network to be trained with traingdm: 1 Set net.trainFcn to 'traingdm'. This will set net.trainParam to traingdm's default parameters. 2 Set net.trainParam properties to desired values. In either case, calling train with the resulting network will train the network with traingdm. See newff, newcf, and newelm for examples. Algorithm traingdm can train any network as long as its weight, net input, and transfer functions have derivative functions. Backpropagation is used to calculate derivatives of performance perf with respect to the weight and bias variables X. Each variable is adjusted according to gradient descent with momentum, dX = mc*dXprev + lr*mc*dperf/dX where dXprev is the previous change to the weight or bias. Training stops when any of these conditions occur: 1 The maximum number of epochs (repetitions) is reached. 2 The maximum amount of time has been exceeded. 3 Performance has been minimized to the goal. 4 The performance gradient falls below mingrad. 5 Validation performance has increase more than max_fail times since the last time it decreased (when using validation). See Also newff, newcf, traingd, traingda, traingdx, trainlm 13-261 traingdx Purpose Syntax traingdx Gradient descent w/momentum & adaptive lr backpropagation [net,tr] = traingdx(net,Pd,Tl,Ai,Q,TS,VV) info = traingdx(code) Description traingdx is a network training function that updates weight and bias values according to gradient descent momentum and an adaptive learning rate. traingdx(net,Pd,Tl,Ai,Q,TS,VV) takes these inputs, net - Neural network. Pd - Delayed input vectors. Tl - Layer target vectors. Ai - Initial input delay conditions. Q - Batch size. TS - Time steps. VV - Either empty matrix [] or structure of validation vectors. and returns, net - Trained network. TR - Training record of various values over each epoch: TR.epoch - Epoch number. TR.perf - Training performance. TR.vperf - Validation performance. TR.tperf - Test performance. TR.lr 13-262 - Adaptive learning rate. traingdx Training occurs according to the traingdx's training parameters shown here with their default values: net.trainParam.epochs 10 net.trainParam.goal 0 Maximum number of epochs to train Performance goal net.trainParam.lr 0.01 Learning rate net.trainParam.lr_inc 1.05 Ratio to increase learning rate net.trainParam.lr_dec 0.7 Ratio to decrease learning rate net.trainParam.max_fail 5 Maximum validation failures net.trainParam.max_perf_inc 1.04 Maximum performance increase net.trainParam.mc Momentum constant. net.trainParam.min_grad 0.9 1e-10 net.trainParam.show net.trainParam.time 25 Minimum performance gradient Epochs between showing progress inf Maximum time to train in seconds Dimensions for these variables are: Pd - No x Ni x TS cell array, each element P{i,j,ts} is a Dij x Q matrix. Tl - Nl x TS cell array, each element P{i,ts} is a Vi x Q matrix. Ai - Nl x LD cell array, each element Ai{i,k} is an Si x Q matrix. where Ni = net.numInputs Nl = net.numLayers LD = net.numLayerDelays Ri = net.inputs{i}.size Si = net.layers{i}.size Vi = net.targets{i}.size Dij = Ri * length(net.inputWeights{i,j}.delays) 13-263 traingdx If VV is not [], it must be a structure of validation vectors, VV.PD - Validation delayed inputs. VV.Tl - Validation layer targets. VV.Ai - Validation initial input conditions. VV.Q - Validation batch size. VV.TS - Validation time steps. which is used to stop training early if the network performance on the validation vectors fails to improve or remains the same for max_fail epochs in a row. traingdx(code) return useful information for each code string: 'pnames' - Names of training parameters. 'pdefaults' - Default training parameters. Network Use You can create a standard network that uses traingdx with newff, newcf, or newelm. To prepare a custom network to be trained with traingdx: 1 Set net.trainFcn to 'traingdx'. This will set net.trainParam to traingdx's default parameters. 2 Set net.trainParam properties to desired values. In either case, calling train with the resulting network will train the network with traingdx. See newff, newcf, and newelm for examples. Algorithm traingdx can train any network as long as its weight, net input, and transfer functions have derivative functions. Backpropagation is used to calculate derivatives of performance perf with respect to the weight and bias variables X. Each variable is adjusted according to gradient descent with momentum, dX = mc*dXprev + lr*mc*dperf/dX where dXprev is the previous change to the weight or bias. 13-264 traingdx For each epoch, if performance decreases toward the goal, then the learning rate is increased by the factor lr_inc. If performance increases by more than the factor max_perf_inc, the learning rate is adjusted by the factor lr_dec and the change, which increased the performance, is not made. Training stops when any of these conditions occur: 1 The maximum number of epochs (repetitions) is reached. 2 The maximum amount of time has been exceeded. 3 Performance has been minimized to the goal. 4 The performance gradient falls below mingrad. 5 Validation performance has increase more than max_fail times since the last time it decreased (when using validation). See Also newff, newcf, traingd, traingdm, traingda, trainlm 13-265 trainlm Purpose trainlm Levenberg-Marquardt backpropagation Syntax [net,tr] = trainlm(net,Pd,Tl,Ai,Q,TS,VV) info = trainlm(code) Description trainlm is a network training function that updates weight and bias values according to Levenberg-Marquardt optimization. trainlm(net,Pd,Tl,Ai,Q,TS,VV) takes these inputs, net - Neural network. Pd - Delayed input vectors. Tl - Layer target vectors. Ai - Initial input delay conditions. Q - Batch size. TS - Time steps. VV - Either empty matrix [] or structure of validation vectors. and returns, net - Trained network. TR - Training record of various values over each epoch: TR.epoch - Epoch number. TR.perf - Training performance. TR.vperf - Validation performance. TR.tperf - Test performance. TR.mu 13-266 - Adaptive mu value. trainlm Training occurs according to the trainlm's training parameters shown here with their default values: net.trainParam.epochs 10 net.trainParam.goal net.trainParam.lr 0 0.01 Maximum number of epochs to train Performance goal Learning rate Maximum validation failures net.trainParam.max_fail 5 net.trainParam.mem_reduc 1 Factor to use for memory/speed trade off net.trainParam.min_grad 1e-10 Minimum performance gradient net.trainParam.show 25 Epochs between showing progress net.trainParam.time inf Maximum time to train in seconds Dimensions for these variables are: Pd - No x Ni x TS cell array, each element P{i,j,ts} is a Dij x Q matrix. Tl - Nl x TS cell array, each element P{i,ts} is a Vi x Q matrix. Ai - Nl x LD cell array, each element Ai{i,k} is an Si x Q matrix. where Ni = net.numInputs Nl = net.numLayers LD = net.numLayerDelays Ri = net.inputs{i}.size Si = net.layers{i}.size Vi = net.targets{i}.size Dij = Ri * length(net.inputWeights{i,j}.delays) If VV is not [], it must be a structure of validation vectors, VV.PD - Validation delayed inputs. VV.Tl - Validation layer targets. VV.Ai - Validation initial input conditions. VV.Q - Validation batch size. VV.TS - Validation time steps. 13-267 trainlm which is used to stop training early if the network performance on the validation vectors fails to improve or remains the same for max_fail epochs in a row. trainlm(code) returns useful information for each code string: 'pnames' - Names of training parameters. 'pdefaults' - Default training parameters. Network Use You can create a standard network that uses trainlm with newff, newcf, or newelm. To prepare a custom network to be trained with trainlm: 1 Set net.trainFcn to 'trainlm'. This will set net.trainParam to trainlm's default parameters. 2 Set net.trainParam properties to desired values. In either case, calling train with the resulting network will train the network with trainlm. See newff, newcf, and newelm for examples. Algorithm trainlm can train any network as long as its weight, net input, and transfer functions have derivative functions. Backpropagation is used to calculate the Jacobian jX of performance perf with respect to the weight and bias variables X. Each variable is adjusted according to Levenberg-Marquardt, jj = jX * jX je = jX * E dX = -(jj+I*mu) \ je where E is all errors and I is the identity matrix. The adaptive value mu is increased by mu_inc until the change above results in a reduced performance value. The change is then made to the network and mu is decreased by mu_dec. The parameter mem_reduc indicates how to use memory and speed to calculate the Jacobian jX. If mem_reduc is 1, then trainlm runs the fastest, but can require a lot of memory. Increasing mem_reduc to 2, cuts some of the memory 13-268 trainlm required by a factor of two, but slows trainlm somewhat. Higher values continue to decrease the amount of memory needed and increase training times. Training stops when any of these conditions occur: 1 The maximum number of epochs (repetitions) is reached. 2 The maximum amount of time has been exceeded. 3 Performance has been minimized to the goal. 4 The performance gradient falls below mingrad. 5 mu exceeds mu_max. 6 Validation performance has increase more than max_fail times since the last time it decreased (when using validation). See Also newff, newcf, traingd, traingdm, traingda, traingdx 13-269 trainoss Purpose Syntax trainoss One step secant backpropagation [net,tr] = trainoss(net,Pd,Tl,Ai,Q,TS,VV) info = trainoss(code) Description trainoss is a network training function that updates weight and bias values according to the one step secant method. trainoss(net,Pd,Tl,Ai,Q,TS,VV,TV) takes these inputs, net - Neural network. Pd - Delayed input vectors. Tl - Layer target vectors. Ai - Initial input delay conditions. Q - Batch size. TS - Time steps. VV - Either empty matrix [] or structure of validation vectors. TV - Either empty matrix [] or structure of test vectors. and returns, net - Trained network. TR - Training record of various values over each epoch: TR.epoch - Epoch number. TR.perf - Training performance. TR.vperf - Validation performance. TR.tperf - Test performance. 