Model-Based Calibration

Model-Based Calibration
Model-Based Calibration
Toolbox
For Use with MATLAB
®
Computation
Visualization
Programming
Model Browser User’s Guide
Version 1
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For contact information about worldwide offices, see the MathWorks Web site.
Model-Based Calibration Toolbox Model Browser User’s Guide
 COPYRIGHT 2002 by The MathWorks, Inc.
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.
FEDERAL ACQUISITION: This provision applies to all acquisitions of the Program and Documentation by
or for the federal government of the United States. By accepting delivery of the Program, the government
hereby agrees that this software qualifies as "commercial" computer software within the meaning of FAR
Part 12.212, DFARS Part 227.7202-1, DFARS Part 227.7202-3, DFARS Part 252.227-7013, and DFARS Part
252.227-7014. The terms and conditions of The MathWorks, Inc. Software License Agreement shall pertain
to the government’s use and disclosure of the Program and Documentation, and shall supersede any
conflicting contractual terms or conditions. If this license fails to meet the government’s minimum needs or
is inconsistent in any respect with federal procurement law, the government agrees to return the Program
and Documentation, unused, to MathWorks.
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TargetBox is a trademark of The MathWorks, Inc.
Other product or brand names are trademarks or registered trademarks of their respective holders.
Printing History: July 2002
Online only
Revised for Version 1.1 (Release 13)
Contents
Getting Started
1
What Is the Model-Based Calibration Toolbox? . . . . . . . . . . 1-3
About the Model Browser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-3
About CAGE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-3
How to Use This Manual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-5
System Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Hardware Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Operating System Requirements . . . . . . . . . . . . . . . . . . . . . . . .
Required MathWorks Products . . . . . . . . . . . . . . . . . . . . . . . . .
Optional MathWorks Products . . . . . . . . . . . . . . . . . . . . . . . . . .
1-6
1-6
1-6
1-7
1-7
Quickstart Tutorial
2
Two-Stage Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-3
Starting the Toolbox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-5
Setting Up the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-9
Setting Up the Local Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-12
Setting Up the Global Model . . . . . . . . . . . . . . . . . . . . . . . . . . 2-16
Selecting Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-20
Specifying the Response Model . . . . . . . . . . . . . . . . . . . . . . . . . 2-23
i
Verifying the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Verifying the Local Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Verifying the Global Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Selecting the Two-Stage Model . . . . . . . . . . . . . . . . . . . . . . . . .
Comparing the Local Model and the Two-Stage Model . . . . . .
Maximum Likelihood Estimation . . . . . . . . . . . . . . . . . . . . . . .
Response Node . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-24
2-24
2-26
2-29
2-34
2-35
2-38
Exporting the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-40
Design of Experiment Tutorial
3
What Is Design of Experiment? . . . . . . . . . . . . . . . . . . . . . . . . . 3-4
Design Styles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-5
Starting the Design Editor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-6
Optimal Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-9
Start Point Tab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-10
Candidate Set Tab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-11
Algorithm Tab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-14
Design Displays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-16
Prediction Error Variance Viewer . . . . . . . . . . . . . . . . . . . . . 3-19
Improving the Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-21
Classical Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-24
Classical Design Browser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-26
Set Up and View a Classical Design . . . . . . . . . . . . . . . . . . . . . 3-27
Design Evaluation Tool . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-30
Improving the Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-32
Space-Filling Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-33
Setting Up a Space-Filling Design . . . . . . . . . . . . . . . . . . . . . . 3-34
ii
Contents
Applying Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-35
Saving Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-40
Data Editor Tutorial
4
Loading the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-3
Entering the Data Editor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-3
Loading a Data File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-4
Viewing and Editing the Data . . . . . . . . . . . . . . . . . . . . . . . . . . 4-6
User-Defined Variables & Filtering . . . . . . . . . . . . . . . . . . . . . . 4-8
Defining a New Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-9
Applying a Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-10
Sequence of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-12
Deleting and Editing Variables and Filters . . . . . . . . . . . . . . . 4-13
Storage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-14
Test Groupings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-15
GUI Reference
5
Project Level: Startup View . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-3
Project Level: Toolbar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-7
Project Level: Menus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-8
Model Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-10
Test Plans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-15
Creating a New Test Plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-15
Creating New Test Plan Templates . . . . . . . . . . . . . . . . . . . . . 5-16
iii
Test Plan Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-19
Block Diagram Representations of Models . . . . . . . . . . . . . . . . 5-20
Test Plan Level: Toolbar and Menus . . . . . . . . . . . . . . . . . . . . 5-23
Setting Up Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Instant One-Stage Model Setup . . . . . . . . . . . . . . . . . . . . . . . .
Instant Two-Stage Model Setup . . . . . . . . . . . . . . . . . . . . . . . .
Setting Up Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Global Model Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Global Linear Models: Polynomials and Hybrid Splines . . . . .
Global Model Class: Radial Basis Function . . . . . . . . . . . . . . .
Global Model Class: Hybrid RBF . . . . . . . . . . . . . . . . . . . . . . .
Global Model Class: Multiple Linear Models . . . . . . . . . . . . . .
Global Model Class: Free Knot Spline . . . . . . . . . . . . . . . . . . .
Local Model Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Local Model Class: Polynomials and Polynomial Splines . . . .
Local Model Class: Truncated Power Series . . . . . . . . . . . . . . .
Local Model Class: Free Knot Spline . . . . . . . . . . . . . . . . . . . .
Local Model Class: Growth Models . . . . . . . . . . . . . . . . . . . . . .
Local Model Class: Linear Models . . . . . . . . . . . . . . . . . . . . . .
Local Model Class: User-Defined Models . . . . . . . . . . . . . . . . .
Local Model Class: Transient Models . . . . . . . . . . . . . . . . . . . .
Covariance Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Correlation Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Designing Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-26
5-27
5-28
5-30
5-32
5-33
5-39
5-42
5-43
5-44
5-46
5-47
5-50
5-51
5-52
5-53
5-54
5-54
5-54
5-56
5-56
5-57
Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Data Editor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Data Loading and Merging . . . . . . . . . . . . . . . . . . . . . . . . . . . .
User-Defined Variables and Filtering . . . . . . . . . . . . . . . . . . . .
Variable and Filter Editors . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Storage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Test Groupings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Data Wizard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Data Selection Window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-59
5-61
5-65
5-69
5-71
5-74
5-75
5-78
5-81
New Response Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-87
Datum Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-88
iv
Contents
Local Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-89
Local Special Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-91
Local Scatter Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-92
Diagnostic Statistics Pane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-93
Response Features List Pane . . . . . . . . . . . . . . . . . . . . . . . . . . 5-93
Test Notes Pane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-94
Local Level: Toolbar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-94
Local Level: Menus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-95
Model Menu (Local Level) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-96
View Menu (Local Level) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-99
Outliers Menu (Local Level) . . . . . . . . . . . . . . . . . . . . . . . . . . 5-100
Outlier Selection Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-100
Data Tab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-103
Global Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Global Special Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Global Scatter Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Global Level: Toolbar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Global Level: Model-Specific Tools . . . . . . . . . . . . . . . . . . . . .
Global Level: Menus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-105
5-107
5-108
5-110
5-111
5-113
Selecting Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Select . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Model Selection Window . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Tests View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Predicted/Observed View . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Response Surface View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Likelihood View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
RMSE View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Residuals View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Cross Section View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-117
5-117
5-118
5-120
5-122
5-124
5-127
5-129
5-132
5-133
MLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-137
MLE Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-139
Response Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-141
Response Level: Toolbar and Menus . . . . . . . . . . . . . . . . . . . . 5-143
v
Model Evaluation Window . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-144
Model Evaluation Using Other Data . . . . . . . . . . . . . . . . . . . 5-146
Exporting Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-148
What Is Exported? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-151
Evaluating Models in the Workspace . . . . . . . . . . . . . . . . . . . 5-152
The Design Editor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-153
Design Styles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-154
Design Editor Displays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-155
Creating a Classical Design . . . . . . . . . . . . . . . . . . . . . . . . . . 5-159
Creating a Space Filling Design . . . . . . . . . . . . . . . . . . . . . . 5-162
Setting Up a Space Filling Design . . . . . . . . . . . . . . . . . . . . . 5-166
Creating an Optimal Design . . . . . . . . . . . . . . . . . . . . . . . . . .
Optimal Design: Start Point Tab . . . . . . . . . . . . . . . . . . . . . .
Optimal Design: Candidate Set Tab . . . . . . . . . . . . . . . . . . . .
Optimal Design: Algorithm tab . . . . . . . . . . . . . . . . . . . . . . . .
Adding Design Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fixing, Deleting, and Sorting Design Points . . . . . . . . . . . . .
Saving and Importing Designs . . . . . . . . . . . . . . . . . . . . . . . .
5-168
5-170
5-172
5-175
5-178
5-181
5-183
Applying Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-184
Constraint Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-185
Importing Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-191
Prediction Error Variance Viewer . . . . . . . . . . . . . . . . . . . . 5-193
Technical Documents
6
Linear Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-3
Stepwise Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-3
Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-5
vi
Contents
Prediction Error Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-7
High-Level Model Building Process Overview . . . . . . . . . . . 6-9
Univariate Model Building Process Overview . . . . . . . . . . . . . 6-10
Stepwise Regression Techniques . . . . . . . . . . . . . . . . . . . . . . 6-13
Stepwise Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-13
Box-Cox Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-18
Linear Model Statistics Displays . . . . . . . . . . . . . . . . . . . . . .
Summary Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ANOVA Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Diagnostic Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
PRESS Statistic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Pooled Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-22
6-22
6-22
6-23
6-23
6-24
Design Evaluation Tool . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Design Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Full FX Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Model Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Z2 Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Alias Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Z2.1 Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Regression Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Coefficient Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Standard Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Hat Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
|X’X| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Raw Residual Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Degrees of Freedom Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Design Evaluation Graphical Displays . . . . . . . . . . . . . . . . . . .
Export of Design Evaluation Information . . . . . . . . . . . . . . . .
6-27
6-29
6-29
6-29
6-29
6-29
6-30
6-30
6-30
6-32
6-32
6-33
6-33
6-34
6-35
6-35
Maximum Likelihood Estimation . . . . . . . . . . . . . . . . . . . . . . 6-36
vii
Two-Stage Models for Engines . . . . . . . . . . . . . . . . . . . . . . . .
Local Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Global Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Two-Stage Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Prediction Error Variance for Two-Stage Models . . . . . . . . . .
Global Model Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-37
6-38
6-40
6-41
6-43
6-45
6-46
Local Model Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Local Models and Associated Response Features . . . . . . . . . .
Polynomial Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Polynomial Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Truncated Power Series Basis (TPSBS) Splines . . . . . . . . . . .
Free Knot Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Three Parameter Logistic Model . . . . . . . . . . . . . . . . . . . . . . . .
Morgan-Mercer-Flodin Model . . . . . . . . . . . . . . . . . . . . . . . . . .
Four-Parameter Logistic Curve . . . . . . . . . . . . . . . . . . . . . . . .
Richards Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Weibul Growth Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exponential Growth Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Gompertz Growth Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-48
6-48
6-48
6-49
6-50
6-51
6-51
6-52
6-53
6-54
6-55
6-55
6-56
Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-57
User-Defined Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
To Begin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Template File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Subfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Optional Subfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Checking into MBC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Check In . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
MBC Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
viii Contents
6-58
6-58
6-58
6-59
6-59
6-62
6-62
6-63
Transient Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Simulink Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Checking the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Transient Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Template File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Subfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Checking into MBC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Check In . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
MBC Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-64
6-64
6-65
6-67
6-67
6-68
6-68
6-70
6-70
6-71
Data Loading Application Programming Interface . . . . . .
Data Function Prototype . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Input Arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Output Arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Data Function Check In . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-73
6-73
6-73
6-73
6-74
Radial Basis Functions
7
Guide to Radial Basis Functions for Model Building . . . . . 7-3
Types of Radial Basis Functions . . . . . . . . . . . . . . . . . . . . . . . . 7-4
Gaussian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-4
Thin-Plate Spline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-6
Logistic Basis Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-7
Wendland’s Compactly Supported Function . . . . . . . . . . . . . . . 7-8
Multiquadrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-10
Reciprocal Multiquadrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-11
Fitting Routines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-12
Center Selection Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . .
Rols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
RedErr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
WiggleCenters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
CenterExchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7-13
7-13
7-14
7-14
7-14
ix
Lambda Selection Algorithms . . . . . . . . . . . . . . . . . . . . . . . . .
IterateRidge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
IterateRols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
StepItRols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7-16
7-16
7-17
7-18
Width Selection Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-19
TrialWidths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-19
WidPerDim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-19
Prune Functionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-21
Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
GCV Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
GCV for Ridge Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
GCV for Rols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7-24
7-24
7-25
7-25
7-27
Hybrid Radial Basis Functions . . . . . . . . . . . . . . . . . . . . . . . .
Width Selection Algorithm: TrialWidths . . . . . . . . . . . . . . . . .
Lambda and Term Selection Algorithms: Interlace . . . . . . . . .
Lambda and Term Selection Algorithms: TwoStep . . . . . . . . .
7-28
7-28
7-28
7-29
Tips for Modeling with Radial Basis Functions . . . . . . . . .
Plan of Attack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
How Many RBFs to Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Width Selection Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Which RBF to Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Lambda Selection Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . .
Center Selection Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . .
General Parameter Fine-Tuning . . . . . . . . . . . . . . . . . . . . . . . .
Hybrid RBFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7-30
7-30
7-31
7-32
7-33
7-33
7-34
7-34
7-34
Index
x
Contents
1
Getting Started
What Is the Model-Based Calibration Toolbox? . . . . 1-3
About the Model Browser . . . . . . . . . . . . . . . 1-3
About CAGE . . . . . . . . . . . . . . . . . . . . 1-3
How to Use This Manual
. . . . . . . . . . . . . . 1-5
System Requirements . . . .
Hardware Requirements . . .
Operating System Requirements
Required MathWorks Products .
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1-6
1-6
1-6
1-7
Optional MathWorks Products . . . . . . . . . . . . . 1-7
1
Getting Started
The following sections introduce the Model-Based Calibration Toolbox.
“What Is the Model-Based Calibration Toolbox?” on page 1-3
“About the Model Browser” on page 1-3
“About CAGE” on page 1-3
“How to Use This Manual” on page 1-5
“System Requirements” on page 1-6
Starting the Model Browser
To start the application, type
mbcmodel
at the command prompt.
For information on learning and using the Model Browser, see “How to Use
This Manual” on page 1-5.
1-2
What Is the Model-Based Calibration Toolbox?
What Is the Model-Based Calibration Toolbox?
The Model-Based Calibration Toolbox contains tools for design of experiment,
statistical modeling, and calibration of complex systems. There are two main
user interfaces:
• Model Browser for design of experiment and statistical modeling
• CAGE Browser for analytical calibration
About the Model Browser
The Model Browser is a flexible, powerful, intuitive graphical interface for
building and evaluating experimental designs and statistical models.
• Design of experiment tools can drastically reduce expensive data collection
time.
• You can create and evaluate optimal, space filling, and classical designs, and
constraints can be designed or imported.
• Hierarchical statistical models can capture the nature of variability inherent
in engine data, accounting for variation both within and between tests.
• The Model Browser has powerful, flexible tools for building, comparing, and
evaluating statistical models and experimental designs.
• There is an extensive library of prebuilt model types and the capability to
build user-defined models.
• You can export models to MATLAB, Simulink, or CAGE.
About CAGE
CAGE (CAlibration GEneration) is an easy-to-use graphical interface for
calibrating lookup tables for your Electronic Control Unit (ECU).
As engines get more complicated, and models of engine behavior more
intricate, it is increasingly difficult to rely on intuition alone to calibrate lookup
tables. CAGE provides analytical methods for calibrating lookup tables.
1-3
1
Getting Started
CAGE uses models of the engine control subsystems to calibrate lookup tables.
With CAGE you fill and optimize lookup tables in existing ECU software using
Model Browser models. From these models, CAGE builds steady-state ECU
calibrations.
CAGE also compares lookup tables directly to experimental data for validation.
1-4
How to Use This Manual
How to Use This Manual
This manual is the Model Browser User’s Guide.
See also the CAGE Browser User’s Guide for information on the other main
interface of the Model-Based Calibration toolbox.
Learning the Model Browser:
For new users there are three tutorial chapters to guide you through using the
Model Browser tools.
• “Quickstart Tutorial” on page 2-1 provides a quick introduction to modeling
with the Toolbox. The tutorial describes how to set up and view a two-stage
model using some engine data.
• “Design of Experiment Tutorial” on page 3-1 covers the Design of
Experiment tools with a step-by-step guide to setting up, viewing, and
comparing one of each of the design types: classical, space-filling, and
optimal. The tutorial also describes how to define and apply constraints and
export designs.
• “Data Editor Tutorial” on page 4-1 is a guide to using the Data Editor to load
and manipulate data for modeling. You can load data from files or the
workspace or custom Excel sheets. You can view plots of the data and define
new variables and filters. You can store and import user-defined variables
and filters, and define test groupings.
Using the Model Browser:
• “GUI Reference” on page 5-1 is a complete guide to the functionality in the
user interfaces of the Model Browser, Design Editor, Data Editor, Model
Selection and Evaluation windows, and all related dialogs.
• “Technical Documents” on page 6-1 covers the modeling process and the
mathematical basis of hierarchical models, including a guide to using the
Stepwise window, the Box-Cox transformation dialog, the Design Evaluation
tool, and user-defined and transient models.
• “Radial Basis Functions” on page 7-1 is a guide to all aspects of using radial
basis functions in modeling, from setup to the mathematical basis.
1-5
1
Getting Started
System Requirements
This section lists the following:
• Hardware requirements
• Operating system requirements
• Required MathWorks products
• Optional MathWorks products
Hardware Requirements
The Model-Based Calibration Toolbox has been tested on the following
processors:
• Pentium, Pentium Pro, Pentium II, Pentium III, and Pentium IV
• AMD Athlon
Minimum memory:
• 256 MB
Minimum disk space:
• 450 MB for the software and the documentation
Operating System Requirements
The Model-Based Calibration Toolbox is a PC-Windows only product.
The product has been tested on Microsoft Windows NT, 2000, and 98.
You can see the system requirements for MATLAB online at
http://www.mathworks.com/products/system.shtml/Windows.
1-6
System Requirements
Required MathWorks Products
Model-Based Calibration requires the following other MathWorks products:
• Simulink
• Optimization Toolbox
• Statistics Toolbox
• Extended Symbolic Math Toolbox
Optional MathWorks Products
The Model-Based Calibration Toolbox can use the following MathWorks
product:
• Neural Network Toolbox
Note If you want to import Excel files or use the custom Excel file facility of
the toolbox, you must also have the Excel application.
1-7
1
Getting Started
1-8
2
Quickstart Tutorial
Two-Stage Models
. . . . . . . . . . . . . . . . . 2-3
Starting the Toolbox . . . . . . . . . . . . . . . . 2-5
Setting Up the Model . . .
Setting Up the Local Model .
Setting Up the Global Model .
Selecting Data . . . . . . .
Specifying the Response Model
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Verifying the Model . . . . . . . . . . . . .
Verifying the Local Model . . . . . . . . . . . .
Verifying the Global Model . . . . . . . . . . . .
Selecting the Two-Stage Model . . . . . . . . . .
Comparing the Local Model and the Two-Stage Model
Maximum Likelihood Estimation . . . . . . . . .
Response Node . . . . . . . . . . . . . . . .
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2-24
2-24
2-26
2-29
2-34
2-35
2-38
Exporting the Model . . . . . . . . . . . . . . . . 2-40
2
Quickstart Tutorial
The following sections give you a quick introduction to the modeling end of the
Model-Based Calibration Toolbox. They do so by showing you how to use the
toolbox to create a statistical model of an automobile engine that predicts the
torque generated by the engine as a function of spark angle and other
variables.
• “Two-Stage Models” on page 2-3
• “Starting the Toolbox” on page 2-5
• “Setting Up the Model” on page 2-9
• “Setting Up the Local Model” on page 2-12
• “Setting Up the Global Model” on page 2-16
• “Selecting Data” on page 2-20
• “Specifying the Response Model” on page 2-23
• “Verifying the Model” on page 2-24
• “Verifying the Local Model” on page 2-24
• “Verifying the Global Model” on page 2-26
• “Selecting the Two-Stage Model” on page 2-29
• “Exporting the Model” on page 2-40
For an explanation of how two-stage models are constructed and how they
differ from one-stage models, see the following section, “Two-Stage Models” on
page 2-3, or see “Two-Stage Models for Engines” on page 6-37 in the Technical
Documents for more depth.
This tutorial is a step-by-step guide to constructing a single two-stage model
for some engine data. In the normal modeling process you might create many
different models for one project and compare them to find the best solution.
2-2
Two-Stage Models
Two-Stage Models
This tutorial is a step-by-step guide to constructing a single two-stage model
for modeling engine brake torque as a function of spark, engine speed, load, and
air/fuel ratio. One-stage modeling fits a model to all the data in one process,
without accounting for the structure of the data. When data has an obvious
hierarchical structure (as here), two-stage modeling is better suited to the task.
The usual way for collecting brake torque data is to fix engine speed, load, and
air/fuel ratio within each test and sweep the spark angle across a range of
angles. For this experimental setup there are two sources of variation. The first
source is variation within tests when the spark angle is changed. The second
source of variation is between tests when the engine speed, load, and air/fuel
ratio are changed. The variation within a test is called local and the variation
between tests, global. Two-stage modeling estimates the local and global
variation separately by fitting local and global models in two stages. A local
model is fitted to each test independently. The results from all the local models
are used to fit global models across all the global variables. Once the global
models have been estimated they can be used to estimate the local models’
coefficients for any speed, load, and air/fuel ratio. The relationship between the
local and global models is shown in the following block diagram.
2-3
2
Quickstart Tutorial
2-4
Starting the Toolbox
Starting the Toolbox
1 Double-click the MATLAB icon to start MATLAB.
2 To start the toolbox from the launch pad, open the Model Based Calibration
(MBC) Toolbox™, then double-click Model Browser.
Alternatively, enter mbcmodel at the command prompt in MATLAB.
3 If you have never used the toolbox before, the User Information dialog
appears. If you want you can fill in any or all of the fields: your name,
company, department, and contact information, or you can click Cancel. The
user information is used to tag comments and actions so that you can track
changes in your files. (It does not collect information for The Mathworks.)
Note You can edit your user information at any time by selecting File –>
Preferences.
4 When you finish with the User Information dialog, click OK.
The Model Browser window appears.
In this window, the left pane, All Models, shows the hierarchy of the models
currently built in a tree. At the start only one node, the project, is in the tree.
As you build models they appear as child nodes of the project. The right panes
change, depending on the tree node selected. You navigate to different views by
selecting different nodes in the model tree. Different tips can appear in the Tip
of the Day pane.
2-5
2
Quickstart Tutorial
Click here to load some
new data
2-6
Starting the Toolbox
Load the example data file holliday.xls.
1 Click the
button on the toolbar, or choose Data –> New Data.
This opens the Data Editor window.
2 Click the Open icon on the toolbar (
) or choose File –> Load data from
file.
3 Use the browse button to the right of the edit box in the Data Import Wizard
to open a file browser and find the file holliday.xls in the mbctraining
directory. Click Open or double-click the file.
Click here to browse for a file
4 The file pathname appears in the Data Import Wizard. Click Next.
5 A summary screen displays information about the data. Click Finish to close
the Data Import Wizard and return to the Data Editor.
You can view plots of the data in the Data Editor by selecting variables and
tests in the lists on the left side. Have a look through the data to get an idea
of the shape of curve formed by plotting torque against spark.
For more details on functionality available within the Data Editor, see “The
Data Editor” on page 5-61.
6 Close the Data Editor to accept the data and return to the Model Browser.
Notice that the new data set appears in the Data Sets pane.
2-7
2
Quickstart Tutorial
This data is from Holliday, T., “The Design and Analysis of Engine Mapping
Experiments: A Two-Stage Approach,” Ph.D. thesis, University of
Birmingham, 1995.
2-8
Setting Up the Model
Setting Up the Model
Now you can use the data to create a statistical model of an automobile engine
that predicts the torque generated by the engine as a function of spark angle
and other variables.
Note It does not matter in which order you set up local and global models, as
both must be completed before you set up the response model.
1 To create a new test plan, do one of the following:
- In the Test Plans list pane at the bottom, click New.
Alternatively, click the New Test Plan button (
) in the toolbar. Note
that this button changes depending on which node is selected in the model
tree. It always creates a child node (not necessarily a test plan node). See
the model tree.
Or select File –> New Test Plan.
The New Test Plan dialog box appears.
2-9
2
Quickstart Tutorial
2 Click the two-stage test plan icon and click OK.
The default name of the model, Two-Stage, appears in the Model Browser
tree, in the All Models pane.
3 Highlight this node of the tree
, Two-Stage, by clicking it. The Model
Browser window displays a diagram representing the two-stage model.
See also “Functions Implemented in the Block Diagram” on page 5-21.
2-10
Setting Up the Model
2-11
2
Quickstart Tutorial
Setting Up the Local Model
Setting up the local model requires that you specify the model’s inputs and
type.
Double-click here
to choose local
model inputs
Double-click here
to choose local
model type
Specifying the Local Model Input
The model you are building is intended to predict the torque generated by an
engine as a function of spark angle at a specified operating point defined by the
engine’s speed, air/fuel ratio, and load. The input to the local model is therefore
the spark angle.
To specify spark angle as the input:
1 Double-click the Local Inputs icon on the model diagram to specify the local
model input.
The Local Input Factor Setup dialog box appears.
2-12
Setting Up the Model
a Set Symbol to S.
b Set Signal to spark. This is optional and matches the raw data.
2 Click OK to dismiss the dialog box.
Notice that the new name of the local model input now appears on the
two-stage model diagram.
Specifying the Local Model Type
The type of a local model is the shape of curve used to fit the test data, for
example, quadratic, cubic, or polyspline curves. In this example, you use
polyspline curves to fit the test data. A spline is a curve made up of pieces of
polynomial, joined smoothly together. The points of the joins are called knots.
In this case, there is only one knot. These polynomial spline curves are very
useful for torque/spark models, where different curvature is required above
and below the maximum.
To specify polyspline as the model type:
1 Double-click the local model icon in the model diagram.
The Local Model Setup dialog box appears.
2-13
2
Quickstart Tutorial
a Select Polynomial Spline from the Local Model Class.
b Set Spline Order to 2 below and 2 above knot.
2 Click OK to dismiss the dialog box.
Notice that the new name of the local model class, PS (for polyspline) 2,2 (for
spline order above and below knot) now appears on the two-stage model
diagram.
2-14
Setting Up the Model
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2-15
2
Quickstart Tutorial
Setting Up the Global Model
Setting up the global model is similar to setting up the local model. You must
specify the model (or curve) type and the inputs used to create the model.
Double-click here to
choose Global Model
Inputs
Double-click here to
choose Global Model
Type
Specifying the Global Model Inputs
The inputs to the global model are the variables that determine the operating
point of the system being modeled. In this example, the operating point of the
engine is determined by the engine’s speed in revolutions per minute (rpm –
often called N), load (L), and air/fuel ratio (afr).
To specify these inputs:
1 Double-click the Global Inputs icon on the model diagram.
The Global Input Factor Setup dialog box appears.
2-16
Setting Up the Model
By default there is one input to the global model. Because this engine model
has three input factors, you need to increase the input factors as follows:
a Click the up arrow button indicated by the cursor above to increase the
number of factors to three.
b Edit the three factors to create the engine model input. In each case,
change the symbols and signals to the following:
c
Symbol
Signal
N
n
L
load
A
afr
Leave the Min and Max boxes at the defaults (you fill them during the
data selection process). You might want to set factor ranges at this stage
if you were designing an experiment, but in this case there is already data
available, so you use the actual range of the data to model instead.
2 Click OK to dismiss the dialog box.
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Quickstart Tutorial
Specifying the Global Model Type
Fitting the local model finds values for each model coefficient or response
feature (for example, knot) for each test. These coefficients then become the
data to which you fit the global model.
By default, quadratic polynomials are used to build the global model for each
response feature. In this case you use the default.
To specify quadratic curves as the global model curve fitting method:
1 Double-click the icon representing the global model in the two-stage model
diagram.
The Global Model Setup dialog box appears.
a Polynomial should already be selected from the Linear Model Subclass
pop-up menu. Under Model options, the order for the three variables N,
L, and A is set by default to 2, which is required.
2-18
Setting Up the Model
b Set Stepwise to Minimize PRESS (PREdicted Sum Square error).
2 Click OK to accept the settings and dismiss the Model Settings dialog box.
You use the Stepwise feature to avoid overfitting the data; that is, you do not
want to use unnecessarily complex models that “chase points” in an attempt to
model random effects. Predicted error sum of squares (PRESS) is a measure of
the predictive quality of a model. Min PRESS throws away terms in the model
to improve its predictive quality, removing those terms that reduce the PRESS
of the model.
This completes the setup of the global model.
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Quickstart Tutorial
Selecting Data
The model you have set up now needs data.
1 Double-click the Responses block in the Test Plan diagram. As no data has
yet been selected for this test plan, this launches the Data Wizard.
Launch Data Wizard by double-clicking here
in the
For the same result, you could also click the Select Data button
toolbar of the model browser (or TestPlan –> Select Data menu item). Also,
if you did not already load a data set at the project node, you can do it at this
point using TestPlan –> Load New Data.
The Data Wizard dialog appears.
2-20
Setting Up the Model
2 Data Object is already selected by default. Click Next.
3 Select S in the Model Input Factors box and Spark under All Data Signals.
4 Select the Copy Range check box, as shown. This makes the model use the
range in the data for that factor.
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Quickstart Tutorial
Select in these lists to match input factor to data signal
Select this box to copy
the data range to the
model inputs
Click here to match each pair of
input and signal
5 Click the large Select Data Signal button, as indicated above.
6 Repeat this process and match the correct data signals to the other three
input factors, N, L, and A (from n, load, and afr).
If the signal name entered during the input factor setup matches a signal
name in the data set, the Wizard automatically selects the correct signal
when the input factor is selected. If the name is not correct, you must select
the correct signal manually by clicking. This autoselect facility can save time
if the data set has a large number of signals.
7 When you have matched all four input factors to the correct data signals (for
both stages of the two-stage model), click Next.
2-22
Setting Up the Model
Specifying the Response Model
The model you just set up now needs a response specified (that is, the factor
you want the model to predict, in this case, Torque).
The next screen of the Data Wizard is for selecting response models.
1 Select tq (torque) as the response.
2 Click Add. Torque appears in the Responses.
3 Select Maximum under Datum.
Only certain model types with a clearly defined maximum or minimum can
support datum models.
4 Click Finish.
The Model-Based Calibration Toolbox now calculates local and global
models using the test plan models you just set up.
Notice that torque appears on the two-stage model diagram, and a new node
appears on the tree in the All Models pane, called PS22.
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Quickstart Tutorial
Verifying the Model
Verifying the Local Model
The first step is to check that the local models agree well with the data.
1 Select PS22 (the local node) on the Model Browser tree.
Click here
The Local Model pane appears, displaying the local model fitting the
torque/spark data for the first test and diagnostic statistics that describe the
fit. The display is flexible in that you can drag, open, and close the divider
bars separating the regions of the screen to adjust the view.
2-24
Verifying the Model
Change test
here
Change plot here
The lower plot shows the data being fitted by the model (blue dots) and the
model itself (line). The red spot shows the position of the polyspline knot, at the
datum (maximum) point.
2 In the upper scatter plot pane, click the y-axis factor pop-up menu and
select Studentized residuals.
3 To display plots and statistics for the other test data, scroll through the tests
using the Test arrows at the top left, or by using the Select Test button.
4 Select Test 588. You see a data point outlined in red. This point has
automatically been flagged as an outlier.
5 Right-click the scatter plot and select Remove Outliers.
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Quickstart Tutorial
Both plots have right-click pop-up menus offering various options such as
removing and restoring outliers and confidence intervals. Clicking any data
point marks it in red as an outlier.
You can use the Test Notes pane to record information on particular tests.
Each test has its own notes pane. Data points with notes recorded against them
are colored in the global model plots, and you can choose the color using the
Test Number Color button in the Test Notes pane.
Verifying the Global Model
The next step is to check through the global models to see how well they fit the
data.
1 Expand the PS22 local node on the Model Browser tree by clicking the plus
sign (+) to the left of the icon. Under this node are four response features of
the local model. Each of these is a feature of the local model of the response,
which is torque.
2 Select the first of the global models, knot.
2-26
Verifying the Model
The Response Feature pane appears, showing the fit of the global model
to the data for knot. Fitting the local model is the process of finding values
for these coefficients or response features. The local models produce a value
of knot for each test. These values are the data for the global model for knot.
The data for each response feature come from the fit of the local model to
each test.
3 The response feature knot has one outlier marked. Points with an absolute
studentized residual value of more than 3 are automatically suggested as
outliers (but included in the model unless you take action). You can use the
right-click menu to remove suggested outliers (or any others you select) in
the same way as from the Local Model plots. Leave this one. If you zoom in
on the plot (Shift-click-drag or middle-click-drag) you can see the value of
the studentized residual of this point more clearly. Double-click to return to
the previous view.
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Quickstart Tutorial
Note Obviously you should never remove outliers as a matter of course.
However, this tutorial is designed to show you how the toolbox helps you to do
this when required. The default outlier selection criterion is a studentized
residual greater than 3, to bring your attention to possible outliers, but you
should never remove data without good reasons. Remove enough points and
the model will simply interpolate the data and become useless for prediction.
You can customize the criteria for outlier selection.
4 Select the other three response features in turn: max, Bhigh 2, and Blow 2.
You will see that Blow 2 has a suggested outlier with a very large
studentized residual; it is a good distance away from all the other data
points for this response feature. All the other points are so clustered that
removing this one could greatly improve the fit of the model to the remaining
points, so remove it.
Return to the Local Model pane by clicking the local node PS22 in the Model
Browser tree.
2-28
Verifying the Model
Selecting the Two-Stage Model
Recall how two-stage models are constructed: two-stage modeling partitions
the variation separately between tests and within tests, by fitting local and
global models separately. A model is fitted to each test independently (local
models). These local models are used to generate global models that are fitted
across all tests.
For each sweep (test) of spark against torque, you fit a local model. The local
model in this case is a spline curve, which has the fitted response features of
knot, max, Bhigh_2 and Blow_2. The result of fitting a local model is a value for
knot (and the other coefficients) for each test. The global model for knot is
fitted to these values (that is, the knot global model fits knot as a function of
the global variables). The values of knot from the global model (along with the
other global models) are then used to construct the two-stage model
The global models are used to reconstruct a model for the local response (in this
case, torque) that spans all input factors. This is the two-stage model across the
whole global space, derived from the global models.
Now you can use the model selection features to view the fit of this two-stage
model in various ways, to compare it with both the data and the local model fit.
Within this tutorial, you use the following:
• “Tests View” on page 2-31
• “Response Surface View” on page 2-32
For more detailed help on all the views available in the Model Selection
window, see “Selecting Models” on page 5-117 in the GUI reference.
Note To construct a two-stage model from the local and global models, you
click the local node in the model tree (with the house icon) and click the Select
button. This is the next step in the tutorial.
Once you are satisfied with the fit of the local and global models, it is time to
construct a two-stage model from them. Return to the Local Model view by
clicking the local node PS22 in the Model Browser tree. The Model Browser
should look like the following example.
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Quickstart Tutorial
Open Model Selection here
Click Select in the Response Features list pane, and the Model Selection
window appears. This window is intended to help you select a Best Model by
comparing several candidate models. There are a number of icons in the toolbar
2-30
Verifying the Model
that enable you to view the fit of the model in various ways. By default the
Tests view appears. These plots show how well the two-stage model agrees with
the data.
Tests View
2-31
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Quickstart Tutorial
Scroll though the tests using the left/right arrows or the Select Test button at
the top left. The plots show the fit of the two-stage model for each test (green
open circles and line), compared with the fit of the local model (black line) and
the data (blue dots). You can left-click (and hold) to see information on each test
or zoom in on points of interest by Shift-click-dragging or
middle-click-dragging. Double-click to return the plot to the original size.
Response Surface View
You view the model as a surface by clicking the Response Surface
the toolbar. You can rotate the plot by click-dragging it.
icon in
1 Click Movie in the Display Type list to see the surface (torque against spark
and speed) vary through different values of load.
2-32
Verifying the Model
2 Dismiss the Model Selection pane, and accept the best model by clicking
Yes in the Model Selection dialog (it is the only two-stage model so far).
2-33
2
Quickstart Tutorial
3 The MLE dialog appears, prompting you to calculate the maximum
likelihood estimate (MLE) for the two-stage model. Click Cancel. You can
calculate MLE later.
Comparing the Local Model and the Two-Stage
Model
Now the lower plots in the Local Model pane show two lines fitted to the test
data: the Local Model line (black), and the Two-Stage Model line (green). The
plots also show the data (in blue), so you can compare how close the two-stage
model fit is to both the data and the local fit for each test.
You can scroll through the various tests (using the arrows at the top left or the
Select Test button) to compare the local and two-stage models for different
tests.
Click the
estimate.
2-34
button in the toolbar to calculate the maximum likelihood
Verifying the Model
Maximum Likelihood Estimation
The global models were created in isolation without accounting for any
correlations between the response features. Using MLE (maximum likelihood
estimation) to fit the two-stage model takes account of possible correlations
between response features. In cases where such correlations occur, using MLE
significantly improves the two-stage model.
1 You reach the MLE dialog from the local node (PS22 in this case) by
- Clicking the
button in the toolbar
- Or by choosing Model –> Calculate MLE
2 Leave the algorithm default settings and click Start to calculate MLE.
3 Watch the progress indicators until the process finishes and a two-stage
RMSE (root mean square error) value appears.
4 Click OK to leave the MLE dialog.
Now the plots on the Local Model pane all show the two-stage model in
purple to indicate that it is an MLE model. This is also indicated in the
legend. Notice that all the model icons in the tree (the response, the local
model, and the response features) have also changed to purple to indicate
that they are MLE models.
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Quickstart Tutorial
5 Click the Select button. This takes you to the Model Selection window.
Here you can compare MLE with the univariate model previously
constructed (without correlations). By default the local fit is plotted against
the MLE model.
6 Select both MLE and the Univariate model for plotting by holding down
Shift while you click the Univariate model in the Model List at the bottom
of the view.
2-36
Verifying the Model
7 Close the Model Selection window. Click Yes to accept the MLE model as
the best.
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Quickstart Tutorial
Response Node
Click the Response node (tq) in the Model Browser tree.
Response node
Now at the Response node in the Model Browser tree (tq), which was
previously blank, you see this:
2-38
Verifying the Model
This shows you the fit of the two-stage model to the data. You can scroll
through the tests, using the arrows at top left, to view the two-stage MLE
model (in green) against the data (in blue) for each test.
You have now completed setting up and verifying a two-stage model.
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2
Quickstart Tutorial
Exporting the Model
All models created in the model browser are exported using the File menu. A
model can be exported to the MATLAB workspace, to a file, or to a Simulink
model.
1 Click the tq node in the model tree.
2 Choose File –> Export Models. The Export Model dialog box appears.
2-40
Exporting the Model
3 Choose File from the Export to pop-up menu. This saves the work as a file
for use within the Model-Based Calibration (MBC) Toolbox, for instance, to
create calibrations.
4 In the Export Options frame, change the destination file to
mbctraining\tutorial1. Do this by typing directly in the edit box. The ...
Browse button could be used if you want to use an existing file.
5 Ensure that Export datum models is selected, as this allows the datum
global model to be exported.
6 Click OK to export the models.
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Quickstart Tutorial
2-42
3
Design of Experiment
Tutorial
What Is Design of Experiment? . . . . . . . . . . . 3-4
Design Styles . . . . . . . . . . . . . . . . . . . . 3-5
Starting the Design Editor
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Optimal Designs
Start Point Tab .
Candidate Set Tab
Algorithm Tab . .
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3-10
3-11
3-14
Design Displays
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Prediction Error Variance Viewer . . . . . . . . . . 3-19
Improving the Design . . . . . . . . . . . . . . . . . 3-21
Classical Designs . . . . . . . . . . . . . . . . . . 3-24
Classical Design Browser . . . . . . . . . . . . . . . 3-26
Set Up and View a Classical Design . . . . . . . . . . . 3-27
Design Evaluation Tool . . . . . . . . . . . . . . . 3-30
Improving the Design . . . . . . . . . . . . . . . . . 3-32
Space-Filling Designs . . . . . . . . . . . . . . . . 3-33
Setting Up a Space-Filling Design . . . . . . . . . . . . 3-34
Applying Constraints . . . . . . . . . . . . . . . . 3-35
Saving Designs . . . . . . . . . . . . . . . . . . . 3-40
3
Design of Experiment Tutorial
Structure of This Design Editor Tutorial
The following sections guide you through constructing optimal, classical, and
space-filling designs; how to compare designs using the prediction error
variance (PEV) viewer and Design Evaluation tool; and how to apply
constraints to designs.
1 The first section introduces the concept of design of experiment, and outlines
the advantages of good experimental design for mapping modern engines.
Then there is a description of the design styles available in the Model-Based
Calibration Toolbox.
- “What Is Design of Experiment?” on page 3-4
- “Design Styles” on page 3-5
2 To start the tutorial you pick a model to design an experiment for, enter the
Design Editor, and construct an optimal design. Once you create a design,
you can use the displays and tools to examine the properties of the design,
save the design, and make changes.
See
- “Starting the Design Editor” on page 3-6
- “Optimal Designs” on page 3-9
- “Design Displays” on page 3-16
- “Prediction Error Variance Viewer” on page 3-19
- “Saving Designs” on page 3-40
- “Improving the Design” on page 3-21
3 Next you create a classical design, and use the PEV viewer to compare it
with the previous design. You can also use the Design Evaluation tool to
view all details of any design; it is introduced in this example.
See
- “Classical Designs” on page 3-24
- “Design Evaluation Tool” on page 3-30
4 Lastly you construct a space-filling design and compare it with the others
using the PEV viewer. Then you construct and apply two different
constraints to this design and view the results. Normally you would design
3-2
constraints before constructing a design, but for the purposes of this tutorial
you make constraints last so you can view the effects on your design.
See
- “Space-Filling Designs” on page 3-33
- “Applying Constraints” on page 3-35
For more details on functionality in the Design Editor, see “The Design Editor”
on page 5-153.
