Foundational and Applied Statistics for Biologists using R Electronic Appendix (Introduction to R) Ken Aho updated 1/16/2017 Foundational and Applied Statistics for Biologists Using R (electronic appendix) INTRODUCTION TO R Ken Aho CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2014 Taylor & Francis Group, LLC Except as permitted under U. S. Copyright Law, no part of this book may be reprinted, reproduced, retransmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, included photocopying, microfilming, and recording in any information storage or retrieval system, without written permission from the publishers. Table of contents ii Contents A.1 Introduction to R �������������������������������������������������������������������������������������������������������������� 1 A.1.1 A.1.2 A.1.3 A.1.4 A.1.5 A brief history��������������������������������������������������������������������������������������������������������������������������������������� 1 The R language������������������������������������������������������������������������������������������������������������������������������������� 2 R copyrights and licenses �������������������������������������������������������������������������������������������������������������������2 R and reliability������������������������������������������������������������������������������������������������������������������������������������2 Installation �������������������������������������������������������������������������������������������������������������������������������������������� 3 A.2 Basics����������������������������������������������������������������������������������������������������������������������������������� 4 A.2.1 A.2.2 A.2.3 A.2.4 A.2.5 A.2.6 First operations������������������������������������������������������������������������������������������������������������������������������������� 4 Use your scroll keys ����������������������������������������������������������������������������������������������������������������������������5 Note to self: # ���������������������������������������������������������������������������������������������������������������������������������������5 Unfinished commands ������������������������������������������������������������������������������������������������������������������������6 Basic R options �������������������������������������������������������������������������������������������������������������������������������������6 Saving and loading your work ���������������������������������������������������������������������������������������������������������� 7 A.3 Getting help in R��������������������������������������������������������������������������������������������������������������� 9 A.3.1 help ������������������������������������������������������������������������������������������������������������������������������������������������������9 A.3.2 Manuals and additional information����������������������������������������������������������������������������������������������10 A.4 Expressions and assignments��������������������������������������������������������������������������������������� 12 A.4.1 Naming objects ���������������������������������������������������������������������������������������������������������������������������������� 12 A.4.2 Combining objects with c ����������������������������������������������������������������������������������������������������������������13 A.5 R objects and R classes�������������������������������������������������������������������������������������������������� 14 A.5.1 Listing objects ������������������������������������������������������������������������������������������������������������������������������������� 15 A.6 Mathematical operations����������������������������������������������������������������������������������������������� 16 A.6.1 Mathematical functions �������������������������������������������������������������������������������������������������������������������� 16 A.6.2 Statistical operators���������������������������������������������������������������������������������������������������������������������������� 20 A.7 Datasets in R packages��������������������������������������������������������������������������������������������������� 22 A.7.1 R and spreadsheets�����������������������������������������������������������������������������������������������������������������������������23 A.7.2 attach and detach ������������������������������������������������������������������������������������������������������������������������ 24 A.7.3 with ����������������������������������������������������������������������������������������������������������������������������������������������������24 A.7.4 remove and rm ���������������������������������������������������������������������������������������������������������������������������������� 25 A.7.5 Cleaning up ����������������������������������������������������������������������������������������������������������������������������������������25 A.8 R graphics������������������������������������������������������������������������������������������������������������������������� 26 A.8.1 plot ����������������������������������������������������������������������������������������������������������������������������������������������������26 A.8.2 Graphics devices����������������������������������������������������������������������������������������������������������������������������������31 A.8.3 par �������������������������������������������������������������������������������������������������������������������������������������������������������32 A.8.4 Exporting graphics�����������������������������������������������������������������������������������������������������������������������������33 A.8.5 Typeface families��������������������������������������������������������������������������������������������������������������������������������34 A.8.6 text, lines, points, paste ��������������������������������������������������������������������������������������������� 36 Table of contents iii A.8.7 plotmath ��������������������������������������������������������������������������������������������������������������������������������������������37 A.8.8 axis �����������������������������������������������������������������������������������������������������������������������������������������������������37 A.8.9 mtext ���������������������������������������������������������������������������������������������������������������������������������������������������38 A.8.10 R data visualization packages����������������������������������������������������������������������������������������������������������41 A.9 Data structures������������������������������������������������������������������������������������������������������������������42 A.10 Data entry at the command line ���������������������������������������������������������������������������������47 A.10.1 scan, cbind, rbind ��������������������������������������������������������������������������������������������������������������������47 A.10.2 Facilitating command line entry: seq and rep ���������������������������������������������������������������������������48 A.11 Importing data into R����������������������������������������������������������������������������������������������������49 A.11.1 A.11.2 A.11.3 A.11.4 A.11.5 read.table �������������������������������������������������������������������������������������������������������������������������������������49 read.csv ������������������������������������������������������������������������������������������������������������������������������������������50 scan ���������������������������������������������������������������������������������������������������������������������������������������������������50 Easy imports: use of file.choose() �����������������������������������������������������������������������������������������50 Additional