13-270 trainoss Training occurs according to the trainoss's training parameters, shown here with their default values: net.trainParam.epochs 100 net.trainParam.show 25 net.trainParam.goal 0 net.trainParam.time inf net.trainParam.min_grad 1e-6 net.trainParam.max_fail 5 net.trainParam.searchFcn 'srchcha' Maximum number of epochs to train Epochs between showing progress Performance goal Maximum time to train in seconds Minimum performance gradient Maximum validation failures Name of line search routine to use Parameters related to line search methods (not all used for all methods): net.trainParam.scal_tol 20 Divide into delta to determine tolerance for linear search. net.trainParam.alpha 0.001 Scale factor which determines sufficient reduction in perf. net.trainParam.beta 0.1 Scale factor which determines sufficiently large step size. net.trainParam.delta 0.01 Initial step size in interval location step. net.trainParam.gama 0.1 Parameter to avoid small reductions in performance. Usually set to 0.1. (See use in srch_cha.) net.trainParam.low_lim 0.1 Lower limit on change in step size. net.trainParam.up_lim 0.5 Upper limit on change in step size. net.trainParam.maxstep 100 Maximum step length. net.trainParam.minstep 1.0e-6 Minimum step length. net.trainParam.bmax 26 Maximum step size. 13-271 trainoss Dimensions for these variables are: Pd - No x Ni x TS cell array, each element P{i,j,ts} is a Dij x Q matrix. Tl - Nl x TS cell array, each element P{i,ts} is a Vi x Q matrix. Ai - Nl x LD cell array, each element Ai{i,k} is an Si x Q matrix. where Ni = net.numInputs Nl = net.numLayers LD = net.numLayerDelays Ri = net.inputs{i}.size Si = net.layers{i}.size Vi = net.targets{i}.size Dij = Ri * length(net.inputWeights{i,j}.delays) If VV is not [], it must be a structure of validation vectors, VV.PD - Validation delayed inputs. VV.Tl - Validation layer targets. VV.Ai - Validation initial input conditions. VV.Q - Validation batch size. VV.TS - Validation time steps. which is used to stop training early if the network performance on the validation vectors fails to improve or remains the same for max_fail epochs in a row. If TV is not [], it must be a structure of validation vectors, TV.PD - Validation delayed inputs. TV.Tl - Validation layer targets. TV.Ai - Validation initial input conditions. TV.Q - Validation batch size. TV.TS - Validation time steps. which is used to test the generalization capability of the trained network. 13-272 trainoss trainoss(code) returns useful information for each code string: 'pnames' - Names of training parameters. 'pdefaults' - Default training parameters. Examples Here is a problem consisting of inputs P and targets T that we would like to solve with a network. P = [0 1 2 3 4 5]; T = [0 0 0 1 1 1]; Here a two-layer feed-forward network is created. The network's input ranges from [0 to 10]. The first layer has two tansig neurons, and the second layer has one logsig neuron. The trainoss network training function is to be used. Create and Test a Network net = newff([0 5],[2 1],{'tansig','logsig'},'trainoss'); a = sim(net,p) Train and Retest the Network net.trainParam.epochs = 50; net.trainParam.show = 10; net.trainParam.goal = 0.1; net = train(net,p,t); a = sim(net,p) See newff, newcf, and newelm for other examples. Network Use You can create a standard network that uses trainoss with newff, newcf, or newelm. To prepare a custom network to be trained with trainoss: 1 Set net.trainFcn to 'trainoss'. This will set net.trainParam to trainoss's default parameters. 2 Set net.trainParam properties to desired values. In either case, calling train with the resulting network will train the network with trainoss. 13-273 trainoss Algorithm trainoss can train any network as long as its weight, net input, and transfer functions have derivative functions. Backpropagation is used to calculate derivatives of performance perf with respect to the weight and bias variables X. Each variable is adjusted according to the following: X = X + a*dX; where dX is the search direction. The parameter a is selected to minimize the performance along the search direction. The line search function searchFcn is used to locate the minimum point. The first search direction is the negative of the gradient of performance. In succeeding iterations the search direction is computed from the new gradient and the previous steps and gradients according to the following formula: dX = -gX + Ac*X_step + Bc*dgX; where gX is the gradient, X_step is the change in the weights on the previous iteration, and dgX is the change in the gradient from the last iteration. See Battiti (Neural Computation, vol. 4, 1992, pp. 141-166) for a more detailed discussion of the one step secant algorithm. Training stops when any of these conditions occur: 1 The maximum number of epochs (repetitions) is reached. 2 The maximum amount of time has been exceeded. 3 Performance has been minimized to the goal. 4 The performance gradient falls below mingrad. 5 Validation performance has increased more than max_fail times since the last time it decreased (when using validation). See Also newff, newcf, traingdm, traingda, traingdx, trainlm, trainrp, traincgf, traincgb, trainscg, traincgp, trainbfg References R. Battiti, “First and second order methods for learning: Between steepest descent and Newton’s method,” Neural Computation, vol. 4, no. 2, pp. 141–166, 1992. 13-274 trainrp Purpose Syntax trainrp RPROP backpropagation [net,tr] = trainrp(net,Pd,Tl,Ai,Q,TS,VV) info = trainrp(code) Description trainrp is a network training function that updates weight and bias values according to the resilient backpropagation algorithm (RPROP). trainrp(net,Pd,Tl,Ai,Q,TS,VV,TV) takes these inputs, net - Neural network. Pd - Delayed input vectors. Tl - Layer target vectors. Ai - Initial input delay conditions. Q - Batch size. TS - Time steps. VV - Either empty matrix [] or structure of validation vectors. TV - Either empty matrix [] or structure of test vectors. and returns, net - Trained network. TR - Training record of various values over each epoch: TR.epoch - Epoch number. TR.perf - Training performance. TR.vperf - Validation performance. TR.tperf - Test performance. 13-275 trainrp Training occurs according to the trainrp's training parameters shown here with their default values: net.trainParam.epochs 100 net.trainParam.show 25 net.trainParam.goal 0 net.trainParam.time inf net.trainParam.min_grad 1e-6 net.trainParam.max_fail 5 net.trainParam.lr 0.01 Maximum number of epochs to train Epochs between showing progress Performance goal Maximum time to train in seconds Minimum performance gradient Maximum validation failures Learning rate net.trainParam.delt_inc 1.2 Increment to weight change net.trainParam.delt_dec 0.5 Decrement to weight change net.trainParam.delta0 0.07 Initial weight change net.trainParam.deltamax 50.0 Maximum weight change Dimensions for these variables are: Pd - No x Ni x TS cell array, each element P{i,j,ts} is a Dij x Q matrix. Tl - Nl x TS cell array, each element P{i,ts} is a Vi x Q matrix. Ai - Nl x LD cell array, each element Ai{i,k} is an Si x Q matrix. where Ni = net.numInputs Nl = net.numLayers LD = net.numLayerDelays Ri = net.inputs{i}.size Si = net.layers{i}.size Vi = net.targets{i}.size Dij = Ri * length(net.inputWeights{i,j}.delays) 13-276 trainrp If VV is not [], it must be a structure of validation vectors, VV.PD - Validation delayed inputs. VV.Tl - Validation layer targets. VV.Ai - Validation initial input conditions. VV.Q - Validation batch size. VV.TS - Validation time steps. which is used to stop training early if the network performance on the validation vectors fails to improve or remains the same for max_fail epochs in a row. If TV is not [], it must be a structure of validation vectors, TV.PD - Validation delayed inputs. TV.Tl - Validation layer targets. TV.Ai - Validation initial input conditions. TV.Q - Validation batch size. TV.TS - Validation time steps. which is used to test the generalization capability of the trained network. trainrp(code) returns useful information for each code string: 'pnames' - Names of training parameters. 'pdefaults' - Default training parameters. Examples Here is a problem consisting of inputs P and targets T that we would like to solve with a network. P = [0 1 2 3 4 5]; T = [0 0 0 1 1 1]; Here a two-layer feed-forward network is created. The network's input ranges from [0 to 10]. The first layer has two tansig neurons, and the second layer has one logsig neuron. The trainrp network training function is to be used. Create and Test a Network net = newff([0 5],[2 1],{'tansig','logsig'},'trainrp'); a = sim(net,p) 13-277 trainrp Train and Retest the Network net.trainParam.epochs = 50; net.trainParam.show = 10; net.trainParam.goal = 0.1; net = train(net,p,t); a = sim(net,p) See newff, newcf, and newelm for other examples. Network Use You can create a standard network that uses trainrp with newff, newcf, or newelm. To prepare a custom network to be trained with trainrp: 1 Set net.trainFcn to 'trainrp'. This will set net.trainParam to trainrp's default parameters. 2 Set net.trainParam properties to desired values. In either case, calling train with the resulting network will train the network with trainrp. Algorithm trainrp can train any network as long as its weight, net input, and transfer functions have derivative functions. Backpropagation is used to calculate derivatives of performance perf with respect to the weight and bias variables X. Each variable is adjusted according to the following: dX = deltaX.*sign(gX); where the elements of deltaX are all initialized to delta0 and gX is the gradient. At each iteration the elements of deltaX are modified. If an element of gX changes sign from one iteration to the next, then the corresponding element of deltaX is decreased by delta_dec. If an element of gX maintains the same sign from one iteration to the next, then the corresponding element of deltaX is increased by delta_inc. See Reidmiller and Braun, Proceedings of the IEEE International Conference on Neural Networks,, 1993, pp. 586-591. 13-278 trainrp Training stops when any of these conditions occur: 1 The maximum number of epochs (repetitions) is reached. 2 The maximum amount of time has been exceeded. 3 Performance has been minimized to the goal. 4 The performance gradient falls below mingrad. 5 Validation performance has increased more than max_fail times since the last time it decreased (when using validation). See Also newff, newcf, traingdm, traingda, traingdx, trainlm, traincgp, traincgf, traincgb, trainscg, trainoss, trainbfg References Riedmiller, M., and H. Braun, “A direct adaptive method for faster backpropagation learning: The RPROP algorithm,” Proceedings of the IEEE International Conference on Neural Networks, San Francisco,1993. 13-279 trainscg Purpose Syntax trainscg Scaled conjugate gradient backpropagation [net,tr] = trainscg(net,Pd,Tl,Ai,Q,TS,VV) info = trainscg(code) Description trainscg is a network training function that updates weight and bias values according to the scaled conjugate gradient method. trainscg(net,Pd,Tl,Ai,Q,TS,VV,TV) takes these inputs, net - Neural network. Pd - Delayed input vectors. Tl - Layer target vectors. Ai - Initial input delay conditions. Q - Batch size. TS - Time steps. VV - Either empty matrix [] or structure of validation vectors. TV - Either empty matrix [] or structure of test vectors. and returns, net - Trained network. TR - Training record of various values over each epoch: TR.epoch - Epoch number. TR.perf - Training performance. TR.vperf - Validation performance. TR.tperf - Test performance. 