3-3
3
Design of Experiment Tutorial
What Is Design of Experiment?
With today’s ever-increasing complexity of models, design of experiment has
become an essential part of the modeling process. The Design Editor within the
MBC Toolbox is crucial for the efficient collection of engine data. Dyno-cell time
is expensive, and the savings in time and money can be considerable when a
careful experimental design takes only the most useful data. Dramatically
reducing test time is growing more and more important as the number of
controllable variables in more complex engines is growing. With increasing
engine complexity the test time increases exponentially.
The traditional method of collecting large quantities of data by holding each
factor constant in turn until all possibilities have been tested is an approach
that quickly becomes impossible as the number of factors increases. A full
factorial design (that is, testing for torque at every combination of speed, load,
air/fuel ratio, and exhaust gas recirculation on a direct injection gasoline
engine with stratified combustion capability) is not feasible for newer engines.
Simple calculation estimates that, for recently developed engines, to calibrate
in the traditional way would take 99 years!
With a five-factor experiment including a multiknot spline dimension and 20
levels in each factor, the number of points in a full factorial design quickly
becomes thousands, making the experiment prohibitively expensive to run.
The Design Editor solves this problem by choosing a set of experimental points
that allow estimation of the model with the maximum confidence using just a
fraction of the number of experimental runs; for the preceding example just 100
optimally-chosen runs is more than enough to fit the model. Obviously this
approach can be advantageous for any complex experimental design, not just
engine research.
The Design Editor offers a systematic, rigorous approach to the data collection
stage. When you plan a sequence of tests to be run on an example engine, you
can base your design on engineering expertise and existing physical and
analytical models. During testing, you can compare your design with the latest
data and optimize the remaining tests to get maximum benefit.
The Design Editor provides prebuilt standard designs to allow a user with a
minimal knowledge of the subject to quickly create experiments. You can apply
engineering knowledge to define variable ranges and apply constraints to
exclude impractical points. You can increase modeling sophistication by
3-4
What Is Design of Experiment?
altering optimality criteria, forcing or removing specific design points, and
optimally augmenting existing designs with additional points.
Design Styles
The Design Editor provides the interface for building experimental designs.
You can make three different styles of design: classical, space-filling, and
optimal.
Optimal designs are best for cases with high system knowledge, where
previous studies have given confidence on the best type of model to be fitted,
and the constraints of the system are well understood. See “Optimal Designs”
on page 3-9.
Space-filling designs are better when there is low system knowledge. In cases
where you are not sure what type of model is appropriate, and the constraints
are uncertain, space-filling designs collect data in such as a way as to maximize
coverage of the factors’ ranges as quickly as possible. See “Space-Filling
Designs” on page 3-33.
Classical designs (including full factorial) are very well researched and are
suitable for simple regions (hypercube or sphere). Engines have complex
constraints and models (high-order polynomials and splines). See “Classical
Designs” on page 3-24.
You can augment any design by optimally adding points. Working in this way
allows new experiments to enhance the original, rather than simply being a
second attempt to gain the necessary knowledge.
3-5
3
Design of Experiment Tutorial
Starting the Design Editor
Setting Up a Model
You must first have a model for which to design an experiment.
1 From the Model Browser at startup, click the
button in the toolbar, or
click New in the Test Plans pane, or choose File –> New Test Plan.
2 Select Two-Stage Model and click OK.
3 Click the new Two-Stage node that appears in the model tree (in the All
Models pane), or double-click Two Stage in the Test Plans list at the bottom.
The Two-Stage Model diagram appears.
If you already have a project open, you can select any existing model within the
test plans in the Model Browser tree. For the purposes of this tutorial, you
design experiments for the default Two-Stage global model, which is a
quadratic.
There is only one input to the global model by default. To increase the number
of input factors:
1 Double-click the Global Model Inputs block in the diagram. The Input
Factors Setup dialog appears.
2 Increase the number of factors to three by clicking the Number of Factors
up/down buttons or entering 3 in the edit box.
3 Change the symbols of the three input factors to N, L, and A. This matches
the global factors modeled in the Quick Start tutorial: speed (n), load (L),
and air/fuel ratio (A).
4 Click OK to leave the Input Factor Setup dialog.
Starting the Design Editor
To access the Design Editor use either of the following methods:
• Right-click the global model in the diagram and choose Design Experiment,
as shown.
• You can also access the Design Editor by selecting the menu item TestPlan
–> Design Experiment.
3-6
Starting the Design Editor
The Design Editor window appears.
Creating a New Design
1 Click the
button in the toolbar or select File –> New. A new node called
Linear Model Design appears.
3-7
3
Design of Experiment Tutorial
2 The new Linear Model Design node is automatically selected. An empty
Design Table appears (see above) because you have not yet chosen a design.
For this example you create an optimal design for the default global model,
which is a quadratic.
You can change the model for which you are designing an experiment from
within the Design Editor window by selecting Edit –> Model.
3 Rename the new node Optimal (you can edit the names by clicking again on
a node when it is already selected, or by pressing F2, as when selecting to
rename in Windows Explorer).
3-8
Optimal Designs
Optimal Designs
Choose an optimal design by clicking the
Design –> Optimal.
button in the toolbar, or choose
Optimal designs are best for cases with high system knowledge, where
previous studies have given confidence on the best type of model to be fitted,
and the constraints of the system are well understood.
The optimal designs in the Design Editor are formed using the following
process:
• An initial starting design is chosen at random from a set of defined candidate
points.
• m additional points are added to the design, either optimally or at random.
These points are chosen from the candidate set.
• m points are deleted from the design, either optimally or at random.
• If the resulting design is better than the original, it is kept.
This process is repeated until either (a) the maximum number of iterations is
exceeded or (b) a certain number of iterations has occurred without an
appreciable change in the optimality value for the design.
The Optimal Design dialog consists of several tabs that contain the settings
for three main aspects of the design:
• Starting point and number of points in the design
• Candidate set of points from which the design points are chosen
3-9
3
Design of Experiment Tutorial
• Options for the algorithm that is used to generate the points
Start Point Tab
The Start Point tab allows you to define the composition of the initial design:
how many points to keep from the current design and how many extra to choose
from the candidate set.
1 Leave the optimality criteria at the default to create a V-Optimal design.
2 Increase the total number of points to 30 by clicking the Optional
Additional Points up/down buttons or by typing directly into the edit box.
3-10
Optimal Designs
Candidate Set Tab
The Candidate Set tab allows you to set up a candidate set of potential test
points. This typically ranges from a few hundred points to several hundred
thousand.
1 Choose Grid for this example. Note that you could choose different schemes
for different factors.
2 This tab also has buttons for creating plots of the candidate sets. Try them
to preview the grid.
3-11
3
Design of Experiment Tutorial
3 Notice that you can see 1-D, 2-D, 3-D, and 4-D displays (the fourth factor is
color, but this example only uses three factors) at the same time as they
appear in separate windows (see example following). Move the display
windows (click and drag the title bars) so you can see them while changing
the number of levels for the different factors. See the effects of changing the
number of levels on different factors, then return them all to the default of
21 levels.
3-12
Optimal Designs
Select variables in this list
Choose Grid from this list
Open display windows with these buttons
Change the number of levels of the selected variable here
3-13
3
Design of Experiment Tutorial
Algorithm Tab
1 Leave the algorithm settings at the defaults and click OK to start optimizing
the design.
When you click the OK button on the Optimal Design dialog, the
Optimizing Design dialog appears, containing a graph. This dialog shows
the progress of the optimization and has two buttons: Accept and Cancel.
Accept stops the optimization early and takes the current design from it.
Cancel stops the optimization and reverts to the original design.
3-14
Optimal Designs
2 Click Accept when iterations are not producing noticeable improvements;
that is, the graph becomes very flat.
3-15
3
Design of Experiment Tutorial
Design Displays
When you press the Accept button, you return to the Design Editor.
When you first see the main display area, it shows the default Design Table
view of the design (see preceding example). There is a context menu, available
by right-clicking, in which you can change the view of the design to 1-D, 2-D,
3-D, and 4-D Projections and Table view (also under the View menu). This
menu also allows you to split the display either horizontally or vertically so
that you simultaneously have two different views on the current design. The
split can also be merged again. After splitting, each view has the same
functionality; that is, you can continue to split views until you have as many as
you want. When you click a view, its title bar becomes blue to show it is the
active view.
3-16
Design Displays
The currently available designs are displayed on the left in a tree structure. For
details, see “The Design Tree” on page 5-156.
Display Options
The Design Editor can display multiple design views at once, so while working
on a design you can keep a table of design points open in one corner of the
window, a 3-D projection of the constraints below it and a 2-D or 3-D plot of the
current design points as the main plot. The following example shows several
views in use at once.
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Design of Experiment Tutorial
The current view and options for the current view are available either through
the context menu or the View menu on the Design Editor window.
1 Change the main display to 3-D Projection view.
2 You can rotate the projection with click-drag mouse movement. View your
design in several projections (singly, or simultaneously by dividing the pane)
by using the right-click context menu in the display pane.
3-18
Prediction Error Variance Viewer
Prediction Error Variance Viewer
A useful measure of the quality of a design is its prediction error variance
(PEV). The PEV hypersurface is an indicator of how capable the design is in
estimating the response in the underlying model. A bad design is either not
able to fit the chosen model or is very poor at predicting the response. The PEV
Viewer is only available for linear models. The PEV Viewer is not available
when designs are rank deficient; that is, they do not contain enough points to
fit the model. Optimal designs attempt to minimize the average PEV over the
design region.
Select Tools –> PEV Viewer.
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Design of Experiment Tutorial
The default view is a 3-D plot of the PEV surface.
This shows where the response predictions are best. This example optimal
design predicts well in the center and the middle of the faces (one factor high
and the other midrange), but in the corners the model has the highest error.
Even in the corners the errors are small; looking at the scale, PEV values under
0.25 are not large.
Try the other display options.
• The Surface menu has many options to change the look of the plots.
3-20
Prediction Error Variance Viewer
• You can change the factors displayed in the 2-D and 3-D plots. The pop-up
menus below the plot select the factors, while the unselected factors are held
constant. You can change the values of the unselected factors using the
buttons and edit boxes in the Input factors list, top left.
• The Movie option shows a sequence of surface plots as a third input factor’s
value is changed. You can change the factors, replay, and change the frame
rate.
• You can change the number, position, and color of the contours on the
contour plot with the Contours button, as shown.
Improving the Design
You can further optimize the design by returning to the Optimal Design
dialog, where you can delete or add points optimally or at random. The most
efficient way is to delete points optimally and add new points randomly —
these are the default algorithm settings. Only the existing points need to be
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Design of Experiment Tutorial
searched for the most optimal ones to delete (the least useful), but the entire
candidate set has to be searched for points to add optimally.
To strengthen the current optimal design:
1 Return to the Design Editor window.
2 Click the Optimal Design button in the toolbar again to reenter the dialog,
and add 60 more points. Keep the existing points (which is the default).
3 Click OK and watch the optimization progress, then click Accept when the
number of iterations without improvement starts increasing.
4 View the improvements to the design in the main displays.
5 Once again select Tools –> PEV Viewer and review the plots of prediction
error variance and the new values of optimality criteria in the optimality
frame (bottom left). The shape of the PEV projection might not change
dramatically, but note the changes in the scales as the design improves. The
values of D, V, and G optimality criteria will also change (you have to click
Calculate to see the values).
To see more dramatic changes to the design, return to the Design Editor
window (no need to close the PEV viewer).
1 Split the display so you can see a 3-D projection at the same time as a Table
view.
2 Choose Edit –> Delete Point.
3 Using the Table and 3-D views as a guide, in the Delete Points dialog, pick
six points to remove along one corner; for example, pick points where N is
100 and L is 0. Add the relevant point numbers to the delete list by clicking
the add (>) button.
4 Click OK to remove the points. See the changes in the main design displays
and look at the new Surface plot in the PEV viewer (see the example
following).
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Prediction Error Variance Viewer
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3
Design of Experiment Tutorial
Classical Designs
1 In the Design Editor window, select the Optimal design in the design tree
by clicking.
2 Add a new design. Use the first toolbar button, or select File –> New.
A new child node appears in the tree, called Optimal_1. Notice that the
parent node now has a padlock on the icon. This indicates it is locked. This
maintains the relationship between designs and their child nodes. The tree
arrangement lets you try out different operations starting from a basic
design, then select the most appropriate one to use. The hierarchy allows
clear viewing of the effects of changes on designs. The locking of parent
designs also gives you the ability to easily reverse out of changes by
retreating back up the tree.
3 Select the new design node in the tree. Notice that the display remains the
same — all the points from the previous design remain, to be deleted or
added to as necessary. The new design inherits all its initial settings from
the currently selected design and becomes a child node of that design.
4 Rename the new node Classical by clicking again or by pressing F2.
5 Click the
Browser.
3-24
button in the toolbar or select Design –> Classical –> Design
Classical Designs
.
Note In cases where the preferred type of classical design is known, you can
go straight to one of the five options under Design –> Classical. Choosing the
Design Browser option allows you to see graphical previews of these same
five options before making a choice.
A dialog appears because there are already points from the previous design.
You must choose between replacing and adding to those points or keeping
only fixed points from the design.
6 Choose the default, replace current points with a new design, and click OK.
The Classical Design Browser appears.
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Design of Experiment Tutorial
Classical Design Browser
In the Design Style drop-down menu there are five classical design options:
• Central Composite
Generates a design that has a center point, a point at each of the design
volume corners, and a point at the center of each of the design volume faces.
You can choose a ratio value between the corner points and the face points
for each factor and the number of center points to add. You can also specify
a spherical design. Five levels are used for each factor.
• Box-Behnken
Similar to Central Composite designs, but only three levels per factor are
required, and the design is always spherical in shape. All the design points
(except the center point) lie on the same sphere, so there should be at least
three to five runs at the center point. There are no face points. These designs
are particularly suited to spherical regions, when prediction at the corners is
not required.
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Classical Designs
• Full Factorial
Generates an n-dimensional grid of points. You can choose the number of
levels for each factor and the number of additional center points to add.
• Plackett Burman
These are "screening" designs. They are two-level designs that are designed
to allow you to work out which factors are contributing any effect to the
model while using the minimum number of runs. For example, for a 30-factor
problem this can be done with 32 runs.
• Regular Simplex
These designs are generated by taking the vertices of a k-dimensional
regular simplex (k = number of factors). For two factors a simplex is a
triangle; for three it is a tetrahedron. Above that are hyperdimensional
simplices. These are economical first-order designs that are a possible
alternative to Plackett Burman or full factorials.
Set Up and View a Classical Design
1 Choose a Box-Behnken design.
2 Reduce the number of center points to 1.
3 View your design in different projections using the tabs under the display.
4 Click OK to return to the Design Editor.
5 Use the PEV Viewer to see how well this design performs compared to the
optimal design created previously; see the following illustration.
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Design of Experiment Tutorial
As you can see, this is not a realistic comparison, as this design has only 13
points (you can find this information in the bottom left of the main Design
Editor display), whereas the previous optimal design had 100, but this is a
good illustration of leverage. A single point in the center is very bad for the
design, as illustrated in the PEV viewer surface plot. This point is crucial and
needs far more certainty for there to be any confidence in the design, as every
other point lies on the edge of the space. This is also the case for Central
Composite designs if you choose the spherical option. These are good designs
for cases where you are not able to collect data points in the corners of the
operating space.
If you look at the PEV surface plot, you should see a spot of white at the center.
This is where the predicted error variance reaches 1. For surfaces that go above
1, the contour at 1 shows as a white line, as a useful visual guide to areas where
prediction error is large.
6 Select Movie, and you see this white contour line as the surface moves
through the plane of value 1.
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Classical Designs
7 Select the Clip Plot check box. Areas that move above the value of 1 are
removed. The edges of the clip are very jagged; you can make it smoother by
increasing the numbers of points plotted for each factor. See the following
example.
Turn clipping on and off
here
Change number of points
plotted here
Change clipping value here
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Design of Experiment Tutorial
Design Evaluation Tool
The Design Evaluation Tool is only available for linear models. See also “Global
Model Class: Multiple Linear Models” on page 5-43.
1 Return to the Design Editor and select Tools –> Evaluate Designs.
2 Choose the Box-Behnken design and click OK in the Select Designs dialog.
The Design Evaluation Tool displays a large amount of statistical
information about the design.
3 Select Hat Matrix from the list on the right.
4 Click the Leverage Values button.
Note that the leverage of the central point is 1.00 (in red) and the leverage
of all other points is less than this. The design would clearly be strengthened
by the addition of more central points. Obviously this is a special case, but
for any kind of design the Design Evaluation Tool is a powerful way to
examine properties of designs.
5 Select Design Matrix from the list box.
6 Click the 3D Surface button in the toolbar.
3-30
Design Evaluation Tool
This illustrates the spherical nature of the current design. As usual, you can
rotate the plot by clicking and dragging with the mouse.
There are many other display options to try in the toolbar, and in-depth details
of the model terms and design matrices can all be viewed. You can export any
of these to the workspace or a .mat file using the Export box.
For a description of all the information available here, see “Design Evaluation
Tool” on page 6-27.
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Design of Experiment Tutorial
Improving the Design
To strengthen the current Box-Behnken design near the center region:
1 Close the Design Evaluation Tool.
2 Return to the Design Editor window.
3 Select Design –> Classical –> Box-Behnken.
4 Click OK to replace the current points with a new design.
5 Increase the number of center points and click OK.
6 Once again select Tools –> PEV Viewer and review the plots of prediction
error variance and the new values of optimality criteria in the optimality
frame (bottom left).
7 Review the leverage values of the center points. From the Design Editor
window, use Tools –> Evaluate Design and go to Hat Matrix.
8 Try other designs from the Classical Design Browser. Compare Full
Factorial with Central Composite designs; try different options and use
the PEV viewer to choose the best design.
Note You cannot use the PEV viewer if there are insufficient points in the
design to fit the model. For example, you cannot fit a quadratic with less than
three points, so the default Full Factorial design, with two levels for each
factor, must be changed to three levels for every factor before you can use the
PEV viewer.
9 When you are satisfied, return to the Design Editor window and choose
Edit –> Select as Best. You will see that this design node is now highlighted
in blue in the tree. This can be applied to any design.
When you are creating designs before you start modeling, the design that
you select as best is the one used to collect data.
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Space-Filling Designs
Space-Filling Designs
Space-filling designs should be used when there is little or no information
about the underlying effects of factors on responses. For example, they are
most useful when you are faced with a new type of engine, with little knowledge
of the operating envelope. These designs do not assume a particular model
form. The aim is to spread the points as evenly as possible around the operating
space. These designs literally fill out the n-dimensional space with points that
are in some way regularly spaced. These designs can be especially useful in
conjunction with nonparametric models such as radial basis functions (a type
of neural network).
1 Add a new design by clicking the
button in the toolbar.
A new Classical child node appears in the tree. Select it by clicking. As
before, the displays remain the same: the child node inherits all points from
the parent design. Notice that in this case the parent node does not acquire
a padlock to indicate it is locked — it is blue and therefore selected as the
best design. Designs are locked when they are selected as best.
2 Rename the new node Space Filling (click again or press F2).
3 Select Design –> Space Filling –> Design Browser, or click the Space
Filling Design button on the toolbar.
4 Click OK in the dialog to replace the current design points with a new
design.
The Space Filling Design Browser appears.
Note As with the Classical Design Browser, you can select the three types
of design you can preview in the Space Filling Design Browser from the
Design –> Space Filling menu in situations when you already know the type
of space-filling design you want.
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Design of Experiment Tutorial
Setting Up a Space-Filling Design
1 Leave the Design drop-down menu at the default Latin Hypercube
Sampling.
2 Choose the default Maximize minimum distance.
3 Select the Enforce Symmetrical Points check box. This creates a design in
which every design point has a mirror design point on the opposite side of
the centre of the design volume and an equal distance away. Restricting the
design in this way tends to produce better Latin Hypercubes.
4 Use the tabs under the display to view 2-D, 3-D, and 4-D previews.
5 Click OK to calculate the Latin Hypercube and return to the main Design
Editor.
6 Use the Design Evaluation Tool and PEV Viewer to evaluate this design.
3-34
Applying Constraints
Applying Constraints
In many cases designs might not coincide with the operating region of the
system to be tested. For example, a conventional stoichiometric AFR
automobile engine normally does not operate with high exhaust gas
recirculation (EGR) in a region of low speed (n) and low load (l). You cannot run
15% EGR at 800 RPM idle with a homogeneous combustion process. There is
no point selecting design points in impractical regions, so you can constrain the
candidate set for test point generation. Only optimal designs have candidate
sets of points; classical designs have set points, and space-filling designs
distribute points between the coded values of (1, -1).
You would usually set up constraints before making designs. Applying
constraints to classical and space-filling designs simply removes points outside
the constraint. Constraining the candidate set for optimal designs ensures that
design points are optimally chosen within the area of interest only.
Designs can have any number of geometric constraints placed upon them. Each
constraint can be one of four types: an ellipsoid, a hyperplane, a 1-D lookup
table, or a 2-D lookup table. For further details, see “Applying Constraints” on
page 5-184 in the “GUI Reference” for a full description of the constraint
functions.
To add a constraint to your currently selected design:
1 Select Edit –> Constraints from the Design Editor menus.
2 The Constraints Manager dialog appears. Click Add.
3 The Constraint Editor dialog with available constraints appears. Select 1D
Table from the Constraint Type drop-down menu.
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Design of Experiment Tutorial
4 You can select the appropriate factors to use. For this example choose speed
(N) and air/fuel ratio (A).
5 Move the large dots (clicking and dragging them) to define a boundary. The
Constraint Editor should look something like the following.
3-36
Applying Constraints
6 Click OK.
Your new constraint appears in the Constraint Manager list box. Click OK
to return to the Design Editor. A dialog appears because there are points in
the design that fall outside your newly constrained candidate set. You can
simply delete them or cancel the constraint. Note that fixed points are not
deleted by this process.
For optimal designs you get the dialog shown, where you also have the
option to replace the points with new ones chosen (optimally if possible)
within the new candidate set.
7 The default is to remove the points outside the new constraint area; choose
this.
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Design of Experiment Tutorial
If you examine the 2-D projection of the hypercube you will notice the effects of
the new constraint on the shape of the design, as shown in the preceding
example.
8 Right-click the display pane to reach the context menu.
9 Select Current View –> 3D Constraints.
These views are intended to give some idea of the region of space that is
currently available within the constraint boundaries.
10 Return to the Constraint Editor, choose Edit –> Constraint, and click Add
in the Constraint Manager.
11 Add an ellipsoid constraint. Choose Ellipsoid from the drop-down menu of
constraint types, and enter values in the table as shown. Note coded values
are used in defining the ellipsoid.
3-38
Applying Constraints
This reduces the space available for the candidate set by a third in the A and
N axes, forming an ellipsoid, as shown below. The L axis, left at 1, is not
constrained at the midpoint of N and A. To leave L unconstrained (a cylinder)
put the value of L=0.
12 Click OK, click OK again in the Constraint Manager, and click Replace to
compensate for design points lost outside the new candidate set. Examine
the new constraint 3-D plot illustrated.
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Design of Experiment Tutorial
Both constraints are applied to this design, but the ellipsoid lies entirely within
the previous 1-D table constraint.
Saving Designs
To save your design:
1 Choose File –> Export Design. The selected design only is exported.
There are three options:
- To File generates a Design Editor file (.mvd).
- To CSV File exports the matrix of design points to a CSV
(comma-separated values) file. You can include factor symbols and/or
convert to coded values by selecting the check boxes.
- To Workspace exports the design matrix to the workspace. You can
convert design points to a range of [-1, 1] by selecting the check box.
2 Choose a Design Editor file.
3 Choose the destination file by typing Designtutorial.mvd in the edit box.
4 Click OK to save the file.
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Data Editor Tutorial
Loading the Data . . . . . . . . . . . . . . . . . . 4-3
Entering the Data Editor . . . . . . . . . . . . . . . 4-3
Loading a Data File . . . . . . . . . . . . . . . . . 4-4
Viewing and Editing the Data . . . .
User-Defined Variables & Filtering . . .
Defining a New Variable . . . . . . .
Applying a Filter . . . . . . . . . . .
Sequence of Variables . . . . . . . . .
Deleting and Editing Variables and Filters
Storage . . . . . . . . . . . . . . .
Test Groupings . . . . . . . . . . .
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4
Data Editor Tutorial
The Data Editor is a GUI for loading data, creating new variables, and creating
constraints for that data.
Data can be loaded from files (Excel files, MATLAB files, Concerto files) and
from the MATLAB workspace. The Data Editor can create an Excel sheet for
data input, set up in the form the Model Browser expects, and can then load
this data. You can merge data in any of these forms with previously loaded data
sets (providing there is no conflict in the form of the data) to produce a new data
set. Test plans can only use one data set, so the merging function allows you to
combine records and variables from different files in one model.
You can define new variables, apply filters to remove unwanted data, and you
can store and retrieve these user-defined variables and filters for any data set.
You can change and add records and apply test groupings, and you can match
data to designs. You can also write your own data loading functions; see “Data
Loading Application Programming Interface” on page 6-73 in the “Technical
Documents”section.
For comprehensive help on all data functions in the Model Browser, see “Data”
on page 5-59 in the “GUI Reference” section.
The following tutorial is a step-by-step guide to the following:
• Loading data from an Excel file
• Viewing and editing the data
• Creating a user-defined variable
• Applying a filter to the data
• Sequence of variables
• Deleting and editing variables and filters
• Placing user-defined variables and filters into storage
• Defining test groupings
4-2
Loading the Data
Loading the Data
Entering the Data Editor
To enter the Data Editor and create a new data object, select one of these
options:
• From the Test Plan level, choose TestPlan –> Load New Data.
• Alternatively, from the Project node, select Tools –> New Data (or click the
New data object toolbar button).
The Data Editor appears.
Note You can delete data objects from the Project node. Select a data set in
the Data Sets list and press the Delete key.
4-3
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Data Editor Tutorial
There is no plot until some data has been loaded.
By default the new data object is called Data Object. You can change this name
by typing in the Name edit box at the top.
You can change the names of data objects at the Project node by select-clicking
a data object in the Data Sets list or pressing F2 (as when selecting to rename
in Windows Explorer) and entering a new name.
Loading a Data File
1 Click the Open File icon in the toolbar
to load data from a file.
The Data Import Wizard appears to select a file.
2 Use the Browse button to find and select the Holliday.xls data file in the
mbctraining folder. Double-click to load the file. You can also enter the file
pathname in the edit box. The pop-up menu contains the file types
recognized by the Model Browser (Excel, concerto, .mat). Leave this at the
4-4
Loading the Data
default, Auto. This setting tries to determine what type of file is selected by
looking at the file extension.
3 Click Next.
4 The Data Import Wizard displays a summary screen showing the total
number of records and variables imported, and you can view each variable’s
range, mean, standard deviation, and units in the list box. Click Finish to
accept the data. (If you have data loaded already, you cannot click Finish
but must continue to the data merging functions.)
The Data Import Wizard disappears and the view returns to the Data Editor,
which now contains the data you just loaded.
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Data Editor Tutorial
Viewing and Editing the Data
The list boxes on the left allow a combination of tests and variables to be plotted
simultaneously. The example shown plots torque against spark for multiple
tests. You can multiple-select tests and y-axes to compare the data in the tests
(hold down Shift or Control).
4-6
Viewing and Editing the Data
Right-click the plot to change the display: you can switch grid lines on and off
and plot lines to join the data points.
Reorder Points in the right-click menu can be useful when record order does
not produce a sensible line joining the data points. For an illustration of this:
1 Choose afr for the y-axis.
2 Choose Load for the x-axis.
3 Make sure lines are plotted between points: choose Show Line from the
right-click plot menu. See Test 537 plotted for these variables, following.
4 Choose Reorder points from the right-click menu.
This command replots the line from left to right instead of in the order of the
records, as shown.
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Data Editor Tutorial
Copy Plot creates a copy of the current plot in a new figure.
The menu View –> Table shows this data in tabular form. You can select and
edit data directly, and you can add records by right-clicking test numbers.
The menu View –> Graph returns the view to the plots shown. There are
toolbar buttons to switch between these views.
User-Defined Variables & Filtering
You can add new variables to the data set, and you can remove records by
imposing constraints on the data.
• Select Tools –> User-Defined Variables & Filtering.
Alternatively, click the
toolbar button.
The following window appears.
4-8
Viewing and Editing the Data
Defining a New Variable
1 Select Variables –> New Variable in the User-Defined Variables &
Filtering window.
The Variable Editor appears.
You can define new variables in terms of existing variables. You define the
new variable by writing an equation in the edit box at the top of the Variable
Editor dialog.
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Data Editor Tutorial
2 Define a new variable called POWER that is defined as the product of two
existing variables, tq and n, by entering POWER=tq*n, as seen in the example
following. You can also double-click variable names and operators to add
them, which can be useful to avoid typing mistakes in variable names, which
must be exact including case.
3 Click OK to add this variable to the current data set.
This new variable appears in the top list (variable expression) of the
User-Defined Variables & Filtering window.
POWER also appears in the Data Information pane, along with the original
variables, and its definition is included in the description field (to the right of
the units column; you might have to scroll or resize to see it).
Applying a Filter
A filter is the name for a constraint on the data set used to exclude some
records. You use the Filter Editor to create new filters.
1 Choose Filters –> New Filter, or click the
Variables & Filtering window.
The Filter Editor dialog appears.
4-10
button in the User-Defined
Viewing and Editing the Data
You define the filter using logical operators on the existing variables.
2 Keep all records with speed (n) greater than 1000. Double-click the variable
n, then the operator >, then type 1000.
3 Click OK to impose this filter on the current data set.
This new filter appears in the center list of the User-Defined Variables &
Filtering window. The Filtration results pie chart shows graphically how
many records were removed by the imposition of this filter, and the number of
records removed appears in the Results column of the Filter Expression pane.
The number of records at the top of the Data Editor window also changes
accordingly (for example, Records 200/264 indicates that 64 records have been
filtered out).
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Data Editor Tutorial
Sequence of Variables
You can change the order of user-defined variables in the list using the arrow
buttons in the toolbar.
Example:
1 Define two new variables, New1 and New2. Notice that New2 is defined in
terms of New1. New variables are added to the data in turn and hence New1
must appear in the list before New2, otherwise New2 is not well defined.
4-12
Viewing and Editing the Data
2 Change the order by clicking the down arrow to produce this erroneous
situation. You see the following:
3 Use the arrows to order user-defined variables in legitimate sequence.
Deleting and Editing Variables and Filters
You can delete user-defined variables and filters.
Example:
1 To delete the added variable New1, select it in the top list of the
User-Defined Variables & Filtering window and press the Delete key.
2 You could also use the menu item Variables –> Delete Variable. Use this to
delete New2 (select it first!).
Similarly, you can delete filters by selecting the unwanted filter in the list and
deleting via the menu or the Delete key.
You can also edit current user-defined variables and filters using the relevant
menu items or toolbar buttons.
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4
Data Editor Tutorial
Storage
Storage allows you to store user-defined variables and filters so that they can
be applied to other data sets loaded later in the session.
You can open the Storage window from the User-Defined Variables &
Filtering window in either of these ways:
• Using the menu File –> Open Storage
• Using the toolbar button
The toolbar buttons allow the following functions:
• Export to File sends Storage Objects to file so they can be sent to another
user or machine. Objects remain in Storage indefinitely unless deleted;
export is only for transporting elsewhere.
• Import from File loads storage objects from file; it does not overwrite
current objects.
• Append Stored Object appends the currently selected Storage Object to
those already in the User-Defined Variables & Filtering window.
• Get Current Variables creates a Storage Object containing definitions of
all current user-defined variables.
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Viewing and Editing the Data
• Get Current Filters creates a Storage Object containing definitions of all
current filters.
• Delete Stored Object deletes the currently selected Storage Object.
1 Use the controls to bring the variable POWER and the filter you just created
into Storage.
2 Close the Storage window (use the X button).
3 This returns you to the User-Defined Variables & Filters window. Close
this window to return to the Data Editor.
Test Groupings
The Define Test Groupings dialog collects records of the current data object
into groups; these groups are referred to as tests.
The dialog is accessed from the Data Editor in either of these ways:
• Using the menu Tools –> Change Test Groupings
• Using the toolbar button
When you enter the dialog, no plot is displayed.
Select a variable to use in defining groups within the data.
1 Select n in the Variables list.
2 Click the
button to add the variable (or double-click n).
The variable n appears in the list view on the left, as seen in the following
example. You can now use this variable to define groups in the data. The
maximum and minimum values of n are displayed. The Tolerance is used to
define groups: on reading through the data, when the value of n changes by
more than the tolerance, a new group is defined. You change the Tolerance
by typing directly in the edit box.
You can define addlitional groups by selecting another variable and choosing
a tolerance. Data records are then grouped by n or by this additional variable
changing outside their tolerances.
3 Add load to the list by selecting it on the right and clicking
.
4-15
4
Data Editor Tutorial
4 Change the tolerance to 0.01 and watch the test grouping change in the plot.
5 Clear the Group By check box. Notice that variables can be plotted without
being used to define groups.
The plot shows the scaled values of all variables in the list view (the color of
the tolerance text corresponds to the color of data points in the plot). Vertical
pink bars show the tests (groups). You can zoom the plot by
Shift-click-dragging or middle-click-dragging the mouse; zoom out again by
double-clicking.
6 Select load in the list view (it becomes highlighted in blue).
7 Remove the selected variable, load, by clicking the
button.
You should return to the view below, when only n was plotted and used to
define groups.
4-16
Viewing and Editing the Data
Reorder records allows records in the data set to be reordered before
grouping. Otherwise the groups are defined using the order of records in the
original data object.
Show original displays the original test groupings if any were defined.
One test/record defines one test per record, regardless of any other
grouping. This is required if the data is to be used in creating one-stage
models.
Test number variable contains a pop-up menu showing all the variables in
the current data set. Any of these can be selected to number the tests.
4-17
4
Data Editor Tutorial
8 Choose logno from the pop-up list of Test number variables.
Test number can be a useful variable for identifying individual tests in
Model Browser views (instead of 1,2,3...) if the data was taken in numbered
tests and you want access to that information during modeling.
If you chose none from the Test number variable list, the tests would be
numbered 1,2,3 and so on in the order in which the records appear in the
data file.
Every record in a test must share the same test number to identify it, so
when you are using a variable to number tests, the value of that variable is
taken in the first record in each test.
Test numbers must be unique, so if any values in the chosen variable are the
same, they are assigned new test numbers for the purposes of modeling (this
does not change the underlying data, which retains the correct test number
or other variable).
9 Click OK to accept the test groupings defined and dismiss the dialog.
You return to the Data Editor window. At the top is a summary of this data
set now that your new variable has been added and a new filter applied
(example shown below).
The number of records shows the number of values left (after filtration) of
each variable in this data set, followed by the original number of records.
The color coded bars also display the number of records removed as a
proportion of the total number. The values are collected into a number of
tests; this number is also displayed. The variables show the original number
of variables plus user-defined variables.You can remove or show the legend
explaining the color coding by clicking the button.
4-18
5
GUI Reference
Project Level: Startup View . . .
Model Tree . . . . . . . . . . .
Test Plans . . . . . . . . . . .
Test Plan Level . . . . . . . . .
Setting Up Models . . . . . . .
Data . . . . . . . . . . . . . .
New Response Models . . . . .
Local Level . . . . . . . . . .
Global Level . . . . . . . . . .
Selecting Models . . . . . . . .
MLE . . . . . . . . . . . . . .
Response Level . . . . . . . . .
Model Evaluation Window . . .
Exporting Models . . . . . . .
The Design Editor . . . . . . .
Creating a Classical Design . . .
Creating a Space Filling Design .
Creating an Optimal Design . . .
Applying Constraints . . . . . .
Prediction Error Variance Viewer
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. 5-3
. 5-10
. 5-15
. 5-19
. 5-26
. 5-61
. 5-89
. 5-91
5-107
5-119
5-140
5-144
5-147
5-151
5-156
5-162
5-165
5-171
5-187
5-196
5
GUI Reference
This is a list of the main sections of this guide. Functionality is described in the
order you see it during the process of model building.
For a quick guide to setting up models, see these overview pages:
“Instant One-Stage Model Setup” on page 5-27
“Instant Two-Stage Model Setup” on page 5-28
The different views of the Model Browser are described in these sections:
“Project Level: Startup View” on page 5-3
“Test Plan Level” on page 5-19
“Local Level” on page 5-89
“Global Level” on page 5-105
“Response Level” on page 5-141
To construct models, you must navigate using the model tree, use test plans,
and load and manipulate data. These topics are covered in these sections:
“Model Tree” on page 5-10
“Test Plans” on page 5-15
“Data” on page 5-59
Model construction, evaluation and export are covered in these sections:
“Setting Up Models” on page 5-26
“Selecting Models” on page 5-117
“Model Evaluation Window” on page 5-144
“Exporting Models” on page 5-148
Design of Experiment is described in this section:
“The Design Editor” on page 5-153
5-2
Project Level: Startup View
Project Level: Startup View
On startup there is a single node, the project (named Untitled), in the model
tree. This node is automatically selected.
When the project node in the model tree is selected at any time, the following
functionality is available. This state is called project level. When you start you
are automatically at project level, as there are not yet any other nodes to select.
Note The node selected in the model tree determines what appears in the
menus and panes of the rest of the Model Browser.
2
1
5
3
4
All Models Pane
(Labeled 1). This pane contains a hierarchical structure showing all the models
created within the current project. See “Model Tree” on page 5-10 for a detailed
description of the information contained in this pane.
5-3
5
GUI Reference
Data Sets Pane
(Labeled 2). All data sets loaded in the current project are displayed in the
Data Sets pane (whether in use for modeling or not).
You can select a data set (by clicking it) and then
• Delete it by pressing the Delete key.
• Rename it, by clicking again or pressing F2 (as when selecting to rename in
Windows Explorer), then editing the name.
• Open it by double-clicking. Double-clicking a data set opens the Data Editor.
Note All data sets loaded are visible at the project node and appear in the
Data Sets pane. However, they are not necessarily used by any test plan child
nodes of that project until you select them within a particular test plan. For
example, with a data set loaded at the project node, when you switch to a new
test plan node, the Data Sets pane at top right displays 'No Data is
selected' until you use the Data Wizard to attach data to that test plan.
The same data set can be used by many test plans within a project, although
each individual test plan can only ever use one data set.
Notes Pane
(Labeled 3). The Notes pane contains a list box showing all previous notes on
the current project. You use notes to record changes within a project, perhaps
by different users, over time.
• You add new notes by clicking the Add new note button
in the toolbar,
or by pressing the Insert key after selecting the Notes pane by clicking.
Notes automatically have the user login name and the date.
• You edit notes (only the user that created them can edit them; user names
must match) by select-clicking or by pressing F2 when selected (as when
selecting to rename in Windows Explorer). Edited notes have updated time
and date fields.
• You remove notes by selecting them and pressing Delete (but only the same
user that created them can delete them).
5-4
Project Level: Startup View
• Notes are automatically added to the project when it is modified (for
example, the initial “Created by <username>” note). These notes (listed as
user “info”) cannot be deleted or edited.
Test Plans List Pane
(Labeled 4). You generate new test plans from the Test Plans list pane by
clicking the New button. See “Test Plans” on page 5-15.
This pane is the Test Plans list pane at startup but changes depending on the
level in the model tree that is selected. The list box always displays all the child
nodes of whichever node is currently selected in the tree in the All Models
pane, and always contains three buttons: New, Delete, and Select.
Double-clicking any item within this pane changes the view directly to that
node. (This is equivalent to selecting that node in the model tree.) You can also
use the Delete and Insert keys to remove or add new test plans (select a test
plan first).
• The Test Plans list becomes the Response Models list, the Local Models
list, the Response Features list, and the Models list as you select the nodes
at subsequent levels of the model tree. In each case this pane displays the
immediate child nodes of the current node selected. You can use the buttons
to delete selected nodes or create new nodes.
• The feature added by clicking New always corresponds to the list items. For
example, clicking New when the pane shows a list of test plans adds a new
test plan. Clicking New when the pane shows a list of response features
opens the New Response Feature dialog, as shown in the following
example. The response features you can add are model-specific. This
example shows the response features available for a polyspline model.
5-5
5
GUI Reference
For example, if you choose f(x + datum) and enter 10 in the Value edit box, the
new response feature tracks the datum +10. For a torque/spark polyspline
model, the datum is MBT (maximum brake torque); so the new response
feature is MBT + 10 degrees of spark angle. This allows you to create response
features that have engineering interest.
The response features available depend on the model type. For more details on
which response features are available, see “Local Models and Associated
Response Features” on page 6-48.
• You can use the Select button to select the best child node, but only when the
child nodes are local models, response features, or submodels of response
features. In each case clicking the Select button takes you to a selection
window where you can compare the child nodes. See “Selecting Models” on
page 5-117.
Tip of the Day
(Labeled 5). Hints about using the Model Browser appear here. You can scroll
through more tips using the buttons at the bottom, and you can snap this pane
closed or open by clicking the “snapper point” where the cursor changes if you
roll the mouse over it.
5-6
Project Level: Startup View
Project Level: Toolbar
New note
New child node
New Project
Open Project
Save Project
Delete current node
Up One Level
Copy data object
Edit data object
New data object
This is how the toolbar appears when you first start the toolbox. The last two
Data buttons are grayed out; the Edit data object and Copy data object
buttons are not enabled until a data set has been loaded.
• All the toolbar items are duplicated under the menus except New Note.
• For the Project buttons, see the File menu.
• For the Data buttons, see the Data menu.
• New Note adds a note to the Notes pane.
• The New and Delete node buttons are duplicated in the File menu. In both
cases, their function depends on the node selected in the model tree. In every
case, New generates a new child node of the one currently selected, and
Delete removes the current node (and all its children).
• The Up One Level button moves the current selection (and hence all the
views) one level up the model tree. For example, if a test plan node is
selected, clicking this button moves one level up to the project node.
Two buttons, Delete and Up One Level, are grayed out at startup because the
default selection in the model tree is the project node, so there are no levels
above, and you cannot delete the project node (although you can replace it with
a new one).