13-280 trainscg Training occurs according to the trainscg's training parameters shown here with their default values: net.trainParam.epochs 100 net.trainParam.show 25 net.trainParam.goal 0 net.trainParam.time inf net.trainParam.min_grad 1e-6 net.trainParam.max_fail 5 net.trainParam.sigma 5.0e-5 net.trainParam.lambda 5.0e-7 Maximum number of epochs to train Epochs between showing progress Performance goal Maximum time to train in seconds Minimum performance gradient Maximum validation failures Determines change in weight for second derivative approximation. Parameter for regulating the indefiniteness of the Hessian. Dimensions for these variables are: Pd - No x Ni x TS cell array, each element P{i,j,ts} is a Dij x Q matrix. Tl - Nl x TS cell array, each element P{i,ts} is a Vi x Q matrix. Ai - Nl x LD cell array, each element Ai{i,k} is an Si x Q matrix. where Ni = net.numInputs Nl = net.numLayers LD = net.numLayerDelays Ri = net.inputs{i}.size Si = net.layers{i}.size Vi = net.targets{i}.size Dij = Ri * length(net.inputWeights{i,j}.delays) If VV is not [], it must be a structure of validation vectors, VV.PD - Validation delayed inputs. VV.Tl - Validation layer targets. VV.Ai - Validation initial input conditions. VV.Q - Validation batch size. VV.TS - Validation time steps. 13-281 trainscg which is used to stop training early if the network performance on the validation vectors fails to improve or remains the same for max_fail epochs in a row. If TV is not [], it must be a structure of validation vectors, TV.PD - Validation delayed inputs. TV.Tl - Validation layer targets. TV.Ai - Validation initial input conditions. TV.Q - Validation batch size. TV.TS - Validation time steps. which is used to test the generalization capability of the trained network. trainscg(code) returns useful information for each code string: 'pnames' - Names of training parameters. 'pdefaults' - Default training parameters. Examples Here is a problem consisting of inputs P and targets T that we would like to solve with a network. P = [0 1 2 3 4 5]; T = [0 0 0 1 1 1]; Here a two-layer feed-forward network is created. The network's input ranges from [0 to 10]. The first layer has two tansig neurons, and the second layer has one logsig neuron. The trainscg network training function is to be used. Create and Test a Network net = newff([0 5],[2 1],{'tansig','logsig'},'trainscg'); a = sim(net,p) Train and Retest the Network net.trainParam.epochs = 50; net.trainParam.show = 10; net.trainParam.goal = 0.1; net = train(net,p,t); a = sim(net,p) See newff, newcf, and newelm for other examples. 13-282 trainscg Network Use You can create a standard network that uses trainscg with newff, newcf, or newelm. To prepare a custom network to be trained with trainscg: 1 Set net.trainFcn to 'trainscg'. This will set net.trainParam to trainscg's default parameters. 2 Set net.trainParam properties to desired values. In either case, calling train with the resulting network will train the network with trainscg. Algorithm trainscg can train any network as long as its weight, net input, and transfer functions have derivative functions. Backpropagation is used to calculate derivatives of performance perf with respect to the weight and bias variables X. The scaled conjugate gradient algorithm is based on conjugate directions, as in traincgp, traincgf and traincgb, but this algorithm does not perform a line search at each iteration. See Moller (Neural Networks, vol. 6, 1993, pp.525-533) for a more detailed discussion of the scaled conjugate gradient algorithm. Training stops when any of these conditions occur: 1 The maximum number of epochs (repetitions) is reached. 2 The maximum amount of time has been exceeded. 3 Performance has been minimized to the goal. 4 The performance gradient falls below mingrad. 5 Validation performance has increased more than max_fail times since the last time it decreased (when using validation). See Also newff, newcf, traingdm, traingda, traingdx, trainlm, trainrp, traincgf, traincgb, trainbfg, traincgp, trainoss References Moller, M. F., “A scaled conjugate gradient algorithm for fast supervised learning,” Neural Networks, vol. 6, pp. 525-533, 1993. 13-283 trainwb Purpose Syntax trainwb By-weight-&-bias 1-vector-at-a-time training function [net,tr] = trainwb(net,Pd,Tl,Ai,Q,TS,VV) info = trainwb(code) Description trainwb is a network training function that updates weight and bias values according to Levenberg-Marquardt optimization. trainwb(net,Pd,Tl,Ai,Q,TS,VV) takes these inputs, net - Neural network. Pd - Delayed input vectors. Tl - Layer target vectors. Ai - Initial input delay conditions. Q - Batch size. TS - Time steps. VV - Either empty matrix [] or structure of validation vectors. and returns, net - Trained network. TR - Training record of various values over each epoch: TR.epoch - Epoch number. TR.perf - Training performance. TR.vperf - Validation performance. TR.tperf - Test performance. Training occurs according to the trainwb1's training parameters, shown here with their default values: net.trainParam.epochs 13-284 100 Maximum number of epochs to train net.trainParam.goal 0 Performance goal net.trainParam.max_fail 5 Maximum validation failures net.trainParam.show 25 Epochs between showing progress net.trainParam.time inf Maximum time to train in seconds trainwb Dimensions for these variables are: Pd - No x Ni x TS cell array, each element P{i,j,ts} is a Dij x Q matrix. Tl - Nl x TS cell array, each element P{i,ts} is a Vi x Q matrix. Ai - Nl x LD cell array, each element Ai{i,k} is an Si x Q matrix. where Ni = net.numInputs Nl = net.numLayers LD = net.numLayerDelays Ri = net.inputs{i}.size Si = net.layers{i}.size Vi = net.targets{i}.size Dij = Ri * length(net.inputWeights{i,j}.delays) If VV is not [], it must be a structure of validation vectors, VV.PD - Validation delayed inputs. VV.Tl - Validation layer targets. VV.Ai - Validation initial input conditions. VV.Q - Validation batch size. VV.TS - Validation time steps. which is used to stop training early if the network performance on the validation vectors fails to improve or remains the same for max_fail epochs in a row. trainwb(code) returns useful information for each code string: 'pnames' - Names of training parameters. 'pdefaults' - Default training parameters. Network Use You can create a standard network that uses trainwb with newp or newlin. 13-285 trainwb To prepare a custom network to be trained with trainwb: 1 Set net.trainFcn to 'trainwb'. (This will set net.trainParam to trainwb's default parameters.) 2 Set each net.inputWeights{i,j}.learnFcn to a learning function. Set each net.layerWeights{i,j}.learnFcn to a learning function. Set each net.biases{i}.learnFcn to a learning function. (Weight and bias learning parameters will automatically be set to default values for the given learning function.) To train the network: 1 Set net.trainParam properties to desired values. 2 Set weight and bias learning parameters to desired values. 3 Call train. See newp and newlin for training examples. Algorithm Each weight and bias updates according to its learning function after each epoch (one pass through the entire set of input vectors). Training stops when any of these conditions occur: 1 The maximum number of epochs (repetitions) is reached. 2 Performance has been minimized to the goal. 3 The maximum amount of time has been exceeded. 4 Validation performance has increase more than max_fail times since the last time it decreased (when using validation). See Also 13-286 newp, newlin, train trainwb1 Purpose Syntax trainwb1 By-weight-and-bias network training function [net,tr] = trainwb1(net,Pd,Tl,Ai,Q,TS,VV) info = trainwb1(code) Description trainwb1 is a network training function which updates each weight and bias according to its learning function. At each epoch trainwb1 randomly chooses just one input vector (or sequence) to present to the network. trainwb1(net,Pd,Tl,Ai,Q,TS,VV) takes these inputs, net - Neural network. Pd - Delayed input vectors. Tl - Layer target vectors. Ai - Initial input delay conditions. Q - Batch size. TS - Time steps. VV - Either empty matrix [] or structure of validation vectors. and returns, net - Trained network. TR - Training record of various values over each epoch: TR.epoch - Epoch number. TR.perf - Training performance. TR.index - Index of presented input vector (or sequence). Training occurs according to the trainwb1's training parameters shown here with their default values: net.trainParam.epochs 100 Maximum number of epochs to train net.trainParam.show 25 Epochs between showing progress net.trainParam.time inf Maximum time to train in seconds 13-287 trainwb1 Dimensions for these variables are: Pd - No x Ni x TS cell array, each element P{i,j,ts} is a Dij x Q matrix. Tl - Nl x TS cell array, each element P{i,ts} is a Vi x Q matrix. Ai - Nl x LD cell array, each element Ai{i,k} is an Si x Q matrix. where Ni = net.numInputs Nl = net.numLayers LD = net.numLayerDelays Ri = net.inputs{i}.size Si = net.layers{i}.size Vi = net.targets{i}.size Dij = Ri * length(net.inputWeights{i,j}.delays) trainwb1 does not implement validation or test vectors, so arguments VV and TV are ignored. trainwb1(code) returns useful information for each code string: 'pnames' - Names of training parameters. 'pdefaults' - Default training parameters. Network Use You can create a standard network that uses trainwb1 with newc or newsom. To prepare a custom network to be trained with trainwb1: 1 Set net.trainFcn to 'trainwb1'. (This will set net.trainParam to trainwb1's default parameters.) 2 Set each net.inputWeights{i,j}.learnFcn to a learning function. Set each net.layerWeights{i,j}.learnFcn to a learning function. Set each net.biases{i}.learnFcn to a learning function. (Weight and bias learning parameters will automatically be set to default values for the given learning function.) To train the network: 1 Set net.trainParam properties to desired values. 2 Set weight and bias learning parameters to desired values. 3 Call train. See newc and newsom for training examples. 13-288 trainwb1 Algorithm For each epoch a vector (or sequence) is chosen randomly and presented to the network and then the weight and bias values are updated accordingly. Training stops when any of these conditions are met: 1 The maximum number of epochs (repetitions) is reached. 2 The maximum amount of time has been exceeded. See Also newp, newlin, train 13-289 tramnmx Purpose tramnmx Transform data using a precalculated min and max Syntax [pn] = tramnmx(p,minp,maxp) Description tramnmx transforms the network input set using minimum and maximum values that were previously computed by premnmx. This function needs to be used when a network has been trained using data normalized by premnmx. All subsequent inputs to the network need to be transformed using the same normalization. tramnmx(P,minp, maxp)takes these inputs P - R x Q matrix of input (column) vectors. minp- R x 1 vector containing original minimums for each input. maxp- R x 1 vector containing original maximums for each input. and returns, PN Examples - R x Q matrix of normalized input vectors Here is the code to normalize a given data set, so that the inputs and targets will fall in the range [-1,1], using premnmx, and also code to train a network with the normalized data. p = [-10 -7.5 -5 -2.5 0 2.5 5 7.5 10]; t = [0 7.07 -10 -7.07 0 7.07 10 7.07 0]; [pn,minp,maxp,tn,mint,maxt] = premnmx(p,t); net = newff(minmax(pn),[5 1],{'tansig' 'purelin'},'trainlm'); net = train(net,pn,tn); If we then receive new inputs to apply to the trained network, we will use tramnmx to transform them first. Then the transformed inputs can be used to simulate the previously trained network. The network output must also be unnormalized using postmnmx. p2 = [4 -7]; [p2n] = tramnmx(p2,minp,maxp); an = sim(net,pn); [a] = postmnmx(an,mint,maxt); Algorithm See Also 13-290 pn = 2*(p-minp)/(maxp-minp) - 1; premnmx, prestd, prepca, trastd, trapca trapca Purpose trapca Principal component transformation Syntax [Ptrans] = trapca(P,TransMat) Description trapca preprocesses the network input training set by applying the principal component transformation that was previously computed by prepca. This function needs to be used when a network has been trained using data normalized by prepca. All subsequent inputs to the network need to be transformed using the same normalization. trapca(P,TransMat) takes these inputs, P - R x Q matrix of centered input (column) vectors. TransMat - Transformation matrix. and returns, Ptrans - Transformed data set. Examples Here is the code to perform a principal component analysis and retain only those components which contribute more than 2 percent to the variance in the data set. prestd is called first to create zero mean data, which are needed for prepca. p = [-1.5 -0.58 0.21 -0.96 -0.79; -2.2 -0.87 0.31 -1.4 -1.2]; t = [-0.08 3.4 -0.82 0.69 3.1]; [pn,meanp,stdp,tn,meant,stdt] = prestd(p,t); [ptrans,transMat] = prepca(pn,0.02); net = newff(minmax(ptrans),[5 1],{'tansig''purelin'},'trainlm'); net = train(net,ptrans,tn); If we then receive new inputs to apply to the trained network, we will use trastd and trapca to transform them first. Then the transformed inputs can be used to simulate the previously trained network. The network output must also be unnormalized using poststd. p2 = [1.5 -0.8;0.05 -0.3]; [p2n] = trastd(p2,meanp,stdp); [p2trans] = trapca(p2n,TransMat) an = sim(net,p2trans); [a] = poststd(an,meant,stdt); 13-291 trapca Algorithm See Also 13-292 Ptrans = TransMat*P; prestd, premnmx, prepca, trastd, tramnmx trastd Purpose trastd Preprocess data using a precalculated mean and standard deviation Syntax [pn] = trastd(p,meanp,stdp) Description trastd preprocesses the network training set using the mean and standard deviation that were previously computed by prestd. This function needs to be used when a network has been trained using data normalized by prestd. All subsequent inputs to the network need to be transformed using the same normalization. trastd(P,T) takes these inputs, P - R x Q matrix of input (column) vectors. meanp - R x 1 vector containing the original means for each input. stdp - R x 1 vector containing the original standard deviations for each input. and returns, PN - R x Q matrix of normalized input vectors. Examples Here is the code to normalize a given data set so that the inputs and targets will have means of zero and standard deviations of 1. p = [-0.92 0.73 -0.47 0.74 0.29; -0.08 0.86 -0.67 -0.52 0.93]; t = [-0.08 3.4 -0.82 0.69 3.1]; [pn,meanp,stdp,tn,meant,stdt] = prestd(p,t); net = newff(minmax(pn),[5 1],{'tansig' 'purelin'},'trainlm'); net = train(net,pn,tn); If we then receive new inputs to apply to the trained network, we will use trastd to transform them first. Then the transformed inputs can be used to simulate the previously trained network. The network output must also be unnormalized using poststd. p2 = [1.5 -0.8;0.05 -0.3]; [p2n] = trastd(p2,meanp,stdp); an = sim(net,pn); [a] = poststd(an,meant,stdt); Algorithm See Also pn = (p-meanp)/stdp; premnmx, prepca, prestd, trapca, tramnmx 13-293 tribas Purpose Syntax tribas Triangular basis transfer function A = tribas(N) info = tribas(code) Description tribas is a transfer function. Transfer functions calculate a layer's output from its net input. tribas(N) takes one input, N - S x Q matrix of net input (column) vectors. and returns each element of N passed through a radial basis function. tribas(code) returns useful information for each code string: 'deriv' - Name of derivative function. 'name' - Full name. 'output' - Output range. 'active' - Active input range. Examples Here we create a plot of the tribas transfer function. n = -5:0.1:5; a = tribas(n); plot(n,a) Network Use To change a network so that a layer uses tribas, set net.layers{i}.transferFcn to 'tribas'. Call sim to simulate the network with tribas. Algorithm tribas(N) calculates its output with according to: tribas(n) = 1-abs(n), if -1 <= n <= 1; = 0, otherwise. See Also 13-294 sim, radbas vec2ind Purpose vec2ind Convert vectors to indices Syntax ind = vec2ind(vec) Description ind2vec and vec2ind allow indices to be represented either by themselves or as vectors containing a 1 in the row of the index they represent. vec2ind(vec) takes one argument, vec - Matrix of vectors, each containing a single 1. and returns the indices of the 1's. Examples Here four vectors (containing only one 1 each) are defined and the indices of the 1's are found. vec = [1 0 0 0; 0 0 1 0; 0 1 0 1] ind = vec2ind(vec) See Also ind2vec 13-295 vec2ind 13-296 A Glossary A Glossary ADALINE - an acronym for a linear neuron: ADAptive LINear Element. adaption - a function that proceeds through the specified sequence of inputs, calculating the output, error and network adjustment for each input vector in the sequence as the inputs are presented. adaptive learning rate - a learning rate that is adjusted according to an algorithm during training to minimize training time. adaptive filter - a network that contains delays and whose weights are adjusted after each new input vector is presented. The network “adapts” to changes in the input signal properties if such occur. This kind of filter is used in long distance telephone lines to cancel echoes. architecture - a description of the number of the layers in a neural network, each layer’s transfer function, the number of neurons per layer, and the connections between layers. backpropagation learning rule -a learning rule in which weights and biases are adjusted by error derivative (delta) vectors backpropagated through the network. Backpropagation is commonly applied to feedforward multilayer networks. Sometimes this rule is called the generalized delta rule. backtracking search - linear search routine which begins with a step multiplier of 1 and then backtracks until an acceptable reduction in the performance is obtained. batch - a matrix of input (or target) vectors applied to the network “simultaneously”. Changes to the network weights and biases are made just once for the entire set of vectors in the input matrix. batching - the process of presenting a matrix (batch) of input vectors for simultaneous calculation of a matrix of output vectors and/or new weights and biases. Bayesian framework - assumes that the weights and biases of the network are random variables with specified distributions. BFGS quasi-Newton algorithm - a variation of Newton’s optimization algorithm, in which an approximation of the Hessian matrix is obtained from gradients computed at each iteration of the algorithm. bias - a neuron parameter that is summed with the neuron’s weighted inputs and passed through the neuron’s transfer function to generate the neuron’s output. A-2 bias vector - a column vector of bias values for a layer of neurons. Brent’s search - a linear search which is a hybrid combination of the golden section search and a quadratic interpolation. Charalambous’ search - a hybrid line search that uses a cubic interpolation, together with a type of sectioning. classification - an association of an input vector with a particular target vector. competitive layer - a layer of neurons in which only the neuron with maximum net input has an output of 1 and all other neurons output 0. Neurons compete with each other for the right to respond to a given input vector. competitive learning - the unsupervised training of a competitive layer with the instar rule or Kohonen rule. Individual neurons learn to become feature detectors. After training, the layer categorizes input vectors among its neurons. competitive transfer function - accepts a net input vector for a layer and returns neuron outputs of 0 for all neurons except for the “winner,” the neuron associated with the most positive element of the net input n. concurrent input vectors - name given to a matrix of input vectors that are to be presented to a network “simultaneously.” All the vectors in the matrix will be used in making just one set of changes in the weights and biases. conjugate gradient algorithm -in the conjugate gradient algorithms a search is performed along conjugate directions, which produces generally faster convergence than a search along the steepest descent directions. connection - a one-way link between neurons in a network. connection strength - the strength of a link between two neurons in a network. The strength, often called weight, determines the effect that one neuron has on another. cycle - a single presentation of an input vector, calculation of output and new weights and biases. dead neurons - a competitive layer neuron that never won any competition during training and so has not become a useful feature detector. Dead neurons do not respond to any of the training vectors. decision boundary - a line, determined by the weight and bias vectors, for which the net input n is zero. A-3 A Glossary delta rule - the Widrow-Hoff rule. delta vector - the delta vector for a layer is the derivative of a network’s output error with respect to that layer’s net input vector. distance - the distance between neurons, calculated from their positions with a distance function. early stopping - a technique based on dividing the data into three subsets. The first subset is the training set used for computing the gradient and updating the network weights and biases. The second subset is the validation set. When the validation error increases for a specified number of iterations, the training is stopped, and the weights and biases at the minimum of the validation error are returned. The third set is the test set. It is used to verify the network design. epoch - the presentation of the set of training (input and/or target) vectors to a network and the calculation of new weights and biases. Note that training vectors may be presented one at a time or all together in a batch. error jumping -a sudden increase in a network’s sum-squared error during training. This is often due to too large a learning rate. error ratio - a training parameter used with adaptive learning rate and momentum training of