• The print icon is only enabled in views with plots, for example, the local node,
response feature nodes, and response nodes after selection of a best two-stage
models (response nodes are blank until then).
5-7
5
GUI Reference
Project Level: Menus
File Menu
Note The File menu remains constant in each Model Browser view. The New
child node function always creates a new child node, and the Delete current
node function always deletes the current node. These change according to
which node in the model tree is currently selected.
• New Project opens a new project file. You are prompted to save or lose the
current project.
• Open Project opens a file browser to select the project to open.
• Save Project and Save Project As save the project with all the models it
contains as a .mat file.
• New Test Plan opens a dialog with the choice of One-Stage or Two-Stage
test plans, or you can browse for other test plans. The New (child node) menu
option always creates a new child node of whichever node is selected in the
model tree.
At startup, the project node is automatically selected, so the appropriate
child node is a new test plan.
Note File –> New changes depending on which node in the model tree is
selected. In each case the option offers a new child node immediately below
the one currently selected, that is, a New Test Plan (if a project node is
selected), a New Response Model (from a test plan) or a New Model child
node (from a one-stage response). For two-stage models you can add a New
Local Model (from a response node), a New Response Feature (from a local
node) and a New Model from a response feature node.
• Export Models brings up the Export Models dialog. This allows you to
export any models selected in the tree (along with their child nodes, in some
cases) to the MATLAB workspace, to file for importing into CAGE, or to
Simulink. See “Exporting Models” on page 5-148.
5-8
Project Level: Startup View
• Delete “Untitled” Like the New item in this menu, this option changes
depending on which node in the model tree is selected. This menu item
deletes whichever node is currently selected in the model tree (along with
any child nodes), and the appropriate name appears here.
• Clean Up Tree From any modeling node where a best model has been
selected (from the child nodes), you can use this to delete all other child
nodes. Only the child nodes selected as best remain.
• Preferences brings up the MBC File Preferences dialog, in which you can
specify default locations for projects, data, models, and templates. You can
also edit and save user information: name, company, department and contact
details. This information is saved with each project level note, and you can
use this to trace the history of a project.
• Print is only enabled in views with plots — not test plan or project level.
• 1,2,3,4: A list of the four most recent project files, including their pathnames.
• Close Exits from the Model Browser part of the toolbox (CAGE and
MATLAB are unaffected).
Data Menu
• New Data — Opens the Data Editor.
• Copy Data — Copies the selected data set.
• Edit Data — Opens the Data Editor to enable data editing.
• Delete Data — Deletes the selected data set.
Window Menu
Depending on which toolbox windows are open, a list appears under this menu
and whichever window is selected is brought to the front. The Window menu
remains constant throughout the Model Browser.
Help Menu
The Help menu remains consistent throughout the Model Browser.
• MBC Help — Opens the Model-Based Calibration Toolbox Roadmap with
links to the help tutorials and the indexed help pages.
• Context Help — Depending on what part of the Model Browser is currently
active, Context Help links to different places in the Help files.
• About MBC — Displays version notes.
5-9
5
GUI Reference
Model Tree
The tree in the All Models pane displays the hierarchical structure of the
models you have built. Views and functionality within the browser differ
according to which node is selected.
The following is an example of a model tree.
1. Project node
2. Test plan node
3. Response node
4. Local node
5. Response feature nodes
The elements of the tree consist of the following:
1 Project node
2 Test plan node
3 Response node
4 Local node
5 Response feature nodes
Note The selected node governs the model that is displayed in the various
other panes of the browser and which menu items are available. The selected
node governs the level displayed: project level, test plan level, and so on. The
functionality of each level is described in the Help.
5-10
Model Tree
You can rename all nodes, as in Windows Explorer, by clicking again or by
pressing F2 when a node is selected.
There is a context menu available. When you right-click any node in the model
tree, you can choose to delete or rename that node, or create a new child node.
Tree Structure
1. Project node
2. Test plan node
3. Response node
4. Two-stage models
5. Response feature nodes
5. Response feature nodes
The preceding example shows a more extensive model tree, with two two-stage
models as child nodes of a single response model.
There can be many models within (or under, as child nodes in the tree) each
response model node.
There can also be many different response nodes within a single test plan, and
each project can contain several different test plans. However, there is only one
project node in the tree at any time.
5-11
5
GUI Reference
Note You can only have one project open at any one time; if you open another,
you are prompted to save or discard your current project.
Response features can themselves have child nodes — several models can be
tried at each response feature node and the best selected. There is an example
showing this at the end of the section on “Icons: Blue Backgrounds and Purple
Worlds” on page 5-13.
Icons: Curves, Worlds, and Houses
The icons are designed to give visual reminders of their function.
• Test plan icons have a tiny representation of the test plan diagram. You can
see the one-stage and two-stage icons in the following example.
• Response features have global models fitted to them, so their icons show a
curve over a globe.
• The local model icon shows a curve over a house.
Test plan nodes
Response node
Local node
Response feature nodes
• The response node (empty until a two-stage model is calculated) has an icon
that combines the local and global motifs — a curve over a house and a globe
— to symbolize the two-stage process.
• When a two-stage model has been calculated, the icon at the local node
changes to show the combination of local and global motifs.
5-12
Model Tree
Icons: Blue Backgrounds and Purple Worlds
1. Project node
2. Test plan node
3. Response node
4. Two-stage models
5. Response
feature nodes
5. Response
feature nodes
Icon changes convey information about modeling processes.
• When a model is selected as the best model, its icon changes color and gains
a blue background, like the BSPLINE1 model in the preceding example.
• When the maximum likelihood estimate (MLE) is calculated and chosen as
the best model, the associated model icon and all its child nodes (along with
the plots of that model) become purple.
You can see this in the preceding example: the B Spline model and all its
response features have purple curves, globes, and house, indicating that they
are MLE models. The Poly3 model and its children have blue curves and
globes and a red house, indicating that they are univariate models.
• Observe the other difference between the B Spline and the Poly3 icons: the
B Spline has a blue background. This indicates that this is selected as best
model, and is used to calculate the two-stage model at the response node, so
the response node is also purple. If an MLE model (with purple worlds) is
selected as best model and is used to create the two-stage model, the
response node always reflects this and is also purple.
5-13
5
GUI Reference
• Notice also that the response features all have blue backgrounds. This shows
they are selected as best and are all being used to calculate the two-stage
model. In this case they are all needed. That is, a B Spline model needs six
response features, and a Poly3 model requires four. If more response features
are added, however, some combination must be selected as best, and the
response features not in use do not have a blue background.
In the following example you can see child nodes of a response feature. You can
try different models within a response feature, and you must select one of the
attempts as best. In this example you can see that Cubic is selected as best,
because it has a blue background, so it is the model being used for the Blow_2
response feature.
When a model is selected as best, it is copied up a level in the tree together with
the outliers for that model fit.
When a new global or local model is created, the parent model and outliers are
copied from the current level to the new child node. This gives a mechanism for
copying outliers around the model tree.
A tree node is automatically selected as best if it is the only child, except
two-stage models which are never automatically selected; you must use the
Model Selection window.
If a best model node is changed the parent node loses best model status (but the
automatic selection process will reselect that best model if it is the only child
node).
5-14
Test Plans
Test Plans
You need to select a test plan to construct one-stage or two-stage models.
You can select the one- or two-stage test plans provided, as described next. You
can also use these to create your own test plan template so you can reuse the
setup for one test plan with another set of data. See “Creating New Test Plan
Templates” on page 5-16.
Creating a New Test Plan
To create a new test plan:
• Click New in the Test Plans pane (visible at startup and whenever the
project node is selected in the model tree).
Alternatively, make sure the project node is selected first, and then do one of
the following:
• Click the New Test Plan icon (
) in the toolbar.
• Select File –> New Test Plan.
• Press Insert immediately after clicking the tree.
These steps all open a dialog with the choice of One Stage or Two Stage test
plans, or you can browse for other test plans (as new templates can be created
and saved). See “Local Level” on page 5-89.
A new test plan node appears in the model tree. To view the new test plan:
Change to test plan level:
• Select the node
in the tree by clicking it.
Alternatively, double-click the new test plan in the Test Plans pane, as in
Windows Explorer.
The Model Browser changes to test plan level, showing the block diagram
representations of models in the main display, and the Test Plans pane
changes to the Response Models pane (empty until models are set up).
For the next steps in model construction, see
• “Setting Up Models” on page 5-26
• “Setting Up Models” on page 5-26
• “Loading Data from File” on page 5-65
5-15
5
GUI Reference
• “Selecting Models” on page 5-117
Creating New Test Plan Templates
You build user-defined templates from existing test plans using the Make
Template toolbar icon
or TestPlan –> Make Template.
The procedures for modeling engines for calibrations are usually repeated for
a number of different engine programs. The test plan template facility allows
you to reuse the setup for one test plan with another set of data. You can alter
the loaded test plan settings without restriction.
A list of test plan templates is displayed when you build a new test plan. There
are built-in templates for one- and two-stage models.
Test plan templates store the following information:
• Hierarchical model — Whether the model is one- or two-stage and the
default models for each level.
• Designs — If they were saved with the template (check box in the Test Plan
Template Setup dialog)
The design for one type of engine might or might not be appropriate for a
similar type of engine. You can redefine or modify the design using the
Design Editor.
• All response models (for example, torque, exhaust temperature, emissions)
— If they were saved with the template (check box in the Test Plan
Template Setup dialog)
• Numbers and names of input factors to models
• Model types (local and global)
• No model child nodes are included, just the top level of the test plan
(response models, and local and global models for two-stage models).
The response models are automatically built after you assign data to the test
plan; see “Using Stored Templates” on page 5-17.
5-16
Test Plans
Saving a New Template
From the test plan node that you want to make into a template:
• Click the Make Template toolbar icon or choose TestPlan –> Make
Template.
The templates are stored in the directory specified in the File –>
Preferences dialog.
The Test Plan Template Setup dialog appears, in which you can change the
name of the new template and choose whether to include designs and/or
response models.
Using Stored Templates
• When you load a new test plan from the project node, any stored templates
appear in the New Test Plan dialog.
From the project node, select File –> New Test Plan, or use the toolbar
button, or click New in the Test Plans pane.
• Selecting templates in the New Test Plan dialog displays all templates
found in the directory specified in the Preferences dialog (File menu). Select
by clicking to see the information on a particular template; the number of
stages and factors is displayed in the Information pane. You can use the
Browse button if the required template is not in the directory specified in the
File –> Preferences dialog.
• Click OK to use the selected test plan template. The new test plan node
appears in the model tree.
Use stored templates in exactly the same way as the default blank one- and
two-stage templates. Models and input factors are already selected (including
the response if that was saved with the template) so you can go straight to
selecting new data to model. You can still change any settings and design
experiments.
Double-click the Responses block to launch the Data Wizard and select data for
the test plan. The response models are automatically built after selection of
data.
5-17
5
GUI Reference
Note The data selection process takes you through the Data Wizard. If any
signal names in the new data do not match the template input factors, you
must select them here, including the responses. If signal names match the
factor names stored in the template, they are automatically selected by the
Data Wizard, and you just click Next all the way to the end of the wizard.
When you click Finish the response models are built automatically.
5-18
Test Plan Level
Test Plan Level
When you select a test plan node (with the icon
appears.
) in the model tree, this view
This example is a two-stage model, the same model used in the Quick Start
tutorial. All test plan nodes (one- and two-stage) bring up this view with a block
diagram of the test plan and the functionality described below.
5-19
5
GUI Reference
• At test plan level, the block diagram representations of models provides a
graphical interface so you can set up inputs and set up models by
double-clicking the blocks in the test plan diagram. These functions can also
be reached using the TestPlan menu.
• You can access the Design Editor via the right-click menus on the model
blocks or the TestPlan menu (for a particular model — you must select a
model or input block before you can design an experiment). View–> Design
Data also opens the Design Editor where you can investigate the design
properties of the data.
• You can attach data to a new test plan by choosing TestPlan –> Load New
Data or by double-clicking the Responses block in the diagram, which
launches the Data Wizard (if the project already has data loaded).
• If a test plan already has data attached to it, you can reach the Data
or the TestPlan
Selection window using the Select Data toolbar button
menu item. Here you can match data to a design. For example, after the
design changes, new data matching might be necessary. See “Data Selection
Window” on page 5-81 for details.
• You can save the current test plan as a template using the TestPlan –> Make
. See “Local Level” on
Template command or the toolbar button
page 5-89.
See also “Test Plan Level: Toolbar and Menus” on page 5-23
Block Diagram Representations of Models
A block diagram in the test plan view represents the hierarchical structure of
models. Following is an example of a two-stage test plan block diagram.
5-20
Test Plan Level
Functions Implemented in the Block Diagram
The diagram has functionality for setting up the stages in hierarchical
modeling. At present MBC only supports one- and two-stage models.
1 “Setting Up Models” on page 5-26 — Setting the number of inputs for each
stage of the model hierarchy.
2 “Setting Up Models” on page 5-26 — Setting up the new default models for
each stage in the model hierarchy.
3 “Designing Experiments” on page 5-57 —Using the Design Editor.
4 “Loading Data from File” on page 5-65 and “New Response Models” —
5 “Viewing Designs” on page 5-57 — Viewing the statistical properties of the
collected data.
The selected Model block is highlighted in yellow if a Setup dialog is open;
otherwise it is indicated by blocks at the corners. The selected Model block
indicates the stage of the model hierarchy that is used by the following
functions:
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• Set Up Model
• Design Experiment
• View Design Data
• View Model
You can reach these functions via the right-click context menu (on each block)
or the menus.
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Test Plan Level
Test Plan Level: Toolbar and Menus
New Project
Open Project
New child node
Delete current node
Make Template
Select Data
Design Experiment
Save Project
Up One Level
The seven buttons on the left (project and node management, plus the print
button) appear in every view level. See “Project Level: Toolbar” on page 5-7 for
details.
The right buttons change at different levels.
In the test plan level view, the right buttons are as follows:
• Design Experiment opens the Design Editor. Only available when a model
or input has been selected in the test plan block diagram. You must specify
the stage (local or global) you are designing for.
• Select Data opens the Data Wizard, or opens the Data Selection window if
data sets have already been selected.
• Make Template opens a dialog to save the current test plan as a template,
including any designs and response models. See “Local Level” on page 5-89.
Test Plan Level: Menus
File Menu
Only the New (child node) and Delete (current node) functions change
according to the node level currently selected. Otherwise the File menu
remains constant. See “File Menu” on page 5-8.
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Window Menu
The Window menu remains the same throughout the Model Browser. It allows
you to switch windows if there is more than one toolbox window open. See
“Window Menu” on page 5-9.
Help Menu
The Help menu remains the same throughout the Model Browser. You can
always reach the MBC Toolbox Help Roadmap by selecting Help –> MBC Help.
The context help takes you to relevant Help pages, and Help –> About MBC
shows the version notes. See “Help Menu” on page 5-9.
TestPlan Menu
• Set Up Inputs — See “Setting Up Models” on page 5-26.
• Set Up Model — See “Setting Up Models” on page 5-26.
You can also reach these two functions by double-clicking the blocks in the
test plan diagram, and both can only be used when a Model block is first
selected in the diagram. You must specify the model to set up, local or global.
• Design Experiment — See “The Design Editor” on page 5-153.
This is also available in the toolbar and in the right-click context menu on
the blocks in the test plan diagram.
• Load New Data — Opens the Data Editor to load new data.
• Select Data — Opens the Data Selection window.
You can reach both these functions with the toolbar Select Data button. If
no data is selected, this button opens the Data Wizard, and if a data set is
already selected, it takes you to the Data Selection window.
• Make Template — Opens a dialog for saving the current test plan as a new
template, with or without designs and response models. Same as the toolbar
button. See “Local Level” on page 5-89.
View Menu (Test Plan Level)
• Design Data — Opens the Design Editor. The view design facility enables
you to investigate the statistical properties of the collected data. This
provides access to all the Design Editor and design evaluation utility
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Test Plan Level
functions with the current design rather than the prespecified design (after
data matching, the data points are used as the new design points).
For two-stage models, viewing level 1 designs creates a separate design for
each test.
• Model — Opens a dialog showing the terms in the current model.
Both of these are only available when a model or input block is selected in the
test plan block diagram.
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Setting Up Models
The following are one-page overviews for a quick guide to setting up one-stage
and two-stage models:
• “Instant One-Stage Model Setup” on page 5-27
• “Instant Two-Stage Model Setup” on page 5-28
These sections cover setting up inputs and models in detail:
• “Setting Up Inputs” on page 5-30
• “Global Model Setup” on page 5-32
- “Global Linear Models: Polynomials and Hybrid Splines” on page 5-33
- “Global Model Class: Radial Basis Function” on page 5-39
- “Global Model Class: Hybrid RBF” on page 5-42
- “Global Model Class: Multiple Linear Models” on page 5-43
- “Global Model Class: Free Knot Spline” on page 5-44
• “Local Model Setup” on page 5-46
- “Local Model Class: Polynomials and Polynomial Splines” on page 5-47
- “Local Model Class: Truncated Power Series” on page 5-50
- “Local Model Class: Free Knot Spline” on page 5-51
- “Local Model Class: Growth Models” on page 5-52
- “Local Model Class: Linear Models” on page 5-53
- “Local Model Class: User-Defined Models” on page 5-54
- “Local Model Class: Transient Models” on page 5-54
- “Covariance Modeling” on page 5-54
- “Correlation Models” on page 5-56
- “Transforms” on page 5-56
• “Designing Experiments” on page 5-57 tells you how to proceed to
experimental design after setting up models.
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Setting Up Models
Instant One-Stage Model Setup
The following steps are necessary to set up a one-stage model:
1 From the project node, create a new one-stage test plan. See “Creating a
New Test Plan” on page 5-15.
2 Select the new node in the model tree to change to the test plan level.
3 Set up the inputs and model type by double-clicking the Inputs block and the
Model block in the test plan diagram. See “Setting Up Models” on page 5-26
and “Setting Up Models” on page 5-26.
4 At this point, you might want to design an experiment. See “The Design
Editor” on page 5-153.
5 From the test plan node, load a new data set to use. Choose TestPlan –>
Load New Data, which opens the Data Editor. See “Loading Data from File”
on page 5-65.
Note Data for one-stage models must be set to one record per test using the
test grouping utility. Select Tools –> Change Test Groupings in the Data
Editor, and select the one test/record radio button. See “Test Groupings” on
page 5-75.
6 Dismissing the Data Editor opens the Data Wizard, where you match data
signals to model variables and then set up the response model.
On completion of these steps the model fit is calculated and the new model node
appears in the model tree. Select the new node to view the model fit.
Functionality available for viewing and refining the model fit is described in
“Global Level” on page 5-105 and “Selecting Models” on page 5-117.
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Instant Two-Stage Model Setup
The following steps are necessary to set up a two-stage model:
1 From the project node, create a new two-stage test plan. See “Creating a
New Test Plan” on page 5-15.
2 From the test plan node, set up the inputs and models at the local and global
stages. See “Setting Up Models” on page 5-26 and “Setting Up Models” on
page 5-26.
3 At this point, you might want to design an experiment. See “The Design
Editor” on page 5-153.
4 From the test plan node, load the data set you want to use. See “Loading
Data from File” on page 5-65.
This opens the Data Wizard, where you match data signals to model
variables and then set up the response model.
Note On completing the Data Wizard, the local and global models are
calculated.
5 At the local node, you can view the fit of the local models to each test, and
you can also view the global models at the response feature nodes (optional).
6 The two-stage model is not calculated until you use the Select button (from
the local node, in the Response Features pane) and choose a model as best
(even if it is the only one so far), unless you go straight to MLE. See below.
See “Selecting Models” on page 5-117.
Note At this point, the two-stage model is calculated, and the icon changes at
the local node to reflect this. See “Icons: Curves, Worlds, and Houses” on
page 5-12.
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Setting Up Models
7 You are prompted to calculate the maximum likelihood estimate (MLE) at
this point. You can do this now, or later by selecting Model –> Calculate
MLE. See “MLE” on page 5-137 for a detailed explanation.
Note If there are exactly enough response features for the model, you can go
straight to MLE calculation without going through the Select process. The
MLE toolbar button and the Model –> Calculate MLE menu item are both
active in this case. If you add new response features, you cannot calculate
MLE until you go through model selection to choose the response features to
use.
See “Two-Stage Models for Engines” on page 6-37 for a detailed explanation of
two-stage models.
Double-click the Model blocks of the block diagram or select the TestPlan –>
Set Up Model menu item. MBC supports a wide range of models.
For model descriptions, see “Global Model Setup” on page 5-32 and “Local
Model Setup” on page 5-46.
For further statistical details, see “Technical Documents” on page 6-1.
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Setting Up Inputs
You set up the number and definition of model input factors for each stage by
double-clicking the inports of the block diagram. See “Functions Implemented
in the Block Diagram” on page 5-21.
The preceding example shows the input setup dialog for the global model. The
dialog for the local model contains exactly the same controls.
Number of Factors
You can change the number of input factors using the buttons at the top.
Symbol
The input symbol is used as a shortened version of the signal name throughout
the application. The symbol should contain a maximum of three characters.
Min and Max Model Range
This setting is important before you design experiments. The default range is
[0.100]. There is usually some knowledge about realistic ranges for variables.
If you are not designing an experiment you can use the data range as the model
range later, in the data selection stage. In some cases you might not want to
use the data range (for example, if the data covers too wide a range, or not wide
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Setting Up Models
enough) if you are interested in modeling a particular region. In that case you
can set the range of interest here.
Transform
You can use input transformations to change the distribution of the input
factor space for designing experiments. The available input transformations
are 1/x, sqrt(x), log10(x), x^2, log(x). See “Transforms” on page 5-56 for a
discussion of the uses of transformations.
Signal
You can set up the signal name in the input factor setup dialog. It is not
necessary to set this parameter at this stage, as it can be defined later at the
data selection stage (as with the range). However, setting the signal name in
this dialog simplifies the data selection procedures, as the Model Browser looks
for matching signal names in loaded data sets. When the number of data set
variables is large this can save time.
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Global Model Setup
The following example shows the default settings for the global model.
Global Model Classes
The default global model is linear polynomial. There are a number of different
global model classes available, covered in the following sections:
• “Global Linear Models: Polynomials and Hybrid Splines” on page 5-33
• “Global Model Class: Radial Basis Function” on page 5-39
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Setting Up Models
• “Global Model Class: Hybrid RBF” on page 5-42
• “Global Model Class: Multiple Linear Models” on page 5-43
• “Global Model Class: Free Knot Spline” on page 5-44
“Linear Model Statistics Displays” on page 6-22 has a description of the
statistics displayed when this type of model is analyzed.
See “Setting Up Models” on page 5-26 for a list of all information about setting
up local and global models.
Global Linear Models: Polynomials and Hybrid
Splines
Global linear models can be
• Polynomial
• Hybrid Spline
Polynomials
Polynomials of order n are of the form
2
3
β 0 + β1 x + β 2 x + β 3 x …β n x
n
You can choose any order you like for each input factor.
Quadratic curve
Cubic curve
2
As shown, a quadratic polynomial y = ax + bx + c can have a single turning
3
2
point, and a cubic curve y = ax + bx + cx + d can have two. As the order of a
polynomial increases, it is possible to fit more and more turning points. The
curves produced can have up to (n-1) bends for polynomials of order n.
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Term Editor
Click the Edit Terms button to see the terms in the model. This opens the
Term Editor dialog. Here you can remove any of the terms (see example).
See also “Interaction” on page 5-37.
Hybrid Splines
You can use the Hybrid Spline model to fit a spline to one factor and
polynomials to all other factors.
A spline is a piecewise polynomial function, where different sections of
polynomials are fitted smoothly together. The locations of the breaks are called
knots. You can choose the required number of knots (up to a maximum of 50)
and their positions. In this case all the pieces of curves between the knots are
formed from polynomials of the same order. You can choose the order (up to 3).
The example following illustrates the shape of a spline curve with one knot and
third-order basis functions. The knot position is marked on the N axis.
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Setting Up Models
You can fit more complicated curves using splines, so they can be useful for the
factor you expect to behave in the most complex way. This allows you to model
detailed fluctuations in the response for one factor, while simpler models are
sufficient to describe the other factors. The following example clearly shows
that the response (Blow_2 in this case) is quadratic in the Load (L) axis and
much more complex in the RPM (N) axis.
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You can choose the order of the polynomial for each factor and the factor to fit
the spline to. The maximum order for each factor is cubic.
The following example shows the options available for the Hybrid Spline linear
model subclass.
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Setting Up Models
This example only shows one input factor. When there are more, you use the
radio buttons to select which factor is modeled with a spline. Select the order
for each factor in the edit boxes.
See also the other linear model subclass, “Polynomials” on page 5-33.
Interaction
You can choose the interaction level on both linear model subclasses
(polynomial and hybrid spline). For polynomials, the maximum interaction
level is the same as the polynomial order (for example, 3 for cubics). For hybrid
splines, the maximum interaction level is one less than the maximum order of
the polynomials.
The interaction level determines the allowed order of cross-terms included.
You can use the Term Editor to see the effects of changing the interaction level.
Click the Edit Terms button. The number of constant, linear, second- and
third-order (and above) terms can be seen in the Model Terms frame.
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For polynomials: With an interaction level of 1, there are no terms in the model
involving more than one factor. For example, for a four-factor cubic, for factor
2
3
L, you see the terms for L, L , and L , but no terms involving L and other
factors. In other words, there are no cross-terms included.
If you increase the interaction level to 2, under second-order terms you see L
and also L multiplied by each of the other factors; that is, second-order
cross-terms (for example, LN, LA, and LS).
2
2
Increase the interaction to 3, and under third-order terms you see L
2
2
2
multiplied by each of the other factors ( L N , L A , L S ), L multiplied by other
pairs of factors (LNA, LNS, LAS), and L multiplied by each of the other factors
2
2
2
squared ( N L , A L , S L ). Interaction level 3 includes all third-order
cross-terms.
The preceding also applies to all four factors in the model, not just L.
For hybrid splines: The interaction function is different: it refers only to
cross-term interactions between the spline term and the other variables. For
example, at interaction order 1, raw spline terms only; interaction 1, raw terms
and the spline terms x the first-order terms; interaction 3, includes spline
terms x the second-order terms; and so on.
Stepwise
Take care not to overfit the data; that is, you do not want to use unnecessarily
complex models that “chase points” in an attempt to model random effects.
The Stepwise feature can help.
Predicted sum of squares error(PRESS) is a measure of the predictive quality
of a model.
Min PRESS throws away terms in the model to improve its predictive quality,
removing those terms that reduce the PRESS of the model.
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Setting Up Models
For a further discussion of the statistical effects of all the menu items in the
Stepwise feature, see “Stepwise Regression Techniques” on page 6-13 and
“Linear Model Statistics Displays” on page 6-22.
Global Model Class: Radial Basis Function
A variety of radial basis functions (RBFs) are available in MBC. They are all
radially symmetrical functions that can be visualized as mapping a flexible
surface across a selection of hills or bowls, which can be circular or elliptical.
Networks of RBFs can model a wide variety of surfaces. You can optimize on
the number of centers and their position, height and width. You can have
different widths of centers in different factors. RBFs can be very useful for
investigating the shapes of surfaces when system knowledge is low. Combining
several RBFs allows complicated surfaces to be modeled with relatively few
parameters. The following example shows a surface of an RBF model.
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There is a detailed user guide for modeling using RBFs in “Radial Basis
Functions” on page 7-1. See especially “Tips for Modeling with Radial Basis
Functions” on page 7-30 and “Types of Radial Basis Functions” on page 7-4.
The statistical basis for each setting in the RBF global models is explained in
detail in “Guide to Radial Basis Functions for Model Building” on page 7-3.
The following example illustrates the types of RBF available.
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Setting Up Models
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Global Model Class: Hybrid RBF
This option combines an RBF model with a linear model.
The RBF kernel drop-down menu offers the same options as for normal RBF.
The Linear Part tab contains the same options as the other global linear
models; see “Global Linear Models: Polynomials and Hybrid Splines” on
page 5-33.
See “Hybrid Radial Basis Functions” on page 7-28.
See also “Radial Basis Functions” on page 7-1 for a detailed guide to the use of
all the available RBFs in modeling.
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Setting Up Models
Global Model Class: Multiple Linear Models
The preceding example shows the defaults for multiple linear models. You can
add linear models (as seen in the single linear model settings).
This is primarily for designing experiments using optimal designs. If you have
no idea what model you are going to fit, you would choose a space-filling design.
However, if you have some idea what to expect, but are not sure exactly which
model to use, you can specify a number of possible models here. The Design
Editor can average optimality across each model.
For example, if you expect a quadratic and cubic for three factors but are
unsure about a third, you can enter several alternative polynomials here. You
can change the weighting of each model as you want (for example, 0.5 each for
two models you think equally likely). This weighting is then taken into account
in the optimization process in the Design Editor.
The model that appears in the model tree is the one you select, listed as
Primary model. Click the model in the list, then click Use Selected. The
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Primary model changes to the desired model. If you do not select a primary
model, the default is the first in the list.
When the model has been fitted, you can view the primary model at the global
node. To compare the fit of all the alternatives, click Build Models in the
toolbar, select Multiple Linear Models in the dialog, and click OK. One of each
model is then built as a selection of child nodes.
See also “Polynomials” on page 5-33.
Global Model Class: Free Knot Spline
This option is only available for global models with only one input factor. See
also “Hybrid Splines” on page 5-34 for a description of splines. The major
difference is that you choose the position of the knots for hybrid splines; here
the optimal knot positions are calculated as part of the fitting routine.
You can set the number of knots and the spline order can be between one and
three.
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Setting Up Models
There are three different algorithms under Optimization settings: Penalized
least squares, Genetic algorithm, and Constrained least squares.
For all three methods, you can set the Initial population. This is the number
of initial guesses at the knot positions. The other settings put limits on how
long the optimization takes.
The example following shows a free knot spline model with three knots. The
position of the knots is marked on the N axis.
See also the local models involving splines:
• “Polynomial Spline” on page 5-48
• “Local Model Class: Truncated Power Series” on page 5-50
• “Local Model Class: Free Knot Spline” on page 5-51
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Local Model Setup
The preceding example shows the default settings for the local model.
Local models can be of the following types:
• Polynomial
• Polynomial Spline
• Truncated Power Series
• Free Knot Spline
• Growth Models
• Linear Models
• User-Defined Models
You can choose additional response features at this stage using the Response
Features tab. These can also be added later. The Model Browser automatically
chooses sufficient response features for the current model.
See also
• “Covariance Modeling” on page 5-54
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Setting Up Models
• “Correlation Models” on page 5-56
• “Transforms” on page 5-56
See “Local Model Definitions” on page 6-48 for statistical details of all local
models.
See “Setting Up Models” on page 5-26 for a list of all information about setting
up local and global models.
Local Model Class: Polynomials and Polynomial
Splines
Polynomials
At the local level, if you have one input factor, you can choose Polynomial
directly from the list of local model classes. Here you can choose the order of
polynomials used, and you can define a datum model for this kind of local model
(see below).
If there is more than one input factor, you can only choose Linear Models or
Transient Models from the Local Model Class list. Under Linear Models you
can choose Polynomial or Hybrid Spline. This is a different polynomial model
where you can change more settings such as Stepwise, the Term Editor (where
you can remove any model terms) and you can choose different orders for
different factors (as with the global level polynomial models). See “Local Model
Class: Linear Models” on page 5-53.
Different response features are available for this polynomial model and the
Linear Models: Polynomial choice. You can view these by clicking the
Response Features tab on the Local Model Setup dialog. Single input
polynomials can have a datum model, and you can define response features
relative to the datum. See “Datum Models” on page 5-88
See “Polynomials” on page 5-33 for a general description of polynomial models.
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Polynomial Spline
A spline is a piecewise polynomial, where different sections of polynomial are
fitted smoothly together. The location of each break is called a knot.
This model has only one knot. You can choose the orders of the polynomials
above and below the knot. See also “Hybrid Splines” on page 5-34. These global
models also use splines, but use the same order polynomial throughout.
Polynomial splines are only available for single input factors. The following
example shows a typical torque/spark curve, which requires a spline to fit
properly. The knot is shown as a red spot at the maximum, and the curvature
above and below the knot is different. In this case there is a cubic basis function
below the knot and a quadratic above.
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Setting Up Models
See “Polynomial Splines” on page 6-49 for statistical details.
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Local Model Class: Truncated Power Series
This is only available for a single input factor.
You can choose the order of the polynomial basis function and the number of
knots for Truncated Power Series Basis Splines. A spline is a piecewise
polynomial, where different sections of polynomial are fitted smoothly
together. The point of each break is called a knot. The truncated power series
changes the coefficient for the highest power when the input passes above the
knot value.
It is truncated because the power series is an approximation to an infinite sum
of polynomial terms. You can use infinite sums to approximate arbitrary
functions, but clearly it is not feasible to fit all the coefficients.
Click Polynomial to see (and remove, if you want) the polynomial terms. One
use of the remove polynomial term function is to make the function linear until
the knot, and then quadratic above the knot. In this case we remove the
quadratic coefficient.
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Setting Up Models
See also
• “Polynomial Spline” on page 5-48, where you can choose different order basis
functions either side of the knot
• “Hybrid Splines” on page 5-34, a global model where you can choose a spline
order for one of the factors (the rest have polynomials fitted)
• “Local Model Class: Free Knot Spline” on page 5-51, free knot splines with
cubic basis functions, where you can choose the number of knots
Local Model Class: Free Knot Spline
These are the same as the “Global Model Class: Free Knot Spline” on page 5-44
(which is also only available for one input factor). See the global free knot
splines for an example curve shape.
A spline is a piecewise polynomial, where different sections of polynomial are
fitted smoothly together. The point of the join is called the knot.
You can choose the number of knots. The order of polynomial fitted (in all curve
sections) is cubic. The “B” in B-spline stands for Basis, after the polynomial
basis function used for the curve sections.
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You can set the number of Random starting points. These are the number of
initial guesses at the knot positions.
See also
• “Polynomial Spline” on page 5-48, where you can choose different order basis
functions for either side of the knot
• “Local Model Class: Truncated Power Series” on page 5-50, where you can
choose the order of the basis function
• “Hybrid Splines” on page 5-34, a global model where you can choose a spline
order for one of the factors (the rest have polynomials fitted)
Local Model Class: Growth Models
Growth models have a number of varieties available, as shown. They are only
available for single input factors.
These are all varieties of sigmoidal curves between two asymptotes, like the
following example.
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Setting Up Models
Growth models are often the most appropriate curve shape for air charge
engine modeling.
See “Local Model Definitions” on page 6-48 for mathematical details on the
differences between these growth models.
Local Model Class: Linear Models
Select Linear Models and then click Setup.
You can now set up polynomial or hybrid spline models. The settings are
exactly the same as the global linear models.
These models are for multiple input factors - for single input factors you can
use a different polynomial model from the Local Model Class list, where you
can only change the polynomial order. See “Local Model Class: Polynomials and
Polynomial Splines” on page 5-47.
If there is more than one input factor, you can only choose Linear Models or
Transient Models from the Local Model Class list. Under Linear Models you
can choose Polynomial or Hybrid Spline. This polynomial is a different model
where you can change more settings such as Stepwise, the Term Editor (where
you can remove any model terms) and you can choose different orders for
different factors (as with the global level polynomial models).
See “Global Linear Models: Polynomials and Hybrid Splines” on page 5-33 for
details.
Different response features are available for this Linear Models: Polynomial
model and the other Polynomial choice (for single input factors). You can view
these by clicking the Response Features tab on the Local Model Setup dialog.
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Single input polynomials can have a datum model, and you can define response
features relative to the datum. See “Datum Models” on page 5-88.
These linear models are labelled Quadratic, Cubic etc on the test plan block
diagram, while the single input type of polynomials are labelled Poly2, Poly3
etc.For higher orders both types are labelled Poly n.
Local Model Class: User-Defined Models
You must check user-defined models into the Model-Based Calibration Toolbox
before you can use them here. The only model checked in by default is
Exponential.
See “User-Defined Models” on page 6-58 of the Technical Documents for
detailed instructions on this.
Local Model Class: Transient Models
These are supported for multiple input factors, where time is one of the factors.
You can define a dynamic model using Simulink and a template file that
describes parameters to be fitted in this model. You must check these into the
Model-Based Calibration Toolbox before you can use them for modeling.
See “Transient Models” on page 6-64 of the Technical Documents for detailed
instructions and an example.
Covariance Modeling
This frame is visible no matter what form of local model is selected in the list.
Covariance modeling is used when there is heteroscedasticity. This means that
the variance around the regression line is not the same for all values of the
predictor variable, for example, where lower values of engine speed have a
smaller error, and higher values have larger errors, as shown in the following
example. If this is the case, data points at low speed are statistically more
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Setting Up Models
trustworthy, and should be given greater weight when modeling. Covariance
models are used to capture this structure in the errors.
y
Engine Speed (N)
You can fit a model by finding the smallest least squares error statistic when
there is homoscedasticity (the variance has no relationship to the variables).
Least squares fitting tries to minimize the least squares error statistic
∑ εi , where εi
2
2
is the error squared at point i.
When there is heteroscedasticity, covariance modeling weights the errors in
favor of the more statistically useful points (in this example, at low engine
speed N). The weights are determined using a function of the form
2
∑
εi
-----Wi
where W i is a function of the predictive variable (in this example, engine speed
N).
There are three covariance model types.
Power
α
These determine the weights using a function of the form W i = ŷ . Fitting the
covariance model involves estimating the parameter α .
Exponential
These determine the weights using W i = e
αŷ
.
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Mixed
β
These determine the weights using W i = α + ŷ . Note that in this case there
are two parameters to estimate, therefore using up another degree of freedom.
This might be influential when you choose a covariance model if you are using
a small data set.
Correlation Models
These are only supported for equally spaced data in the Model-Based
Calibration Toolbox. When data is correlated with previous data points, the
error is also correlated.
There are three methods available.
MA(1) – The Moving Average method has the form ε n = α 1 ξ n – 1 + ξ n .
AR(1) – The Auto Regressive method has the form ε n = α 1 ε n – 1 + ξ n .
AR(2) – The Auto Regressive method of the form ε n = α 1 ε n – 1 + α 2 ε n – 2 + ξ n
2
ξ is a stochastic input, ξ n ∼ N ( 0,σ ξ ) .
Transforms
The following example shows the transforms available.
Input transformation can be useful for modeling. For example, if you are trying
to fit
y = e
a + bx + cx
2
using the log transform turns this into a linear regression:
log ( y ) = a + bx + cx
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Setting Up Models
1
2
Transforms available are logarithmic, exponential, square root, y , --y- , and
Other. If you choose Other, an edit box appears and you can enter a function.
Apply transform to both sides is only available for nonlinear models, and
transforms both the input and the output. This is good for changing the error
structure without changing the model function. For instance, a log transform
might make the errors relatively smaller for higher values of x. Where there is
heteroscedasticity, as in the Covariance Modeling example, this transform can
sometimes remove the problem.
Designing Experiments
You can design experiments after setting up models. You can design
experiments for both stages, local and global. You invoke the Design Editor in
several ways from the test plan level:
• Right-click a Model block in the test plan diagram and select Design
Experiment.
You must select (by clicking) a stage to design for (first or second stage) or
the following two options are not possible.
• Click the Design Experiment toolbar button
.
• Select TestPlan –> Design Experiment.
For an existing design, View –> Design Data also launches the Design Editor
(also in the right-click menu on each Model block). In this case you can only
view the current data being used as a design at this stage. If you enter the
Design Editor by the other routes, you can view all alternative designs for that
stage.
See the “Design of Experiment Tutorial” on page 3-1.
Viewing Designs
The view design facility enables the user to investigate the statistical
properties of the current data.
• From the test plan node, select the model stage you are interested in by
clicking, then choose View –> Design Data. Alternatively, use the right-click
menu on a Model block.
This provides access to all the Design Editor and design evaluation utility
functions with the current data rather than the prespecified design. If you have
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done some data-matching to a design, each data point is used as a design point.
You can now investigate the statistical properties of this design.
For two-stage models, viewing stage one (local model) designs creates a
separate design for each test.
See “The Design Editor” on page 5-153 or the step-by-step guide in the “Design
of Experiment Tutorial” on page 3-1.
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Data
This section describes all aspects of loading, manipulating, and selecting data
in the Model Browser. There are three main graphical interfaces for dealing
with data:
• You use the Data Editor for loading, filtering, grouping, and editing data,
and you can define new variables. The Data Editor contains various
graphical interfaces for these tasks: the Data Import Wizard for loading and
merging data; the User-Defined Variables and Filters window (with the
Variable Editor, Filters Editor, and Storage dialog) for creating and storing
new variables and data filters; and the Test Groupings dialog for plotting
and manipulating data groups. You can reach the Data Editor from project
or test plan level.
• You use the Data Wizard to select data for modeling and set up matching
data to designs by setting tolerances and opening the Data Selection
window. You reach the Data Wizard from test plan level.
• You use the Data Selection window for matching data to experimental
designs. You can set tolerances for automatic selection of the nearest data
points to the specified design points, or select data points manually. You
reach the Data Selection window from test plan level.
You can load and merge data from the following:
• From files (Excel, Concerto, MATLAB)
• From the workspace
• From tailor-made Excel sheets
See “Data Loading and Merging” on page 5-65.
Within the Data Editor, you can do the following:
• View plots, edit and add data records.
• Define new variables. See “User-Defined Variables and Filtering” on
page 5-69 and “Variable Editor” on page 5-71.
• Apply filters to remove unwanted records. See “Filter Editor” on page 5-72.
• Store and retrieve user-defined variables and filters. See “Storage” on
page 5-74.
• Define test groupings to collect data into groups.
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You use the Data Wizard to do the following:
• Select the data set and design to use for modeling.
• Select the data signals to use for model input factors (one-stage, or local and
global for two-stage).
• Select matching tolerances (if matching data to a design).
• Select data signals for response model input factors.
The Data Selection window matches data to experimental designs.
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The Data Editor
To reach the Data Editor:
• From the test plan node, choose TestPlan –> Load New Data.
• Alternatively, from the project node, do one of the following:
- Choose Data –> New Data, Copy Data, or Edit Data.
- Select any of the equivalent toolbar buttons.