backpropagation networks. error vector - the difference between a network’s output vector in response to an input vector and an associated target output vector. feedback network - a network with connections from a layer’s output to that layer’s input. The feedback connection may be direct or pass through several layers. feedforward network - a layered network in which each layer only receives inputs from previous layers. Fletcher-Reeves update - a method developed by Fletcher and Reeves for computing a set of conjugate directions. These directions are used as search directions as part of a conjugate gradient optimization procedure. function approximation - the task performed by a network trained to respond to inputs with an approximation of a desired function. generalization - an attribute of a network whose output for a new input vector tends to be close to outputs for similar input vectors in its training set. A-4 generalized regression network - approximates a continuous function to an arbitrary accuracy, given a sufficient number of hidden neurons. global minimum - the lowest value of a function over the entire range of its input parameters. Gradient descent methods adjust weights and biases in order to find the global minimum of error for a network. golden section search - a linear search which does not require the calculation of the slope. The interval containing the minimum of the performance is subdivided at each iteration of the search, and one subdivision is eliminated at each iteration. gradient descent - the process of making changes to weights and biases, where the changes are proportional to the derivatives of network error with respect to those weights and biases. This is done to minimize network error. hard limit transfer function - a transfer that maps inputs greater-than or equal-to 0 to 1, and all other values to 0. Hebb learning rule - historically the first proposed learning rule for neurons. Weights are adjusted proportional to the product of the outputs of pre- and post-weight neurons. hidden layer - a layer of a network that is not connected to the network output. (For instance, the first layer of a two layer feedforward network.) home neuron - a neuron at the center of a neighborhood. hybrid bisection-cubicsearch - a line search that combines bisection and cubic interpolation. input layer - a layer of neurons receiving inputs directly from outside the network. initialization - the process of setting the network weights and biases to their original values. input space - the range of all possible input vectors. input vector - a vector presented to the network. input weights - the weights connecting network inputs to layers. input weight vector - the row vector of weights going to a neuron. Jacobian matrix - contains the first derivatives of the network errors with respect to the weights and biases. A-5 A Glossary Kohonen learning rule - a learning rule that trains selected neuron’s weight vectors to take on the values of the current input vector. layer - a group of neurons having connections to the same inputs and sending outputs to the same destinations. layer diagram - a network architecture figure showing the layers and the weight matrices connecting them. Each layer’s transfer function is indicated with a symbol. Sizes of input, output, bias and weight matrices are shown. Individual neurons and connections are not shown. (See Chapter 2.) layer weights - the weights connecting layers to other layers. Such weights need to have non-zero delays if they form a recurrent connection (i.e. a loop). learning - the process by which weights and biases are adjusted to achieve some desired network behavior. learning rate - a training parameter that controls the size of weight and bias changes during learning. learning rules - methods of deriving the next changes that might be made in a network OR a procedure for modifying the weights and biases of a network. Levenberg-Marquardt - an algorithm that trains a neural network 10 to 100 faster than the usual gradient descent backpropagation method. It will always compute the approximate Hessian matrix, which has dimensions n × n . line search function - procedure for searching along a given search direction (line) to locate the minimum of the network performance. linear transfer function - a transfer function that produces its input as its output. link distance - the number of links, or steps, that must be taken to get to the neuron under consideration. local minimum - the minimum of a function over a limited range of input values. A local minimum may not be the global minimum. log-sigmoid transfer function - a squashing function of the form shown below that maps the input to the interval (0,1). (The toolbox function is logsig.) 1 f (n) = -----------------1 + e –n A-6 Manhattan distance - The Manhattan distance between two vectors x and y is calculated as: D = sum(abs(x-y)) maximum performance increase - the maximum amount by which the performance is allowed to increase in one iteration of the variable learning rate training algorithm. maximum step size - the maximum step size allowed during a linear search. The magnitude of the weight vector is not allowed to increase by more than this maximum step size in one iteration of a training algorithm. mean square error function - the performance function that calculates the average squared error between the network outputs a and the target outputs t. momentum - a technique often used to make it less likely for a backpropagation networks to get caught in a shallow minima. momentum constant - A training parameter that controls how much “momentum” is used. mu parameter - the initial value for the scalar µ. neighborhood - a group of neurons within a specified distance of a particular neuron. The neighborhood is specified by the indices for all of the neurons that lie within a radius d of the winning neuron i∗ : N i ( d ) = { j, d ij ≤ d } net input vector - the combination, in a layer, of all the layer’s weighted input vectors with its bias. neuron - the basic processing element of a neural network. Includes weights and bias, a summing junction and an output transfer function. Artificial neurons, such as those simulated and trained with this toolbox, are abstractions of biological neurons. neuron diagram - a network architecture figure showing the neurons and the weights connecting them. Each neuron’s transfer function is indicated with a symbol. ordering phase - period of training during which neuron weights are expected to order themselves in the input space consistent with the associated neuron positions. A-7 A Glossary output layer - a layer whose output is passed to the world outside the network. output vector - the output of a neural network. Each element of the output vector is the output of a neuron. output weight vector - the column vector of weights coming from a neuron or input. (See outstar learning rule.) outstar learning rule - a learning rule that trains an neuron’s (or input’s) output weight vector to take on the values of the current output vector of the post-weight layer. Changes in the weights are proportional to the neuron’s output. overfitting - a case in which the error on the training set is driven to a very small value, but when new data is presented to the network, the error is large. pass - each traverse through all of the training input and target vectors. pattern - a vector. pattern association - the task performed by a network trained to respond with the correct output vector for each presented input vector. pattern recognition - the task performed by a network trained to respond when an input vector close to a learned vector is presented. The network “recognizes” the input as one of the original target vectors. performance function - commonly the mean squared error of the network outputs. However, the toolbox also considers other performance functions. Type nnets and look under performance functions. perceptron - a single-layer network with a hard limit transfer function. This network is often trained with the perceptron learning rule. perceptron learning rule - a learning rule for training single-layer hard limit networks. It is guaranteed to result in a perfectly functioning network in finite time given that the network is capable of doing so. Polak-Ribiére update - a method developed by Polak and Ribiére for computing a set of conjugate directions. These directions are used as search directions as part of a conjugate gradient optimization procedure. positive linear transfer function - a transfer function that produces an output of zero for negative inputs and an output equal to the input for positive inputs. postprocessing - converts normalized outputs back into the same units which were used for the original targets. A-8 Powell-Beale restarts - a method developed by Powell and Beale for computing a set of conjugate directions. These directions are used as search directions as part of a conjugate gradient optimization procedure. This procedure also periodically resets the search direction to the negative of the gradient. preprocessing - perform some transformation of the input or target data before it is presented to the neural network. principal component analysis - orthogonalize the components of network input vectors. This procedure can also reduce the dimension of the input vectors by eliminating redundant components. quasi-Newton algorithm - class of optimization algorithm based on Newton’s method. An approximate Hessian matrix is computed at each iteration of the algorithm based on the gradients. radial basis networks - a neural network that can be designed directly by fitting special response elements where they will do the most good. radial basis transfer function - the transfer function for a radial basis neuron is: radbas ( n ) = e –n 2 regularization - involves modifying the performance function, which is normally chosen to be the sum of squares of the network errors on the training set, by adding some fraction of the squares of the network weights. resilient backpropagation - a training algorithm that eliminates the harmful effect of having a small slope at the extreme ends of the sigmoid “squashing” transfer functions. saturating linear transfer function - a function that is linear in the interval (-1,+1) and saturates outside this interval to -1 or +1. (The toolbox function is satlin.) scaled conjugate gradient algorithm - avoids the time consuming line search of the standard conjugate gradient algorithm. sequential input vectors - a set of vectors that are to be presented to a network “one after the other.” The network weights and biases are adjusted on the presentation of each input vector. A-9 A Glossary sigma parameter - determines the change in weight for the calculation of the approximate Hessian matrix in the scaled conjugate gradient algorithm. sigmoid - monotonic S-shaped function mapping numbers in the interval (-∞,∞) to a finite interval such as (-1,+1) or (0,1). simulation - takes the network input p, and the network object net, and returns the network outputs a. spread constant - the distance an input vector must be from a neuron’s weight vector to produce an output of 0.5. squashing function - a monotonic increasing function that takes input values between -∞ and +∞ and returns values in a finite interval. star learning rule - a learning rule that trains a neuron’s weight vector to take on the values of the current input vector. Changes in the weights are proportional to the neuron’s output. sum-squared error - The sum of squared differences between the network targets and actual outputs for a given input vector or set of vectors. supervised learning - a learning process in which changes in a network’s weights and biases are due to the intervention of any external teacher. The teacher typically provides output targets. symmetric hard limit transfer function - a transfer that maps inputs greater-than or equal-to 0 to +1, and all other values to -1. symmetric saturating linear transfer function - produces the input as its output as long as the input i in the range -1 to 1. Outside that range the output is -1 and +1 respectively. tan-sigmoid transfer function - a squashing function of the form shown below that maps the input to the interval (-1,1). (The toolbox function is tansig.) 