- Double-click a data set in the Data Sets pane.
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As can be seen in the preceding example, in the default Graph view you can
select combinations of variables from the list boxes on the left and view all
tests. Multiple selection of tests and y-axis variables is possible — in the
example, multiple tests are selected to view several tests simultaneously. In
the Table view (toolbar button and View menu), you can edit and add records.
See “Data Editor Toolbar and Menus” on page 5-63.
The list box at the top right contains the source file information for the data,
and other information is displayed on the left: the name of the data set and the
numbers of records, variables, and tests it contains. See the preceding example.
The bars and figures on the left show the proportion of records removed by any
filters applied, and the number of user-defined variables is shown. For this
example with two user-defined variable added to a data set originally
containing seven variables, you see ‘Variables 7 + 2’.
By default the new data set is called Data Object. You can change this name in
the Name edit box at the top of the window.
See “Data Editor Toolbar and Menus” on page 5-63 for other controls.
You can also change the names of data sets at the project node by
select-clicking a data set in the Data Sets list, or by pressing F2 (as when
selecting to rename in Windows Explorer).
There is a right-click menu on the plot, to show the legend (only applicable for
multiple selections), the grid, and the line. This line joins the data points.
Reorder points redraws the line joining the points in order from left to right.
The line might not make sense when drawn in the order of the records.
Note Dismissing the Data Editor automatically brings up the Data Wizard if
you entered it from the test plan level.
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Data Editor Toolbar and Menus
Load Data from
Workspace
Load Data from File
User-Defined Variables
and Filtering
Change Test Grouping
View Table
View Graph
• Load Data from Workspace — See “Loading Data from the Workspace” on
page 5-66; also in the File menu.
• Load Data from File — See “Loading Data from File” on page 5-65; also in
the File menu.
• User-Defined Variables and Filtering — Opens the User-Defined
Variables and Filtering window; also in the Tools menu.
• View Graph — The default display in the Data Editor is graphical; also in
the View menu.
• View Table — Changes the main display to a table of the data; also in the
View menu. In the Table view, you can edit data records, and you can access
the Add Record command by right-clicking a test number.
• Change Test Grouping — Opens the Test Groupings dialog; also in the
Tools menu.
The Window and Help menus are the same as everywhere else in the Model
Browser. See “Window Menu” on page 5-9 and “Help Menu” on page 5-9.
File Menu
• Load Data from File — See “Loading Data from File” on page 5-65.
• Load Data from Workspace — See “Loading Data from the Workspace” on
page 5-66.
• Export Data — Exports data to Excel or to the workspace.
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• Close — Closes the Data Editor.
View Menu
• Graph
• Table
See Toolbar above.
Tools Menu
• User-Defined Variables & Filtering — You can add new variables to the
data set and remove records by imposing constraints on the data. Also in the
toolbar.
• Change Test Groupings — Opens the Test Groupings dialog; also in the
toolbar.
• Make Excel Sheet for Import — Opens an Excel sheet, prepared for data
input in the form the Model Browser expects, with Name, Unit, and Data the
headings for the first three rows (variables are expected to be column
headings with data records in each row).
• Import Excel sheet — Only active after the Make Excel Sheet for Import
command. This command closes the Excel sheet and loads the data into the
Data Editor (unless there is already data loaded, in which case it first opens
the Data Merging Wizard).
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Data Loading and Merging
Data can be loaded and merged from files, from the workspace, and from
tailor-made Excel sheets. A test plan can only use a single data set, so you need
to use the merge functions to combine data variables from more than one
source file in order to incorporate desired variables into one model.
Loading Data from File
1 From the test plan node, choose TestPlan –> Load New Data.
Alternatively, from the project node choose Data –> New Data.
The Data Editor appears.
2 Click the Open File button in the toolbar
to load data from a file. See
“Data Editor Toolbar and Menus” on page 5-63.
3 Choose File –> Load data from file.
The Data Import Wizard appears, to help you select a file.
Note If you already have some data loaded, the Data Import Wizard can
merge new data into the existing data set.
The Data Import Wizard
1 To import data from a file, enter the file pathname in the edit box, or use the
Browse button to find and select the data file. Double-click to load the file.
The drop-down menu contains the file types recognized by the Model
Browser (Excel, Concerto, MATLAB). The default Auto tries to decide the
type of file it is by looking at the file extension.
2 For Excel files with multiple sheets, you must next select the sheet you want
to use and click OK.
3 The Import Wizard now displays a summary screen showing the total
number of records and variables imported, and you can view each variable’s
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range, mean, standard deviation, and units in the list box. Click Finish to
accept the data, unless you have data to merge. See the following.
The Data Import Wizard disappears and the view returns to the Data Editor.
Merging Data
4 If you already have some data loaded, you cannot click Finish but must click
Next instead. This brings you to the data merging screen of the Wizard.
5 Here you must choose one of the radio buttons:
- Merge extra channels (more variables)
- Merge extra records (more data, same variables)
- Overwrite old data (use only the new data)
6 Click Next. The Wizard summary screen appears, showing the results of the
merge. You can click <Prev to go back and change merging options.
7 Click Finish to accept the data and return to the Data Editor.
Note Obviously the merge might not succeed if the data sets do not contain
the same variables. A message appears if the merge is impossible when you
click Next (step 6), and you must make another choice.
Loading Data from the Workspace
1 From the test plan node, choose TestPlan –> Load New Data.
Alternatively, from the project node choose Data –> New Data.
The Data Editor appears.
You can import variables from the MATLAB workspace by choosing File ->
Load data from the MATLAB workspace.
Alternatively, click the equivalent button in the toolbar
(the left button
with the MATLAB membrane on it — look for the tooltip).
See “Data Editor Toolbar and Menus” on page 5-63.
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The Loading Data from Workspace dialog appears.
Variables in the workspace are displayed in hierarchical form in the top left
pane. Select a variable here, and information about that variable is displayed
in the pane on the right.
1 Select a variable to import in the tree at top left.
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2 Click the Add variable to output data button.
The number of records and variables appears in the Output Data pane. You
can add variables (one at a time) as many times as you like (as long as there
are no name conflicts).
You can add comments in the edit box; pressing Return adds them. Note that
Return does not automatically click OK.
3 Click Clear to remove all data from the Output Data pane and start again.
4 Click OK to accept the data to import and return to the Data Editor.
Data Merging
If you already have data loaded, the Data Merging Wizard appears, where you
must choose one of three radio buttons:
• Merge extra channels (more variables)
• Merge extra records (more data, same variables)
• Overwrite old data (use only the new data)
Click Next. The Wizard summary screen appears, showing the results of the
merge. You can click <Prev to go back and change merging options.
Note Obviously the merge might not succeed if the data sets do not contain
the same variables. A message appears if the merge is impossible when you
click Next, and you must make another choice.
Click Finish to accept the data and return to the Data Editor.
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User-Defined Variables and Filtering
• These functions are available via the menu Tools –> User-Defined
Variables & Filtering in the Data Editor.
• There is an equivalent toolbar button
.
The preceding example shows the User-Defined Variables and Filtering
window after the creation of a new variable and a new filter (as in the “Data
Editor Tutorial” on page 4-1).
New Variables
New variables you create in the Variable Editor appear in the Variable
Expression pane and the Data Information pane.
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• You create variables by doing the following:
- Clicking the
toolbar button
- Selecting Variable –> New Variable
- Alternatively, by selecting the Variable Expression pane by clicking in it,
then pressing Insert
The Variable Editor appears.
You can also load user-defined variables from Storage.
• You can edit variables:
- Directly, by select-clicking in the Variable Expression pane or pressing
F2 (as in renaming in Windows Explorer)
- By double-clicking, which opens the Variable Editor
- By choosing Variables –> Edit Variable
• You can also delete variables:
- By selecting them in the Variable Expression pane and pressing Delete
- By selecting Variables –> Delete Variable
New Filters
New filters you create in the Filter Editor appear in the Filter Expression
pane, and their effects are shown graphically in the pie chart in the Filtration
Results pane.
• You can create filters:
- By clicking the toolbar button
- By selecting Filters –> New Filter
- Alternatively, by selecting the Filter Expression pane by clicking in it,
then pressing Insert
The Filter Editor appears.
You can also load user-defined filters from Storage.
• Filters can be edited in the same way as variables:
- Directly, after you select-click them in the Filters Expression pane, or by
pressing F2
- Using Filters –> Edit Filter, which opens the Filter Editor
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- By double-clicking, which also opens the Filter Editor
• Delete filters by selecting them and pressing Delete, or choose Filters –>
Delete Filter.
User-Defined Variables and Filters Toolbar
New Variable
Open Storage Container
Edit Variable
Edit Filter
Move selection up/down
New Filter
You can change the order of user variables in the list using the arrow buttons
in the toolbar. You can define new variables with reference to each other, and
they are added to the data in turn, in which case you can use the up/down
arrows to reorder variables in legitimate sequence. See “Sequence of Variables”
on page 4-12 in the “Data Editor Tutorial” for an example.
Variable and Filter Editors
Variable Editor
You can define new variables in terms of existing variables:
1 Click the
toolbar button in the User-Defined Variables and Filtering
window.
2 Choose the menu item Variable –> New Variable.
Alternatively, select the Variable Expression pane by clicking in it, then
press Insert.
The Variable Editor dialog appears.
• Define the new variable by writing an equation in the edit box at the top of
the Variable Editor dialog.
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You can type directly in the edit box or add variable names and operators by
double-clicking them. In the case of variable names especially, this latter
method avoids typing mistakes. Variable names are case sensitive.
The preceding example shows a definition for a new variable called POWER that
is defined as the product of two existing variables, tq and n, by entering POWER
= tq x n.
• Click OK to add the new variable to the current data set.
New variables you create in the Variable Editor then appear in the Variable
Expression pane of the User-Defined Variables and Filtering window.
New variables, along with the original variables, appear in the list view (in the
Data Information pane) and the new variable definition is included in the
description field (to the right of the units column; you might have to scroll or
resize to see it).
Filter Editor
A filter is the name for a constraint on the data set used to exclude some
records. To reach the Filter Editor:
1 Click the toolbar button
window.
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in the User-Defined Variables and Filtering
Data
2 Select the menu item Filters –> New Filter.
Alternatively, select the Filter Expression pane by clicking in it, then press
Insert.
The Filter Editor dialog appears.
You define the filter using logical operators on the existing variables.
In the preceding example, n>1000, the effect of this filter is to keep all records
with speed (n) greater than 1000.
3 Click OK to impose new filters on the current data set.
This new filter appears in the Filter Expression pane of the User-Defined
Variables and Filtering window.
The Filtration results pie chart shows how many records have been
removed by the imposition of this filter, and the number of records changes
accordingly (for example, Records 200/264 indicates that 64 records have
been filtered out).
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Storage
Storage allows user-defined variables and filters to be stored so that you can
apply them to other data sets at any time.
You can open the Storage window from the User-Defined Variables and
Filtering window by either
• Using the menu item File –> Open Storage
• Using the toolbar button
Append Stored Object
Get Current Variables
Get Current Filters
Import from File
Delete Stored Object
Export to File
The button Append Stored Object adds the selected user-defined variable or
filter in Storage to the current session. It appears in the Variables and Filters
window. Double-clicking an object in Storage appends it to the current session.
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The two buttons Get Current Variables and Get Current Filters allow you to
put all user-defined variables and filters from the current session into Storage.
They appear in the Storage window. All stored user-defined variables and
filters appear here regardless of which project is open — once created and
brought into Storage, they remain there. If you do not delete them, they are
there indefinitely.
You can select Export to File to send the stored objects to a file. You might do
this to move the objects to a different user or machine. Select Import from File
to bring such variables and filters into Storage.
You can edit the names of stored objects, by select-clicking as in Windows
Explorer, or by clicking once and pressing F2.
Test Groupings
The Define Test Groupings dialog collects records of the current data set into
groups; these groups are referred to as tests. Test groupings are used to define
hierarchical structure in the data for two-stage modeling.
You access the dialog from the Data Editor by doing one of the following:
• Using the menu Tools –> Change Test Groupings
• Using the toolbar button
When you enter the dialog, no plot is displayed.
1 Click to select a variable in the list box to use in defining groups within the
data.
2 The Add Variable button (
) adds the currently selected variable in the
Variables list to the list view on the left. Alternatively, double-click
variables to add them.
You can now use this variable to define groups in the data.
In the following example, the variable n is being used to define groups. The
maximum and minimum values of n are displayed.
The Tolerance is used to define groups: on reading through the data, when the
value of n changes by more than the tolerance, a new group is defined. You can
change the Tolerance by typing directly in the edit box.
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You can define additional groups by selecting another variable and choosing a
tolerance. Data records are then grouped by n or by this additional variable
changing outside their tolerances.
You can plot variables without using them to define groups by clearing the
Group By check box.
You can remove variables from consideration by selecting the unwanted
variable in the list view (the selection is highlighted in blue) and clicking the
Remove variable button
.
Remove variable
The plot shows the scaled values of all variables in the list view (the color of the
Tolerance text corresponds to the color of data points in the plot). Vertical pink
bars show the tests (groups). You can zoom the plot by Shift-click-dragging or
middle-click-dragging the mouse.
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Test number variable contains a drop-down menu showing all the variables
in the current data set. You can select any of these to number the tests (for
example, lognumber could be useful (instead of 1,2,3...) if the data was taken in
numbered tests and you want access to that information during modeling).
Every record in a test must share the same test number to identify it, so when
you are using a variable to number tests, the value of that variable is taken in
the first record in each test.
Test numbers must be unique, so if any values in the chosen variable are the
same, they are assigned new test numbers for the purposes of modeling. (This
does not change the underlying data, which retains the correct lognumber or
other variable.)
Reorder records allows you to reorder records in the data set. This sorts
records before grouping. Otherwise, the groups are defined using the order of
records in the original data set.
Show original displays the original test groupings if any were defined.
One test/record defines one test per record, regardless of any other grouping.
This is required if the data is to be used in creating one-stage models.
Clicking OK accepts the test groupings defined and dismisses the dialog.
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Data Wizard
Dismissing the Data Editor after loading data from the test plan node
automatically brings up the Data Wizard.
Alternatively, after setting up a new test plan, if no data has been selected in
that test plan, then either of the following also brings up the Data Wizard.
• Choosing Select Data from the test plan node (toolbar button or TestPlan
menu item)
• Double-clicking the Responses block in the test plan diagram
Step 1: Select Data Set
Use the first screen of the wizard to select the data set to build models from.
You can also select whether to use all the data set or to match the data to a
design, if any designs are in use in the test plan. Designs appear in the left list
box.
Note The check box at bottom left opens the Data Selection window on
closing the Data Wizard, for matching data to a design. This check box
appears on each screen of the Data Wizard.
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Data
Step 2: Select Input Signals
Select the input signals for the model (on all stages of the hierarchical model)
from the list box of data signals on the right, and match them to the correct
model variables using the big button with the arrow. Double-clicking an item
in the data signals list selects that signal for the currently selected input factor
(with the range if the Copy range check box is selected) and then moves to the
next input.
These options reappear when you select Data –> Input Signals in the Data
Selection window.
If you entered the correct signal name at the model setup stage, the
appropriate signal is automatically selected as each model input factor is
selected. This can be time-saving if there are many data signals in a data set.
If the signal name is not correct, you must select the correct variable for each
input by clicking.
Select the check box Copy range if you want to use the range of the selected
data signal for the model input range. Ranges are not automatically copied,
although stored templates have the ranges that were set when the template
was saved.
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Step 3: Select Response Models
Starting from scratch (with an empty Responses list box), select the desired
response in the Data Variables list and click Add.
In the preceding example, the test plan template specified torque as the
response model, so it already appears in the Responses list box. If you want to
change the response, select another variable and click the large button with the
arrow. This replaces the current selected response. The previous response
appears in brackets to inform you what has changed.
When there is already a response in the list box, clicking Add does not replace
the selection but increases the number of responses. The replace button (with
the arrow) is not available when the Responses box is empty.
You can use Delete to remove selected responses. You can select datum models
(if the local model supports them), and you can change the local and global
models by using the Set Up buttons. See “Global Model Setup” on page 5-32
and “Local Model Setup” on page 5-46 for details.
Step 4: Set Tolerances
Setting tolerances is only relevant if you are matching data to a design. This
screen only appears if you selected the radio button option Match selected
data to design in step 1. You can also set tolerances later using the Data menu
in the Data Selection window.
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Data
The Data Selection window appears by default when you close the Data
Wizard while you are matching data to designs. If you are not matching data,
it does not appear unless you select the check box.
The range tolerance sets the matching tolerance on signal range within a
particular test. Setting the range tolerance to Inf, the default, causes matching
to be performed using test means only.
Data Selection Window
The Data Selection window is primarily for matching data to designs.
To reach the Data Selection window:
• From the test plan node, click the Select Data
button in the toolbar.
• Alternatively, choose TestPlan –> Select Data.
The Data Selection window appears.
Note If the current test plan has no data loaded (in which case you can see
‘No Data is selected in the top right Data Sets pane) these actions open
the Data Wizard instead.
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You can also select the check box Open data selection figure on finish in the
Data Wizard. This check box is selected by default when you select Matching
selected data to design.
There is an option to match data to a design or select all available data from a
particular data set.
Data Menu
• New — Opens the Data Wizard to select a new data set to match to the
design.
• Augment — If there are data left over (that is, not matched to design points
using current tolerances), this command opens a dialog where you can select
tests to add. More data might be useful despite not matching original design
points.
• Tolerances — Opens the Matching Tolerances dialog, where you can set
limits for matching data points to design points. See also the relevant Data
Wizard help: “Step 4: Set Tolerances” on page 5-80.
• Input Signals — Returns to the input signal setup (step 2 of the Data
Wizard).
• Setup Plots — Opens a dialog where you can select variables to plot
(including any variables in the data set, not just those used in modeling).
Matching to designs always occurs at the outer level.
Matching to Design (Two-Stage)
All matching is performed at the second stage.
• Tests with fewer observations than the number of local model parameters
are not selected.
• The ten closest matches are displayed in the data points list.
• You can select data points with the check mark button and clear them with
the cross button.
Icon information
• A selected data or design point has a check mark icon.
• If a point is within the matching tolerance it is colored blue.
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Data
• If a point is available to be matched it has a smiley face icon.
• If a point is not available to be matched (already matched) it has a smiley
face icon with a line through it.
• If a point has not been matched it has a cross icon.
Plots
• A plot of the second stage inputs for the selected data point is shown on the
first tab.
• The Data Plots tab allows the user to screen the data using other variables
(for example, response or other unused variables).
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Data
Selecting All the Data Set
Only one data point is shown in the data point list.
Tests with fewer observations than the number of stage 1 model parameters
are not selected.
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Selecting Data for One-Stage Models
This shows plots of all the tests at once, with one point per test (to plot each
test individually would only plot one point). The selected test is highlighted red
in the plot.
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New Response Models
New Response Models
When you first set up a test plan, the Data Wizard automatically contains the
response model setup after the data matching functions.
The following applies when you return to a previously setup test plan to add a
new response model, or when you click the New button at test plan level to add
new response models to an existing test plan.
Double-click the Responses outport of the block diagram or use the New
Response Model item in the File menu, or use the toolbar icon. (None of these
is available unless you are in the Test Plan view, that is, have the test plan
node selected in the model tree. This should be obvious: you can only see the
test plan block diagram with the test plan node selected.)
The Response Model Setup dialog has a list box containing all the variables
in the selected data set except the inputs to the local and global models; you
cannot use an input also as a response.
You can reach the controls for setting up models using the Set Up buttons to
change the local and global models also, and you can add datum models
(maximum or minimum) if the local model supports this.
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You can return to the local or global setup options individually at any time by
double-clicking the block in the test plan diagram.
Datum Models
Under Datum you can choose a datum model, but only for some local models —
polysplines and polynomials (but see Linked datum model below). Other local
models cannot have a datum model, as they do not necessarily have a unique
turning point.
The datum model tracks the maximum or minimum of the local models. This is
equivalent to adding the maximum or minimum as a response feature, which
can be very useful for analysis if those points are interesting from an
engineering point of view.
The Datum options are
• None
• Maximum — This can be useful in cases using polyspline modeling of torque
against spark. The maximum is often a point of engineering interest.
• Minimum — This can be useful for cases where the object is to minimize
factors such as fuel consumption or emissions.
• Linked datum model — This is only available to subsequent two-stage
models within a test plan in which the first two-stage model has a datum
model defined. In this case you can make use of that datum model. The
linked datum option carries the name of the response of the first two-stage
model, where it originated.
If the maximum and minimum are at points of engineering interest, like MBT
or minimum fuel consumption, you can add other response features later using
the datum model (for example, MBT plus or minus 10 degrees of spark angle)
and track these across local models too. It can be useful to know the value of
MBT when modeling exhaust temperature, so you can use a linked datum
model from a previous torque/spark model. Having responses relative to datum
can also be a good thing as it means the response features are more likely to
relate to a feature within the range of the data points.
You can also export the datum model along with local, global, and response
models if required. See “Exporting Models” on page 5-148.
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Local Level
Local Level
When you select a local node (with the
appears.
icon) in the model tree, this view
Note that after the two-stage model is calculated the local node icon changes to
a two-stage icon (
) to reflect this. See the model tree for clarification. The
response node also has a two-stage icon, but produces the response level view
instead.
The following example shows a local model of torque/spark curves.
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Click here to change test
Click-and-drag anywhere along this
line to change the size of the
Response Features pane
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Click-and-drag or double-click
here to expand plots/hide
statistics pane
Local Level
See “Local Level: Toolbar” on page 5-94 and “Local Level: Menus” on page 5-95
for details on these controls.
The default view is the Model tab, described below. See also “Data Tab” on
page 5-103.
Upper scatter plots are replaced by an icon if you resize the Browser too small.
Local Special Plots
The lower plots are referred to as special plots as they can be different for
different models.
The lower plot at the local level shows the local model fit to the data for the
current test only, with the datum point if there is a datum model. If there are
multiple inputs to the local model, a predicted/observed plot is displayed. In
this case to examine the model surface in more detail you can use Model –>
Evaluate. See “Model Evaluation Window” on page 5-144.
You can scroll through all the local models by using the up/down test buttons,
type directly in the edit box, or go directly to test numbers by clicking Select
Test.
To examine the local fit in more detail, double-click the arrows (indicated in the
preceding figure) to hide the scatter plot and expand the lower plot. You can
zoom in on parts of the plot by Shift-click-dragging or middle-click-dragging on
the place of interest on the plot. Return to full size by double-clicking.
You can change the lower plot from the Local Response to a Normal Plot by
using the drop-down menu at the top of the plot.
Special plot right-click
context menu
Scatter plot right-click
context menu
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Above are the right-click context menus for both plots. On both plots you can
manipulate outliers with all the same commands available in the Outliers
menu. See “Outliers Menu (Local Level)” on page 5-100 for details.
On both plots the Print to Figure command opens a MATLAB figure plot
showing the current plot. On the special plots you can switch the confidence
intervals and legend on and off, and hide or show removed data.
Local Scatter Plots
The upper plots are referred to as scatter plots. They can show various scatter
plots of statistics for assessing goodness-of-fit for the current local model
shown.
The statistics available for plotting are model dependent.
The preceding is an example drop-down menu on the scatter plot for changing
x and y factors. In this case spark is the local input factor and torque is the
response. The local inputs, the response, and the predicted response are always
available in these menus. The observation number is also always available.
The other options are statistics that are model dependent, and can include
residuals, weighted residuals, studentized residuals, and leverage.
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Diagnostic Statistics Pane
The Diagnostic Statistics pane drop-down menu is shown, where you can
select the information to be displayed in the pane. If there is not enough room
there are scroll bars.
• Local Parameters — Shows the values and standard errors of the
parameters in the local model for the current test selected.
• Local Correlations — Shows a table of the correlations between the
parameters.
• Response Features — Shows the values and standard errors of the response
features defined for this local model, from the current test (often some or all
of them are the same as the parameters; others are derived from the
parameters).
• Global Variables — Shows the values and standard errors of the global
variables at the position of the current test.
• Local Diagnostics — S.i. (the standard error for the current test), number of
observations, degrees of freedom on the error, R squared, Cond(J or Sigma):
the condition index for the Jacobian matrix or the covariance matrix.
• Global Covariance — For MLE models, shows a covariance matrix for the
response features at the global level.
For information on the Pooled Statistics, see “Definitions” on page 6-5.
Response Features List Pane
See the “Test Plans List Pane” on page 5-5 for information on the lower pane,
which here contains a list of response features (with New, Delete, and Select
buttons). The contents of this pane change in different views; it always
contains the child nodes of the node selected in the model tree.
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Here is a list of all the response features calculated for the local model. A
two-stage model using the local and global models is formed by using Select.
Click the Select button here to enter the Model Selection window. This step
is required before two-stage models can be calculated.
Test Notes Pane
You can use the Test Notes pane to record information on particular tests.
Each test has its own notes pane. Data points with notes recorded against them
are colored in the global model plots. You choose the color using the Test
Number Color button in the Test Notes pane.
Local Level: Toolbar
This toolbar appears when a local node is selected in the model tree.
New Project
Open Project
Save Project
New child node
RMSE plots
Delete current node
Up One Level
Print
Calculate MLE
Update Fit
View Model
The seven left icons remain constant throughout the levels. They are for project
and node management, and here the print icon is enabled as there are plots in
this view. See “Project Level: Toolbar” on page 5-7 for details on these buttons.
• View Model — Opens a dialog displaying the terms in the current model.
• Update Fit — This button is only enabled when data has been excluded from
the plot using the Remove Outliers command (in the right-click context
menu or the Outliers menu). At this point the local fit in the view is updated
to fit only the remaining data, but this change also affects a point in all the
global models. You can make this update to all the global models by using
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Local Level
this toolbar button, or it happens automatically when another node is
selected.
Note Update fit can affect several models. Removing an outlier from a best
local model changes all the response features for that two-stage model. The
global models all change; therefore the two-stage model must be recalculated.
For this reason the local model node returns to the local (house) icon and the
response node becomes blank again. If the two-stage model has a datum
model defined, and other models within the test plan are using a datum link
model, they are similarly affected.
• Calculate MLE — Calculates the two-stage model using maximum
likelihood estimation. This takes correlations between response features into
account. See “MLE” on page 5-137 for details.
• RMSE Plots — Opens the RMSE Explorer dialog, where you can view plots
of the standard errors of all the response features. There is one value of
standard error per test for each response feature. You can also plot these
standard errors against the global variables to examine the global
distribution of error.
Local Level: Menus
File
• Only the New (child node) and Delete (current node) functions change
according to the node level currently selected. Otherwise the File menu
remains constant.
See “File Menu” on page 5-8.
Window and Help Menus
• The Window and Help menus remain the same throughout the Model
Browser.
See “Window Menu” on page 5-9 and “Help Menu” on page 5-9.
See also
“Model Menu (Local Level)” on page 5-96
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“View Menu (Local Level)” on page 5-99
“Outliers Menu (Local Level)” on page 5-100
“Outlier Selection Criteria” on page 5-100
Model Menu (Local Level)
• Set Up — Opens the Local Model Setup dialog where you can change the
model type. See “Local Model Setup” on page 5-46.
• Fit Local— Opens the Local Model Fit Tool dialog. Without covariance
modeling, you see the following controls. This example shows the results
after clicking Fit once. The optimization process can be stopped early by
clicking Stop or you can wait until it finishes. The Ordinary Least Squares
(OLS) parameters are displayed. You can click Fit to run the process again
as many times as required, or Close to exit the dialog. You can enter a
different change in parameters in the edit box.
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• For covariance models this offers three different algorithms: REML
(Restricted Maximum Likelihood – the default), Pseudo-likelihood, and
Absolute residuals. The following example shows that there is also an
additional button, One Step. Using the Fit button might take several steps
to converge, but if you use the One Step button only one step in the
optimization process is taken. Every time you run a process, the initial and
final Generalized Least Squares parameter values are displayed for each
iteration.
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• Update Fit — Only enabled when outliers have been removed. The Update
Fit option updates the global models to take these changes into account. This
happens automatically when you select a different node. Duplicated in the
toolbar. This can affect many models; see “Local Level: Toolbar” on
page 5-94.
• Calculate MLE — Calculates the two-stage model using maximum
likelihood estimation. This takes interactions between response features
into account. Duplicated in the toolbar. See “MLE” on page 5-137 for details.
• Evaluate — Opens the Model Evaluation window.
• Select — Available whenever the Select button is also enabled in the lower
right pane (when it is titled Local Models, Response Features, or Models).
This item opens the Model Selection window to allow you to choose the best
model. See “Select” on page 5-117.
• Assign Best selects the current model as best. If it is one of several child node
models of a response model, selecting it as best means that this local model
(and associated response features) is used for the two-stage model. Note that
this is only enabled if the local node selected has a two-stage model
calculated; that is, if the local node still has a local icon (a house) you cannot
use Assign Best. See “Model Tree” on page 5-10.
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Local Level
View Menu (Local Level)
• Model Definition — Opens the Model Viewer dialog, showing the terms in
the model.
• Data Plots — Opens the Data tab on the display and the Plot Variables
Setup dialog, where you can choose to view any of the data signals in the
data set for the current test (including signals not being used in modeling).
Choose variables from the list on the left and use the buttons to move them
into the Y Variable(s) list or the X variable edit box. You can use the No X
Data button to plot a variable against record number only.
• RMSE Plots — Opens the RMSE Explorer where you can view plots of the
standard errors of all the response features. There is one value of standard
error per sweep for each response feature. You can also plot these standard
errors against the global variables to examine the global distribution of
error.
• Next Test, Previous Test, and Select Test — Duplicate the buttons for
changing tests above the plots in the top left of the Local Model display
pane.
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Outliers Menu (Local Level)
All the commands except Remove All Data are also available in the right-click
context menus on all plots.
• Select Multiple Outliers — Use this item to draw a selection box around as
many data points as required to select them all as outliers. This is useful for
removing many data points at once.
• Clear Outliers — Returns all data points to the unselected state (that is, no
points outlined in red) as possible outliers.
• Remove Outliers — Removes red-outlined data points from the fit and refits
the current local fit only. Use the Update Fit toolbar button or Model –>
Update Fit to refit all the global models also. This also happens
automatically when another node is selected.
• Restore Removed Data — Refits the local model, including all data points
previously removed as outliers. Use Update Fit once more to refit the global
models, or it happens automatically when a new node is selected.
• Select Criteria — Opens the Outlier Selection Criteria dialog where you
can set the criteria for the automatic selection of outliers. This is disabled for
MLE models.
• Remove All Data — Leaves the current local model with no data, so entirely
removes the current test. This test is removed from all the global models.
Outlier Selection Criteria
You can select outliers as those satisfying a condition on the value of some
statistic (for example, residual>3), or by selecting those points that fall in a
region of the distribution of values of that statistic. For example, assume that
residuals are normally distributed and select those with p-value>0.9. You can
also select outliers using the values of model input factors.
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Local Level
The drop-down menu labeled Select using contains all the available criteria,
shown in the following example.
The options available in this menu change depending on the type of model
currently selected. The options are exactly the same as those found in the
drop-down menus for the x- and y-axis factors of the scatter plots in the Model
Browser (local level and global level views).
In the preceding example, the model selected is the knot response feature, so
knot and Predicted knot appear in the criteria list, plus the global input
factors; and it is a linear non-MLE model, so Cook s Distance and Leverage
are also available.
The range of the selected criteria (for the current data) is indicated above the
Value edit box, to give an indication of suitable values. You can type directly in
the edit box. You can also use the up/down buttons on this box to change the
value (incrementing by about 10% of the range).
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Distribution
You can use the Distribution drop-down menu to remove a proportion of the
tail ends of the normal or t distribution. For example, to select residuals found
in the tails of the distribution making up 10% of the total area:
• Select Normal in the Distribution drop-down menu.
• Select the operator >.
• Enter 10 as the α % value in the edit box.
Residuals found in the tails of the distribution that make up 10% of the total
area are selected. If you had a vast data set, approximately 10% of the residuals
would be selected as outliers.
As shown, residuals found beyond the value of Z α in the distribution are
selected as outliers. α is a measure of significance; that is, the probability of
finding residuals beyond Z α is less than 10%. Absolute value is used (the
modulus) so outliers are selected in both tails of the distribution.
The t distribution is used for limited degrees of freedom.
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If you select None in the Distribution drop-down menu, you can choose
whether or not to use the absolute value. That is, you are selecting outliers
using the actual values rather than a distribution. Using absolute value
allows you to select using magnitude only without taking sign into account (for
example, both plus and minus ranges). You can select No here if you are only
interested in one direction: positive or negative values, above or below the
value entered. For example, selecting only values of speed below 2000 rpm.
The Select using custom m-file check box enables the adjacent edit box. Here
you can choose an m-file that selects outliers. Type the name of the file and
path into the edit box, or use the browse button.
In this M-file you define a MATLAB function of the form:
function outIndices = funcname (Model, Data, Names)
Model is the current MBC model.
Data is the data used in the scatter plots. For example, if there are currently
10 items in the drop-down menus on the scatter plot and 70 data points, the
data make up a 70 x 10 array.
Names is a cell array containing the strings from the drop-down menus on the
scatter plot. These label the columns in the data (for example, spark, residuals,
leverage, and so on).
The output, outIndices, must be an array of logical indices, the same size as
one column in the input Data, so that it contains one index for each data point.
Those points where index = 1 in outIndices are highlighted as outliers; the
remainder are not highlighted.
Data Tab
• When you click the Data tab at the local level or select View –> Data Plots,
you can view plots of the data for the current test.
• Use the right-click menu item Set Up Plot Variables to open the Plot
Variables Setup dialog. The dialog appears automatically if you open the
tab using the View menu.
Here you can choose to view any of the data signals in the data set for the
current test (including signals not being used in modeling). Choose variables
from the list on the left and use the buttons to move them into the Y
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Variable(s) list or the X variable edit box. You can use the No X Data button
to plot a variable against record number only.
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Global Level
When you select a response feature node or one-stage model node in the model
tree, this view appears. Both kinds of models have a global icon (
), so this
is referred to as global level. Plots shown here are referred to as global plots.
Child nodes of these models also have global icons. Any node with a global icon
produces this view.
For one-stage models, this view shows the functionality available at all model
nodes. For two-stage models there are other levels with different functionality
for local and response models.
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Click-and-drag anywhere along this line to
change the size of the Models pane
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Click-and-drag or double-click here to
expand plots/hide statistics panes
Global Level
See “Global Level: Toolbar” on page 5-110 and “Global Level: Menus” on
page 5-113 for details on these controls.
The upper scatter plots are replaced by an icon if you resize the Browser too
small.
This view is similar in format to the local level view, which also contains scatter
plots above special plots. The statistical information panes on the right side are
different, there is a Removed Data pane, and there are no test number
controls. There is an edit box for model comments.
For information on the displayed statistics, see “Linear Model Statistics
Displays” on page 6-22 and “Definitions” on page 6-5. Here you can find
explanations of the information found in the Summary table, ANOVA table,
Diagnostic Statistics pane and (for two-stage models) the Pooled Statistics
pane.
Global Special Plots
The lower plots in the global level view are referred to as special plots, as they
can be different for different models (for example, none at all for neural net
models).
The special plot at the global level shows a Predicted/Observed plot. Where
there is only one input factor, the plot shows the model fit and the data against
the input factor (as in most local model special plots, which often have only one
input factor).
For response feature models, each data point is the value taken by this
response feature for some local model fit (of this two-stage model). Note that
response features are not necessarily coefficients of the local curves, but are
always derived from them in some way.
When there is more than one input factor it becomes impossible to display the
fit in the same way, so the data for the response feature is plotted against the
values predicted by the global model. The line of predicted=observed is
shown. With a perfect fit, each point would be exactly on this line. The
distances of the points from the line (the residuals) show how well the model
fits the data.
To examine the fit in more detail, double-click the arrows (indicated in the
figure in “Global Level” on page 5-105) to hide the scatter plot and expand the
lower plot. You can also zoom in on parts of the plot by Shift-click-dragging or
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middle-click-dragging on the place of interest on the plot. Return to full size by
double-clicking.
Note Right-click a point in either the special or scatter plot to open a figure
plot of that particular test (for example, torque against spark).
You can change the lower plot from the Predicted/Observed to a Normal Plot
by using the drop-down menu at the top of the plot.
Global Scatter Plots
The upper plots in the global level view are referred to as scatter plots. They
can show various scatter plots of statistics for assessing goodness-of-fit for the
current model shown.
The statistics available for plotting are model dependent.
Special plot right-click
context menu
Scatter plot right-click
context menu
The preceding are the context menus for both plots. On both plots you can
manipulate outliers with all the same commands available in the Outliers
menu except Outlier Selection Criteria. See “Outliers Menu (Local Level)” on
page 5-100 for details.
You can choose the x- and y-axis factors using the drop-down menus. The
available statistics and factors are model dependent. Following is an example.
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Shown is an example drop-down menu on the scatter plot for changing x and y
factors. In this example knot is the response feature node selected. Therefore
the model output is knot, so knot and Predicted knot are available in the
menu. (For child nodes of knot, the model output is still knot.) The global
inputs, the model output, and the predicted model output are always available
in these menus. The observation number is also always available.
The other options are statistics that are model dependent, and can include:
Residuals, weighted residuals, Studentized Residuals, Leverage, and Cook’s
Distance. These statistics and the other factors are also used as the available
criteria for selection of outliers, so the options in the Outlier Selection
Criteria dialog are similarly model dependent.
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Global Level: Toolbar
This toolbar appears when a response feature node or one-stage model node
(both have a global icon) is selected in the model tree. Note that for one-stage
models all model child nodes of the one-stage test plan are of this type.
Further buttons appear on the right depending on the type of model at the node
selected. See “Global Level: Model-Specific Tools” on page 5-111.
New Project
Open Project
Save Project
New child node
Delete current node
Up One Level
Print
Build Models
Box-Cox Transform
View Model
The seven left icons remain constant throughout the levels. They are for project
and node management, and here the print icon is enabled, as there are plots in
this view. See “Project Level: Toolbar” on page 5-7 for details on these buttons.
• View Model — Opens a dialog displaying the terms in the current model.
• Box-Cox Transform — Opens the Box-Cox Transformation plots, where you
can minimize SSE to try to improve the fit. See “Box-Cox Transformation” on
page 6-18 for statistical details.
• Build Models — Opens the Build Models dialog, where you can choose a
template for the type of models you want to build. There are predefined
templates for polynomials, RBF kernels, and free knot splines. You can also
save templates of whatever models you choose using the Model –> Make
Template menu item. User-defined templates can then be found via the
Build Models dialog. You can use the Browse button to find stored
templates that are not in the default directory.
These three toolbar icons appear for every global model node (although
Box-Cox is not enabled for neural net models). The icons that appear to the
right are model specific.
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Global Level: Model-Specific Tools
All nine left buttons (up to Box-Cox Transform) appear for all response feature
models and one-stage models. See “Global Level: Toolbar” on page 5-110 for
details on these buttons. The right buttons change according to model type.
Linear Model and Multiple Linear Models
Prediction Error Surface
View Model
Box-Cox Transform
Build models
Design Evaluation
Stepwise
• Stepwise — This opens the Stepwise Regression window, where you can
view the effects of removing and restoring model terms on the PRESS
statistic (Predicted Error Sum of Squares), which is a measure of the
predictive quality of a model. You can also use Min PRESS to remove all at
once model terms that do not improve the predictive qualities of the model.
See “Stepwise Regression Techniques” on page 6-13 for further discussion of
the statistical effects of the Stepwise feature.
• Design Evaluation – Opens the Design Evaluation tool, where you can view
properties of the design. See “Design Evaluation Tool” on page 6-27.
• Prediction Error Surface – Opens the Prediction Error Variance Viewer.
See “Prediction Error Variance Viewer” on page 5-193.
Free-Knot Spline Models
Free knot spline models do not have any model-specific tools, just the standard
View Model, Box-Cox Transform, and Build Models.
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Radial Basis Function Models
Prediction Error
Surface
View Model
Box-Cox Transform
Design Evaluation
Build models
Stepwise
Update Fit
View centers
Prune
• Update Fit refits the RBF widths and centers. See “Fitting Routines” on
page 7-12 in the Radial Basis Functions chapter.
• View Centers opens a dialog where you can view the position of the radial
basis function’s centers graphically and in table form.
• Prune opens the Number of Centers Selector where you can minimize
various error statistics by decreasing the number of centers. See “Prune
Functionality” on page 7-21.
• Stepwise opens the Stepwise Regression window.
• Design Evaluation – Opens the Design Evaluation tool, where you can view
properties of the design. See “Design Evaluation Tool” on page 6-27.
• Prediction Error Surface – Opens the Prediction Error Variance Viewer.
Hybrid RBFs have the same toolbar buttons as linear models.
MLE Models
This toolbar appears when you select any response feature that is an MLE
model (purple icon). See “Global Level” on page 5-105 for other functionality in
this view.
View Model
Box-Cox Transform
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Recalculate MLE
Build Models
Global Level
At this point the New child node and Box-Cox buttons are disabled.
• Recalculate MLE returns to the MLE dialog, where you can perform more
iterations to try to refine the MLE model fit. See “MLE” on page 5-137 for
more details.
Neural Networks
Neural net models have the View Model, Build Models, and Update Fit tools.
Global Level: Menus
File
Only the New (child node) and Delete (current node) functions change
according to the node level currently selected. Otherwise the File menu
remains constant.
See “File Menu” on page 5-8.
Window and Help Menus
The Window and Help menus have the same form throughout the Model
Browser.
See “Window Menu” on page 5-9 and “Help Menu” on page 5-9.
Model Menu (Global Level)
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• Set Up opens the Global Model Setup dialog, where you can change the
model type. See “Global Model Setup” on page 5-32.
• Reset returns to the global model defaults, that is, the global model specified
at the test plan stage, restoring any removed outliers and removing any
transforms.
• Evaluate opens the Model Evaluation window.
• Box-Cox Transform opens the Box-Cox Transformation plots, where you
can minimize SSE to try to improve the fit. See “Box-Cox Transformation” on
page 6-18 for statistical details.
• Utilities –> opens a submenu showing the model-specific options available,
duplicating the model-specific toolbar buttons (for example, Stepwise,
Design Evaluation, View Centers, Prediction Error Surface, and so on).
• Make Template is available when child nodes exist. This opens a file
browser where you can choose to save all the current child node models as a
template, which you can then access using the Build Models menu item or
toolbar button.
• Build Models opens the Build Models dialog. Here you can select a template
and build a selection of models as child nodes of the current node.
Choose a template for the type of models you want to build. There are
predefined templates for polynomials, RBF kernels, and free knot splines.
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You can also save templates of whatever models you choose using the Model
–> Make Template menu item.
User-defined templates can then be found via the Build Models dialog. You
can use the Browse button to find stored templates that are not in the
default directory.
When you select a template, a dialog opens where you can specify the model
settings:
- Polynomials – The Model Building dialog opens, where you can choose the
initial and final order of the polynomials you want to build, and whether
to use Stepwise settings. For example, if you choose 1 and 5 as the
minimum and maximum polynomial order, 5 child node models are built
(linear, quadratic, cubic, and so on).
- RBF Kernels – the Radial Basis Function Options dialog opens, where
you can choose all the RBF settings. See “Types of Radial Basis Functions”
on page 7-4. When you click OK, a family of RBF child node models are
built using one of each kind of RBF kernel.
- Free Knot Splines – The Model Building dialog opens, where you can
choose the initial and final number of knots. For example if you specify the
initial and final numbers of knots as 1 and 5, five child nodes are built, one
with one knot, one with two, and so on.
• Select is available whenever the Select button is also enabled in the lower
right pane (when it is titled Local Models, Response Features, or Models).
This item opens the Model Selection window to allow you to choose the best
model. See “Select” on page 5-117.
• Assign Best selects the current model as best. If it is one of several child node
models of a global model, selecting it as best means that it is duplicated at
the parent global model. See “Model Tree” on page 5-10.
View Menu (Global Level)
At this level there are only two items:
• Model Definition opens the Model Viewer dialog displaying the model
terms.
• Test Numbers turns test numbers on and off on both the special and scatter
plots. Also available in the right-click plot menus.
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Outliers Menu (Global Level)
This is the same as the local level Outliers menu, except that there is no
Remove All Data command. All items are duplicated in the right-click context
menu on the plots, except Selection Criteria. See “Outliers Menu (Local
Level)” on page 5-100.
At global level, the Restore Removed Data item opens the Restore Removed
Data dialog, where you can choose the points to restore from the list, or restore
all available points. Select points in the left list and use the buttons to move
points between the lists. Note that entire tests removed at the local level (using
the Remove All Data item) cannot be restored at global level.
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Selecting Models
The Model Selection window appears when you click the Select… button. This
window is intended to help you select a Best Model by comparing several
candidate models.
The Model Selection window allows visual comparison of several models in
many ways, depending on the type of model:
• “Tests View” on page 5-120
• “Predicted/Observed View” on page 5-122
• “Response Surface View” on page 5-124
• “Likelihood View” on page 5-127
• “RMSE View” on page 5-129
• “Residuals View” on page 5-132
• “Cross Section View” on page 5-133
Select
The Select button is under the list view in the pane at the bottom of the
display. This pane is the Test Plans list pane at startup and changes title
depending on the level in the model tree that is selected. The list box in this
pane always contains the child nodes of whichever node in the tree is selected.
The pane also always contains three buttons: New, Delete, and Select.
Select is only available when the lower pane lists local models, response
models, or models.
You can select among the following:
• Local models
• Response features
• Submodels of response features
But you cannot select between response models or test plans.
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Note To get a two-stage model at the response node, you must use the Select
button at the local node (when the lower pane is the Response Features
pane) to assign a model (even if it is the only one) as best. This step then
calculates the two-stage model.
Select might not be available if you are not ready to choose among the child
nodes. For example, at the response node, the child nodes must have models
assigned as best (using the Select feature at those levels) before you can select
among them. Also, if a response feature has child nodes of alternate models,
you must select the best, or the Browser cannot tell which to use to calculate
that response feature. After calculating MLE, Select compares the MLE model
with the previous univariate model, and you can choose the best.
The Model Selection window allows visual comparison of several models.
From the response level you can compare several two-stage models. From the
local level, if you have added new response features you can compare the
different two-stage models (constructed using different combinations of
response feature models). If you have added child nodes to response feature
models, you can compare them all using the Model Selection window.
When a model is selected as best it is copied up a level in the tree together with
the outliers for that model fit.
A tree node is automatically selected as best if it is the only child, except
two-stage models which are never automatically selected - you must use the
Model Selection window.
If a best model node is changed the parent node loses best model status (but the
automatic selection process will reselect that best model if it is the only child
node).
Model Selection Window
The Model Selection window comprises several different views depending on
the type of models being compared:
• “Tests View” on page 5-120
• “Predicted/Observed View” on page 5-122
• “Response Surface View” on page 5-124
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• “Likelihood View” on page 5-127
• “RMSE View” on page 5-129
• “Residuals View” on page 5-132
• “Cross Section View” on page 5-133
The Assign Best button at the bottom of the window marks the currently
selected model as best.
To print the current view, use the Figure/Print menu item or its hot key
equivalent Ctrl+P.
To close the Model Selection window, use the Figure/Close menu item or its
hot key equivalent Ctrl+W. This window is intended to help you select a best
model by comparing several candidate models. On closing the figure, you are
asked to confirm the model you chose as best.
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Tests View
For a two-stage model the initial view is as follows.
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The tests view shows the data being modeled (blue dots) and two models that
have been fitted to this data. The black line shows the local model that has been
fitted to each test separately. The green line shows the two-stage model: you
can see the local model curve reconstructed using response feature values
taken from the global models.
If the local input has more than one factor, a predicted/observed plot appears.
This view allows you to compare several models simultaneously. Using
standard Windows multiselect behavior (Shift+click and Ctrl+click) in the list
view, or by clicking the Select All button, you can view several two-stage
models together. A maximum of five models can be selected at once. The legend
allows you to identify the different plot lines.
Clicking one of the plots (and holding the mouse button down) displays
information about the data for that test. For example:
Here you see the values of the global variables for this test and some diagnostic
statistics describing the model fit. Also displayed are the values (for this test)
of the response features used to build this two-stage model and the two-stage
model’s estimation of these response features.
The controls allow navigation between tests.
You can change the size of the confidence intervals; these are displayed using
a right-click menu on the plots themselves.
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The prediction type allows a choice of Normal or PRESS (Predicted Error Sum of
Squares) — although not if you entered this view through model evaluation
(rather than model selection). PRESS predictions give an indication of the
model fit if that test was not used in fitting the model. For more on PRESS see
“Linear Model Statistics Displays” on page 6-22 and “Stepwise Regression
Techniques” on page 6-13.
Predicted/Observed View
For a one-stage model, or when you are comparing different models for one
Response Feature, the initial view is as follows:
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The plot shows the data used to fit this model (values of knot), against the
values found by evaluating the model (here Predicted knot) at these data
points. The straight black line is the plot of y=x. If the model fitted the data
exactly, all the blue points would lie on this line. The error bars show the 95%
confidence interval of the model fit.
For single inputs, the response is plotted directly against the input.
The Predicted/Observed view only allows single selection of models for display.
Response Surface View
You can change the view using the View menu or by clicking the buttons of the
toolbar. The next view along the toolbar shows the model surface in a variety
of ways.
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The default view is a 3-D plot of the model surface. This model has four
dependent factors; you can see these in the controls at the top left and in the
menus below the plot.
You use the drop-down menus to select the input factors for the plot. The
unselected input factors are held constant and you can change their values
using the controls at the top left of the view (either by clicking the arrow
buttons or by typing directly in the edit box).
Display using (SPK - datum) — If a datum model is being displayed, this
check box appears. The datum variable here is spark angle, SPK. When you
select this box, the model is displayed in terms of spark angle relative to the
datum. The appropriate local variable name appears here. See “Datum Models”
on page 5-88.
Display Type — Changes the model plot. Display options are available for
some of these views and are described under the relevant view. The choices are
as follows:
• A table showing the model evaluated at a series of input factor values.
• A 2-D plot against one input factor.
• A 2-D plot with several lines on it (called a multiline plot); this shows
variation against two input factors.
• A contour plot.
The Contours… button opens the Contour Values dialog. Here you can set
the number, position, and coloring of contour lines.
Fill Contour colors each space between contours a different color.
Contour Labels toggles the contour value numbers on and off. Without
labels a color bar is shown to give you a scale.
Auto (the default) automatically generates contours across the model range.
N Contour Lines opens an edit box where you can enter any number of
contour lines you want.
Specify values opens an edit box where you can enter the start and end
values where you want contour lines to appear, separated by a colon. For
example, entering 5:15 gives you 10 contour lines from 5 to 15. You can also
enter the interval between the start and end values; for example 1:100:600
gives you contour lines between 1 and 600 at intervals of 100.
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• A surface (shown above).
Prediction error shading — Colors the surface in terms of the prediction
error (sqrt(Prediction Error Variance)) of the model at each point. A color bar
appears, to show the value associated with each color.
Note For datum models, Prediction Error shading is only available when the
Display using (local variable - datum) check box is not selected.
P. E. Threshold — To see good color contrast in the range of PE of interest,
you can set the upper limit of the coloring range. All values above this
threshold are colored as maximum P.E.
• A movie: this is a sequence of surfaces as a third input factor’s value changes.
- Replay replays the movie.
- Frame/sec selects the speed of movie replay.
- The number of frames in the movie is defined by the number of points in
the input factor control (in the array at the top left) that corresponds to the
time factor below the plot.
Export allows the currently displayed model surface to be saved to a MAT file
or to the MATLAB workspace.
Likelihood View
The likelihood view shows two plots relating to the log likelihood function
evaluated at each test. It is useful for identifying problem tests for maximum
likelihood estimation (MLE).
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Each plot has a right-click menu that allows test numbers to be displayed on
the plots and also offers autoscaling of the plots. You can also Print to Figure.
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The likelihood view allows several models to be displayed simultaneously; click
the Select All button at the bottom of the window or, in the model list view,
Shift+click or Ctrl+click to select the models for display.
The upper plot shows values of the negative log likelihood function for each
test. This shows the contribution of each test to the overall negative log
likelihood function for the model, as compared with the average, as indicated
by the horizontal green line.
The lower plot shows values of the T-squared statistic for each test. This is a
weighted sum squared error of the response feature models for each test. As
above, the purpose of this plot is to show how each test contributes to the
overall T-squared statistic for this model. The horizontal line indicates the
average.
RMSE View
The Root Mean Square Errors view has three different plots, each showing
standard errors in the model fit for each test.
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Each plot has a right-click menu that allows test numbers to be displayed on
the plots, and you can Print to Figure.
The X variable menu allows you to use different variables as the x-axis of these
plots.
The RMSE view allows several models to be displayed simultaneously; click the
Select All button at the bottom of the window or, in the model list view,
Shift+click or Ctrl+click to select the models for display.
Local RMSE shows the root mean squared error in the local model fit for each
test.
Two-Stage RMSE shows the root mean squared error in the two-stage model
fit to the data for each test. You should expect this to be higher than the local
RMSE.
PRESS RMSE is available when all response feature models are linear. This
plot shows the root mean squared error in the PRESS two-stage model fit at
each test.
For information on PRESS RMSE see “Linear Model Statistics Displays” on
page 6-22.
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Residuals View
The residuals view shows the scatter plots of observation number, predicted
and observed response, input factors, and residuals.
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Selecting Models
This view allows several models to be displayed simultaneously (as shown in
this example); click the Select All button at the bottom of the window or, in the
model list view, Shift+click or Ctrl+click to select the models for display.
A right-click menu allows the test number of each point to be displayed when
only one model is being displayed.
The x-axis factor and y-axis factor menus allow you to display various
statistics.
Cross Section View
The cross-section view shows an array of cross sections through the model
surface. You can choose the point of cross section in each factor. Data points
near cross sections are displayed, and you can alter the tolerances to determine
how much data is shown. The only exception is when you evaluate a model
without data; in this case no data points are displayed.
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The number of plots is the same as the number of input factors to the model.
The plot in S shows the value of the model for a range of values of S while the
other input factors (L, N, A) are held constant. Their values are displayed in
the controls at the top left, and are indicated on the plots by the vertical orange
bars.
• You can change the values of the input factors by dragging the orange bars
on the plots, using the buttons on the controls, or by typing directly into the
edit boxes.
• For example, changing the value of N to 1000 (in any of these ways) does
nothing to the graph of N, but all the other factor plots now show cross
sections through the model surface at N = 1000 (and the values of the other
variables shown in the controls).
On the plots, the dotted lines indicate a confidence interval around the model.
You define the confidence associated with these bounding lines using the Plot
confidence level (%) edit box. You can toggle confidence intervals on and off
using the check box on this control.
For each model displayed, the value of the model and the confidence interval
around this are recorded in the legend at the lower left. The text colors match
the plot colors. You can select multiple models to display in the list at the
bottom using Ctrl+click, or click Select All. The values of the input factors (for
which the model is evaluated) can be found in the controls (in the Input
Factors pane) and seen as the orange lines on the plots.
Data points are displayed when they fall within the tolerance limit near each
cross section. You can set the tolerance in the Tol edit boxes.
• For example, if N is set to 1000, and the tolerance for N is set to 500, all data
points with values between N = 500 and N = 1500 appear on the plots of the
other factors.
• This means that changing the tolerance in one factor affects the data points
that appear on the plots of all the other factors. It does not affect the plots of
that factor.
• You can click data points in the plots to see their values. Several points can
mask each other; in this case the values of all coincident data points are
displayed.
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The following example illustrates how the tolerance level determines which
data points are displayed. The tolerance for TP_REL (500) includes all points in
the data set (this is an extreme example). The plot for N therefore shows the
data points for all the tests. Note that you can see the structure of the data as
each test shows as a vertical line of points.
You can see that the orange line on the N plot passes through a test. This orange
line shows the value of N for the cross-section plot of TP_REL. You can also read
the value in the edit box (N=1753.3). The tolerance for N (200) only includes
data points of this test. Data in adjacent tests fall outside this tolerance.
Therefore the TP_REL plot shows the data points from one test only.
Increasing the tolerance on N will mean that more data points fall within the
tolerance and so would appear on the TP_REL plot.
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MLE
MLE
For an ordinary (univariate) two-stage model, the global models are created in
isolation without accounting for any correlations between the response
features.
• Using MLE (maximum likelihood estimation) to fit the two-stage model
takes account of possible correlations between response features.
• In cases where such correlations occur, using MLE significantly improves
the two-stage model.
Calculating MLE
When you close the Model Selection window, a dialog asks if you want to
calculate MLE. If you click Cancel at this point, you can calculate MLE later
as follows:
1 From the local node, click the MLE icon in the toolbar
.
Alternatively, choose Model –> Calculate MLE.
2 The MLE dialog appears. Click Start.
You can alter various MLE settings on this dialog.
3 After you click Start a series of progress messages appears, and when
finished a new Two-Stage RMSE (root mean square error) value is reported.
4 You can perform more iterations by clicking Start again to see how the
RMSE value changes, or you can click Stop at any time.
5 Clicking OK returns you to the Model Browser, where you can view the new
MLE model fit.
Note After calculating MLE, you will notice that the plots and the icons in
the model tree for the whole two-stage model (response node, local node, and
all response feature nodes) have turned purple. See “Icons: Blue Backgrounds
and Purple Worlds” on page 5-13.
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You can select all response features in turn to inspect their properties
graphically; the plots are all purple to symbolize MLE. At the local node the
plots show the purple MLE curves against the black local fit and the blue data.
• From the response feature nodes, at any time, you can use the MLE toolbar
icon to Recalculate MLE and perform more iterations.
• From the local node, you can click Select to enter the Model Selection
window, compare the MLE model with the previous univariate model
(without correlations), and choose the best. Here you can select the
univariate model and click Assign Best to “undo” MLE and return to the
previous model.
Note If there are exactly enough response features for the model, you can go
straight to MLE calculation after model setup without going through the
Select process. The MLE toolbar button and the Model –> Calculate MLE
item are both active in this case. If you add new response features, you cannot
create MLE until you go through model selection to choose the response
features to use.
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MLE
MLE Settings
Algorithm
The algorithm drop-down menu offers a choice between two covariance
estimation algorithms, Quasi-Newton (the default) and Expectation
Maximization. These are algorithms for estimating the covariance matrix for
the global models.
Quasi-Newton is recommended for smaller problems (< 5 response features
and < 100 tests). Quasi-Newton usually produces better answers (smaller
values of -logL) and hence is the default for small problems.
Expectation Maximization is an iterative method for calculating the global
covariance (as described in Davidian and Giltinan (1995); see References in
“Two-Stage Models for Engines” on page 6-37). This algorithm has slow
convergence, so you might want to use the Stop button.
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Tolerance
You can edit the tolerance value. Tolerance is used to specify a stopping
condition for the algorithm. The default values are usually appropriate, and if
calculation is taking too long you can always click Stop.
Initialize with previous estimate
When you recalculate MLE (that is, perform more iterations), there is a check
box you can use to initialize with the previous estimate.
Predict missing values
The other check box (selected by default) predicts missing values. When it is
selected, response features that are outliers for the univariate global model are
replaced by the predicted value. This allows tests to be used for MLE even if
one of the response features is missing. If all the response features for a
particular test are missing or the check box is unselected, the whole test is
removed from MLE calculation.
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Response Level
Response Level
Note The response node remains empty until you have used the Model
Selection window at the local level. Exiting this window then copies the best
two-stage model to the response node.
Selecting a response model node (with a two-stage icon
globe) in the model tree produces this view.
— a house and a
Note that local model nodes also have the same icon after calculation of the
two-stage model (see the model tree for clarification) but selecting them
produces the local level view instead.
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These plots show the data and the two-stage model fit. You can scroll through
the tests using the test controls, as at the local level: by clicking the up/down
page number buttons, typing directly in the edit box, or clicking Select Test to
go directly to a particular test.
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Response Level: Toolbar and Menus
View Model
• View Model is the only toolbar icon after the standard project and node
management and print buttons (see “Project Level: Toolbar” on page 5-7 for
details on these buttons). View Model opens the Model Viewer dialog
displaying the model terms.
• File, Window, and Help menus remain constant throughout the Model
Browser. See “Project Level: Menus” on page 5-8.
• Model –> Evaluate opens the Model Evaluation window, for examining the
fit without data, against current data, or against additional data.
• Model –> Select is the same as the Select button in the Local Models pane
and opens the Model Selection window. Here you can examine the fit of the
two-stage model against the local fit and the data.
• View –> View Model Definition duplicates the toolbar button and opens the
Model Viewer dialog.
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Model Evaluation Window
You can access the Model Evaluation window via the menu item
Model -> Evaluate… (or the hot key Ctrl+E) from any of the modeling nodes:
the one-stage or two-stage model node, local model node, response feature
nodes, or any child model nodes.
Recall that the Model Selection window allows you to compare different
models with each other and with the data used to create these models. The
Model Evaluation window also allows you either to examine a model without
data or to validate it against data other than that used in creating the model.
For any model node, model evaluation is a quick way to examine the model in
more ways than those available in the main Browser views. For example, local
models with more than one input factor can only be viewed in the Predicted/
Observed view in the main Browser, and the Model Selection window only
shows you the two-stage model, so you go to Model Evaluation to view the
local model itself in more detail. For other models such as childless response
feature nodes, or their child nodes, Select is not available, so Model
Evaluation is the only way to view these models in detail.
The Model Evaluation window comprises some of the same views you see in
the Model Selection window. The views available depend on what kind of
model you are evaluating (two-stage, local, global, or one-stage) and the
evaluation mode you choose.
The following example illustrates the evaluation process using other data to
validate a model. From a model node, you reach the Model Evaluation window
via the Model –> Evaluate… menu item. The Data Selection dialog appears.
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Model Evaluation Window
There are three modes for evaluation, each represented by a radio button
choice:
• View model without data — The evaluation window appears with only the
model surface view and the cross-section view. These do not show the model
fit against data. Here you can investigate the shape of the model surface.
• Evaluate using fit data — The evaluation window appears, and the data
shown along with the model surface is the data that was used to create this
model. The views available are residuals, response surface, and cross section.
If you are evaluating a two-stage model, you can also have the tests view,
where the local model fit is shown for each test. If you are evaluating a
one-stage model or a two-stage response feature model, the predicted/
observed view is also available.
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• Evaluate using other data — The data that is shown along with the model
surface is chosen in this dialog. The evaluation window then appears with
the residuals, response surface, and cross-section view. If you are evaluating
a two-stage model, the tests view is also available, although the local fit is
not shown, as the local model was fitted to different data.
In the tests view (two-stage models only) you can click and hold on a plot to see
the values of the global variables for that test.
Model Evaluation Using Other Data
Prediction performance creates a resource dilemma for the experimenter. You
could keep some data back to use later on to get a measure of the predictive
capacity of the model, but why withhold data from the model building process
if its addition is likely to improve the predictive capability of the model?
There is always a tradeoff in statistics when you decide how much of your data
to use for model fitting and whether to leave some data to test your model
against.
The Model Evaluation window is intended to help you validate your model
against other data, although you can also evaluate the fit against the original
data or without any data. The rest of the Model Evaluation dialog is only
enabled when you choose Evaluate using other data, the last of three choices
in the Data Selection dialog.
When you select the Evaluate using other data choice, if the input factors
required to evaluate the model do not appear in the selected data set, another
dialog appears to match signal names in the selected data set to those required
to evaluate the model.
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Model Evaluation Window
Match signal names from the data set (on the right) to the symbols required to
evaluate the model (on the left) and click OK to accept this assignment.
In the example shown in “Model Evaluation Window” on page 5-144, the Data
Selection dialog shows three data sets that you can use to evaluate this model:
Data Object, Data Object_1, and New Data. Traceability information about
the data set currently selected in the list box is always displayed in the Data
info pane on the right.
In the lower Tests frame you can choose the tests from this data set to use.
Certain test numbers might have been used to create the model (as in this case)
and you can eliminate these test numbers from the evaluation data set if you
want, by selecting the Filter out used log numbers check box.
Of the remaining tests, you can select some and reject others using the
Selected Tests/Unselected Tests list boxes. You move tests between them
using the arrow buttons. For the currently selected test, the mean test values
of all variables in this data set are displayed on the right.
Click OK to use the selected tests to evaluate this model. The Model
Evaluation window appears.
For information about the available views, see “Selecting Models” on
page 5-117.
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Exporting Models
You can export all models created in MBC using the menu item File ->Export
Models.
The Export Model dialog appears.
You can export a model to the MATLAB workspace, to an EXM file for CAGE,
or to a Simulink model (an MDL file). You choose the export format using the
Export to drop-down menu. If a file format is chosen (export to file or to
Simulink), the Destination file controls are enabled, and the “…” button
allows you to locate a destination file using a file browser.
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Exporting Models
The EXM file format is specifically designed for loading into the CAGE part of
MBC Toolbox, for example, to create calibrations.
Export As — Defines the name that the model has when loaded. For example,
if the model is exported to the MATLAB workspace, a workspace variable
appears called PS22 (in the preceding example figure).
Export global models — When a two-stage model is being exported (from the
response node) the constituent response feature models can also be exported.
Multiple models are exported to the workspace as a cell array.
Export all local models — When exporting at the local node, the single local
model for the current test is exported. Selecting this control exports the local
models for all tests (to the workspace as a cell array).
Export datum models — When exporting a two-stage model that has a datum
defined, this control allows you to export the datum global model (without
exporting all other response feature models).
Constraints exported — Where design constraints exist, they are always
exported.
Export PEV blocks — When exporting to Simulink, you can create a PEV
block as part of the Simulink diagram so that the prediction error variance can
be evaluated along with the model. This is not available for models where PEV
cannot be calculated.
Export Preview — Displays the models that are exported with the current
choice of options. For example:
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See “What Is Exported?” on page 5-151 for details on what to expect here.
Export information — Traceability information is exported with the models.
You can add/edit/delete further comments using the buttons on the right.
Tradeoff… — This button appears when you are exporting from the test plan
node. This button exports all local models in a .mat file that can then be loaded
into the Multimodel Tradeoff tool within CAGE.
Note The Tradeoff button is only enabled when the current test plan has at
least one two-stage model available for export, and when this model has
exactly two global input factors.
Clicking Tradeoff… creates a file browser. When you click Save, all local
models are saved to the specified file, regardless of the main dialog's OK or
Cancel buttons.
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Exporting Models
Click OK to export the current selection of models and close the dialog.
What Is Exported?
Note At any point you can use the Export Preview button to check the
models that have been selected for export. This displays the Models Export
List dialog.
At the test plan node:
• You cannot export from the test plan node unless all response models within
that test plan have a best model chosen (that is, you used the Select process
at response level to assign the best model and calculate the two-stage model).
All models within the test plan are exported.
• If the datum model check box is selected, the datum model is exported.
• If the check box for global models is selected, all the response features are
also exported.
At the response node:
• The response model is exported. You cannot export from a response node
until it contains a best model (it is empty before that).
• If the check box for global models is selected, all the response features are
also exported. Note that the datum model is not necessarily a response
feature.
• There is also a datum model check box. As at the test plan node, this exports
the datum model along with the two-stage model (without exporting all other
response feature models).
At the local node:
• The local model for the current test only is exported.
• If the check box for all local models is selected, all the local models are
exported for all the tests.
• If the node is purely a local model (with a house icon, that is; no two-stage
model has yet been calculated), the model is exported under its own name
(for example, PS2,3); if a two-stage model has been calculated (that is, the
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local node has a two-stage house-and-globe icon), the local model is exported
under the name of the response node.
At response feature nodes and all child nodes from here:
• The current response feature (or other selected subnode) only is exported.
Note When you are exporting to Simulink, linear models support PEV, so the
Export PEV blocks check box is active. This is only the case when you are
exporting from response features that are linear functions of the local
parameters. See Export PEV blocks under “Exporting Models” on page 5-148.
Evaluating Models in the Workspace
If a model is exported to the workspace as MyMod and has four input factors, it
can be evaluated at a point as follows:
Y = MyMod([3.7, 89.55, -0.005, 1]);
If column vectors p1,p2,p3,p4 (of equal length) are created for each input
factor, the model can be evaluated to give a column vector output
Y = MyMod([p1,p2,p3,p4]);
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The Design Editor
The Design Editor provides prebuilt standard designs to allow a user with a
minimal knowledge of the subject to quickly create experiments. You can apply
engineering knowledge to define variable ranges and apply constraints to
exclude impractical points. You can increase modeling sophistication by
altering optimality criteria, forcing or removing specific design points, and
optimally augmenting existing designs with additional points.
There is a step-by-step guide to using the Design Editor in the “Design of
Experiment Tutorial” on page 3-1.
The functionality in the Design Editor is covered in the following sections:
“Design Styles” on page 5-154
“Design Editor Displays” on page 5-155
“The Design Tree” on page 5-156
“Display Options” on page 5-156
“Adding Design Points” on page 5-178
“Fixing, Deleting, and Sorting Design Points” on page 5-181
“Saving and Importing Designs” on page 5-183
“Creating a Classical Design” on page 5-159
“Creating a Space Filling Design” on page 5-162
“Creating an Optimal Design” on page 5-168
“Applying Constraints” on page 5-184
You can design experiments at both stages, local and global. You can invoke the
Design Editor in several ways from the test plan level:
1 First you must select the stage (first/local or second/global) for which you
want to design an experiment. Click the appropriate Model block in the test
plan diagram.
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2 Right-click the model block and select Design Experiment.
Alternatively, click the Design Experiment toolbar icon
.
You can also select TestPlan –> Design Experiment.
For an existing design, View –> Design Data also launches the Design Editor
(also in the right-click menu on each Model block). This shows the selected data
as a design.
Design Styles
The Design Editor provides the interface for building experimental designs.
You can make three different styles of design: Classical, Space-Filling, and
Optimal.
Optimal designs are best for cases with high system knowledge, where
previous studies have given confidence in the best type of model to be fitted,
and the constraints of the system are well understood. See “Creating an
Optimal Design” on page 5-168.
Space-filling designs are better when there is low system knowledge. In cases
where you are not sure what type of model is appropriate, and the constraints
are uncertain, space-filling designs collect data in such as a way as to maximize
coverage of the factors’ ranges as quickly as possible. See “Creating a Space
Filling Design” on page 5-162.
Classical designs (including full factorial) are very well researched and are
suitable for simple regions (hypercube or sphere). See “Creating a Classical
Design” on page 5-159.
Any design can be augmented by optimally adding points. Working in this way
allows new experiments to enhance the original, rather than simply being a
second attempt to gain the necessary knowledge. See “Adding Design Points”
on page 5-178.
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Design Editor Displays
The following example shows the display after creating an optimal design.
When you first see the main display area, it shows the default Design Table
view of the design (see example above). There is a context menu, available by
right-clicking, in which you can change the view of the design to 1-D, 2-D, 3-D,
and 4-D Projections and Table view (also under View menu). This menu also
allows you to split the display either horizontally or vertically so that you
simultaneously have two different views on the current design. The split can
also be merged again. After splitting, each view has the same functionality;
that is, you can continue to split views until you have as many as you want.
When you click a view, its title bar becomes blue to show it is the current active
view.
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The Design Tree
The currently available designs are displayed on the left in a tree structure.
The tree displays three pieces of information:
• The name of the design, which you can edit by clicking it.
• The state of the design:
- The icon changes from
if it is empty, to the appropriate icon for the
design type when it has design points (for example,
optimized, as in
the toolbar buttons for Optimal, Classical, and Space-Filling designs).
- The icon changes to
when design points have been added using a
different method (for example, augmenting a classical design with
optimally chosen points). It becomes a custom design style. You can mix
and match all design options in this way.
- A padlock appears (
) if the design is locked. This happens when it has
child nodes (to maintain the relationship between designs, so you can
retreat back up the design tree to reverse changes).
• The design that is selected as best. This is the design that is used for
matching against experimental data. The icon for the selected design is the
normal icon turned blue. When you have created more than one design, you
must select as best the design to be used in modeling, using the Edit menu.
Blue icons are also locked designs, and do not acquire padlocks when they
have child nodes.
• You can reach a context menu by right-clicking in the design tree pane. Here
you can delete or rename designs and add new designs. Choose Evaluate
Design to open the Design Evaluation window. Properties opens the Design
Properties dialog, which displays information about the size, constraints,
properties (such as optimality values), and modification dates of the selected
design.
The information pane, bottom left, displays pieces of information for the
current design. The amount of information in this pane can change depending
on what the design is capable of; for example, only certain models can support
the optimal designs and only these can show current optimal values.
Display Options
The Design Editor can display multiple design views at once, so while working
on a design you can keep a table of design points open in one corner of the
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window, a 3-D projection of the constraints below it, and a 2-D or 3-D plot of
the current design points as the main plot. The following example shows
several views in use at once.
The current view and options for the current view are available either through
the context menu or the View menu on the Design Editor window. Also in
these menus is the Print to Figure command. This option copies the current
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view into its own figure, allowing you to use the standard MATLAB plotting
tools to annotate and print the display.
The Viewer Options item in the View menu opens windows for configuring
details of the current display. You can change basic properties such as color on
all the projections (1-D, 2-D, 3-D, and 4-D). For the table view, you can alter the
precision used for displaying data and set up a filter to selectively display
certain ranges of values. You can rotate all 3-D views as usual. You can
double-click the color bar to edit the colormap.
You can also view projections of constraints; see “Applying Constraints” on
page 5-184.
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Creating a Classical Design
1 Add a new design by clicking the
button in the toolbar or select File –>
New.
2 Select the new design node in the tree. An empty Design Table appears if
you have not yet chosen a design. Otherwise if this is a new child node the
display remains the same, because child nodes inherit all the parent design’s
properties. All the points from the previous design remain, to be deleted or
added to as necessary. The new design inherits all of its initial settings from
the currently selected design and becomes a child node of that design.
3 Click the
button in the toolbar or select Design –> Classical –> Design
Browser.
Note In cases where the preferred type of classical design is known, you can
go straight to one of the five options under Design –> Classical. Choosing the
Design Browser option allows you to see graphical previews of these same
five options before making a choice.
• A dialog appears if there are already points from the previous design. You
must choose between replacing and adding to those points or keeping only
fixed points from the design. The default is replacement of the current points
with a new design. Click OK to proceed, or Cancel to change your mind.
The Classical Design Browser appears.
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Classical Design Browser
In the Design Style drop-down menu there are five classical design options:
• Central Composite
Generates a design that has a center point, a point at each of the design
volume corners, and a point at the center of each of the design volume faces.
The options are Face-center cube, Spherical, Rotatable, or Custom. If you
choose Custom, you can then choose a ratio value ( α ) between the corner
points and the face points for each factor and the number of center points to
add. Five levels for each factor are used. You can set the ranges for each
factor. Inscribe star points scales all points within the coded values of 1 and
-1 (instead of plus or minus α outside that range). When this box is not
selected, the points are circumscribed.
• Box-Behnken
Similar to Central Composite designs, but only three levels per factor are
required, and the design is always spherical in shape. All the design points
(except the center point) lie on the same sphere, so you should choose at least
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three to five runs at the center point. There are no face points. These designs
are particularly suited to spherical regions, when prediction at the corners is
not required. You can set the ranges of each factor.
• Full Factorial
Generates an n-dimensional grid of points. You can choose the number of
levels for each factor, the number of additional center points to add, and the
ranges for each factor.
• Plackett Burman
These are "screening" designs. They are two-level designs that are designed
to allow you to work out which factors are contributing any effect to the
model while using the minimum number of runs. For example, for a 30-factor
problem this can be done with 32 runs. They are constructed from Hadamard
matrices and are a class of two-level orthogonal array.
• Regular Simplex
These designs are generated by taking the vertices of a k-dimensional
regular simplex (k = number of factors). For two factors a simplex is a
triangle; for three it is a tetrahedron. Above that are hyperdimensional
simplices. These are economical first-order designs that are a possible
alternative to Plackett Burman or full factorials.
You can always toggle coded values by selecting the check box at the top.
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Creating a Space Filling Design
Space-filling designs should be used when there is little or no information
about the underlying effects of factors on responses. For example, they are
most useful when you are faced with a new type of engine, with little knowledge
of the operating envelope. These designs do not assume a particular model
form. The aim is to spread the points as evenly as possible around the operating
space. These designs literally fill out the n-dimensional space with points that
are in some way regularly spaced. These designs can be especially useful in
conjunction with nonparametric models such as radial basis function (a type of
neural network).
1 Add a new design by clicking the
button in the toolbar or select File –>
New.
2 Select the node in the tree by clicking. An empty Design Table appears if you
have not yet chosen a design. Otherwise, if this is a new child node the
display remains the same, because child nodes inherit all the parent design’s
properties.
3 Select Design –> Space Filling –> Design Browser, or click the
Space
Filling Design button on the toolbar.
4 A dialog appears if there are already points from the previous design. You
must choose between replacing and adding to those points or keeping only
fixed points from the design. The default is replacement of the current points
with a new design. Click OK to proceed, or Cancel to change your mind.
The Space Filling Design Browser appears.
Note As with the Classical Design Browser, you can select the three types
of design you can preview in the Space Filling Design Browser from the
Design –> Space Filling menu in situations when you already know the type
of space-filling design you want.
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Space Filling Design Styles
In the Design drop-down menu you can see the three design styles available:
• Latin Hypercube Sampling
• Lattice
• Stratified Latin Hypercube
Latin Hypercube Sampling
Latin Hypercube Sampling (LHS) are sets of design points that, for an N point
design, project onto N different levels in each factor. Here the points are
generated randomly. You choose a particular Latin Hypercube by trying
several such sets of randomly generated points and choosing the one that best
satisfies user-specified criteria.
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Lattice
Lattice designs project onto N different levels per factor for N points. The
points are not randomly generated but are produced by an algorithm that uses
a prime number per factor. If good prime numbers are chosen, the lattice
spreads points evenly throughout the design volume. A poor choice of prime
numbers results in highly visible lines or planes in the design projections. If all
the design points are clustered into one or two planes, it is likely that you
cannot estimate all the effects in a more complex model. When design points
are projected onto any axes, there are a large number of factor levels.
For a small number of trials (relative to the number of factors) LHS designs are
preferred to Lattice designs. This is because of the way Lattice designs are
generated. Lattice designs use prime numbers to generate each successive
sampling for each factor in a different place. No two factors can have the same
generator, because in such cases the lattice points all fall on the main diagonal
of that particular pairwise projection, creating the visible lines or planes
described above. When the number of points is small relative to the number of
factors, the choice of generators is restricted and this can lead to Lattice
designs with poor projection properties in some pairwise dimensions, in which
the points lie on diagonals or double or triple diagonals. This means that Latin
Hypercube designs are a better choice for these cases.
Stratified Latin Hypercube
Stratified Latin Hypercubes separate the normal hypercube into N different
levels on user-specified factors. This can be useful for situations where the
preferred number of levels for certain factors might be known; more detail
might be required to model the behavior of some factors than others. They can
also be useful when certain factors can only be run at given levels.
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The preceding example shows the different properties of a poor lattice (left) and
a good lattice (right), with a similar number of points.The poorly chosen prime
number produces highly visible planes and does not cover the space well. An
example of an LHS design of the same size is shown for comparison.
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Setting Up a Space Filling Design
You can toggle coded units for all space-filling designs by selecting the check
box at the top.
The default Design drop-down menu item is Latin Hypercube Sampling.
For both Latin Hypercube Sampling and Stratified Latin Hypercube, you
can choose from several selection criteria available in a drop-down menu:
• Maximize minimum distance (between points). This is the default.
• Minimize maximum distance (between points)
• Minimize discrepancy — Minimizes the deviation from the average point
density
• Minimize RMS variation from CDF — Minimizes the Root Mean Square
(RMS) variation of the Cumulative Distribution Function (CDF) from the
ideal CDF
• Minimize maximum variation from CDF — Minimizes the maximum
variation of the CDF from the ideal CDF
The final two (CDF variation) options scale best with the number of points and
it is advisable to choose one of these options for large designs.
The same criteria are available for the Stratified Latin Hypercube.
• You can set the number of points using the controls or by typing in the edit
box.
• You can set the ranges for each factor.
• If you select the Enforce Symmetrical Points check box, you create a design
in which every design point has a mirror design point on the opposite side of
the center of the design volume and an equal distance away. Restricting the
design in this way tends to produce better Latin Hypercubes.
• You can use the tabs under the display to view 2-D, 3-D, and 4-D previews.
• Click OK to calculate the Latin Hypercube and return to the main Design
Editor.
For a Lattice space-filling design:
• You can choose the lattice size by using the buttons or typing in the edit box.
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• You can choose the prime number generator by using the up/down buttons
on the Prime number for X edit box.
• You can choose the range for each factor.
• Click OK to calculate the lattice and return to the Design Editor.
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Creating an Optimal Design
Optimal designs are best for cases with high system knowledge, where
previous studies have given confidence on the best type of model to be fitted,
and the constraints of the system are well understood. Optimal designs require
linear models.
The Design Editor can average optimality across several linear models. This is
a flexible way to design experiments using optimal designs. If you have no idea
what model you are going to fit, you would choose a space-filling design.
However, if you have some idea what to expect, but are not sure which model
to use, you can specify a number of possible models. The Design Editor can
average an optimal design across each model.
For example, if you expect a quadratic and cubic for three factors but are
unsure about a third, you can specify several alternative polynomials. You can
change the weighting of each model as you want (for example, 0.5 each for two
models you think equally likely). This weighting is then taken into account in
the optimization process in the Design Editor. See “Global Model Class:
Multiple Linear Models” on page 5-43.
1 Click the
button in the toolbar or select File –> New. A new node appears
in the design tree. It is named according to the model for which you are
designing, for example, Linear Model Design.
2 Select the node in the tree by clicking. An empty Design Table appears if you
have not yet chosen a design. Otherwise, if this is a new child node the
display remains the same, because child nodes inherit all the parent design’s
properties.
3 Set up any constraints at this point. See “Applying Constraints” on
page 5-184.
4 Choose an Optimal design by clicking the
button in the toolbar, or choose
Design –> Optimal.
The optimal designs in the Design Editor are formed using the following
process:
• An initial starting design is chosen at random from a set of defined candidate
points.
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• m additional points are added to the design, either optimally or at random.
These points are chosen from the candidate set.
• m points are deleted from the design, either optimally or at random.
• If the resulting design is better than the original, it is kept.
This process is repeated until either (a) the maximum number of iterations is
exceeded or (b) a certain number of iterations has occurred without an
appreciable change in the optimality value for the design.
The Optimal Design dialog consists of three tabs that contain the settings for
three main aspects of the design:
• Start Point tab: Starting point and number of points in the design
• Candidate Set tab: Candidate set of points from which the design points are
chosen
• Algorithm tab: Options for the algorithm that is used to generate the points
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Optimal Design: Start Point Tab
The Start Point tab allows you to define the composition of the initial design:
how many points to keep from the current design and how many extra to choose
from the candidate set.
1 Choose the type of the optimal design, using the Optimality Criteria
drop-down menu:
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• D-Optimal designs — Aims to reduce the volume of the confidence ellipsoid
to obtain accurate coefficients. This is set up as a maximization problem, so
the progress graph should go up with time.
The D-optimality value used is calculated using the formula
loge ( det ( X'X ) )
D eff = -------------------------------------k
where X is the regression matrix and k is the number of terms in the
regression matrix.
• V-Optimal designs — Minimizes the average prediction error variance, to
obtain accurate predictions. This is better for calibration modeling problems.
This is a minimization process, so the progress graph should go down with
time.
The V-optimality value is calculated using the formula
1
V eff = ------nC
∑ xj' ( XC'XC )
–1
xj
j
where xj are rows in the regression matrix, XC is the regression matrix for
all candidate set points and nC is the number of candidate set points.
• A-Optimal designs — Minimizes the average variance of the parameters and
reduces the asphericity of the confidence ellipsoid. The progress graph also
goes down with this style of optimal design.
The A-optimality value is calculated using the formula
A eff = trace ( ( X'X )
–1
)
where X is the regression matrix.
2 You might already have points in the design (if the new design node is a
child node, it inherits all the properties of the parent design). If so, choose
from the radio buttons:
- Keep current design points
- Keep current fixed design points
- Do not keep any current points
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3 Choose the number of additional points to add by clicking the Optional
Additional Points up/down buttons or by typing directly into the edit box.