1 f (n) = -----------------1 + e –n tapped delay line - a sequential set of delays with outputs available at each delay output. target vector - the desired output vector for a given input vector. topology functions - ways to arrange the neurons in a grid, box, hexagonal, or random topology. A-10 training - a procedure whereby a network is adjusted to do a particular job training vector - an input and/or target vector used to train a network. transfer function - the function that maps a neuron’s (or layer’s) net output n to its actual output. tuning phase - period of SOFM training during which weights are expected to spread out relatively evenly over the input space while retaining their topological order found during the ordering phase. underdetermined system - a system that has more variables than constraints. unsupervised learning - a learning process in which changes in a network’s weights and biases are not due to the intervention of any external teacher. Commonly changes are a function of the current network input vectors, output vectors, and previous weights and biases. update - make a change in weights and biases. The update can occur after presentation of a single input vector or after accumulating changes over several input vectors. weighted input vector - the result of applying a weight to a layer's input, whether it is a network input or the output of another layer. weight matrix - a matrix containing connection strengths from a layer’s inputs to its neurons. The element wi,j of a weight matrix W refers to the connection strength from input j to neuron i. Widrow-Hoff learning rule - a learning rule used to trained single layer linear networks. This rule is the predecessor of the backpropagation rule and is sometimes referred to as the delta rule. A-11 A Glossary A-12 B Notation Mathematical Notation for Equations and Figures Basic Concepts . . . . . . . . . . . . . . . . . Language . . . . . . . . . . . . . . . . . . . . Weight Matrices . . . . . . . . . . . . . . . . . Layer Notation . . . . . . . . . . . . . . . . . Figure and Equation Examples . . . . . . . . . . . Mathematics and Code Equivalents . . . . . . . . . . . . B-2 B-2 B-2 B-2 B-2 B-3 . . . . . . . . . B-4 B Notation Mathematical Notation for Equations and Figures Basic Concepts Scalars-small italic letters.....a,b,c Vectors - small bold non-italic letters.....a,b,c Matrices - capital BOLD non-italic letters.....A,B,C Language Vector means a column of numbers. Weight Matrices Scalar Element w i, j ( t ) i - row, j - column, t - time or iteration Matrix W ( t ) Column Vector w j ( t ) Row Vector iw ( t ) ...vector made of ith row of weight matrix W Bias Vector Scalar Element b i ( t ) Vector b ( t ) Layer Notation A single superscript will be used to identify elements of layer. For instance, the net input of layer 3 would be shown as n3. Superscripts k, l will be used to identify the source (l) connection and the destination (k) connection of layer weight matrices ans input weight matrices. For instance, the layer weight matrix from layer 2 to layer 4 would be shown as LW4,2. B-2 Mathematical Notation for Equations and Figures Input Weight Matrix IW k, l Layer Weight Matrix LW k, l Figure and Equation Examples The following figure, taken from Chapter 11, “Advanced Topics,” illustrates notation used in such advanced figures. Inputs p1(k) 2x1 Layers 1 and 2 AA AA AA AAAA AA AAAA AA IW1,1 4x2 1 Layer 3 n1(k) 4x1 TDL 1 4 0,1 IW2,1 3 x (2*2) p2(k) 5x1 5 TDL 1 IW2,2 3 x (1*5) IW3,1 4x1 1x4 a1(k) = tansig (IW1,1p1(k) +b1) TDL LW3,3 1 x (1*1) a1(k) b1 4x1 2 AA AA A A AA AA A AA AA AA AA A AA AA A AA A A 1 b3 1x1 a2(k) n2(k) 3x1 3x1 Outputs n3(k) a3(k) y2(k) 1x1 1x1 1x1 1 LW3,2 1x3 y1(k) 3x1 3 a2(k) = logsig (IW2,1 [p1(k);p1(k-1) ]+ IW2,2p2(k-1)) a3(k)=purelin(LW3,3a3(k-1)+IW3,1 a1 (k)+b3+LW3,2a2 (k)) B-3 B Notation Mathematics and Code Equivalents The transition from mathematics to code or vice versa can be made with the aid of a few rules. They are listed here for future reference. To change from Mathematics notation to MATLAB notation the user needs to: • Change superscripts to cell array indices 1 For example, p → p { 1 } • Change subscripts to parentheses indices 1 For example, p 2 → p ( 2 ) , and p 2 → p { 1 } ( 2 ) • Change parentheses indices to a second cell array index 1 For example, p ( k – 1 ) → p { 1, k – 1 } • Change mathematics operators to MATLAB operators and toolbox functions For example, ab → a*b The following Equations illustrate the notation used in figures. n = w 1, 1 p 1 + w 1, 2 p 2 + ... + w 1, R p R + b w 1, 1 w 1, 2 … w 1, R W = w 2, 1 w 2, 2 … w 2, R w S, 1 w S, 2 … w S, R B-4 C Bibliography C Bibliography [Batt92] Battiti, R., “First and second order methods for learning: Between steepest descent and Newton’s method,” Neural Computation, vol. 4, no. 2, pp. 141–166, 1992. [Beal72] Beale, E. M. L., “A derivation of conjugate gradients,” in F. A. Lootsma, ed., Numerical methods for nonlinear optimization, London: Academic Press, 1972. [Bren73] Brent, R. P., Algorithms for Minimization Without Derivatives, Englewood Cliffs, NJ: Prentice-Hall, 1973. [Caud89] Caudill, M., Neural Networks Primer, San Francisco, CA: Miller Freeman Publications, 1989. This collection of papers from the AI Expert Magazine gives an excellent introduction to the field of neural networks. The papers use a minimum of mathematics to explain the main results clearly. Several good suggestions for further reading are included. [CaBu92] Caudill, M., and C. Butler, Understanding Neural Networks: Computer Explorations, Vols. 1 and 2, Cambridge, MA: the MIT Press, 1992. This is a two volume workbook designed to give students “hands on” experience with neural networks. It is written for a laboratory course at the senior or first-year graduate level. Software for IBM PC and Apple Macintosh computers is included. The material is well written, clear and helpful in understanding a field that traditionally has been buried in mathematics. [Char92] Charalambous, C.,“Conjugate gradient algorithm for efficient training of artificial neural networks,” IEEE Proceedings, vol. 139, no. 3, pp. 301–310, 1992. [ChCo91] Chen, S., C. F. N. Cowan, and P. M. Grant, “Orthogonal least squares learning algorithm for radial basis function networks,” IEEE Transactions on Neural Networks, vol. 2, no. 2, pp. 302-309, 1991. This paper gives an excellent introduction to the field of radial basis functions. The papers use a minimum of mathematics to explain the main results clearly. Several good suggestions for further reading are included. [DARP88] DARPA Neural Network Study, Lexington, MA: M.I.T. Lincoln Laboratory, 1988. This book is a compendium of knowledge of neural networks as they were known to 1988. It presents the theoretical foundations of neural networks and C-2 discusses their current applications. It contains sections on associative memories, recurrent networks, vision, speech recognition, and robotics. Finally, it discusses simulation tools and implementation technology. [DeSc83] Dennis, J. E., and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Englewood Cliffs, NJ: Prentice-Hall, 1983. [Elma90] Elman, J. L.,“Finding structure in time,” Cognitive Science, vol. 14, pp. 179-211, 1990. This paper is a superb introduction to the Elman networks described in Chapter 10, “Recurrent Networks.” [FlRe64] Fletcher, R., and C. M. Reeves, “Function minimization by conjugate gradients,” Computer Journal, vol. 7, pp. 149-154, 1964. [FoHa97] Foresee, F. D., and M. T. Hagan, “Gauss-Newton approximation to Bayesian regularization,” Proceedings of the 1997 International Joint Conference on Neural Networks,, pages 1930-1935, 1997. [GiMu81] Gill, P. E., W. Murray, and M. H. Wright, Practical Optimization, New York: Academic Press, 1981. [Gros82] Grossberg, S., Studies of the Mind and Brain, Drodrecht, Holland: Reidel Press, 1982. This book contains articles summarizing Grossberg’s theoretical psychophysiology work up to 1980. Each article contains a preface explaining the main points. [HaMe94] Hagan, M. T., and M. Menhaj, “Training feedforward networks with the Marquardt algorithm,” IEEE Transactions on Neural Networks, vol. 5, no. 6, pp. 989–993, 1994. This paper reports the first development of the Levenberg-Marquardt algorithm for neural networks. It describes the theory and application of the algorithm, which trains neural networks at a rate 10 to 100 times faster than the usual gradient descent backpropagation method. [HDB96] Hagan, M. T., H. B. Demuth, and M. H. Beale, Neural Network Design, Boston, MA: PWS Publishing, 1996. [Hebb49] Hebb, D. O., The Organization of Behavior, New York: Wiley, 1949. C-3 C Bibliography This book proposed neural network architectures and the first learning rule. The learning rule is used to form a theory of how collections of cells might form a concept. [Himm72] Himmelblau, D. M., Applied Nonlinear Programming, New York: McGraw-Hill, 1972. [Joll86] Jolliffe, I. T., Principal Component Analysis, New York: Springer-Verlag, 1986. [Koho87] Kohonen, T., Self-Organization and Associative Memory, 2nd Edition, Berlin: Springer-Verlag, 1987. This book analyzes several learning rules. The Kohonen learning rule is then introduced and embedded in self-organizing feature maps. Associative networks are also studied. [LiMi89] Li, J., A. N. Michel, and W. Porod, “Analysis and synthesis of a class of neural networks: linear systems operating on a closed hypercube,” IEEE Transactions on Circuits and Systems, vol. 36, no. 11, pp. 1405-1422, 1989. This paper discusses a class of neural networks described by first order linear differential equations that are defined on a closed hypercube. The systems considered retain the basic structure of the Hopfield model but are easier to analyze and implement. The paper presents an efficient method for determining the set of asymptotically stable equilibrium points and the set of unstable equilibrium points. Examples are presented. The method of Li et. al. is implemented in Chapter 9 of this Users Guide. [Lipp87] Lippman, R. P., “An introduction to computing with neural nets,” IEEE ASSP Magazine, pp. 4-22, 1987. This paper gives an introduction to the field of neural nets by reviewing six neural net models that can be used for pattern classification.The paper shows how existing classification and clustering algorithms can be performed using simple components that are like neurons. This is a highly readable paper. [MacK92] MacKay, D. J. C., “Bayesian interpolation,” Neural Computation, vol. 4, no. 3, pp. 415-447, 1992. [McPi43] McCulloch, W. S., and W. H. Pitts, “A logical calculus of ideas immanent in nervous activity,” Bulletin of Mathematical Biophysics, vol. 5, pp. 115-133, 1943. C-4 A classic paper that describes a model of a neuron that is binary and has a fixed threshold. A network of such neurons can perform logical operations. [Moll93] Moller, M. F., “A scaled conjugate gradient algorithm for fast supervised learning,” Neural Networks, vol. 6, pp. 525-533, 1993. [NgWi89] Nguyen, D., and B. Widrow, “The truck backer-upper: An example of self-learning in neural networks,” Proceedings of the International Joint Conference on Neural Networks, vol 2, pp. 357-363, 1989. This paper describes a two-layer network that first learned the truck dynamics and then learned how to back the truck to a specified position at a loading dock. To do this, the neural network had to solve a highly nonlinear control systems problem. [NgWi90] Nguyen, D., and B. Widrow, “Improving the learning speed of 2-layer neural networks by choosing initial values of the adaptive weights,” Proceedings of the International Joint Conference on Neural Networks, vol 3, pp. 21-26, 1990. Nguyen and Widrow demonstrate that a 2-layer sigmoid/linear network can be viewed as performing a piecewise linear approximation of any learned function. It is shown that weights and biases generated with certain constraints will result in an initial network better able to form a function approximation of an arbitrary function. Use of the Nguyen-Widrow (instead of purely random) initial conditions often shortens training time by more than an order of magnitude. [Powe77] Powell, M. J. D., “Restart procedures for the conjugate gradient method,” Mathematical Programming, vol. 12, pp. 241-254, 1977. [Pulu92] N. Purdie, E.A. Lucas and M.B. Talley, "Direct measure of total cholesterol and its distribution among major serum lipoproteins," Clinical Chemistry, vol. 38, no. 9, pp. 1645-1647, 1992. [RiBr93] Riedmiller, M., and H. Braun, “A direct adaptive method for faster backpropagation learning: The RPROP algorithm,” Proceedings of the IEEE International Conference on Neural Networks, 1993. [Rose61] Rosenblatt, F., Principles of Neurodynamics, Washington D.C.: Spartan Press, 1961. This book presents all of Rosenblatt’s results on perceptrons. In particular, it presents his most important result, the perceptron learning theorem. C-5 C Bibliography [RuHi86a] Rumelhart, D. E., G. E. Hinton, and R. J. Williams, “Learning internal representations by error propagation,”, in D. E. Rumelhart and J. L. McClelland, eds. Parallel Data Processing, vol.1, Cambridge, MA: The M.I.T. Press, pp. 318-362, 1986. This is a basic reference on backpropagation. [RuHi86b] Rumelhart, D. E., G. E. Hinton, and R. J. Williams, “Learning representations by back-propagating errors,” Nature, vol. 323, pp. 533–536, 1986. [RuMc86] Rumelhart, D. E., J. L. McClelland, and the PDP Research Group, eds., Parallel Distributed Processing, Vols. 1 and 2, Cambridge, MA: The M.I.T. Press, 1986. These two volumes contain a set of monographs that present a technical introduction to the field of neural networks. Each section is written by different authors. These works present a summary of most of the research in neural networks to the date of publication. [Scal85] Scales, L. E., Introduction to Non-Linear Optimization, New York: Springer-Verlag, 1985. [VoMa88] Vogl, T. P., J. K. Mangis, A. K. Rigler, W. T. Zink, and D. L. Alkon, “Accelerating the convergence of the backpropagation method,” Biological Cybernetics, vol. 59, pp. 256-264, 1988. Backpropagation learning can be speeded up and made less sensitive to small features in the error surface such as shallow local minima by combining techniques such as batching, adaptive learning rate, and momentum. [Wass93] Wasserman, P. D., Advanced Methods in Neural Computing, New York: Van Nostrand Reinhold, 1993. [WiHo60] Widrow, B., and M. E. Hoff, “Adaptive switching circuits,” 1960 IRE WESCON Convention Record, New York IRE, pp. 96-104, 1960. [WiSt85] Widrow, B., and S. D. Sterns, Adaptive Signal Processing, New York: Prentice-Hall, 1985. This is a basic paper on adaptive signal processing. C-6 D Demonstrations and Applications Tables of Demonstrations and Applications . Chapter 2 Neuron Model & Network Architectures . Chapter 3 Perceptrons . . . . . . . . . . . . Chapter 4 Adaptive Linear Filters . . . . . . . . Chapter 5 Backpropagation . . . . . . . . . . Chapter 6 Radial Basis Networks . . . . . . . . Chapter 7 Self-Organizing Networks . . . . . . . Chapter 8 Learning Vector Quantization . . . . . Chapter 9 Recurrent Networks . . . . . . . . . Chapter 10 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-2 D-2 D-2 D-3 D-3 D-4 D-4 D-4 D-5 D-5 D Demonstrations and Applications Tables of Demonstrations and Applications Chapter 2 Neuron Model & Network Architectures File Name Page Simple neuron and transfer functions nnd2n1 page 2-6 Neuron with vector input nnd2n2 page 2-9 File Name Page Decision Boundaries nnd4db page 3-5 Perceptron learning rule. Pick boundaries nnd4pr page 3-14 Classification with a 2-input perceptron demop1 page 3-20 Outlier input vectors demop4 page 3-22 Normalized perceptron rule demop5 page 3-22 Linearly non-separable vectors demop6 page 3-21 Chapter 3 Perceptrons D-2 Tables of Demonstrations and Applications Chapter 4 Adaptive Linear Filters File Name Page Pattern association showing error surface demolin1 page 4-10 Training a linear neuron demolin2 page 4-15 Linear classification system nnd10lc page 4-15 Adaptive noise cancellation, Toolbox Example demolin8 page 4-10 Adaptive noise cancellation in airplane cockpit nnd10nc page 4-21 Linear fit of nonlinear problem demolin4 page 4-15 Underdetermined problem demolin5 page 4-25 Linearly dependent problem demolin6 page 4-26 Too large a learning rate demolin7 page 4-26 File Name Page Generalization nnd11gn page 5-37 Steepest descent backpropagation nnd12sd1 page 5-13 nnd12mo page 5-15 nnd12vl page 5-17 nnd12cg page 5-22 nnd12m page 5-33 Chapter 5 Backpropagation Momentum backpropagation Variable learning rate backpropagation Conjugate gradient backpropagation Marquardt backpropagation D-3 D Demonstrations and Applications Chapter 6 Radial Basis Networks File Name Page Radial basis approximation demorb1 page 6-8 Radial basis underlapping neurons demorb3 page 6-8 Radial basis overlapping neurons demorb4 page 6-8 GRNN Function Approximation demogrn1 page 6-11 PNN Classification demopnn1 page 6-14 Chapter 7 Self-Organizing Networks File Name Page Competitive learning democ1 page 7-9 One-Dimensional Self-organizing map demosm1 page 7-25 Two-Dimensional Self-organizing map demosm2 page 7-25 Chapter 8 Learning Vector Quantization Learning vector quantization D-4 File Name Page demolvq1 page 8-12 Tables of Demonstrations and Applications Chapter 9 Recurrent Networks File Name Page Hopfield two neuron design demohop1 page 9-15 Hopfield unstable equilibria demohop2 page 9-15 Hopfield three neuron design demohop3 page 9-15 Hopfield spurious stable points demohop4 page 9-15 File Name Page Linear design applin1 page 10-3 Adaptive linear prediction applin2 page 10-7 Linear system identification applin3 page 10-11 Adaptive linear system identification applin4 page 10-15 Elman amplitude detection appelm1 page 10-19 Nonlinear control system identification appcs1 page 10-24 Model Reference Control appcs2 page 10-30 Character Recognition appcr1 page 10-38 Chapter 10 Applications D-5 D Demonstrations and Applications D-6 E Simulink Block Set . . . . . . . Transfer Function Blocks Net Input Blocks . . . . Weight Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-2 E-3 E-3 E-4 Block Generation . . . . . . . . . . . . . . . . . . E-5 Example . . . . . . . . . . . . . . . . . . . . . . E-5 Exercises . . . . . . . . . . . . . . . . . . . . . . E-7 E Simulink Block Set The Neural Network Toolbox provides a set of blocks you can use to build neural networks in Simulink or which can be used by the function gensim to generate the Simulink version of any network you have created in MATLAB. Bring up the Neural Network Toolbox block set with this command: neural The result will be a window, which contains three blocks. Each of these block contains additional blocks. E-2 Block Set Transfer Function Blocks Double click on the Transfer Functions block in the Neural window to bring up a window containing several transfer function blocks. Each of these blocks takes a net input vector and generates a corresponding output vector whose dimensions are the same as the input vector. Net Input Blocks Double click on the Net Input Functions block in the Neural window to bring up a window containing two net input function blocks. Each of these blocks takes any number of weighted input vectors, weight layer output vectors, and bias vectors, and returns a net input vector. E-3 E Simulink Weight Blocks Double click on the Weight Functions block in the Neural window to bring up a window containing three weight function blocks. Each of these blocks takes a neuron’s weight vector and applies it to an input vector (or a layer output vector) to get a weighted input value for a neuron. It is important to note that the blocks above expect the neuron’s weight vector to be defined as a column vector. This is because Simulink signals can be column vectors, but cannot be matrices or row vectors. It is also important to note that because of this limitation you will have to create S weight function blocks, one for each row, to implement a weight matrix going to a layer with S neurons. This contrasts with the other two kinds of blocks. Only one net input function and one transfer function block are required for each layer. E-4 Block Generation Block Generation The function gensim generates block descriptions of networks so you can simulate them in Simulink. gensim(net,st) The second argument to gensim determines the sample time, which is normally chosen to be some positive real value. If a network has no delays associated with its input