Optimal Design: Candidate Set Tab
The Candidate Set tab allows you to set up a candidate set of points for your
optimal design. Candidate sets are a set of potential test points. This typically
ranges from a few hundred points to several hundred thousand. The set
generation schemes are as follows:
• Grid — Full factorial grids of points, with fully customizable levels.
• Lattice — These have the same definition as the space-filling design lattices,
but are typically used with about 10,000 points. The advantage of a lattice is
that the number of points does not increase as the number of factors
increases; however ,you do have to try different prime number generators to
achieve a good lattice. See “Creating a Space Filling Design” on page 5-162.
• Grid/Lattice — A hybrid set where the main factors are used to generate a
lattice, which is then replicated over a small number of levels for the
remaining factors.
• Stratified Lattice — Another method of using a lattice when some factors
cannot be set to arbitrary values. Stratified lattices ensure that the required
number of levels is present for the chosen factor. Note that you cannot set
more than one factor to stratify to the same N levels.
• User-defined — Import custom matrices of points from MATLAB or
MAT-files.
For each factor you can define the range and number of different levels within
that range to select points.
There is an edit box where you can choose the maximum number of points to
display. This is a limiter, as candidate sets with several factors soon become
very large if each factor has, say, 21 levels, and the computation and display
could be very slow.
1 Choose a type of generation algorithm from the drop-down menu. Note that
you could choose different parameters for different factors (within an overall
scheme such as Grid).
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2 This tab also has buttons for creating plots of the candidate sets. Try them
to preview your design. If you have added custom points, you can check them
here.
3 Notice that you can see 1-D, 2-D, 3-D, and 4-D displays (fourth factor is
color) all at the same time as they appear in separate windows (see the
example following). Move the display windows (click and drag the title bars)
so you can see them while changing the number of levels for the different
factors.
4 You can change the factor ranges and the number of levels using the edit
boxes or buttons.
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Select variables in this list
Choose algorithm type from this list
Open display windows with these buttons
Change the number of levels of the selected variable here
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Optimal Design: Algorithm tab
The Algorithm tab has the following algorithm details:
• Augmentation method - Random/Optimal — Optimal can be very slow
(searches the entire candidate set for points) but converges using fewer
iterations. Random is much faster per iteration, but requires a larger
number of iterations. The Random setting does also have the ability to lower
the optimal criteria further when the Optimal setting has found a local
minimum.
• Deletion method - Random/Optimal — Optimal deletion is much faster
than augmentation, because only the design points are searched.
• p value — The number of points added/removed per iteration. For optimal
augmentation this is best kept smaller (~5); for optimal deletion only it is
best to set it larger.
• Delta — This is the size of change below which changes in the optimality
criteria are considered to be not significant.
• q value — Number of consecutive iterations to allow that do not increase the
optimality of the design. This only has effect if random augmentation or
deletion is chosen.
• Maximum number of iterations to perform — Overall maximum number
of iterations.
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1 Choose the augmentation and deletion methods from the drop-down menus
(or leave at the defaults).
2 You can alter the other parameters by using the buttons or typing directly
in the edit boxes.
3 Click OK to start optimizing the design.
When you click the OK button on the Optimal Design dialog another window
appears that contains a graph. This window shows the progress of the
optimization and has two buttons: Accept and Cancel. Accept stops the
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optimization early and takes the current design from it. Cancel stops the
optimization and reverts to the original design.
4 You can click Accept at any time, but it is most useful to wait until
iterations are not producing noticeable improvements; that is, the graph
becomes very flat.
You can always return to the Optimal Design dialog (following the same steps)
and choose to keep the current points while adding more.
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Adding Design Points
In any design, you can add points using the Edit menu. You can specify how
many points to add and how to do so: optimally, randomly, or at specified
values.
1 Select Edit –> Add Point or click the
button. A dialog appears, as shown.
2 Choose an augmentation method from the drop-down menu: optimal (D,V,
or A), random, or user-specified.
Note You can add points optimally to any design based on a linear or
multilinear model, as long as it has the minimum number of points required
to fit that model. This means that after adding a constraint you might remove
so many points that a subsequent replace operation does not allow optimal
addition.
3 Choose the number of points to add, using the buttons or typing into the edit
box. For user-specified custom points, you also enter the values of each
factor for each point you want to add.
4 If you choose an optimal augmentation method and click Edit, the
Candidate Set dialog appears, as shown in the following example. Here you
can edit the ranges and levels of each factor and which generation algorithm
to use. These are the same controls you see on the Candidate Set tab of the
Optimal Design dialog.
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5 Click OK to add the points and return to the main display.
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Creating an Optimal Design
Fixing, Deleting, and Sorting Design Points
You can fix or delete points using the Edit menu. You can also sort points or
clear all points from the design.
Fixed points become red in the main Design Editor table display. If you have
matched data to a design or used experimental data as design points, those
points are automatically fixed. You already have the data points, so you do not
want them to be changed or deleted. Once fixed, they are not moved by design
optimization processes. This automatic fixing makes it easy for you to
optimally augment the fixed design points.
1 Select Edit –> Fix/Free Points or Edit –> Delete Point (there is also a
toolbar button
)
A dialog appears in which you can choose the points to fix or delete.
The example above shows the dialog for fixing or freeing points. The dialog for
deleting points has the same controls for moving points between the Keep
points list and the Delete points list.
2 In order to see where these points are in the design, you must change the
main Design Editor display pane to the Table view. This gives a numbered
list of every point in the design.
3 Move points from the Free points list to the Fixed points list; or from the
Keep points list to the Delete points list, by using the buttons.
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4 Click OK to complete the changes specified in the list boxes, or click Cancel
to return to the unchanged design.
Design points that are matched to experimental data are set to fixed points so
you can redesign unmatched points easily.
• Selecting Edit –> Clear deletes all points in the current design.
• Edit –> Sort opens a dialog (see example following) for sorting the current
design — by ascending or descending factor values, randomly, or by a custom
expression.
To sort by custom expression you can use MATLAB expressions (such as
abs(N) for the absolute value of N) using the input symbols. Note that sorts are
made using coded units (from -1 to 1) so remember that points in the centre of
your design space will be near zero in coded units, and those near the edge will
tend to 1 or -1.
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Saving and Importing Designs
You can save any design by choosing File –> Export Design. The selected
design only is exported.
There are three options:
• To File generates a Design Editor file (.mvd).
• To CSV File exports the matrix of design points to a CSV (comma separated
values) file. and/or include factor symbols by selecting the check boxes.
• To Workspace exports the design matrix to the workspace. You can convert
design points to a range of (1, -1) by selecting the check box.
You can choose the destination file by typing in the edit box or using the browse
button.
Import designs by selecting File –> Import Design. The controls on the dialog
are very similar to the Export Design dialog: you can import from Design
Editor files, CSV files, or the workspace, and you can convert design points
from a (1,-1) range.
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Applying Constraints
In many cases designs might not coincide with the operating region of the
system to be tested. For example, an automobile engine normally does not
operate in a region of low speed (n) and high exhaust gas recirculation (EGR).
You cannot run 15% EGR at 1000 RPM. There is no point selecting design
points in impractical regions, so you can constrain the candidate set for test
point generation.
Designs can have any number of geometric constraints placed upon them. Each
constraint can be one of four types: an ellipsoid, a hyperplane, a 1-D lookup
table, or a 2-D lookup table.
To add a constraint to a design:
1 Select Edit –> Constraints from the Design Editor menus.
2 The Constraints Manager dialog appears.
The example shows the Constraints Manager dialog with two constraints.
Here you can add new constraints, and delete, edit, or duplicate existing
constraints. If there are no constraints yet, the Constraints Manager is empty
and you can only click Add to construct a new constraint.
3 Click Add.
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Applying Constraints
4 The Constraint Editor dialog with available constraints appears. You can
select Linear, Ellipsoid, 1-D Table, or 2-D Table from the Constraint Type
drop-down menu, as shown.
Constraint Types
Linear Constraints
You specify the coefficients of the equation for an (N-1) dimensional hyperplane
in the N-factor space. The form of the equation is A.x = b where A is your
defined coefficient vector, x is the vector of factor settings, and b is a scalar. The
equation is applied by substituting design point settings (in coded values) for
x. For example:
In two dimensions: A=(1, 2), x=(L, A), b=3
Then A.x = b expands to
1*L + 2*A = 3
Rearranging this, you can write it as
A = -L/2 + 3/2
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which corresponds to the traditional equation of a 2-D straight line, y = mx + c,
with m = -1/2 and c = 3/2. A.x = b is thus the higher dimensional extension of
this equation.
The linear constraints work by selecting the region below the defined plane
(that is, A.x <= b). To select a region above the plane, multiply all your values
by -1: A -> -A, b -> -b.
For example, to select a simple plane where N<0.8 as a constraint boundary,
enter 8 under N and set all the other factors to 0.
Ellipsoid Constraints
The ellipsoid constraint allows you to define an N-dimensional ellipsoid. You
can specify the center of the ellipsoid, the length of each axis, and the rotation
of the ellipsoid.
Ellipsoid center. You specify the center of the ellipsoid by entering values in the
column marked Xc. These are the values, in coded units, that mark where you
want the ellipsoid to be centered in each of the factor dimensions.
Axis length. You specify the size of the ellipsoid by entering values along the
diagonal of the matrix to the right of Xc. The default values of 1 create an
ellipsoid that touches the edge of the design space in each of the factor
dimensions. Changing an entry to less than 1 extends the ellipsoid edge outside
the design space along that factor axis (the extreme in this direction, 0, creates
a cylinder). Changing an entry to greater than 1 contracts the ellipsoid edge to
be inside the design space. In general, for an entry value X, the ellipsoid size in
that factor is sqrt(1/X) times the size of the design space in that factor.
Rotation. The matrix entries that are not on the main diagonal control rotation
of the ellipsoid.
The following example shows a defined ellipsoid constraint.
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Applying Constraints
• You must enter values in the table to define the ellipsoid. If you leave the
values at the defaults, the candidate set is a sphere.
If you change a 1 to a 3, you reduce that axis to 1 ⁄ ( 3 ) times its original size.
A value of 2 reduces that axis to 1 ⁄ ( 2 ) , and so on.
The example above reduces the space available for the candidate set by a third
in the A and N axes, forming an ellipsoid, as shown. A 3-D display of this
constraint can be seen below.
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1-D Table Constraints
1-D table constraints limit the maximum or minimum setting of one factor as
a function of another factor. Linear interpolation between user-defined points
is used to specify the constraint.
• You can select the appropriate factors to use.
• Move the large dots (by clicking them and dragging) to define a boundary.
You must click the dots within the boundary of the space to select them. The
following example shows a 1-D Table constraint. You can choose whether to
constrain your design above or below the defined boundary using the
Inequality drop-down menu.
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Applying Constraints
2-D Table Constraints
2-D table constraints are an extension of the 1-D table. Constraint boundary
values for a factor are specified over a 2-D grid of two other factors.
• You can specify these grid locations by entering values in the top row and left
column, while the matrix of values for the third factor is entered in the rest
of the edit boxes. To specify grid values, you can enter values directly or just
choose the number of breakpoints for your grid and space them over the
factors’ ranges, using the controls decribed below.
• You can specify the number of breakpoints for the X and Y factors by using
the buttons or typing directly into the edit boxes.
• You can click Span Range to space your breakpoints evenly over the range
of X or Y. This is useful if you add some breakpoints as new points are often
all at the maximum value for that factor. It is much easier to use the Span
Range button than to change points manually.
• You can specify to keep the region below (<=) or above (>=) the constraint
boundary, as for the 1-D table. Do this by choosing above or below from the
Inequality drop-down menu for the Z factor.
• You can switch to coded values using the check box. See the example.
The constraint boundary between the defined grid points is calculated using
bilinear interpolation.
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• After defining any constraint, click OK. Your new constraint appears in the
Constraint Manager list box. Click OK to return to the Design Editor, or
Add to define more constraints.
A dialog appears if there are points in the design that fall outside your newly
constrained candidate set. You can simply continue (delete them) or cancel the
constraint. Fixed points are not removed by this process. For optimal designs
you can also replace them with new random points within the new candidate
set, as shown in the preceding example dialog.
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Applying Constraints
Note You only get the Replace points option for optimal designs. If you want
to replace points removed by constraints from other designs, you can always
use Edit –> Add Point to add points optimally, randomly, or at chosen places.
However, if so many points have been removed by a constraint that there are
not enough left to fit the current model, optimal addition is not possible. See
“Adding Design Points” on page 5-178.
To view constraints:
1 Right-click the Design Editor display pane to reach the context menu.
2 Select Current View –> 3D Constraints. An example is shown.
These views are intended to give some idea of the region of space that is
currently available within the constraint boundaries.
Importing Constraints
Select File –> Import Constraints. The Import Constraints dialog appears,
as shown in the following example.
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Here you can import constraints for the currently selected design, from any
existing constraints in the design tree.
Note You can only import constraints from designs that have the same
number of factors and have the same coded range for each factor.
Select constraints in the list by clicking, or Ctrl-click to select multiple
constraints.
You can choose Design Editor file (.mvd) from the Import from drop-down
menu, and type the file name in the edit box or use the browse button to find
another design file. In this way you can extract constraints from any other
design file.
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Prediction Error Variance Viewer
Prediction Error Variance Viewer
You can use the Prediction Error Variance (PEV) viewer to examine the quality
of the model predictions. You can examine the properties of designs or global
models. When you open it from the Design Editor, you can see how well the
underlying model predicts over the design region. When you open it from a
global model, you can view how well the current global model predicts. A low
PEV (tending to zero) means that good predictions are obtained at that point.
The PEV viewer is only available for linear models and radial basis functions.
When designs are rank deficient the PEV Viewer appears but is empty; that is,
the PEV values cannot be evaluated because there are not enough points to fit
the model.
• From the Design Editor, select Tools –> PEV Viewer.
• From the global level of the Model Browser, if the selected global model is
linear or a radial basis function:
- You can click the
toolbar button to open the PEV viewer.
- Alternatively, you can select Model –> Utilities –> Prediction Error
Surface.
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Turn clipping on and off here
Change number of points plotted here
Change clipping value here
The default view is a 3-D plot of the PEV surface.
The plot shows where the model predictions are best. This example shows an
MBT model response feature. The model predicts well where the PEV values
are lowest.
Display Options
• The Surface menu has many options to change the look of the plots.
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Prediction Error Variance Viewer
• You can change the factors displayed in the 2-D and 3-D plots. The
drop-down menus below the plot select the factors, while the unselected
factors are held constant. You can change the values of the unselected factors
using the buttons or edit boxes in the frame, top left.
• The Movie option shows a sequence of surface plots as a third input factor’s
value is changed. You can change the factors, replay, and change the frame
rate.
• You can change the number, position, and color of the contours on the
contour plot with the Contours button. See the contour plot section (in the
Response Surface view of model selection and evaluation) for a description of
the controls.
• You can select the Clip Plot check box, as shown in the preceding example.
Areas that move above the value in the Clipping envelope edit box are
removed. You can enter the value for the clipping envelope. The edges of the
clip are very jagged; you can make it smoother by increasing the numbers of
points plotted for each factor (enter values in the Pts edit boxes in the top
frame).
When you use the PEV viewer to see design properties, optimality values for
the design appear in the Optimality criteria frame.
Note that you can choose Prediction Error shading in the Response Feature
view (in Model Selection or Model Evaluation). This shades the model surface
according to Prediction Error values (sqrt(PEV)). This is not the same as the
PEV viewer, which shows the shape of a surface defined by the PEV values. See
“Response Surface View” on page 5-124.
Optimality Criteria
No optimality values appear in the Optimality criteria frame until you click
Calculate. Clicking Calculate opens the Optimality Calculations dialog.
Here iterations of the optimization process are displayed.
In the Optimality criteria frame in the PEV viewer are listed the values of the
input factors at the point of maximum PEV (Gmax). This is the point where the
G optimality value is calculated. The D and V values are calculated for the
entire design, not just at the point Gmax.
For statistical information about how PEV is calculated, see “Prediction Error
Variance” on page 6-7 and “Prediction Error Variance for Two-Stage Models”
on page 6-43 in the Technical Documents section.
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6
Technical Documents
Linear Regression
. . . . . . . . . . . . . . . . . 6-3
Definitions . . . . . . . . . . . . . . . . . . . . . 6-5
Prediction Error Variance
. . . . . . . . . . . . . 6-8
High-Level Model Building Process Overview . . . . . 6-10
Stepwise Regression Techniques
Box-Cox Transformation
. . . . . . . . . . 6-14
. . . . . . . . . . . . . . 6-19
Linear Model Statistics Displays . . . . . . . . . . . 6-23
Design Evaluation Tool . . . . . . . . . . . . . . . 6-29
Maximum Likelihood Estimation
. . . . . . . . . . 6-38
Two-Stage Models for Engines . . . . . . . . . . . . 6-39
Local Model Definitions . . . . . . . . . . . . . . . 6-50
Neural Networks . . . . . . . . . . . . . . . . . . 6-60
User-Defined Models . . . . . . . . . . . . . . . . 6-61
Transient Models . . . . . . . . . . . . . . . . . . 6-67
Data Loading Application Programming Interface
. . 6-76
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Technical Documents
The technical documents are divided into the following sections:
• “Linear Regression” on page 6-3
• “Definitions” on page 6-5
• “High-Level Model Building Process Overview” on page 6-9
• “Stepwise Regression Techniques” on page 6-13
• “Box-Cox Transformation” on page 6-18
• “Linear Model Statistics Displays” on page 6-22
• “Design Evaluation Tool” on page 6-27
• “Maximum Likelihood Estimation” on page 6-36
• “Two-Stage Models for Engines” on page 6-37
• “Local Model Definitions” on page 6-48
• “Neural Networks” on page 6-57
• “User-Defined Models” on page 6-58
• “Transient Models” on page 6-64
• “Data Loading Application Programming Interface” on page 6-73
6-2
Linear Regression
Linear Regression
This introduction to linear regression in the Model-Based Calibration Toolbox
is divided into the following sections:
• “Stepwise Regression” on page 6-3
• “Definitions” on page 6-5
• “Prediction Error Variance” on page 6-7
Stepwise Regression
Building a regression model that includes only a subset of the total number of
available terms involves a tradeoff between two conflicting objectives:
• Increasing the number of model terms always reduces the Sum Squared
Error.
• However, you do not want so many model terms that you overfit by chasing
points and trying to fit the model to signal noise. This reduces the predictive
value of your model.
The best regression equation is the one that provides a satisfactory tradeoff
between these conflicting goals, at least in the mind of the analyst. It is well
known that there is no unique definition of best. Different model building
criteria (for example, forward selection, backward selection, PRESS search,
stepwise search, Mallows Cp Statistic…) yield different models. In addition,
even if the optimal value of the model building statistic is found, there is no
guarantee that the resulting model will be optimal in any other of the accepted
senses.
Principally the purpose of building the regression model for calibration is for
predicting future observations of the mean value of the response feature.
Therefore the aim is to select the subset of regression terms such that PRESS,
defined below, is minimized. Minimizing PRESS is consistent with the goal of
obtaining a regression model that provides good predictive capability over the
experimental factor space. This approach can be applied to both polynomial
and spline models. In either case the model building process is identical.
1 The regression matrix can be viewed in the Design Evaluation Tool. Terms
in this matrix define the full model. In general, the stepwise model is a
subset of this full term set.
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Technical Documents
2 All regressions are carried out with the factors represented on their coded
scales (-1,1).
3 All factors and response features are identified by appropriate symbols
defined by the test plan. These symbols are automatically carried across to
the model building tool.
6-4
Definitions
Definitions
Symbol
Definition
N
Number of data points
p
Number of terms currently included in the model
q
Total number of possible model parameters (q=p+r)
r
Number of terms not currently included from the model
y
(Nx1) response vector
X
Regression matrix. X has dimensions (Nxq)
Xp
(Nxp) model matrix corresponding to terms currently excluded from the model
Xr
(Nxr) matrix corresponding to terms currently excluded from the model
βp
(px1) vector of model coefficients β p = { β 1, β 2, …, β p }
–1 T
T
""βˆ = ( X X ) X y
–1
T
varβˆ = ( X X ) MSE
PEV
Prediction Error Variance
T
–1 T
PEV ( x ) = vary ( ŷ ) = x ( X X ) x MSE
α
User-defined threshold criteria for automatically rejecting terms
ŷ
(Nx1) vector of predicted responses. ŷ = X p β p
e
(Nx1) residual vector. e = ( y – ŷ )
e(i)
(Nx1) vector of PRESS residuals. e ( i ) = e i ⁄ ( 1 – H ii )
H
Hat matrix. X' ( X'X ) X
L
(Nx1) vector of leverage values. L = { l 1, l 2, …, l N }' = { H 11, H 22, …, H NN }
VIF
Variance Inflation Factors
–1
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Technical Documents
SSE
Error Sum of Squares. SSE = e’e
SSR
Regression Sum of Squares. SSE =
SST
Total Sum of Squares. SST = y’y - N y
MSE
Mean Square Error. MSE = SSE/(N-p)
MSR
Mean Square of Regression. MSR = SSR/P
F
F-statistic. F = MSR/MSE
MSE(i)
MSE calculated with ith point removed from the data set.
∑ ei
2
2
( N – p )MSE – e i ⁄ ( 1 – H ii )
MSE ( i ) = ---------------------------------------------------------------------N–p–1
RMSE
Root Mean Squared Error: the standard deviation of regression. RMSE =
si
ith R-Student or Externally Scaled Studentized Residual.
MSE
ei
s i = ----------------------------------------------MSE ( i ) ( 1 – H ii )
ri
ith Standardized or Internally Scaled Studentized Residual.
ei
r i = ----------------------------------------MSE ( 1 – H ii )
D
Cook’s D Influence Diagnostic.
2
r i H ii
Di = ------------------------p ( 1 – H ii )
SEBETA
(px1) vector of model coefficient standard errors.
–1
T
SEBETA = MSE { c 11, c 22, …, c pp }' where c = ( X X )
PRESS
Predicted Error Sum of Squares. PRESS = e’(i)e(i)
For more on PRESS and other displayed statistics, see “Linear Model Statistics
Displays” on page 6-22.
6-6
Prediction Error Variance
Prediction Error Variance
Prediction Error Variance (PEV) is a very useful way to investigate the
predictive capability of your model. It gives a measure of the precision of a
model’s predictions.
You start with the regression (or design) matrix, for example:
X =
1
L1
N1
L
1
L2
N2
L
…
…
…
1
Ln
Nn
2
2
1
L1 N1 N
2
L2 N2 N
…
L
2
n
…
2
2
1
2
…
Ln Nn N
2
n
If you knew the actual model, you would know the actual model coefficients β .
In this case the observations would be
y = Xβ + ε
where ε is the measurement error with variance
T
–1
var ( ε ) = ( X X ) MSE
However you can only ever know the predicted coefficients:
–1
T
T
β̂ = ( X X ) X y
which have variance
–1
T
varβˆ = ( X X ) MSE
Let x be the regression matrix for some new point where you want to evaluate
the model, for example:
x =
1
L new
N new
L
2
new
L new N new
N
2
new
Then the model prediction for this point is
–1 T
T
ŷ = xβˆ = x ( X X ) X y
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Technical Documents
Now you can calculate PEV as follows:
T
–1
T
T
–1 T
PEV ( x ) = var ( ŷ ) = ( x ( X X ) X ) ( X ( X X ) x )MSE
T
–1 T
PEV ( x ) = x ( X X ) x MSE
Note the only dependence on the observed values is in the variance (MSE) of
the measurement error. You can look at the PEV(x) for a design (without MSE,
as you don’t yet have any observations) and see what effect it will have on the
measurement error - if it is greater than 1 it will magnify the error, and the
closer it is to 0 the more it will reduce the error.
You can examine PEV for designs or global models using the Prediction Error
Variance viewer. When you open it from the Design Editor, you can see how
well the underlying model predicts over the design region. When you open it
from a global model, you can view how well the current global model predicts.
A low PEV (tending to zero) means that good predictions are obtained at that
point. See “Prediction Error Variance Viewer” on page 5-193.
For information on the calculation of PEV for two-stage models, see “Prediction
Error Variance for Two-Stage Models” on page 6-43.
6-8
High-Level Model Building Process Overview
High-Level Model Building Process Overview
The recommended overall process is best viewed graphically, as shown in the
following flow chart.
Start
Transform
Response
Change
Model
Stepwise
PRESS Search
Stepwise PRESS
Search
Interrogate
Model
Diagnostics
Interrogate Model
Diagnostics
Bad
Good
Bad
Stop
Good
Stop
Note that the process depicted in the preceding diagram should be carried out
for each member of the set of response features associated with a given
response and then repeated for the remaining responses.
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Univariate Model Building Process Overview
For each response feature,
1 Begin by conducting a stepwise search.
You can do this automatically or by using the Stepwise window.
The goal of the stepwise search is to minimize PRESS. The precise nature of
this process is discussed in future sections. What is important to appreciate
about the output from this step is that usually not one but several candidate
models per response features arise, each with a very similar PRESS R2. The
fact is that the predictive capability of a model with a PRESS R2 of 0.91
cannot be assumed superior in any meaningful engineering sense to a model
with a PRESS R2 of 0.909. Further, the nature of the model building process
is that the “improvement” in PRESS R2 offered by the last few terms is often
very small. Consequently, several candidate models can arise. You can store
each of the candidate models and associated diagnostic information
separately for subsequent review. Do this by making a selection of child
nodes for the response feature.
However, experience has shown that a model with a PRESS R2 of less than
0.8, say, is of little use as a predictive tool for engine mapping purposes. This
criteria must be viewed with caution. Low PRESS R2 values can result from
a poor choice of the original factors but also from the presence of outlying or
influential points in the data set. Rather than relying on PRESS R2 alone, a
safer strategy is to study the model diagnostic information in order to
discern the nature of any fundamental issues and then take appropriate
corrective action.
2 Once the stepwise process is complete, the diagnostic data should be
reviewed for each candidate model.
It might be that these data alone are sufficient to provide a means of
selecting a single model. This would be the case given that one model clearly
exhibited more ideal behavior than the others. Remember that the
interpretation of diagnostic plots is subjective.
3 You should also remove outlying data at this stage, using the mouse to select
the offending point. You can set criteria for detecting outlying data. The
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High-Level Model Building Process Overview
default criterion is any case where the absolute value of the external
studentized residual is greater than 3.
4 Given that outlying data has been removed, you might want to continue the
model building process in an attempt to remove further terms.
This seems reasonable because high-order terms might have been retained
in the model in an attempt to follow the outlying data. Even after removing
outlying data, there is no guarantee that the diagnostic data will suggest
that a suitable candidate model has been found. Under these circumstances,
5 A transform of the response feature might prove beneficial.
A useful set of transformations is provided by the Box and Cox family, which
are discussed in the next section. Note that the Box-Cox algorithm is model
dependent and as such is always carried out using the (Nxq) regression
matrix X.
6 After you select a transform, you should repeat the stepwise PRESS search
and select a suitable subset of candidate models.
7 After this you should analyze the respective diagnostic data for each model
in the usual manner.
At this juncture it might not be apparent why the original stepwise search
was carried out in the natural metric. Why not proceed directly to taking a
transformation? This seems sensible when it is appreciated that the
Box-Cox algorithm often, but not always, suggests that a contractive
transform such as the square root or log be applied. There are two main
reasons for this:
- The primary reason for selecting response features is that they possess a
natural engineering interpretation. It is unlikely that the behavior of a
transformed version of a response feature is as intuitively easy to
understand.
- Outlying data can strongly influence the type of transformation selected.
Applying a transformation to allow the model to fit bad data well does not
seem like a prudent strategy. By “bad” data it is assumed that the data is
truly abnormal and a reason has been discovered as to why the data is
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Technical Documents
outlying; for example, “The emission analyser was purging while the
results were taken.”
Finally, if you cannot find a suitable candidate model on completion of the
stepwise search with the transformed metric, then a serious problem exists
either with the data or with the current level of engineering knowledge of the
system. Model augmentation or an alternative experimental or modeling
strategy should be applied in these circumstances.
After these steps it is most useful to validate your model against other data (if
any is available). See “Model Evaluation Window” on page 5-144.
6-12
Stepwise Regression Techniques
Stepwise Regression Techniques
You can open the stepwise regression window through the
toolbar icon,
when in the global level view (that is, with a response feature selected in the
model tree). The Stepwise tool provides a number of methods of selecting the
model terms that should be included.
Minimizing Predicted Error Sum of Squares (PRESS) is a good method for
working toward a regression model that provides good predictive capability
over the experimental factor space.
The use of PRESS is a key indicator of the predictive quality of a model. The
predicted error uses predictions calculated without using the observed value
for that observation. PRESS is known as Delete 1 statistics in the Statistics
Toolbox. See also “Linear Model Statistics Displays” on page 6-22.
Stepwise Table
Term
Label for Coefficient
Status
Always. Stepwise does not remove this term.
Never. Stepwise does not add this term.
Step. Stepwise considers this term for addition or removal.
B
Value of coefficient. When the term is not in the model the value of the
coefficient if it is added to the model is displayed in red.
stdB
Standard Error of coefficient.
t
t value to test whether the coefficient is statistically different from zero.
The t value is highlighted in blue if it is less than the critical value
specified in the α % edit box (at bottom right).
Next PRESS
The value of PRESS if the status of this term is changed at the next
iteration. The smallest Next PRESS is highlighted with a yellow
background, The column header is also yellow if there is a model change
that results in a smaller PRESS value.
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The preceding table describes the meanings of the column headings in the
Stepwise Regression window, shown in the following example.
6-14
Stepwise Regression Techniques
4
1
2
3
7
6
5
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Technical Documents
1 The confidence intervals for all the coefficients are shown to the right of the
table. Note that the interval for the constant term is not displayed, as the
value of this coefficient is often significantly larger than other coefficients.
2 Terms that are currently not included in the model are displayed in red.
3 Terms can be included or removed from the model by clicking on the Term,
Next PRESS, or coefficient error bar line.
4 A history of the PRESS and summary statistics is shown on the right of the
stepwise figure. You can return to a previous model by selecting an item in
the list box or a point on the stepwise plot.
5 The ANOVA table and diagnostic statistics for the current model are shown
on the right side of the stepwise figure.
6 The critical values for testing whether a coefficient is statistically different
from zero at the α % level are displayed at the bottom right side of the
stepwise figure. The value of α can be entered in the edit box to the left of
the critical values. The default is 5%.
7 A number of further stepwise commands are provided through the buttons
at the bottom of the figure (and duplicated in the Regression menu):
- Min. PRESS includes or remove terms to minimize PRESS. This
procedure provides a model with improved predictive capability.
- Include All terms in the model (except the terms flagged with Status as
Never). This option is useful in conjunction with Min. PRESS and
backward selection. For example, first click Include All, then Min.
PRESS. Then you can click Include All again, then Backwards, to
compare which gives the best result.
- Remove All terms in the model (except the terms flagged with Status as
Always). This option is useful in conjunction with forward selection (click
Remove All, then Forwards).
- Forwards selection adds all terms to the model that would result in
statistically significant terms at the α % level. The addition of terms is
repeated until all the terms in the model are statistically significant.
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Stepwise Regression Techniques
- Backwards selection removes all terms from the model that are not
statistically significant at the α % level. The removal of terms is repeated
until all the terms in the model are statistically significant.
Any changes made in the stepwise figure automatically update the diagnostic
plots in the Model Browser.
You can revert to the starting model when closing the Stepwise window. When
you exit the Stepwise window, the Confirm Stepwise Exit dialog asks Do you
want to update regression results? You can click Yes (the default), No (to
revert to the starting model), or Cancel (to return to the Stepwise window).
You can set the Minimize PRESS, Forward, and Backward selection routines
to run automatically without the need to enter the stepwise figure. These
options are selected through the Global Model Setup dialog.
From the global level, select Model –> Set Up. The Global Model Setup dialog
has a drop-down menu Stepwise, with the options None, Minimize PRESS,
Forward selection, and Backward selection. You can set these options when
you initially set up your test plan.
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Box-Cox Transformation
You might want to transform a response feature either to correct for
nonnormality and/or a heteroscedastic variance structure. A useful class of
transformations for this purpose is the power transform yλ, where λ is a
parameter to be determined. Box and Cox (1964) showed how λ and the
regression coefficients themselves could be estimated simultaneously using the
method of maximum likelihood. The procedure consists of conducting a
standard least squares fit using
y
y
λ
(λ)
y – 1 for λ ≠ 0 λ
= ---------------λ–1
λy·
(λ)
= y· ln ( y ) for λ = 0
where the so called geometric mean of the observations is given by
N
∑ ln ( yi )
i=1
y· = exp -------------------------N
The maximum likelihood estimate of λ corresponds to the value for which the
SSE(λ) from the fitted model is a minimum. This value of λ is determined by
fitting a model (assumed throughout to be defined by the regression matrix for
the full model - X) for various levels of λ and choosing the value corresponding
to the minimum SSE(λ). A plot of SSE(λ) versus λ is often used to facilitate this
choice.
The parameter λ is swept between the range of -3 to 3 in increments of 0.5.
• You can enter a value for lambda in the edit box that approaches the point
on the plot with the smallest SSE.
Although SSE(λ) is a continuous function of λ, simple choices for λ are
recommended. This is because the practical difference between 0.5 and 0.593,
say, is likely to be very small but a simple transform like 0.5 is much easier to
interpret.
6-18
Box-Cox Transformation
You can also find an approximate 100(1- α ) confidence interval on l by
computing
t α ⁄ ( 2, υ )
*
SS = SSE ( λ ) 1 + ------------------υ
where υ is the number of residual degrees of freedom equal to (N-q).
In this formula λ is understood to be the value that minimizes SSE(λ). Note
that this confidence interval might encompass more than one incremental
value for λ. In this case, any of these values is as valid as any other and you can
select any of these transformations from which to develop trial models.
• You should always look at the residuals plots at the top to see the effect of
different transforms.
• You can create several child nodes of a single model and choose different
transforms for each in order to compare them using the rest of the Model
Browser tools.
For the sake of clarity, consider the example following, which illustrates the
outcome of applying the Box-Cox method.
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The preceding example shows the results of applying the Box-Cox algorithm to
a polyspline torque model.
In this example the minimum value of SSE(λ) occurs near to λ=0. The
minimum is marked in green. The 95% confidence limit has been calculated
and drawn on the figure as a red solid line. It is apparent in this example that,
after rounding to the nearest incremental value contained within the
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Box-Cox Transformation
0≤λ≤1
confidence interval, any λ in the range
is appropriate. Of the
three possible increments, 0, 0.5, and 1, λ = 0.5 is the closest to the minimum
SSE.
You can select any point on the plot by clicking. The chosen point (current
lambda) is then outlined in red. You can also enter values of lambda directly in
the edit box and press Return.
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Linear Model Statistics Displays
Summary Statistics
Observations
Number of observations used to estimate model
Parameters
Number of parameters in model
Box-Cox
Power transform used for Box-Cox transformation. A value of zero means a
log transform is used. A value of 1 means there is no transformation.
PRESS RMSE
Root mean squared error of predicted errors. The divisor used for PRESS
RMSE is the number of observations. The residuals are in untransformed
values to enable comparison between alternative models with different
Box-Cox transformations.
RMSE
Root mean squared error. The divisor used for RMSE is the number of
observations minus the number of parameters. The residuals are in
untransformed values, to enable comparison between alternative models
with different Box-Cox transformations.
ANOVA Table
6-22
SS
df
MS
Regression
SSR
p-1
SSR/(p-1)
Error
SSE
n-p
SSE/(N-p)
Total
SST
n-1
Linear Model Statistics Displays
Diagnostic Statistics
PRESS
Predicted Error Sum of Squares.
PRESS R^2
R^2 value calculated using PRESS = (SST-PRESS)/SST.
Note that this figure is sometimes negative.
R^2
R^2= (SST-SSE)/SST.
s
Root Mean Squares Error for regression = sqrt(SSE/
(N-p)).
PRESS Statistic
With n runs in the data set, the model equation is fitted to n-1 runs and a
prediction taken from this model for the remaining one. The difference between
the recorded data value and the value given by the model (at the value of the
omitted run) is called a prediction residual. PRESS is the sum of squares of the
prediction residuals. The square root of PRESS/n is PRESS RMSE (root mean
square prediction error).
Note that the prediction residual is different from the ordinary residual, which
is the difference between the recorded value and the value of the model when
fitted to the whole data set.
The PRESS statistic gives a good indication of the predictive power of your
model, which is why minimizing PRESS is desirable. It is useful to compare
PRESS RMSE with RMSE as this can indicate problems with overfitting.
RMSE is minimized when the model gets very close to each data point;
“chasing” the data will therefore improve RMSE. However, chasing the data
can sometimes lead to strong oscillations in the model between the data points;
this behavior can give good values of RMSE but is not representative of the
data and does not give reliable prediction values where you do not already have
data. The PRESS RMSE statistic guards against this by testing how well the
current model would predict each of the points in the data set (in turn) if they
were not included in the regression. To get a small PRESS RMSE usually
indicates that the model is not overly sensitive to any single data point.
For more information, see “Stepwise Regression Techniques” on page 6-13 and
“Definitions” on page 6-5.
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Note that calculating PRESS for the two-stage model applies the same
principle (fitting the model to n-1 runs and taking a prediction from this model
for the remaining one) but in this case the predicted values are first found for
response features instead of data points. The predicted value, omitting each
test in turn, for each response feature is estimated. The predicted response
features are then used to reconstruct the local curve for the test and this curve
is used to obtain the two-stage predictions. This is applied as follows:
To calculate two stage PRESS:
1 For each test, S, do the following steps:
- For each of the response features, calculate what the response feature
predictions would be for S (with the response features for S removed from
the calculation).
- This gives a local prediction curve C based on all tests except S.
- For each data point in the test, calculate the difference between the
observed value and the value predicted by C.
2 Repeat for all tests.
3 Sum the square of all of the differences found and divide by the total number
of data points.
Pooled Statistics
6-24
Local
RMSE
Root mean squared error, using the local model fit to the
data for the displayed test. The divisor used for RMSE is the
number of observations minus the number of parameters.
Two-Stage
RMSE
Root mean squared error, using the two-stage model fit to
the data for the displayed test. You want this error to be
small for a good model fit.
Linear Model Statistics Displays
PRESS
RMSE
Root mean squared error of predicted errors, see “PRESS
Statistic” on page 6-23 above. The divisor used for PRESS
RMSE is the number of observations. Not displayed for MLE
models because the simple univariate formula cannot be
used.
Two-Stage
T^2
T^2 is a normalized sum of squared errors for all the
response features models. You can see the basic formula on
the Likelihood view of the Model Selection window.
2
T –1
T = ( y rf – ŷ ) Σ ( y rf – ŷ )
Where Σ = blockdiag ( C i + D ) , where Ci is the local
covariance for test i. See blockdiag diagram following.
A large T^2 value indicates that there is a problem with the
response feature models.
-log L
Log-likelihood function: the probability of a set of
observations given the value of some parameters. You want
the likelihood to be large, tending towards -infinity, so large
negative is good.
For n observations x1,x2,..xn, with probability distribution
f ( x, θ ) , the likelihood is
L =
∏n
f ( x i, θ )
i=1
This is the basis of “Maximum Likelihood Estimation” on
page 6-36.
T –1
log L = log ( det ( Σ ) ) + ( y rf – ŷ ) Σ ( y rf – ŷ )
which is the same as
log L = log ( det ( Σ ) ) + T
2
This assumes a normal distribution.
You can view plots of -log L in the Model Selection window,
see “Likelihood View” on page 5-127.
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To explain blockdiag as it appears under T^2 in the Pooled statistics table:
Σ = blockdiag ( C i + D ) , where Ci is the local covariance for test i, is calculated
as shown below.
C1 + D
blockdiag ( C i + D ) =
C2 + D
C3 + D
Ci + D
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Design Evaluation Tool
Design Evaluation Tool
The Design Evaluation tool is only available for linear models.
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You can open the Design Evaluation tool from the Design Editor or from the
Model Browser windows. From the Design Editor select Tools –> Evaluate
Designs and choose the design you want to evaluate. From the Model Browser
global view, you can click the
button.
In the Design Evaluation tool you can view all the information on correlations,
covariance, confounding, and variance inflation factors (VIFs). You can
investigate the effects of including or excluding model terms aided by this
information (you must remove them in the Stepwise window). Interpretation is
aided by color-coded values based on the magnitude of the numbers. You can
specify changes to these criteria.
When you open the Design Evaluation tool, the default view is a table, as
shown in the preceding example. You choose the elements to display from the
list on the right. Click any of the items in the list described below to change the
display. Some of the items have a choice of buttons that appear underneath the
list box.
To see information about each display, click the
toolbar button or select
View –> Matrix Information.
Table Options
You can apply color maps and filters to any items in a table view, and change
the precision of the display.
To apply a color map or edit an existing one:
1 Select Options –> Table –> Colors. The Table Colors dialog appears.
2 Select the check box Use a colormap for rendering matrix values.
3 Click the Define colormap button. The Colormap dialog appears, where
you can choose how many levels to color map, and the colors and values to
use to define the levels. Some tables have default color maps to aid analysis
of the information, described below.
You can also use the Options –> Table menu to change the precision (number
of significant figures displayed) and to apply filters that remove specific values
or values above or below a specific value from the display.
The status bar at bottom left displays whether color maps and filters are active
in the current view.
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Design Evaluation Tool
When evaluating several designs, you can switch between them with the Next
design toolbar button or the Design menu.
Design Matrix
Xn/Xc: design/actual factor test points matrix for the experiment, in natural or
coded units. You can toggle between natural and coded units with the buttons
on the right.
Full FX Matrix
Full model matrix, showing all possible terms in the model. You can include
and exclude terms from the model here, by clicking on the green column
headings. When you click one to remove a term, the column heading becomes
red and the whole column is grayed.
Model Terms
You can select terms for inclusion in or exclusion from the model here by
clicking. You can toggle the button for each term by clicking. This changes the
button from in (green) to out (red) and vice versa. You can then view the effect
of these changes in the other displays.
Note Removal of model terms only affects displays within the Design
Evaluation tool. If you decide the proposed changes would be beneficial to your
model, you must return to the Stepwise window and make the changes there
to fit the new model.