weights or layer weights this value can be set to -1. A value of -1 tells gensim to generate a network with continuous sampling. Example Here is a simple problem defining a set of inputs p and corresponding targets t: p = [1 2 3 4 5]; t = [1 3 5 7 9]; The code below designs a linear layer to solve this problem. net = newlind(p,t) We can test the network on our original inputs with sim. y = sim(net,p) The results returned show the network has solved the problem. y = 1.0000 3.0000 5.0000 7.0000 9.0000 Call gensim as follows to generate a Simulink version of the network: gensim(net,-1) The second argument is -1 so the resulting network block will sample continuously. The call to gensim results in the following screen appearing. It contains a Simulink system consisting of the linear network connected to a sample input and a scope. E-5 E Simulink To test the network double click on the Input 1 block at left. The input block is actually a standard Constant block. Change the constant value from the initial randomly generated value to 2, then select Close. Select Start from the Simulation menu. Simulink will momentarily pause as it simulates the system. When the simulation is over double click the scope at the right to see the following display of the network’s response. E-6 Block Generation Note that the output is 3, which is the correct output for an input of 2. Exercises Here are a couple of exercises you might try. Changing Input Signal Replace the constant input block with a signal generator from the standard Simulink block set Sources. Simulate the system and view the network’s response. Discrete Sample Time Recreate the network, but with a discrete sample time of 0.5, instead of continuous sampling. gensim(net,0.5) Again replace the constant input with a signal generator. Simulate the system and view the network’s response. E-7 E Simulink E-8 Index A ADALINE network decision boundary 4-6 Adaption 5-9 custom function 11-32 definition 3-15 function 12-10 parameters 12-13 Adaptive filter 4-17 example 4-18 noise cancellation example 4-21 prediction application 10-7 prediction example 4-20 system identification application 10-15 training 2-20 Adaptive linear networks 4-2 Amplitude detection 10-19 Applications adaptive filtering 4-16 aerospace 1-16 automotive 1-16 banking 1-16 defense 1-16 electronics 1-16 entertainment 1-17 financial 1-17 insurance 1-17 manufacturing 1-17 medical 1-17, 5-49 oil and gas exploration 1-17 robotics 1-17 speech 1-18 telecommunications 1-18 transportation 1-18 Architecture bias connection 11-6, 12-3 input connection 11-6, 12-4 input delays 12-5 layer connection 11-6, 12-4 layer delays 12-6 number of inputs 11-5, 12-2 number of layers 11-5, 12-2 number of outputs 11-7, 12-5 number of targets 11-7, 12-5 output connection 11-7, 12-4 target connection 11-7, 12-5 B Backpropagation 5-2 algorithm 5-9 example 5-49 Backtracking search 5-28 Batch training 2-20, 2-22, 5-12 dynamic networks 2-24 static networks 2-22 Bayesian framework 5-39 BFGS quasi-Newton algorithm 5-29 Bias connection 11-6 definition 2-4 initialization function 12-26 learning 12-26 learning function 12-27 learning parameters 12-27 subobject 11-11, 12-26 value 11-13, 12-16 Box distance 7-17 Brent’s search 5-27 I-1 Index C Cell array 2-3, 2-17, 2-18, 2-19, 2-21, 2-22, 2-24, 11-8, 11-9, 11-13, 11-14, 11-29 Charalambous’ search 5-28 Classification 6-12 input vectors 3-4 linear 4-13 regions 3-5 Code mathematical equivalents 2-2, 2-12 perceptron network 3-7 writing 2-7 Competitive layer 7-3 Competitive neural network 7-4 example 7-8 Competitive transfer function 6-12, 7-3, 7-19 Concurrent inputs 2-15, 2-18 Conjugate gradient algorithm 5-20 Fletcher-Reeves update 5-21 Polak-Ribiere update 5-22 Powell-Beale restarts 5-24 scaled 5-25 Custom neural network 11-2 D Dead neurons 7-5 Decision boundary 4-6 definition 3-5 Demonstrations appcs1 10-24 appelm1 10-19 applin3 10-11 applin4 10-15 definition 1-4 demohop1 9-15 I-2 demohop2 9-15 demolin4 4-25 demorb4 6-8 nnd10lc 4-15 nnd11gn 5-37 nnd12cg 5-22 nnd12m 5-33 nnd12mo 5-15 nnd12sd1 5-13, 5-26 nnd12vl 5-17 Distance 7-10, 7-16 box 7-17 custom function 11-40 Euclidean 7-16 link 7-18 Manhattan 7-18 tuning phase 7-20 Dynamic network 2-16, 2-18 Dynamic networks training 2-22, 2-24 E Early stopping 1-9, 5-41 Elman network 9-3 recurrent connection 9-3 Euclidean distance 7-16 F Feedforward network 5-5 Finite impulse response filter 4-17 Fletcher-Reeves update 5-21 Index G Generalization 5-37 regularization 5-38 Generalized regression network 1-11, 6-9 Golden section search 5-26 Gradient descent algorithm 5-2, 5-9 batch 5-12, 5-33 with momentum 5-11, 5-13 Gridtop topology 7-12 H Hard limit transfer function 2-5, 2-28, 3-4 Heuristic techniques 5-16 Hextop topology 7-12 Hidden layer definition 2-14 Home neuron 7-16 Hopfield network architecture 9-9 design equilibrium point 9-10 solution trajectories 9-15 stable equilibrium point 9-11 target equilibrium points 9-11 Hybrid bisection-cubic search 5-27 I Identification adaptive 10-15 linear 10-11 nonlinear system application 10-24 Incremental training 2-20 Initial step size function 5-19 Initialization additional functions 11-17 custom function 11-26 definition 3-9 function 5-7, 12-10 parameters 12-13 Input concurrent 2-15 connection 11-6 number 11-5 range 12-17 size 12-17 subobject 11-8, 11-9, 12-17 Input vector classification 3-4 dimension reduction 5-46 distance 7-10 outlier 3-21 topology 7-10 Input weights definition 2-12 Inputs concurrent 2-18 sequential 2-15, 2-16 Installation guide 1-4 J Jacobian matrix 5-32 K Kohonen learning rule 7-5 I-3 Index L Lambda parameter 5-25 Layer connection 11-6 dimensions 12-18 distance function 12-19 distances 12-19 initialization function 12-20 net input function 12-20 number 11-5 positions 12-21 size 12-22 subobject 12-18 topology function 12-22 transfer function 12-23 Layer weights definition 2-12 Learning rate 5-10 adaptive 5-17 maximum stable 4-12 optimal 5-16 ordering phase 7-20 too large 4-26 tuning phase 7-20 Learning rules 3-2 custom 11-36 Hebb 11-18 Hebb with decay 11-18 instar 11-18 Kohonen 7-5 outstar 11-18 supervised learning 3-11 unsupervised learning 3-11 Widrow-Hoff 4-2, 4-5, 4-11, 5-2 Learning vector quantization 8-2 creation 8-5 learning rule 8-9 I-4 LVQ network 8-3 subclasses 8-3 target classes 8-3 union of two sub-classes 8-7 Least mean square error 4-9 Levenberg-Marquardt algorithm 5-31 definition 1-6 reduced memory 5-33 Line search function 5-21 backtracking search 5-28 Brent’s search 5-27 Charalambous’ search 5-28 Golden section search 5-26 hybrid bisection-cubic search 5-27 Linear networks design 4-10 Linear transfer function 2-6, 2-28, 4-4, 5-4 Linearly dependent vectors 4-26 Link distance 7-18 Log-sigmoid transfer function 2-6, 2-29, 5-3 M MADALINE 4-5 Manhattan distance 7-18 Maximum performance increase 5-14 Maximum step size 5-19 Mean square error function 5-8 least 4-9 Memory reduction 5-34 Model reference control 10-30 Momentum constant 5-11 Mu parameter 5-32 Index N Neighborhood 7-10 Net input function custom 11-22 Network definition 11-5 dynamic 2-16, 2-18 static 2-15 Network Function 11-12 Network layer competitive 7-3 definition 2-8 Neural network adaptive linear 4-2 competitive 7-4 custom 11-2 definition 1-2 feedforward 5-5 generalized regression 6-9 multiple layer 2-13, 4-23, 5-2 one layer 2-10, 3-6, 4-5 probabilistic 6-12 radial basis 6-2 self organizing 7-2 self-organizing feature map 7-10 Neuron dead (not allocated) 7-5 definition 2-4 home 7-16 Newton’s method 5-32 Normalization inputs and targets 5-44 mean and standard deviation 5-45 Notation abbreviated 2-7, 4-24 layer 2-13 . mathematical 2-2 transfer function symbols 2-6, 2-9 Numerical optimization 5-16 O One step secant algorithm 5-30 Ordering phase learning rate 7-20 Outlier input vector 3-21 Output connection 11-7 number 11-7 size 12-25 subobject 11-10, 12-25 Output layer definition 2-14 linear 5-5 Overdetermined systems 4-25 Overfitting 5-37 P Pass definition 3-15 pattern recognition 10-38 Perceptron learning rule 3-2, 3-12 normalized 3-22 Perceptron network 3-2 code 3-7 creation 3-3 limitations 3-21 I-5 Index S Performance function 12-11 custom 11-34 modified 5-38 parameters 12-13 Polak-Ribiere update 5-22 Postprocessing 5-44 Post-training analysis 5-47 Powell-Beale restarts 5-24 Preprocessing 5-44 Principal component analysis 5-46 Probabilistic neural network 6-12 design 6-13 Q Quasi-Newton algorithm 1-7, 5-28 BFGS 1-7, 5-29 R Radial basis design 6-11 efficient network 6-7 function 6-2 network 6-2 network design 6-5 Radial basis transfer function 6-4 Randtop topology 7-12 Recurrent connection 9-3 Recurrent networks 9-2 Regularization 1-9, 5-38 automated 5-39 Resilient backpropagation 5-18 training algorithm 1-7 I-6 Self-organizing feature map (SOFM) network 7-10 neighborhood 7-10 one-dimensional example 7-25 two-dimensional example 7-27 Self-organizing networks 7-2 Sequential inputs 2-15, 2-16 S-function 13-2 Sigma parameter 5-25 Simulation 5-8 definition 3-8 Simulink generating networks E-5 NNT block set E-2 Simulink support 1-13 Speed comparison 1-8, 5-35 Spread constant 6-5 Squashing functions 5-18 Static network 2-15 Static networks batch training 2-22 training 2-20 Subobject bias 11-11, 12-8, 12-26 input 11-8, 11-9, 12-7, 12-17 layer 12-7, 12-18 output 11-10, 12-7, 12-25 target 11-10, 12-8, 12-25 weight 11-11, 12-9, 12-28, 12-32 Supervised learning 3-11 target output 3-11 training set 3-11 Index T Tan-sigmoid transfer function 5-4 Tapped delay line 4-16 Target connection 11-7 number 11-7 size 12-25 subobject 11-10, 12-25 Target output 3-11 Topologies gridtop 7-12 hextop 7-12 randtop 7-12 Topologies for SOFM neuron locations 7-12 Topology 7-10 custom function 11-39 Training 5-8 batch 2-20, 5-12 competitive networks 7-6 custom function 11-29 definition 2-4, 3-2 efficient 5-44 faster 5-16 function 12-11 incremental 2-20, 5-9 ordering phase 7-22 parameters 12-14 post-training analysis 5-47 self organizing feature map 7-22 styles 2-20 tuning phase 7-22 Training set 3-11 Training styles 2-20 Training with noise 10-41 Transfer function competitive 6-12, 7-3, 7-19 custom 11-20 definition 2-4 derivatives 5-4 hard limit 2-5, 3-4 linear 4-4, 5-4 log-sigmoid 2-6, 2-29, 5-3 radial basis 6-4 saturating linear 11-17 soft maximum 11-17 tan-sigmoid 5-4 triangular basis 11-17 Transformation matrix 5-46 Tuning phase learning rate 7-20 Tuning phase neighborhood distance 7-20 U Underdetermined systems 4-25 Unsupervised learning 3-11 V Variable learning rate algorithm 5-17 Vectors linearly dependent 4-26 I-7 Index W Weight definition 2-4 delays 12-28, 12-32 initialization function 12-29, 12-32 learning 12-29, 12-33 learning function 12-30, 12-33 learning parameters 12-31, 12-35 size 12-31, 12-35 subobject 11-11, 12-28, 12-32 value 11-13, 12-14, 12-15 weight function 12-32, 12-36 I-8 Weight function custom 11-24 Weight matrix definition 2-10 Widrow-Hoff learning rule 4-2, 4-5, 4-11, 5-2

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