Z2 Matrix
Z2 : Matrix of terms that have been removed from the model. If you haven’t
removed any terms, the main display is blank apart from the message “All
terms are currently included in the model.”
Alias Matrix
Like the Z2 matrix, the alias matrix also displays terms that are not included
in the model (and is therefore not available if all terms are included in the
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model). The purpose of the alias matrix is to show the pattern of confounding
in the design.
A zero in a box indicates that the row term (currently included in the model) is
not confounded with the column term (currently not in the model). A complete
row of zeros indicates that the term in the model is not confounded with any of
the terms excluded from the model. A column of zeros also indicates that the
column term (currently not in the model) could be included (but at the cost of a
reduction in the residual degrees of freedom).
A: the alias matrix is defined by the expression
–1
A = ( X'X ) X'Z 2
Z2.1 Matrix
As this matrix also uses the terms not included in the model, it is not available
if all terms are included.
Z2.1 : matrix defined by the expression Z 2.1 = Z 2 – XA
Regression Matrix
Regression matrix. Consists of terms included in the model. n × p matrix
where n is the number of test points in the design and p is the number of terms
in the model.
Coefficient Information
When you select Coefficient information, six buttons appear below the list
box. Covariance is displayed by default; click the buttons to select any of the
others for display.
Covariance
Cov(b): variance-covariance matrix for the regression coefficient vector b.
Cov ( b ) = ( X'X )
–1
Correlation
Corr(b): correlation matrix for the regression coefficient vector b.
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Design Evaluation Tool
Cov ( b )ij
Corr ( b ) ij = ---------------------------------------------------------------------( Cov ( b ) ii ) ( Cov – ( b ) jj )
By default Correlation has an active color map to aid analysis. Values below
-0.9 are red, -0.9 to -0.7 are orange, -0.7 to 0.707 are black, 0.707 to 0.9 are
orange, and greater than 0.9 are red. You can view and edit the color map
using Options –> Table –> Colors.
Partial VIFs
Variance Inflation Factors (VIFs) are a measure of the nonorthogonality of the
design with respect to the selected model. A fully orthogonal design has all
VIFs equal to unity.
The Partial VIFs are calculated from the off-diagonal elements of Corr(b) as
1
VIF ij = ---------------------------------------- for p ≥ i > j > 1
2
( 1 – Corr ( b ) ij )
Partial VIFs also has a default color map active (<1.2 black, >1.2<1.4 orange,
>1.4 red). A filter is also applied, removing all values within 0.1 of 1. In
regular designs such as Box-Behnken, many of the elements are exactly 1 and
so need not be displayed; this plus the color coding makes it easier for you to
see the important VIF values. You can always edit or remove color maps and
filters.
Multiple VIFs
Measure of the nonorthogonality of the design. The Multiple VIFs are defined
as the diagonal elements of Corr(b):
–1
VIF i = { Corr ( b ) } ii
Multiple VIFs also has a default color map active (<8 black, 8><10 orange, >10
red). A filter is also applied, removing all values within 0.1 of 1. Once again
this makes it easier to see values of interest.
2 Column Corr.
Corr(X); correlation for two columns of X.
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x ij – x j
w ij = ------------------------------------N
∑ ( xij – x )
2
i=1
Let W denote the matrix of wij values. Then the correlation matrix for the
columns of X (excluding column 1) is Corr(X), defined as
Corr(X) = W W
2 Column Correlation has the same default color map active as Correlation.
Single Term VIFs
Measure of the nonorthogonality of the design. The Single Term VIFs are
defined as
1
VIF ij = ----------------------------------------- for p ≥ i > j > 1
2
( 1 – Corr ( X ) ij )
Single term VIFs have a default color map active (<2 black, 2>red) and values
within 0.1 of 1 are filtered out, to highlight values of interest.
Standard Error
σ j : standard error of the jth coefficient relative to the RMSE.
Hat Matrix
Full Hat matrix
H: the Hat matrix.
H = QQ
where Q results from a QR decomposition of X. Q is an n × n orthonormal
matrix and R is an n × p matrix.
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Design Evaluation Tool
Leverage values
The leverage values are the terms on the leading diagonal of H (the Hat
matrix). Leverage values have a color map active (<0.8 black, 0.8>orange<0.9,
>0.9 red).
|X’X|
D; determinant of X’X.
D can be calculated from the QR decomposition of X as follows:
2
p
D =
∏ ( R1 ) ii
,
i=1
where p is the number of terms in the currently selected model.
This can be displayed in three forms:
X'X
log ( X'X )
X'X
(1 ⁄ p)
’
Raw Residual Statistics
Covariance
Cov(e): variance-covariance matrix for the residuals.
Cov(e) = (I-H)
Correlation
Corr(e) : correlation matrix for the residuals.
Cov ( e ) ij
Corr ( e ) ij = --------------------------------------------------------------( Cov ( e ) ii ) ( Cov ( e )jj )
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Degrees of Freedom Table
To see the Degrees of Freedom table (and the information about each display),
click the
toolbar button or select View –> Matrix Information.
Source
D.F.
Model
p
Residual
n-p
Replication
by calculation
Lack of fit
by calculation
Total
n
Replication is defined as follows:
Let there be nj (>1) replications at the jth replicated point. Then the degrees of
freedom for replication are
∑ ( nj – 1 )
j
and Lack of fit is given by n - p - degrees of freedom for replication.
Note that replication exists where two rows of X are identical. In regular
designs the factor levels are clearly spaced and the concept of replication is
unambiguous. However, in some situations spacing can be less clear, so a
tolerance is imposed of 0.005 (coded units) in all factors. Points must fall within
this tolerance to be considered replicated.
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Design Evaluation Tool
Design Evaluation Graphical Displays
The Design Evaluation tool has options for 1-D, 2-D, 3-D, and 4-D displays. You
can switch to these by clicking the toolbar buttons or using the View menu.
Which displays are available depends on the information category selected in
the list box. For the Design matrix, (with sufficient inputs) all options are
available. For the Model terms, there are no display options.
You can edit the properties of all displays using the Options menu. You can
configure the grid lines and background colors. In the 2-D image display you
can click points in the image to see their values. All 3-D displays can be rotated
as usual. You can edit all color map bars by double-clicking.
Export of Design Evaluation Information
All information displayed in the Design Evaluation tool can be exported to the
workspace or to a .mat file using the radio buttons and Export button at the
bottom right. You can enter a variable name in the edit box.
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Maximum Likelihood Estimation
Maximum Likelihood Estimation for Nonlinear Repeated Measures
This section contains an overview of the mathematics of two-stage models. A
comprehensive reference for two-stage modeling is Davidian and Giltinan [3].
The information is divided into the following sections:
• “Two-Stage Models for Engines” on page 6-37
• “Local Models” on page 6-38
- “Local Covariance Modeling” on page 6-38
- “Response Features” on page 6-40
• “Global Models” on page 6-40
• “Two-Stage Models” on page 6-41
- “Prediction Error Variance for Two-Stage Models” on page 6-43
- “Global Model Selection” on page 6-45
- “Initial Values for Covariances” on page 6-45
- “Quasi-Newton Algorithm” on page 6-45
- “Expectation Maximization Algorithm” on page 6-45
• “References” on page 6-46
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Two-Stage Models for Engines
Two-Stage Models for Engines
Lindstrom and Bates [6] define repeated measurements as data generated by
observing a number of individuals repeatedly under various experimental
conditions, where the individuals are assumed to constitute a random sample
from a population of interest. An important class of repeated measurements is
longitudinal data where the observations are ordered by time or position in
space. More generally, longitudinal data is defined as repeated measurements
where the observations on a single individual are not, or cannot be, randomly
assigned to the levels of a treatment of interest.
Modeling data of this kind usually involves the characterization of the
relationship between the measured response, y, and the repeated
measurement factor, or covariate x. Frequently, the underlying systematic
relationship between y and x is nonlinear. In some cases the relevant nonlinear
model can be derived on physical or mechanistic grounds. However, in other
contexts a nonlinear relationship might be imposed simply to provide a
convenient empirical description for the data. The presence of repeated
observations on an individual requires particular care in characterizing the
variation in the experimental data. In particular, it is important to represent
two sources of variation explicitly: random variation among measurements
within a given individual (intraindividual) and random variation among
individuals (interindividual). Inferential procedures accommodate these
different variance components within the framework of an appropriate
hierarchical statistical model. This is the fundamental idea behind the analysis
of repeated measurement data.
Holliday [1,2] was perhaps the first to apply nonlinear repeated measurements
analysis procedures to spark injection engine data. The focus of Holliday's work
was the modeling of data taken from engine mapping experiments. In these
experiments, engine speed, load, and air/fuel ratio were held constant while
spark was varied. Various engine response characteristics, for example, torque
or emission quantities, were measured at each spark setting. Holliday modeled
the response characteristics for each sweep as a function of spark advance.
Variations in the individual sweep parameters were then modeled as a function
of the global engine operating variables speed, load, and air/fuel ratio.
Conceptually, variations in the measurements taken within a sweep represent
the intraindividual component of variance. Similarly, variation in the
sweep-specific parameters between sweeps represents the interindividual
component of variance. You can generalize these principles to other
steady-state engine modeling exercises where the nature of data collection
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usually involves sweeping a single engine control variable while the remainder
are held at fixed values. These points suggest that nonlinear repeated
measurements analysis represents a general approach to the parameterization
of mean value engines models for controls-oriented development.
Another application for models of this form is the flow equations for a throttle
body. Assuming the flow equations are based upon the usual one-dimensional
isentropic flow principle, then they must be modified by an effective area term,
Ae, which accounts for the fact that the true flow is multidimensional and
irreversible. You can map the throttle flow characteristics by sweeping the
throttle position at fixed engine speed. This data collection methodology
naturally imposes a hierarchy the analysis of which is consistent with the
application of nonlinear repeated measures. Experience in modeling effective
area suggests that free knot spline or biological growth models provide good
local predictions. The global phase of the modeling procedure is concerned with
predicting the systematic variation in the response features across engine
speed. A free knot spline model has proven useful for this purpose.
Local Models
Modeling responses locally within a sweep as a function of the independent
variable only. That is,
j
j
j
y i = f i (s i,θ i) + ε i for j = 1, 2, …m i
(6-1)
j
where the subscript i refers to individual tests and j to data within a test, s i is
j
the jth independent value, θi is a (rx1) parameter vector, y i is the j th response,
j
and ε i is a normally distributed random variable with zero mean and variance
σ2. Note that Equation 6-1 can be either a linear or a nonlinear function of the
curve fit parameters. The assumption of independently normally distributed
errors implies that the least squares estimates of θ are also maximum
likelihood parameters.
Local Covariance Modeling
The local model describes both the systematic and random variation associated
with measurements taken during the ith test. Systematic variation is
characterized through the function f while variation is characterized via the
distributional assumptions made on the vector of random errors ei. Hence,
specification of a model for the distribution of ei completes the description of the
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Two-Stage Models for Engines
intratest model. The Model-Based Calibration Toolbox allows a very general
specification of the local covariance,
2
e i ∼ N (0,σ C i (β i,ξ i))
(6-2)
2
where Ci is an (ni x ni) covariance matrix, σ is the coefficient of variation, and
ξi is a (q-by-1) vector of dispersion parameters that account for heterogeneity
of variance and the possibility of serially correlated data. The specification is
very general and affords considerable flexibility in terms of specifying a
covariance model to adequately describe the random component of the
intratest variation.
The Model-Based Calibration Toolbox supports the following covariance
models:
• Power Variance Model

ξ 
C i = diag  f (x i,β i) 1 


(6-3)
• Exponential Variance Model
C i = diag { exp ( f (x i,βi)ξ 1 ) }
(6-4)
• Mixed Variance Model

ξ2 
C i = diag  ξ 1 + f (x i,θ i) 


(6-5)
where diag{x} is a diagonal matrix.
Correlation models are only available for equispaced data in the Model-Based
Calibration Toolbox. It is possible to combine correlation models with models
with the variance models such as power.
One of the simplest structures that can be used to account for serially
correlated errors is the AR(m) model (autoregressive model with lag m). The
general form of the AR(m) model is
ej = φ1 ej – 1 + φ2 ej – 2 + … + φm ej – m + vj
(6-6)
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Technical Documents
where φ k is the kth lag coefficient and vj is an exogenous stochastic input
2
identically and independently distributed as N (0,σ v) . First- and second-order
autoregressive models are implemented in the Model-Based Calibration
Toolbox.
Another possibility is a moving average model (MA). The general structure is
ej = φ 1 vj – 1 + φ 2 vj – 2 + … + φm v j – m + v j
(6-7)
where φ k is the kth lag coefficient and vj is an exogenous stochastic input
2
identically and independently distributed as N (0,σ v ) . Only a first-order
moving average model is implemented in the Model-Based Calibration
Toolbox.
Response Features
From an engineering perspective, the curve fit parameters do not usually have
any intuitive interpretation. Rather characteristic geometric features of the
curve are of interest. The terminology “response features” of Crowder and
Hand [7] is used to describe these geometric features of interest. In general, the
response feature vector pi for the ith sweep is a nonlinear function (g) of the
corresponding curve fit parameter vector θi, such that
pi = g ( θi )
(6-8)
Global Models
Modeling the variation in the response features as a function of the global
variables. The response features are carried through to the second stage of the
modeling procedure rather than the curve fit parameters because they have an
engineering interpretation. This ensures that the second stage of the modeling
process remains relatively intuitive. It is much more likely that an engineer
will have better knowledge of how a response feature such as MBT behaves
throughout the engine operating range (at least on a main effects basis) as
opposed to an esoteric curve fit parameter estimate.
The global relationship is represented by one of the global models available in
the Model-Based Calibration Toolbox. In this section we only consider linear
models that can be represented as
p i = X i β + γi for i = 1, 2, …, r
6-40
(6-9)
Two-Stage Models for Engines
where the Xi contains the information about the engine operating conditions at
the ith spark sweep, β is the vector of global parameter estimates that must be
estimated by the fitting procedure, and γi is a vector of normally distributed
random errors. It is necessary to make some assumption about the error
distribution for γ, and this is typically a normal distribution with
γ i ∼ N r (0,D)
(6-10)
where r is the number of response features. The dimensions of D are (rxr) and,
being a variance-covariance matrix, D is both symmetric and positive definite.
Terms on the leading diagonal of D represent the test-to-test variance
associated with the estimate of the individual response features. Off-diagonal
terms represent the covariance between pairs of response features. The
estimation of these additional covariance terms in a multivariate analysis
improves the precision of the parameter estimates.
Two-Stage Models
To unite the two models, it is first necessary to review the distributional
assumptions pertaining to the response feature vector pi. The variance of pi
(Var(pi)) is given by
∂g ( θ i ) 2
∂g ( θ i )
Var ( p i ) = ---------------- σ C i ---------------∂θ
∂θ
T
(6-11)
For the sake of simplicity, the notation σ2Ci is to denote Var(pi). Thus, pii is
distributed as
2
p i ∼ N r ( p i, σ C i )
(6-12)
where Ci depends on fi through the variance of θi and also on gi through the
conversion of θi to the response features pi. Two standard assumptions are used
in determining Ci: the asymptotic approximation for the variance of maximum
likelihood estimates and the approximation for the variance of functions of
maximum likelihood estimates, which is based on a Taylor series expansion of
gi. In addition, for nonlinear f i or gi, Ci depends on the unknown θi ; therefore,
we will use the estimate θ̂ i in its place. These approximations are likely to be
good in the case where σ2 is small or the number of points per sweep (mi) is
large. In either case we assume that these approximations are valid
throughout.
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We now return to the issue of parameter estimation. Assume that the γi are
j
independent of the ε i . Then, allowing for the additive replication error in
response features, the response features are distributed as
2
p i ∼ N (X i,β, σ C i + D)
(6-13)
When all the tests are considered simultaneously, equation (6-13) can be
written in the compact form
P ∼ N ( Zβ, W ( ϖ ) )
(6-14)
where P is the vector formed by stacking the n vectors pi on top of each other,
Z is the matrix formed by stacking the n Xi matrices, W is the block diagonal
weighting matrix with the matrices on the diagonal being σ2Ci+D , and ω is a
vector of dispersion parameters. For the multivariate normal distribution
(6-14) the negative log likelihood function can be written
–1
log L ( β, ϖ ) = log W + ( P – Zβ )'W ( P – Zβ )
(6-15)
Thus, the maximum likelihood estimates are the vectors βML and ωML that
minimize logL(β,ω).Usually there are many more fit parameters than
dispersion parameters; that is, the dimension of β is much larger than ω. As
such, it is advantageous to reduce the number of parameters involved in the
minimization of logL(β,ω). The key is to realize that equation (6-15) is
conditionally linear with respect to β. Hence, given estimates of ω, equation
(6-15) can be differentiated directly with respect to β and the resulting
expression set to zero. This equation can be solved directly for β as follows:
–1
–1
–1
β = ( Z'W Z ) ( Z'W P )
(6-16)
The key point is that now the likelihood depends only upon the dispersion
parameter vector ω, which as already discussed has only modest dimensions.
Once the likelihood is minimized to yield ωML , then, since W(ωML) is then
known, equation (6-16) can subsequently be used to determine βML.
6-42
Two-Stage Models for Engines
Prediction Error Variance for Two-Stage Models
It is very useful to evaluate a measure of the precision of the model’s
predictions. You can do this by looking at Prediction Error Variance (PEV).
Prediction error variance will tend to grow rapidly in areas outside the original
design space. The following section describes how PEV is calculated for
two-stage models.
For linear global models applying the variance operator to Equation 6-15 yields
T
–1
–1 T
T
–1
–1
–1
–1
T
–1
Var ( β ) = ( Z W Z ) Z W Var(P)W Z ( Z W Z )
Var ( β ) = ( Z W Z )
,
–1
so
(6-17)
since Var(P) = W. Assume that it is required to calculate both the response
features and their associated prediction error variance for the ith test. the
predicted response features are given by
ˆ
p i = z i β̂
(6-18)
where z i is an appropriate global covariate matrix. Applying the variance
operator to Equation 6-18 yields
T –1 –1 T
T
Var ( pˆ i ) = z i Var ( β̂ )z i = z i ( Z W Z ) z i
(6-19)
In general, the response features are non-linear functions of the local fit
coefficients. Let g denote the non-linear function mapping θ onto p .
i
i
Similarly let h denote the inverse mapping.
θ̂ i = h ( pˆ i )
(6-20)
Approximating h using a first order Taylor series expanded about p i (the true
and unknown fixed population value) and after applying the variance operator
to the result,
·
·T
Var ( θ̂ i ) = h Var ( pˆ i )h
(6-21)
where the dot notation denotes the Jacobian matrix with respect to the
·
response features, p i . This implies that h is of dimension (pxp). Finally the
predicted response values are calculated from
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Technical Documents
yˆ i = f ( θ i )
(6-22)
Again, after approximating f by a first order Taylor series and applying the
variance operator to the result,
∂f
ˆ
Var ( y i ) = -----∂θ
θ̂
·
ˆ · T ∂f
h Var ( p i )h -----∂θ
T
(6-23)
θ̂
After subsituting Equation 6-19 into Equation 6-23 the desired result is
obtained:
∂f
ˆ
Var ( y i ) = -----∂θ
θ̂
T – 1 – 1 T · T ∂f
·
h z i ( Z W Z ) z i h -----∂θ
T
(6-24)
θ̂
This equation gives the value of Prediction Error Variance.
See also the introduction to “Prediction Error Variance” on page 6-7 for details
about PEV for one-stage models.
6-44
Two-Stage Models for Engines
Global Model Selection
Before undertaking the minimization of Equation 6-15 (see “Two-Stage
Models” on page 6-41) it is first necessary to establish the form of the Xi matrix.
This is equivalent to establishing a global expression for each of the response
features a priori. Univariate stepwise regression is used to select the form of
the global model for each response feature. Minimization of the appropriate
PRESS statistic is used as a model building principle, as specified in
“High-Level Model Building Process Overview” on page 6-9. The underlying
principle is that having used univariate methods to establish possible models,
maximum likelihood methods are subsequently used to estimate their
parameters.
Initial Values for Covariances
An initial estimate of the global covariance is obtained using the standard
two-stage estimate of Steimer et al. [10],
r
D STS
1
= -----------r–1
∑ ( pi – Xi β )( pi – Xi β )
T
(6-25)
i=1
where β are the estimates from all the univariate global models. This estimate
is biased.
Quasi-Newton Algorithm
Implicit to the minimization of equation (6-17) is that D is positive definite. It
is a simple matter to ensure this by noting that D is positive definite if and only
if there is an upper triangular matrix, G, say, such that
D = G'G
(6-26)
This factorization is used in the Quasi-Newton algorithm. Primarily, the
advantage of this approach is that the resulting search in G, as opposed to D,
is unconstrained.
Expectation Maximization Algorithm
The expectation maximization algorithm is an iterative method that converges
toward the maximal solution of the likelihood function. Each iteration has two
steps:
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1 Expectation Step — Produce refined estimates of the response features
given the current parameter estimates.
2 Maximization Step — Obtain new estimates of the parameters (global model
parameters and covariance matrix) for the new response features.
These steps are repeated until the improvement in value of the log likelihood
function is less than the tolerance. Details of the algorithm can be found in [3,
ch. 5].
References
1 Holliday, T., The Design and Analysis of Engine Mapping Experiments: A
Two-Stage Approach, Ph.D. thesis, University of Birmingham, 1995.
2 Holliday, T., Lawrance, A. J., Davis, T. P., Engine-Mapping Experiments: A
Two-Stage Regression Approach, Technometrics, 1998, Vol. 40, pp 120-126.
3 Davidian, M., Giltinan, D. M., Nonlinear Models for Repeated Measurement
Data, Chapman & Hall, First Edition, 1995.
4 Davidian, M., Giltinan, D. M., Analysis of repeated measurement data using
the nonlinear mixed effects model, Chemometrics and Intelligent Laboratory
Systems, 1993, Vol. 20, pp 1-24.
5 Davidian, M., Giltinan, D. M., Analysis of repeated measurement data using
the nonlinear mixed effects model, Journal of Biopharmaceutical Statistics,
1993, Vol. 3, part 1, pp 23-55.
6 Lindstrom, M. J., Bates, D. M., Nonlinear Mixed Effects Models for
Repeated Measures Data, Biometrics, 1990, Vol. 46, pp 673-687.
7 Davidian, M., Giltinan, D. M., Some Simple Methods for Estimating
Intraindividual Variability in Nonlinear Mixed Effects Models, Biometrics,
1993, Vol. 49, pp 59-73.
8 Hand, D. J., Crowder, M. J., Practical Longitudinal Data Analysis,
Chapman and Hall, First Edition, 1996.
6-46
Two-Stage Models for Engines
9 Franklin, G.F., Powell, J.D., Workman, M.L., Digital Control of Dynamic
Systems, Addison-Wesley, Second Edition, 1990.
10 Steimer, J.-L., Mallet, A., Golmard, J.L., and Boisvieux, J.F., Alternative
approaches to estimation of population pharmacokinetic parameters:
Comparison with the nonlinear mixed effect model. Drug Metabolism
Reviews, 1984, 15, 265-292.
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Local Model Definitions
Local Models and Associated Response Features
In the following sections are listed the model classes and their associated
response features available for modeling at the local node in MBC.
“Polynomial Models” on page 6-48
“Polynomial Splines” on page 6-49
“Truncated Power Series Basis (TPSBS) Splines” on page 6-50
“Free Knot Splines” on page 6-51
“Three Parameter Logistic Model” on page 6-51
“Morgan-Mercer-Flodin Model” on page 6-52
“Four-Parameter Logistic Curve” on page 6-53
“Richards Curves” on page 6-54
“Weibul Growth Curve” on page 6-55
“Exponential Growth Curve” on page 6-55
“Gompertz Growth Model” on page 6-56
Polynomial Models
MBC includes extensive capabilities for using polynomials of arbitrary order to
model local phenomena. The following response features are permitted for the
polynomial model class:
• Location of the maximum or minimum value (when using datum models;
note that the datum model is not used in reconstructing).
• Value of the fit function at a user-specified x-ordinate. When datum models
are used, the value is relative to the datum (for example, mbt - x).
• The nth derivative at a user-specified x-ordinate, for n = 1, 2, …, d where d is
the degree of the polynomial.
6-48
Local Model Definitions
Polynomial Splines
These are essential for modeling torque/spark curves. To model responses that
are characterized in appearance by a single and well defined stationary point
with asymmetrical curvature either side of the local minimum or maximum, we
define the following spline class,
yij = β o + ∑ β Low _ a (x j − k )+ + ∑ β High _ b (x j − k )+
c
h
a
a=2
b
b=2
where k is the knot location, β denotes a regression coefficient,
(x
j
−k
)
−
{(
= min 0, x j − k
)} , ( x
j
−k
)
+
{(
= max 0, x j − k
)} ,
where c is the user-specified degree for the left polynomial, h is the
user-specified degree for the right polynomial, and the subscripts Low and
High denote to the left (below) and right of (above) the knot, respectively.
(
)
(
)
Note that by excluding terms in x j − k and x j − k we ensure that the
−
+
first derivative at the knot position is continuous. In addition, by definition the
constant β o must be equal to the value of the fit function at the knot, that is,
the value at the stationary point.
For this model class, response features can be chosen as
• Fit constants
{β
o
, β Low _ 2 , … , β Low _ p , β High _ 2 , β High _ q }
• Knot position { k }
• Value of the fit function at a user-specified delta
{∆a
j
= xj − k
}
{(
from the knot position f ± ∆a j
value is absolute.
)} if the datum is defined, otherwise the
• Difference between the value of the fit function at a user-specified delta from
the knot position and the value of the fit function at the knot
{ f ( ± ∆a ) − f ( k )}
j
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Truncated Power Series Basis (TPSBS) Splines
A very general class of spline functions with arbitrary (but strictly increasing)
knot sequence:
k = {k 1 , k 2 ,… , k k } :
T
( )
m −1
k
i=0
i =1
(
f x j ≅ ∑ β i x mj −1 + ∑ β m −1+ i x j − k i
)
m −1
+
This defines a spline of order m with knot sequence
k = {k 1 , k 2 ,… , k k }
T
For this model class, response features can be chosen as
{
• Fit constants β o , β 1 ,… , β m−1+ k
• Knot position vector { k} .
• Value of the fit function
}
{f (a )} at a user-specified value {a }
j
j
{( )}
• Value of the n derivative of the fit function with respect to xj f a j
a user-specified value a j , with n = 1, 2, …, m-2
th
{ }
Any of the polynomial terms can be removed from the model.
6-50
n
at
Local Model Definitions
Free Knot Splines
(
)
m −1
The x j − k i + basis is not the best suited for the purposes of estimation and
evaluation, as the design matrix might be poorly conditioned. In addition, the
number of arithmetic operations required to evaluate the spline function
depends on the location of x j relative to the knots. These properties can lead
to numeric inaccuracies, especially when the number of knots is large. You can
reduce such problems by employing B-splines.
The most important aspect is that for moderate m the design matrix expressed
in terms of B-splines is relatively well conditioned.
For this model class, response features can be chosen as
{
• Fit constants β −( m −1) , β − m ,… , β k
• Knot position vector { k} .
• Value of the fit function
}
{f (a )} at a user specified value {a }
j
j
Three Parameter Logistic Model
The three parameter logistic curve is defined by the equation
yj =
α
( (
1 + exp − κ x j − γ
)) r
where α is the final size achieved, κ is a scale parameter, and γ is the
x-ordinate of the point of inflection of the curve.
The curve has asymptotes yj = 0 as xj → -∞ and y j = α as xj → ∞. Growth rate
is at a maximum when yj = α/2, which occurs when xj = γ. Maximum growth
rate corresponds to
κα
------4
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The following constraints apply to the fit coefficients:
α > 0, κ > 0, γ > 0
The response feature vector g for the 3 parameter logistic function is defined as
g = α γ κ κα
------4
T
Morgan-Mercer-Flodin Model
The Morgan-Mercer-Flodin (MMF) growth model is defined by
yj = α −
α−β
( )
 1 + κx δ 
j


where α is the value of the upper asymptote, β is the value of the lower
asymptote, κ is a scaling parameter, and δ is a parameter that controls the
location of the point of inflection for the curve. The point of inflection is located
at
δ–1
x = -----------δ+1
1⁄δ
δ–2
y = -----------2δ
for δ ≥ 1
There is no point of inflection for δ < 1. All the MMF curves are sublogistic, in
the sense that the point of inflection is always located below 50% growth (0.5α).
The following constraints apply to the fit coefficient values:
α > 0, β > 0, κ > 0, δ > 0
α>β
The response feature vector g is given by
δ–1
g = α β κ δ ----------2δ
6-52
T
Local Model Definitions
Four-Parameter Logistic Curve
The four-parameter logistic model is defined by
yj = β +
α −β
( (
1 + exp − κ log( x j ) − γ
))
with constraints α , β , κ , γ > 0 , β < α and β < γ < α . Again, α is the value of
the upper asymptote, κ is a scaling factor, and γ is a factor that locates the
x-ordinate of the point of inflection at

1+ κ 

 κγ − log
 κ − 1 

exp


k




The following constraints apply to the fit coefficient values:
• α > 0, β > 0, κ > 0, γ > 0
•α > β
•α > γ > β
This the available response feature vector:
+κ
 κα – log  1
------------- 
g =

 κ – 1 
α β κ γ ---------------------------------------------κ
T
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Richards Curves
The Richards curves family of growth models is defined by the equation
[
( (
y j = α 1 + (δ − 1) exp − κ x j − γ
) )]
1
( 1− δ )
δ≠1
where α is the upper asymptote, γ is the location of the point of inflection on the
x axis, κ is a scaling factor, and δ is a parameter that indirectly locates the point
of inflection. The y-ordinate of the point of inflection is determined from
α
---------------------- δ > 0
1 ⁄ (δ – 1)
δ
Richards also derived the average normalized growth rate for the curve as
κ
--------------------2( δ + 1)
The following constraints apply to the fit coefficient values:
• α > 0, γ > 0, κ > 0, δ > 0
•α >γ
•δ ≠ 1
Finally, the response feature vector g for Richards family of growth curves is
defined as
δκ
g = α γ κ δ -----------------δ2δ + 1
6-54
T
Local Model Definitions
Weibul Growth Curve
The Weibul growth curve is defined by the equation
( )
δ
y j = α − ( α − β) exp − κx j 
where α is the value of the upper curve asymptote, β is the value of the lower
curve asymptote, κ is a scaling parameter, and δ is a parameter that controls
the x-ordinate for the point of inflection for the curve at
δ–1 1⁄δ
1
---  ------------
 κ  δ 
The following constraints apply to the curve fit parameters:
• α > 0, β > 0, κ > 0, δ > 0
•α > β
The associated response feature vector is
δ–1
g = α β κ δ 1
---  ------------
 κ  δ 
1⁄δ
T
Exponential Growth Curve
The exponential growth model is defined by
(
y j = α − ( α − β) exp − κx j
)
where α is the value of the upper asymptote, β is the initial size, and κ is a scale
parameter (time constant controlling the growth rate). The following
constraints apply to the fit coefficients:
• α > 0, β > 0, κ > 0
•α > β
The response feature vector g for the exponential growth model is defined as
[
g= α β κ
]
T
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Gompertz Growth Model
Another useful formulation that does not exhibit a symmetric point of
inflection is the Gompertz growth model. The defining equation is
−κ( x j−γ ) 

y j = α exp − e

where α is the final size achieved, κ is a scaling factor, and γ is the x-ordinate
of the point of inflection. The corresponding y-ordinate of the point of inflection
occurs at
α
--e
With maximum growth rate
κα
------e
The following constraints apply to the selection of parameter values for the
Gompertz model:
α > 0, κ > 0, γ > 0
The response feature vector g for the Gompertz growth model is defined as
κα
g = αγκ ------e
6-56
T
Neural Networks
Neural Networks
For help on the neural net models implemented in the MBC Toolbox, see the
documentation in the Neural Network Toolbox. At the MATLAB command
line, enter
doc nnet
The training algorithms available in MBC are traingdm, trainlm, and
trainbr.
These algorithms are a subset of the ones available in the Neural Network
Toolbox. (The names indicate the type: gradient with momentum, named after
the two authors, and bayesian reduction). Neural networks are inspired by
biology, and attempt to emulate learning processes in the brain.
Neural nets contain no preconceptions of what the model shape will be, so they
are ideal for cases with low system knowledge. They are useful for functional
prediction and system modeling where the physical processes are not
understood or are highly complex.
The disadvantage of neural nets is that they require a lot of data to give good
confidence in the results, so they are not suitable for small data sets. Also, with
higher numbers of inputs, the number of connections and hence the complexity
increase rapidly.
MBC provides an interface to some of the neural network capability of the
Neural Network Toolbox. Therefore these functions are only available if the
Neural Network Toolbox is installed. See the Neural Network Toolbox
documentation for more help.
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User-Defined Models
Throughout this section we refer to the template file
<MATLAB root>\toolbox\mbc\mbcmodel\@xregusermod\weibul.m
This file defines the local model Weibul in MBC and hence should not be
changed.
The xregusermod class allows the user to define new local models. When the
data being modeled at the local level matches the profile of (checked-in)
user-defined local models, these models appear as options on the Local Model
Setup dialog. At the end of the following section MBC offers the user-defined
model weibul under the category User-defined in the Local Model Setup
dialog.
To Begin
Decide on the function that you want to use. We will be using the Weibul
function:
y = alpha - (alpha - beta).*exp(-(kappa.*x).^delta)
Template File
Open the file
<MATLAB root>\toolbox\mbc\mbcmodel\@xregusermod\weibul.m
The m-file is called using
varargout= weibul(U,X,varargin)
Where the variables are given by
U = the xregusermod object
X = input data as a column vector for fast eval -ORX = specifies what sub-fcn to evaluate (not usually called
directly)
The first function in the template file is a vectorized evaluation of the function.
First the model parameters are extracted:
b= double(U);
6-58
User-Defined Models
Then the evaluation occurs:
y = b(1) - (b(1)-b(2)).*exp(-(b(3).*x).^b(4));
Note that the parameters are always referred to and declared in the same
order.
Subfunctions
Those subfunctions that must be edited are as follows:
function n= i_nfactors(U,b,varargin);
n= 1;
This is the number of input factors. For functions y = f(x) this is 1.
function n= i_numparams(U,b,varargin);
n= 4;
This is the number of fitted parameters. In this example there are four
parameters in the Weibul model.
function [param,OK]= i_initial(U,b,X,Y)
param= [2 1 1 1]';
OK=1;
This subfunction returns a column vector of initial values for the parameters
to be fitted. The initial values can be defined to be data dependent; hence there
is a flag to signal if the data is not worth fitting. In the template file weibul.m
there is a routine for calculating a data-dependent initial parameter estimate.
If no data is supplied, the default parameter values are used.
Optional Subfunctions
function [LB,UB,A,c,nlcon,optparams]=i_constraints(U,b,varargin)
LB=[0 0 0 0]';
UB=[1e10 1e10 1e10 1e10]';
A= [-1 1 0 0];
c= [0];
nlcon= 0;
optparams= [];
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Lower and upper bounds are stated for the model parameters. These are in the
same order that the parameters are declared and used throughout the
template file.
A linear constraint is defined:
b = the vector of model parameters, then the constraint is A*b <c
We define A and c in the subfunction above
The number of nonlinear constraints is declared to be zero. If the number of
nonlinear constraints is not zero, the nonlinear constraints are calculated in
i_nlconstraints.
No optional parameters are declared for the cost function.
function fopts= i_foptions(U,b,fopts)
fopts= optimset(fopts,'Display','iter');
The fit options are always based on the input fopts. See MATLAB help on the
function optimset for more information on fit options. When there are no
constraints the fitting is done using the MATLAB function lsqnonlin,
otherwise fmincon is used.
function J= i_jacobian(U,b,x)
x = x(:);
J= zeros(length(x),4);
a=b(1); beta=b(2); k=b(3); d=b(4);
ekd= exp(-(k.*x).^d);
j2= (a-beta).*(k.*x).^d.*ekd;
J(:,1)=
J(:,2)=
J(:,3)=
J(:,4)=
1-ekd;
ekd;
j2.*d./k;
j2.*log(k.*x);
To speed up the fitting algorithm an analytic Jacobian can be supplied, as it is
here.
function c= i_labels(U,b)
c={'\alpha','\beta','\kappa','\delta'};
6-60
User-Defined Models
These labels are used on plots and so on. Latex notation can be used and is
formatted.
function str= i_char(U,b,fopts)
s= get(U,'symbol');
str=sprintf('%.3g - (%.3g-%.3g)*exp(-(%.3g*x)^{%.3g})',...
b([1 1 2 3 4]));
This is the display equation string and can contain Latex expressions. The
current values of model parameters appear.
function str= i_str_func(U,b)
s= get(U,'symbol');
lab= labels(U);
str= sprintf('%s - (%s - %s)*exp(-(%s*x)^{%s})',...
lab{1},lab{1},lab{2},lab{3},lab{4});
This displays the function definition with labels appearing in place of the
parameters (not numerical values).
function rname= i_rfnames(U,b)
rname= {'INFLEX'};
This does not need to be defined (can return an empty array). Here we define a
response feature that is not one of the parameters (here it is also nonlinear).
function [rf,dG]= i_rfvals(U,b)
% response feature definition
rf= (1/b(3))*((b(4)-1)/b(4))^(1/b(4))
if nargout>1
% delrf/delbi
dG= [0, 0, -((b(4)-1)/b(4))^(1/b(4))/b(3)^2,...
1/b(3)*((b(4)-1)/b(4))^(1/b(4))*...
(-1/b(4)^2*log((b(4)-1)/b(4))+(1/b(4)-...
(b(4)-1)/b(4)^2)/(b(4)-1))];
end
The response feature (labeled as INFLEX above) is defined. The Jacobian is also
defined here as dG.
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function p= i_reconstruct(U,b,Yrf,dG,rfuser)
p= Yrf/dG';
f= find(rfuser>size(p,2));
if any(rfuser==4)
% need to use delta
p(:,3)= ((p(:,4)-1)./p(:,4)).^(1./p(:,4))./Yrf(:,f);
end
If all response features are linear in the parameters this function does not need
to be defined. Here we must first find out which response features (if any) are
user-defined. This subfunction allows the model parameters to be
reconstructed from the response features we have been given.
Checking into MBC
Having created a model template file, save it somewhere on the path.
To ensure that the model you have defined provides an interface that allows
MBC to evaluate and fit it, we check in the model. If this procedure succeeds,
the model is registered by MBC and is thereafter available for fitting at the
local level whenever appropriate input data is being used.
Check In
At the command line, with the template file on the path, create a model and
some input data, then call checkin. For the user-defined Weibul function, the
procedure is as follows:
Create an xregusermod using the template model file called weibul.
m = xregusermod('name','weibul');
Call checkin with the transient model, its name, and some appropriate data.
checkin(m, 'weibul', [0.1:0.01:0.2]');
This creates some command line output, and a figure appears with the model
name and variable names displayed over graphs of input and model evaluation
output. The final command line output (if checkin is called with a semicolon as
above) is
Model successfully registered for MBC
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User-Defined Models
The figure displayed for weibul is
MBC Modeling
Start up MBC and load data that has the necessary format to evaluate your
user-defined model.
Set up a Test Plan with one Stage1 input factor. The Local Model Setup now
offers the following options:
Here two user-defined models exist: exponential and weibul, as checked in.
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Transient Models
Transient models are supported in the Model-Based Calibration Toolbox, for
multiple input factors where time is one of the factors. You can define a
dynamic model using Simulink and a template file that describes parameters
to be fitted in this model. You must check these into the Model-Based
Calibration Toolbox before you can use them for modeling.
The following sections describe the process uing an example called fuelpuddle.
Throughout this section you use this Simulink model:
<MATLAB root>\toolbox\mbc\mbcsimulink\fuelPuddle.mdl
and the template file
<MATLAB root>\toolbox\mbc\mbcmodel\@xregtransient\fuelPuddle.m
The Simulink Model
Create a Simulink model to represent your transient system. This has some
free parameters that MBC fits by regression. There are three places to declare
these variables and they all need to match (see Parameters section below).
Any subsystems can be masked so that, at the highest level, the Simulink
model looks like this:
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Transient Models
The labels are not important. The model inputs are represented by Simulink
inports at the top level. The output is represented by a Simulink output. Only
one output is supported in MBC.
The model returns a single vector output (here labeled "Y").
Parameters
Inside the subsystem, some blocks depend on parameters that have not been
quantified. In fuelPuddle.mdl there is a constant block whose value is 'x' and
a gain block with the value '1/tau'.
These are the parameters we will be fitting by regression using MBC. The
following setup needs to be followed:
1 Having masked the subsystem, you need to pass the unknown parameters
in to the subsystem. We need to pass them in from the workspace. This is
done via a Block Parameters dialog. To create such a dialog, right-click the
masked subsystem and select Edit mask to see the Mask Editor. Use the
Add button to add fields. Here we require two parameters. The Variable
field must match the name of the corresponding parameter in the
subsystem. Having declared the necessary parameters (all the unknowns of
the subsystem), click OK to close and save.
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2 On double-clicking on the subsystem mask, the Block Parameters dialog
appears with the Prompt and variable names declared in the Mask Editor.
By default the variable names appear in the edit boxes, and these should not
be changed. It is these names that the Simulink model tries to find in the
workspace.
3 The same variable names must also be used in the template m-file that we
shall discuss later. For this example the template m-file is fuel Puddle.m.
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Transient Models
Checking the Model
Open your Simulink model and clear the MATLAB workspace.
Define the masked block parameters in the workspace. Here, for example, we
could enter at the command line
tau = 0.5;
x = 0.3;
In the subsystem, connect sources to the input factors (for example, use a pulse
generator at the input) and run the model (press play). If the model is taking
parameters from the workspace it should run okay.
Clear the workspace again and, without defining tau and x, try to run the
model; it should fail to run.
Transient Models
You can create a transient model at the command line:
m = xregtransient
m = functemplate([X1,X2],b=[0.5,0.5])
The default transient model is created. This model can be evaluated and fitted,
but it uses the default Simulink model, probably not a model that you are
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interested in! In order to create a transient model that uses your Simulink
model, you must first define a template file for your Simulink model and this
template file must have the same name as your Simulink model. The template
file for fuelPuddle.mdl is therefore called fuelPuddle.m.
Template File
Note that the template file has a common layout for transient models and for
user-defined models. Several sections of the template file are often not actively
used but must be filled in with sensible defaults.
Open the file
<MATLAB root>\toolbox\mbc\mbcmodel\@xregtransient\fuelPuddle.m
The fast eval at the top of the file is not used for transient models and can be
left as is.
The next section of commented code gives a summary of the functions in the
template file. The function definitions follow this. The m-file is called using
varargout= fuelPuddle(U,X,varargin)
U = the transient model
X = input data e.g. X = [[0:100] , sin([0:100] )]
Subfunctions
Those subfunctions that must be edited are as follows:
function vars= i_simvars(U,b,varargin);
vars = {'tau','x'};
These are the parameters of the Simulink model that will be fitted.
This subfunction must return a cell array of strings. These are the parameter
names that the Simulink model requires from the workspace. These strings
must match the parameter names declared in the Simulink model (see
“Parameters” on page 6-65).
function [vars,vals]= i_simconstants(U,b,varargin);
vars = {};
vals = [];
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Transient Models
These are constant parameters required by the Simulink model from the
workspace. They are not fitted. These parameters must be the same as those in
the Simulink model and all names must match. Here fuelPuddle requires no
such parameters, and hence we return an empty call array and empty matrix.
function [ic]= i_initcond(U,b,X);
ic=[];
Initial conditions (for the integrators) are based on the current parameters,
and inputs could be calculated here. The steady state can be passed in and you
can calculate this from supplied data.
function n= i_numparams(U,b,varargin);
n= 2;
This is the number of fitted parameters. For fuelPuddle we have two
parameters, x and tau.
function n= i_nfactors(U,b,varargin);
n= 2;
This is the number of input factors, including time. For fuelPuddle we input X
= [t, u(t)] and hence the number of input factors is 2.
function [param,OK]= i_initial(U,b,X,Y)
param= [.5 .5]';
OK=1;
This subfunction returns a column vector of initial values for the parameters
that are to be fitted. The initial values can be defined to be data-dependent,
hence there is a flag to signal if the data is not worth fitting. Here we simply
define some default initial values for x and tau.
Optional Subfunctions
The remaining subfunctions need not be edited unless they are required. The
comments in the code describe the role of each function. Mostly these functions
are used when you are creating a template for user-defined models. There is
only one of these subfunctions relevant here.
function c= i_labels(U,b)
b= {'\tau','x'};
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These labels are used on plots and so on. You can use Latex notation, and it is
correctly formatted.
Checking into MBC
Having created a Simulink model and the corresponding template file, save
each somewhere on the path. One place to put these files is
<MATLAB root>\toolbox\mbc\mbcsimulink\
To ensure that the transient model you have defined provides an interface that
allows MBC to evaluate and fit it, we check in the model. If this procedure
succeeds, the model is registered by MBC and is thereafter available for fitting
at the local level whenever appropriate input data is being used.
Check In
At the command line, with both template file and Simulink models on the path,
create a model and some input data, then call checkin. For fuelPuddle the
procedure would be
1 Create appropriate input data. The data is not important; it is to check if the
model can be evaluated.
t = [0:0.1:4*pi]'; u = sin(t); %% creates a sine wave input factor
X = [t,u];
2 Create a transient model using the template/Simulink model called
fuelPuddle:
m = xregtransient('name','fuelPuddle');
3 Call checkin with the transient model, its name, and the data.
checkin(m, 'fuelpuddle', X);
This creates some command line output, and a figure appears with the model
name and variable names displayed over graphs of input and model evaluation
output. The final command line output (if checkin is called with a semicolon as
above) is
Model successfully registered for MBC
A figure is displayed for fuelPuddle.
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Transient Models
MBC Modeling
With the fuelPuddle model successfully checked in, start up MBC modeling
and load data that has the necessary format to evaluate this transient model.
Set up a Test Plan with two Stage1 input factors (the same number of input
factors required by the fuelPuddle model). The Local Model Setup now offers
the following options:
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Select the fuelpuddle model and click OK.
On building a response model with fuelPuddle as the local model, MBC fits the
two parameters tau and x across all the tests.
Notes
In @xregtransient there is a template for fuelPuddleDelay, and the Simulink
model can be found in mbc\mbcsimulink. You can check this model in using
t = [0:30]'; u = sin(t);
X = [t,u];
m = xregtransient('name','fuelPuddleDelay');
checkin(m, 'fuelpuddledelay', X);
And now at the MBC Local Model Setup dialog, this model is also available.
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Data Loading Application Programming Interface
Data Loading Application Programming Interface
Data Loading API Specification
You can use the data loading API (application programming interface) to write
your own data loading function, plug these functions into the toolbox, and
subsequently use data loaded by these functions within the toolbox. To allow
this, there are several stages that need to be followed as described below. For
an example, see xregReadConcerto.m (in the mbctools directory).
Data Function Prototype
A function to successfully load data has the following prototype:
[OK, msg, out] = dataLoadingFcn(filename, protoOut)
Input Arguments
filename is the full path to the file to be loaded.
protoOut is an empty structure with the fields expected in the return
argument out. This allows the data loading API to be easily extended without
the need for data loading functions to change when MBC changes.
Output Arguments
The first return argument, OK, allows the function to signal that it has
successfully loaded the data file. A value of 1 signals success, and 0 failure. If
the function fails, it can return a message, msg, to indicate the reason for the
failure. This message is displayed in a warning dialog box if the function fails.
If the function is successful, the return argument out contains the data
necessary for MBC.
out.varNames is a cell array of strings that hold the names of the variables in
the data (1 x n or n x 1).
out.varUnits is a cell array of strings that hold the units associated with the
variables in varNames (1 x n or n x 1). This array can be empty, in which
case no units are defined.
out.data is an array that holds the values of the variables (m x n).
out.comment is an optional string holding comment information about the
data.
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Data Function Check In
Once you have written the function, you need to check it into the MBC Toolbox,
using the xregCheckinDataLoadingFcn function. This function has the
following prototype:
OK= xregCheckinDataLoadingFcn(fcn, filterSpec, fileType)
fcn is either a function handle or a string that calls the data loading function
via feval. This function must be on the MATLAB path.
filterSpec is a 1-by-2 element cell array that contains the extensions that this
function loads and the descriptions of those files. This cell array is used in the
uigetfile function, for example, {'*.m;*.fig;*.mat;*.mdl', 'All MATLAB
Files'}. MBC attempts to decide automatically which sort of file is loaded,
based on the extension. In the case of duplicate extensions, the first in the list
is selected; however, it is always possible to override the automatic selection
with a user selection. You will see a warning message if there is any ambiguity.
fileType is a string that describes the file type, for example, 'MATLAB file' or
'Excel file'.
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Radial Basis Functions
Guide to Radial Basis Functions for Model Building . . 7-3
Types of Radial Basis Functions . . . . . . . . . . . 7-4
Fitting Routines . . . . . . . . . . . . . . . . . . 7-12
Center Selection Algorithms . . . . . . . . . . . . . 7-13
Lambda Selection Algorithms . . . . . . . . . . . . 7-16
Width Selection Algorithms . . . . . . . . . . . . . 7-19
Prune Functionality
Statistics
. . . . . . . . . . . . . . . . 7-21
. . . . . . . . . . . . . . . . . . . . . 7-24
Hybrid Radial Basis Functions
. . . . . . . . . . . 7-28
Tips for Modeling with Radial Basis Functions . . . . 7-30
7
Radial Basis Functions
This Radial Basis Functions chapter is divided into the following sections:
• “Guide to Radial Basis Functions for Model Building” on page 7-3
• “Types of Radial Basis Functions” on page 7-4
• “Fitting Routines” on page 7-12
• “Center Selection Algorithms” on page 7-13
• “Lambda Selection Algorithms” on page 7-16
• “Width Selection Algorithms” on page 7-19
• “Prune Functionality” on page 7-21
• “Statistics” on page 7-24
• “Hybrid Radial Basis Functions” on page 7-28
• “Tips for Modeling with Radial Basis Functions” on page 7-30
7-2
Guide to Radial Basis Functions for Model Building
Guide to Radial Basis Functions for Model Building
A radial basis function has the form
z( x) = Φ( x – µ )
where x is a n-dimensional vector, µ is an n-dimensional vector called the
center of the radial basis function, ||.|| denotes Euclidean distance, and Φ is
a univariate function, defined for positive input values, that we shall refer to
as the profile function.
The model is built up as a linear combination of N radial basis functions with
N distinct centers. Given an input vector x, the output of the RBF network is
the activity vector ŷ given by
N
ŷ ( x ) =
∑ β j zj ( x )
j=1
where β j is the weight associated with the jth radial basis function, centered
at µ j , and z j = Φ ( x – µ j ) . The output ŷ approximates a target set of values
denoted by y.
A variety of radial basis functions are available in MBC, each characterized by
the form of Φ . All the radial basis functions also have an associated width
parameter, σ , which is related to the spread of the function around its center.
Selecting the box in the model setup provides a default setting for the width.
The default width is the average over the centers of the distance of each center
to its nearest neighbor. This is a heuristic given in Hassoun (see “References”
on page 7-27) for Gaussians, but it is only a rough guide that provides a
starting point for the width selection algorithm.
Another parameter associated with the radial basis functions is the
regularization parameter λ . This (usually small) positive parameter is used in
most of the fitting algorithms. The parameter λ penalizes large weights, which
tends to produce smoother approximations of y and to reduce the tendency of
the network to overfit (that is, to fit the target values y well, but to have poor
predictive capability).
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Radial Basis Functions
Types of Radial Basis Functions
Within the model setup, you can choose which RBF kernel to use. Kernels are
the types of RBF (multiquadric, gaussian, thinplate, and so on).
Gaussian
This is the radial basis function most commonly used in the neural network
community. Its profile function is
Φ(r) = e
( –r ⁄ σ )
2
2
This leads to the radial basis function
 x – µ 2
z ( x ) = exp  --------------------
 σ2 
In this case, the width parameter is the same as the standard deviation of the
Gaussian function.
7-4
Types of Radial Basis Functions
7-5
7
Radial Basis Functions
Thin-Plate Spline
This radial basis function is an example of a smoothing spline, as popularized
by Grace Wahba (http://www.stat.wisc.edu/~wahba/). They are usually
supplemented by low-order polynomial terms. Its profile function is
2
Φ ( r ) = ( r ⁄ σ ) log ( r ⁄ σ )
7-6
Types of Radial Basis Functions
Logistic Basis Function
These radial basis functions are mentioned in Hassoun (see “References” on
page 7-27). They have the profile function
1
Φ ( r ) = -----------------------------r
1 + exp(------)
2
σ
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7
Radial Basis Functions
Wendland’s Compactly Supported Function
These form a family of radial basis functions that have a piecewise polynomial
profile function and compact support [Wendland, see “References” on
page 7-27]. The member of the family to choose depends on the dimension of
the space (n) from which the data is drawn and the desired amount of
continuity of the polynomials.
Dimension
Continuity
Profile
n=1
0
Φ ( r )= (1 − r ) +
2
Φ (r )= (1 − r ) 3+ (3r + 1)
4
Φ(r )= (1 − r ) 5+ (8r 2 + 5r + 1)
0
Φ ( r )= (1 − r ) 2+
2
Φ(r )= (1 − r ) 4+ (4r + 1)
4
Φ(r )= (1 − r ) 6+ (35r 2 + 18r + 3)
0
Φ ( r )= (1 − r ) 3+
2
Φ(r )= (1 − r ) 5+ (5r + 1)
4
Φ(r )= (1 − r ) 7+ (16r 2 + 7 r + 1)
n=3
n=5
 a, a > 0
We have used the notation a + := 
for the positive part of a.
 0, a ≤ 0
When n is even, the radial basis function corresponding to dimension n+1 is
used.
Note that each of the radial basis functions is nonzero when r is in [0,1]. It is
possible to change the support to be [0,σ] by replacing r by r ⁄ σ in the
preceding formula. The parameter σ is still referred to as the width of the
radial basis function.
Similar formulas for the profile functions exist for n>5, and for even continuity
> 4. Wendland’s functions are available up to an even continuity of 6, and in
any space dimension n.
7-8
Types of Radial Basis Functions
Notes on Use
• Better approximation properties are usually associated with higher
continuity.
• For a given data set the width parameter for Wendland’s functions should be
larger than the width chosen for the Gaussian.
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Radial Basis Functions
Multiquadrics
These are a popular tool for scattered data fitting. They have the profile
function
Φ(r) =
7-10
2
r +σ
2
Types of Radial Basis Functions
Reciprocal Multiquadrics
These have the profile function
2
Φ( r) = 1 ⁄ r + σ
2
Note that a width σ of zero is invalid.
7-11
7
Radial Basis Functions
Fitting Routines
There are four characteristics of the RBF that need to be decided: weights,
centers, width, and λ . Each of these can have significant impact on the quality
of the resulting fit, and good values for each of them need to be determined. The
weights are always determined by specifying the centers, width, and λ , and
then solving an appropriate linear system of equations. However, the problem
of determining good centers, width, and λ in the first place is far from simple,
and is complicated by the strong dependencies among the parameters. For
example, the optimal λ varies considerably as the width parameter changes. A
global search over all possible center locations, width, and λ is computationally
prohibitive in all but the simplest of situations.
To try to combat this problem, the fitting routines come in three different
levels.
At the lowest level are the algorithms that choose appropriate centers for given
values of width and λ . The centers are chosen one at a time from a candidate
set (usually the set of data points or a subset of them). The resulting centers
are therefore ranked in a rough order of importance.
At the middle level are the algorithms that choose appropriate values for λ and
the centers, given a specified width.
At the top level are the algorithms that aim to find good values for each of the
centers, width, and λ . These top-level algorithms test different width values.
For each value of width, one of the middle-level algorithms is called that
determines good centers and values for λ .
7-12
Center Selection Algorithms
Center Selection Algorithms
Rols
This is the basic algorithm as described in Chen, Chng, and Alkadhimi [See
“References” on page 7-27]. In Rols (Regularized Orthogonal Least Squares)
the centers are chosen one at a time from a candidate set consisting of all the
data points or a subset thereof. It picks new centers in a forward selection
procedure. Starting from zero centers, at each step the center that reduces the
regularized error the most is selected. At each step the regression matrix X is
decomposed using the Gram-Schmidt algorithm into a product X = WB where
W has orthogonal columns and B is upper triangular with ones on the diagonal.
This is similar in nature to a QR decomposition. Regularized error is given by
e'e + λg'g where g = Bw and e is the residual, given by e = y – ŷ . Minimizing
regularized error makes the sum square error e'e small, while at the same time
not letting g'g get too large. As g is related to the weights by g = Bw, this has
the effect of keeping the weights under control and reducing overfit. The term
g'g rather than the sum of the squares of the weights w'w is used to improve
efficiency.
The algorithm terminates either when the maximum number of centers is
reached, or adding new centers does not decrease the regularized error ratio
significantly (controlled by a user-defined tolerance).
Fit Parameters
Maximum number of centers: The maximum number of centers that the
algorithm can select. The default is the smaller of 25 centers or ¼ of the number
of data points. The format is min(nObs/4, 25). You can enter a value (for
example, entering ten produces ten centers) or edit the existing formula (for
example, (nObs/2, 25) produces half the number of data points or 25,
whichever is smaller).
Percentage of data to be candidate centers: The percentage of the data
points that should be used as candidate centers. This determines the subset of
the data points that form the pool to select the centers from. The default is
100%, that is, to consider all the data points as possible new centers. This can
be reduced to speed up the execution time.
Regularized error tolerance: Controls how many centers are selected before
the algorithm stops. See Chen, Chng, and Alkadhimi [“References” on
page 7-27] for details. This parameter should be a positive number between 0
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Radial Basis Functions
and 1. Larger tolerances mean that fewer centers are selected. The default is
0.0001. If less than the maximum number of centers is being chosen, and you
want to force the selection of the maximum number, then reduce the tolerance
to epsilon (eps).
RedErr
RedErr stands for Reduced Error. This algorithm also starts from zero centers,
and selects centers in a forward selection procedure. The algorithm finds
(among the data points not yet selected) the data point with the largest
residual, and chooses that data point as the next center. This process is
repeated until the maximum number of centers is reached.
Fit Parameters
Only has Number of centers.
WiggleCenters
This algorithm is based on a heuristic that you should put more centers in a
region where there is more variation in the residual. For each data point, a set
of neighbors is identified as the data points within a distance of sqrt(nf) divided
by the maximum number of centers, where nf is the number of factors. The
average residuals within the set of neighbors is computed, then the amount of
wiggle of the residual in the region of that data point is defined to be the sum
of the squares of the differences between the residual at each neighbor and the
average residuals of the neighbors. The data point with the most wiggle is
selected to be the next center.
Fit Parameters
Almost as in the Rols algorithm, except no Regularized error.
CenterExchange
This algorithm takes a concept from optimal Design of Experiments and
applies it to the center selection problem in radial basis functions. A candidate
set of centers is generated by a Latin hypercube, a method that provides a
quasi-uniform distribution of points. From this candidate set, n centers are
chosen at random. This set is augmented by p new centers, then this set of n+p
centers is reduced to n by iteratively removing the center that yields the best
7-14
Center Selection Algorithms
PRESS statistic (as in stepwise). This process is repeated the number of times
specified in Number of augment/reduce cycles.
This is the only algorithm that permits centers that are not located at the data
points. The algorithm has the potential to be more flexible than the other
center selection algorithms that choose the centers to be a subset of the data
points; however, it is significantly more time-consuming and not recommended
on larger problems.
Fit Parameters
Number of centers: The number of centers that will be chosen.
Number of augment/reduce cycles: The number of times that the center set
is augmented, then reduced.
Number of centers to augment by: How many centers to augment by.
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Radial Basis Functions
Lambda Selection Algorithms
Lambda is the regularization parameter.
IterateRidge
For a specified width, this algorithm optimizes the regularization parameter
with respect to the GCV criterion (generalized cross-validation; see the
discussion under GCV criterion).
The initial centers are selected either by one of the low-level center selection
algorithms or the previous choice of centers is used (see discussion under the
parameter Do not reselect centers). You can select an initial start value for λ
by testing an initial number of values for lambda (set by the user) that are
equally spaced on a logarithmic scale between 10-10 and 10 and choosing the
one with the best GCV score. This helps avoid falling into local minima on the
GCV – λ curve. The parameter λ is then iterated to try to minimize GCV using
the formulas given in the GCV criterion section. The iteration stops when
either the maximum number of updates is reached or the log10(GCV) value
changes by less than the tolerance.
Fit Parameters
Center selection algorithm: The center selection algorithm to use.
Maximum number of updates: Maximum number of times that the update of
λ is made. The default is 10.
Minimum change in log10(GCV): Tolerance. This defines the stopping
criterion for iterating λ ; the update stops when the difference in the
log10(GCV) value is less than the tolerance. The default is 0.005.
Number of initial test values for lambda: Number of test values of λ to
determine a starting value for λ . Setting this parameter to 0 means that the
best λ so far is used.
Do not reselect centers for new width: This check box determines whether
the centers are reselected for the new width value, and after each lambda
update, or if the best centers to date are to be used. It is cheaper to keep the
best centers found so far, and often this is sufficient, but it can cause premature
convergence to a particular set of centers.
7-16
Lambda Selection Algorithms
Display: When you select this check box, this algorithm plots the results of the
algorithm. The starting point for λ is marked with a black circle. As λ is
updated, the new values are plotted as red crosses connected with red lines.
The best λ found is marked with a green asterisk.
A lower bound of 10-12 is placed on λ , and an upper bound of 10.
IterateRols
For a specified width, this algorithm optimizes the regularization parameter in
the Rols algorithm with respect to the GCV criterion. An initial fit and the
centers are selected by Rols using the user-supplied λ . As in IterateRidge, you
select an initial start value for λ by testing an initial number of start values
for lambda that are equally spaced on a logarithmic scale between 10-10 and 10,
and choosing the one with the best GCV score.
λ is then iterated to improve GCV. Each time that λ is updated, the center
selection process is repeated. This means that IterateRols is much more
computationally expensive than IterateRidge.
A lower bound of 10-12 is placed on λ , and an upper bound of 10.
Fit Parameters
Center selection algorithm: The center selection algorithm to use. For
IterateRols the only center selection algorithm available is Rols.
Maximum number of updates: The same as for IterateRidge.
Minimum change in log10(GCV): The same as for IterateRidge.
Number of initial test values for lambda: The same as for IterateRidge.
Do not reselect centers for new width: This check box determines whether
the centers are reselected for the new width value or if the best centers to date
are to be used.
Display: When you select this check box, this algorithm plots the results of the
algorithm. The starting point for λ is marked with a black circle. As λ is
updated, the new values are plotted as red crosses connected with red lines.
The best λ found is marked with a green asterisk.
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Radial Basis Functions
StepItRols
This algorithm combines the center-selection and lambda-selection processes.
Rather than waiting until all centers are selected before λ is updated (as with
the other lambda-selection algorithms), this algorithm offers the ability to
update λ after each center is selected. It is a forward selection algorithm that,
like Rols, selects centers on the basis of regularized error reduction. The
stopping criterion for StepItRols is on the basis of log10(GCV) changing by less
than the tolerance more than a specified number of times in a row (given in the
parameter Maximum number of times log10(GCV) change is minimal).
Once the addition of centers has stopped, the intermediate fit with the smallest
log10(GCV) is selected. This can involve removing some of the centers that
entered late in the algorithm.
Fit Parameters
Maximum number of centers: As in the Rols algorithm.
Percentage of data to candidate centers: As in the Rols algorithm.
Number of centers to add before updating: How many centers are selected
before iterating λ begins.
Minimum change in log10(GCV): Tolerance. It should be a positive number
between 0 and 1. The default is 0.005.
Maximum number of times log10(GCV) change is minimal: Controls how
many centers are selected before the algorithm stops. The default is 5. Left at
the default, the center selection stops when the log10(GCV) values change by
less than the tolerance five times in a row.
7-18
Width Selection Algorithms
Width Selection Algorithms
TrialWidths
This routine tests several width values by trialing different widths. A set of
trial widths equally spaced between specified initial upper and lower bounds is
selected. The width with the lowest value of log10(GCV) is selected. The area
around the best width is then tested in more detail - this is referred to as a
zoom. Specifically, the new range of trial widths is centered on the best width
found at the previous range, and the length of the interval from which the
widths are selected is reduced to 2/5 of the length of the interval at the previous
zoom. Before the new set of trial widths is tested, the center selection is
updated to reflect the best width and λ found so far. This can mean that the
location of the optimum width changes between zooms because of the new
center locations.
Fit Parameters
Lambda selection algorithm: Mid level fit algorithm that you test with the
various trial values of λ . The default is IterateRidge.
Number of trial widths in each zoom: Number of trials made at each zoom.
The widths tested are equally spaced between the initial upper and lower
bounds. Default is 10.
Number of zooms: Number of times you zoom in. Default is 5.
Initial lower bound on width: Lower bound on the width for the first zoom.
Default is 0.01.
Initial upper bound on width: Upper bound on the width for the first zoom.
Default is 20.
Display: If you select this check box, a stem plot of log10(GCV) against width
is plotted. The best width is marked by a green asterisk.
WidPerDim
In the WidPerDim algorithm (Width Per Dimension), the radial basis
functions are generalized. Rather than having a single width parameter, a
different width in each input factor can be used; that is, the level curves are
elliptical rather than circular (or spherical, with more factors). The basis
functions are no longer radially symmetric.
7-19
7
Radial Basis Functions
This can be especially helpful when the amount of variability varies
considerably in each input direction. This algorithm offers more flexibility than
TrialWidths, but is more computationally expensive.
Fit Parameters
As for the TrialWidths algorithm.
7-20
Prune Functionality
Prune Functionality
You can use the Prune function to reduce the number of centers in a radial
basis function network. This helps you decide how many centers are needed.
To use the Prune facility:
1 Select an RBF global model in the model tree.
2 Either click the
toolbar button or select the menu item Model –>
Utilities –> Prune.
The Number of Centers Selector dialog appears.
The graphs show how the fit quality of the network builds up as more RBFs are
added. It makes use of the fact that most of the center selection algorithms are
greedy in nature, and so the order in which the centers were selected roughly
reflects the order of importance of the basis functions.
The four fit criteria are the logarithms of PRESS, GCV, RMSE, and Weighted
PRESS. Weighted PRESS penalizes having more centers, and choosing
number of centers to minimize weighted PRESS is often a good option.
7-21
7
Radial Basis Functions
All four criteria in this typical example indicate the same minimum at eight
centers.
If the graphs all decrease, as in the preceding example, this suggests that the
maximum number of centers is too small, and the number of centers should be
increased.
7-22
Prune Functionality
Clicking the Minimize button selects the number of centers that minimizes the
criterion selected in the drop-down menu on the left. It is good if this value also
minimizes all the other criteria. The Clear button returns to the previous
selection.
Note that reducing the number of centers using prune only refits the linear
parameters (RBF weights). The nonlinear parameters (center locations, width,
and lambda) are not adjusted. You can perform a cheap width refit on exiting
the dialog by selecting the Refit widths on close check box. If a network has
been pruned significantly, you should use the update model fit toolbar button.
This performs a full refit of all the parameters.
7-23
7
Radial Basis Functions
Statistics
–1
Let A be the matrix such that the weights are given by β = A X'y where X is
the regression matrix. The form of A varies depending on the basic fit
algorithm employed.
In the case of ordinary least squares, we have A = X’X.
For ridge regression (with regularization parameter λ ), A is given by
A = X’X + λ I.
The most complicated situation is for the Rols algorithm. Recall that during the
Rols algorithm X is decomposed using the Gram-Schmidt algorithm to give X =
WB, where W has orthogonal columns and B is upper triangular. The
corresponding matrix A for Rols is then A = X'X + λB'B .
–1
The matrix H:=XA X' is called the hat matrix, and the leverage of the ith data
point hi is given by the ith diagonal element of H. All the statistics derived from
the hat matrix, for example, PRESS, studentized residuals, confidence
intervals, and Cook’s distance, are computed using the hat matrix appropriate
to the particular fit algorithm.
GCV Criterion
Generalized cross-validation(GCV) is a measure of the goodness of fit of a
model to the data that is minimized when the residuals are small, but not so
small that the network has overfitted the data. It is easy to compute, and
networks with small GCV values should have good predictive capability. It is
related to the PRESS statistic.
The definition of GCV is given by Orr (4, see “References” on page 7-27).
2
p ( y'P y )
GCV = ------------------------------2
( trace ( P ) )
where y is the target vector, and P is the projection matrix, given by I - XA-1XT.
An important feature of using GCV as a criterion for determining the optimal
network in our fit algorithms is the existence of update formulas for the
regularization parameter λ . These update formulas are obtained by
differentiating GCV with respect to λ and setting the result to zero. That is,
they are based on gradient-descent.
This gives the general equation (from Orr, 6, “References” on page 7-27)
7-24
Statistics
∂( Py )
2 ∂( trace ( P ) )
y'P --------------- trace ( P ) = ( y'P y ) -------------------------------∂λ
∂λ
We now specialize these formulas to the case of ridge regression and to the Rols
algorithm.
GCV for Ridge Regression
It is shown in Orr (4), and stated in Orr (5, see “References” on page 7-27) that
for the case of ridge regression GCV can be written as
p ( e'e )
GCV = -----------------2
(p – γ)
where γ is the “effective number of parameters” that is given by
–1
γ = N – λtrace ( A )
The formula for updating λ is given by
η
( e'e )
λ = ----------- ------------------------p–γ
–1 2
( β'A β )
where η = tr ( A
–1
–2
– λA )
In practice, the preceding formulas are not used explicitly in Orr (5, see
“References” on page 7-27). Instead a singular value decomposition of X is
made, and the formulas are rewritten in terms of the eigenvalues and
eigenvectors of the matrix XX’. This avoids taking the inverse of the matrix A,
and it can be used to cheaply compute GCV for many values of λ .
GCV for Rols
In the case of Rols, the components for the formula
2
p ( y'P y )
GCV = ------------------------------2
( trace ( P ) )
are computed using the formulas given in Orr [6, see “References” on
page 7-27]. Recall that the regression matrix is factored during the Rols
7-25
7
Radial Basis Functions
algorithm into the product X = WB. Let wj denote the jth column of W, then we
have
p
2
y'P y = y'y –
2
∑
j=1
( 2λ + w j''w j ) ( y'w j )
--------------------------------------------------2
( λ + w j''w j )
and
p
Trace ( P ) = N –
w j''w j
∑ ---------------------------( λ + w j''w j )
j=1
The reestimation formula for λ is given by
2
y'P y
η
λ = -------------------------- ------------------------- where additionally
Trace ( P )
–1 2
( β'A β )
p
η =
∑
j=1
w j''w j
–1
------------------------------- and β'A β =
2
( λ + w j''w j )
p
∑
j=1
2
( y'w j )
------------------------------3
( λ + w j''wj )
Note that these formulas for Rols do not require the explicit inversion of A.
General Points
1 You can view the location of the centers in graphical and table format by
using the
(View Centers) toolbar button. If a center coincides with a data
point, it is marked with a magenta asterisk on the Predicted/Observed plot.
2 You can alter the parameters in the fit by clicking the Set Up button in the
Model Selection dialog.
3 An estimation of the time for each of the top-level algorithms is computed.
This is given as a number of time units (as it depends on the machine). A
time estimate of over 10 but less than 100 generates a warning. A time
estimate of over 100 might take a prohibitively long amount of time
(probably over five minutes on most machines). You have the option to stop
execution and change some of the parameters to reduce the run time.
7-26
Statistics
4 If too many graphs are likely to be produced, because of the Display check
box’s being activated on a lower level algorithm, a warning is generated, and
you have the option to stop execution.
References
1 Chen S, Chng E.S., Alkadhimi, Regularized Orthogonal Least Squares
Algorithm for Constructing Radial Basis Function Networks, Int J. Control,
1996, Vol. 64, No. 5, pp. 829-837.
2 Hassoun, M., Fundamentals of Artificial Neural Networks, MIT, 1995.
3 Orr, M., Introduction to Radial Basis Function Networks, available from
http://www.anc.ed.ac.uk/~mjo/rbf.html.
4 Orr, M., Optimising the Widths of Radial Basis Functions, available from
http://www.anc.ed.ac.uk/~mjo/rbf.html.
5 Orr, M., Regularisation in the Selection of Radial Basis Function Centers,
available from http://www.anc.ed.ac.uk/~mjo/rbf.html.
6 Wendland, H., Piecewise Polynomials, Positive Definite and Compactly
Supported Radial Basis Functions of Minimal Degree, Advances in
Computational Mathematics 4 (1995), pp. 389-396.
7-27
7
Radial Basis Functions
Hybrid Radial Basis Functions
Hybrid RBFs combine a radial basis function model with more standard linear
models such as polynomials or hybrid splines. The two parts are added together
to form the overall model. This approach offers the ability to combine a priori
knowledge, such as the expectation of quadratic behavior in one of the
variables, with the nonparametric nature of RBFs.
The model setup GUI for hybrid RBFs has a top Set Up button, where you can
set the fitting algorithm and options. The interface also has two tabs, one to
specify the radial basis function part, and one for the linear model part.
Width Selection Algorithm: TrialWidths
This is the same algorithm as is used in ordinary RBFs, that is, a guided search
for the best width parameter.
Lambda and Term Selection Algorithms: Interlace
This algorithm is a generalization of StepItRols for RBFs. The algorithm
chooses radial basis functions and linear model terms in an interlaced way,
rather than in two steps. At each step a forward search procedure is performed
to select the radial basis function (with a center chosen from within the set of
data points) or the linear model term (chosen from the ones specified in the
linear model setup pane) that decreases the regularized error the most. This
process continues until the maximum number of terms is chosen. The first few
terms are added using the stored value of lambda. After StartLamUpdate
terms have been added, lambda is iterated after each center is added to
improve GCV.
The fit options for this algorithm are as follows:
• Maximum number of terms: Maximum number of terms that will be
chosen. The default is a quarter of the data points, or 25, whichever is
smaller.
• Maximum number of centers: Maximum number of terms that can be
radial basis functions. The default is the same as the maximum number of
terms.
• Percentage of data to be candidate centers: Percentage of the data points
that are available to be chosen as centers. The default is 100% when the
number of data points is <=200.
7-28
Hybrid Radial Basis Functions
• Number of terms to add before updating: How many terms to add before
updating lambda begins.
• Minimum change in log10(GCV): Tolerance.
• Maximum no. times log10(GCV) change is minimal: Number of steps in a
row that the change in log10(GCV) can be less than the tolerance before the
algorithm terminates.
Lambda and Term Selection Algorithms: TwoStep
This algorithm starts by fitting the linear model specified in the linear model
pane, and then fits a radial basis function network to the residual. You can
specify the linear model terms to include in the usual way using the term
selector. If desired, you can activate the stepwise options. In this case, after the
linear model part is fitted, some of the terms are automatically added or
removed before the RBF part is fitted. You can choose the algorithm and
options that are used to fit the nonlinear parameters of the RBF by pressing
the Set Up button in the RBF training options.
7-29
7
Radial Basis Functions
Tips for Modeling with Radial Basis Functions
Plan of Attack
Determine which parameters have the most impact on the fit by following
these steps:
1 Fit the default RBF. Remove any obvious outliers.
2 Get a rough idea of how many RBFs are going to be needed.
3 Τry with more than one kernel.
4 Decide on the main width selection algorithm. Try with both TrialWidths
and WidPerDim algorithms.
5 Determine which types of kernel look most hopeful.
6 Narrow the corresponding width range to search over.
7 Decide on the center selection algorithm.
8 Decide on the lambda-selection algorithm.
9 Try changing the parameters in the algorithms.
10 If any points appear to be possible outliers, try fitting the model both with
and without those points.
If at any stage you decide on a change that has a big impact (such as removal
of an outlier), then you should repeat the previous steps to determine whether
this would affect the path you have chosen.
The Model Browser has a quick option for comparing all the different RBF
kernels.
1 After fitting the default RBF, select the RBF global model in the model tree.
2 Click the
(Build Models) toolbar icon.
3 Select the RBF Kernels icon in the Build Models dialog that appears and
click OK.
7-30
Tips for Modeling with Radial Basis Functions
One of each kernel is built as a selection of child nodes of the current RBF
model.
How Many RBFs to Use
• The main parameter that you must adjust in order to get a good fit with an
RBF is the maximum number of centers. This is a parameter of the center
selection algorithm, and is the maximum number of centers/RBFs that is
chosen.
• Usually the maximum number of centers is the number of RBFs that are
actually selected. However, sometimes fewer RBFs are chosen because the
(regularized) error has fallen below the tolerance before the maximum was
reached.
• You should use a number of RBFs that is significantly less than the number
of data points, otherwise there are not enough degrees of freedom in the error
to estimate the predictive quality of the model. That is, you cannot tell if the
model is useful if you use too many RBFs. We would recommend an upper
bound of 60% on the ratio of number of RBFs to number of data points.
Having 80 centers when there are only 100 data points might seem to give a
good value of PRESS, but when it comes to validation, it can sometimes
become clear that the data has been overfitted, and the predictive capability
is not as good as PRESS would suggest.
• One strategy for choosing the number of RBFs is to fit more centers than you
think are needed (say 70 out of 100), then use the
(prune) toolbar button
to reduce the number of centers in the model. After pruning the network,
make a note of the reduced number of RBFs. Try fitting the model again with
the maximum number of centers set to this reduced number. This
recalculates the values of the nonlinear parameters (width and lambda) to be
optimal for the reduced number of RBFs.
• One strategy for the use of Stepwise is to use it to minimize PRESS as a final
fine-tuning for the network, once pruning has been done. Whereas Prune
only allows the last RBF introduced to be removed, Stepwise allows any RBF
to be taken out.
• Do not focus solely on PRESS as a measure of goodness of fit, especially at
large ratios of RBFs to data points. Take log10(GCV) into account also.
7-31
7
Radial Basis Functions
Width Selection Algorithms
• Try both TrialWidths and WidPerDim. The second algorithm offers more
flexibility, but is more computationally expensive. View the width values in
each direction to see if there is significant difference, to see whether it is
worth focusing effort on elliptical basis functions (use the
View Model
toolbar button).
• If with a variety of basis functions the widths do not vary significantly
between the dimensions, and the PRESS/GCV values are not significantly
improved using WidPerDim than TrialWidths, then focus on TrialWidths,
and just return to WidPerDim to fine-tune in the final stages.
• Turn the Display option on in TrialWidths to see the progress of the
algorithm. Watch for alternative regions within the width range that have
been prematurely neglected. The output log10(GCV) in the final zoom should
be similar for each of the widths trialed; that is, the output should be
approximately flat. If this is not the case, try increasing the number of
zooms.
• In TrialWidths, for each type of RBF, try to narrow the initial range of
widths to search over. This might allow the number of zooms to be reduced.
7-32
Tips for Modeling with Radial Basis Functions
Which RBF to Use
• It is hard to give rules of thumb on how to select the best RBF, as the best
choice is highly data-dependent. The best guideline is to try all of them with
both top-level algorithms (TrialWidths and WidPerDim) and with a
sensible number of centers, compare the PRESS/GCV values, then focus on
the ones that look most hopeful.
• If multiquadrics and thin-plate splines give poor results, it is worth trying
them in combination with low-order polynomials as a hybrid spline. Try
supplementing multiquadrics with a constant term and thin-plate splines
with linear (order 1) terms. See “Hybrid Radial Basis Functions” on
page 7-28.
• Watch out for conditioning problems with Gaussian kernels (say condition
number > 10^8).
• Watch out for strange results with Wendland’s functions when the ratio of
the number of parameters to the number of observations is high. When these
functions have a very small width, each basis function only contributes to the
fit at one data point. This is because its support only encompasses the one
basis function that is its center. The residuals will be zero at each of the data
points chosen as a center, and large at the other data points. This scenario
can indicate good RMSE values, but the predictive quality of the network
will be poor.
Lambda Selection Algorithms
Lambda is the regularization parameter.
• IterateRols updates the centers after each update of lambda. This makes it
more computationally intensive, but potentially leads to a better
combination of lambda and centers.
• StepItRols is sensitive to the setting of Number of centers to add before
updating. Switch the Display option on to view how log10(GCV) reduces as
the number of centers builds up.
• Examine the plots produced from the lambda selection algorithm, ignoring
the warning “An excessive number of plots will be produced.” Would
increasing the tolerance or the number of initial test values for lambda lead
to a better choice of lambda?
7-33
7
Radial Basis Functions
Center Selection Algorithms
• On most problems, Rols seems to be the most effective.
• If fewer than the maximum number of centers are being chosen, and you
want to force the selection of the maximum number, reduce the tolerance to
epsilon (eps).
• CenterExchange is very expensive, and you should not use this on large
problems. In this case, the other center selection algorithms that restrict the
centers to be a subset of the data points might not offer sufficient flexibility.
General Parameter Fine-Tuning
• Try Stepwise after pruning, then update the model fit with the new
maximum number of centers set to the number of terms left after Stepwise.
• Update the model fit after removal of outliers; use the toolbar button.
Hybrid RBFs
• Go to the linear part pane and specify the polynomial or spline terms that
you expect to see in the model.
Fitting too many non-RBF terms is made evident by a large value of lambda,
indicating that the underlying trends are being taken care of by the linear part.
In this case, you should reset the starting value of lambda (to say 0.001) before
the next fit.
7-34
Index
B
Box-Cox transformation 6-18
C
Design Editor 5-153
display options 5-156
displays 5-155
designing experiments 5-57
constraints manager 5-184
correlation models 5-56
covariance modeling 5-54
F
filter editor 5-72
filters 5-69
D
data 5-59
import wizard 5-65
loading from file 5-65
loading from the Workspace 5-66
loading user-defined functions 6-73
matching to designs 5-81
merging 5-66, 5-68
wizard 5-78
data editor 5-61
data selection window 5-81
datum models 5-88
definitions 6-5
design
adding points 5-178
classical 5-159
constraints 5-184
deleting points 5-181
fixing points 5-181
importing 5-183
optimal 5-168
saving 5-183
sorting points 5-181
space filling 5-162
styles 5-154
tree 5-156
tutorial 3-1
G
global level 5-105
global models
free knot spline 5-44
hybrid RBF 5-42
linear
hybrid spline 5-34
multiple 5-43
polynomial 5-33
radial basis functions 5-39
specifying inputs to 2-16
specifying type of 2-18
global regression 6-3
H
hierarchical models 6-37
I
inputs
setting up 5-30
interaction 5-37
I-1
Index
L
N
linear model statistics 6-22
local level 5-89
local model definitions 6-48
local model setup 5-46
local models
B-spline 5-51
growth models 5-52
linear 5-53
polynomial 5-47
polynomial spline 5-48
transient 5-54, 6-64
truncated power series 5-50
user-defined 5-54, 6-58
neural nets 6-57
new test plan templates 5-16
O
one-stage model
instant setup 5-27
P
Prediction Error Variance (PEV) viewer 5-193
project level 5-3
R
M
Maximum Likelihood Estimation 6-36
MLE 5-137
model building process overview 6-9
model evaluation 5-144
model selection 5-117
model setup
global 5-32
local 5-46
response 5-87
model tree 5-10
models
setting up 5-26
I-2
radial basis functions
fitting routines 7-12
guide 7-3
hybrid RBFs 7-28
prune functionality 7-21
statistics 7-24
tips for modeling 7-30
requirements, system 1-6
response level 5-141
S
stepwise 5-38
stepwise regression techniques 6-13
storage 5-74
system requirements 1-6
Index
T
V
test groupings 5-75
test plan level 5-19
test plans 5-15
transforms 5-56
tutorial
data editor 4-1
design of experiment 3-1
quick start 2-1
two-stage model
instant setup 5-28
variable editor 5-71
variables
user-defined 5-69
view
cross-section 5-133
likelihood 5-127
predicted/observed 5-122
residuals 5-132
response surface 5-124
RMSE 5-129
tests 5-120
viewing designs 5-57
I-3
Index
I-4
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