comments�����������������������������������������������������������������������������������������������������������������������50 A.12 Exporting data from R���������������������������������������������������������������������������������������������������52 A.13 Subsetting matrix, dataframe and array components���������������������������������������������53 A.14 Operations on matrices and dataframes ������������������������������������������������������������������55 A.14.1 A.14.2 A.14.3 A.14.4 A.14.5 apply �������������������������������������������������������������������������������������������������������������������������������������������������56 tapply ����������������������������������������������������������������������������������������������������������������������������������������������57 outer �������������������������������������������������������������������������������������������������������������������������������������������������57 lower.tri, upper.tri and diag ���������������������������������������������������������������������������������������������58 stack and unstack �����������������������������������������������������������������������������������������������������������������������58 A.15 Logical commands���������������������������������������������������������������������������������������������������������60 A.15.1 ifelse ����������������������������������������������������������������������������������������������������������������������������������������������61 A.15.2 if, else, any, and all ������������������������������������������������������������������������������������������������������������������61 A.16 Simple functions for data management �������������������������������������������������������������������62 A.16.1 A.16.2 A.16.3 A.16.4 A.16.5 A.16.6 replace ��������������������������������������������������������������������������������������������������������������������������������������������62 which �������������������������������������������������������������������������������������������������������������������������������������������������62 sort ���������������������������������������������������������������������������������������������������������������������������������������������������63 rank ���������������������������������������������������������������������������������������������������������������������������������������������������63 order �������������������������������������������������������������������������������������������������������������������������������������������������63 unique ����������������������������������������������������������������������������������������������������������������������������������������������64 A.16.7 match �����������������������������������������������������������������������������������������������������������������������������������������������65 A.16.8 which and %in% �������������������������������������������������������������������������������������������������������������������������������66 A.16.9 strsplit and strtrim �����������������������������������������������������������������������������������������������������������66 A.16.10 Complex pattern matching: gsub, grep and metacharacters �������������������������������������������������67 A.17 Testing and coercing������������������������������������������������������������������������������������������������������69 Table of contents iv A.18 Binary numbers, bits and bytes����������������������������������������������������������������������������������71 A.18.1 Floating point arithmetic������������������������������������������������������������������������������������������������������������������72 A.18.2 Binary characters�������������������������������������������������������������������������������������������������������������������������������74 A.19 NA, NaN, and NULL ���������������������������������������������������������������������������������������������������������75 A.20 Writing functions�����������������������������������������������������������������������������������������������������������78 A.20.1 An introductory example ����������������������������������������������������������������������������������������������������������������79 A.20.2 Global variables versus local variables ������������������������������������������������������������������������������������������80 A.20.3 uniroot ��������������������������������������������������������������������������������������������������������������������������������������������83 A.20.4 switch ����������������������������������������������������������������������������������������������������������������������������������������������83 A.20.5 Triple dot ... argument �����������������������������������������������������������������������������������������������������������������84 A.20.6 Looping ����������������������������������������������������������������������������������������������������������������������������������������������90 A.20.7 Looping without for �����������������������������������������������������������������������������������������������������������������������94 A.20.8 Calling and receiving code from other languages������������������������������������������������������������������������95 A.20.9 Functions with animation ���������������������������������������������������������������������������������������������������������������98 A.20.10 Building GUIs����������������������������������������������������������������������������������������������������������������������������������99 A.20.11 Functions with class and method ������������������������������������������������������������������������������������������102 A.20.12 Documenting functions and workflow��������������������������������������������������������������������������������������103 A.21 Auxiliary packages �������������������������������������������������������������������������������������������������������108 A.22 Exploration of biological data using R��������������������������������������������������������������������110 Exercises�����������������������������������������������������������������������������������������������������������������������������������127 References��������������������������������������������������������������������������������������������������������������������������������134 Index�����������������������������������������������������������������������������������������������������������������������������������������137 Introduction to R 1 I believe that R currently represents the best medium for quality software in support of data analysis. John Chambers, a developer of S R is a real demonstration of the power of collaboration, and I don’t think you could construct something like this any other way. Ross Ihaka, original co-developer of R A.1 Introduction to R R is a computer language and an open source setting for statistics, data management, computation, and graphics. The outward mien of the R-environment is minimalistic, with only low-level interactive facilities. This is in contrast to conventional statistical packages consisting of extensive, high-level, often inflexible tools. The simplicity of R allows one to easily evaluate, edit, and build procedures for data analysis. R is useful to biologists for two major reasons. First, it provides access to a large number of existing statistical, graphical, and organizational procedures, many of which have been designed specifically for biological research. Second, it allows one to “get under the hood”, look at the code, and check to see what algorithms are doing. If, after examining an R-algorithm we are unsatisfied, we can generally modify its code or create new code to meet our specific needs. A.1.1 A brief history R was created in the early 1990s by Australian computational statisticians Ross Ihaka and Robert Gentleman to address scoping and memory use deficiencies in the language S (Ihaka and Gentleman 1996). At the insistence of Swiss statistician Martin Mächler, Ihaka and Gentleman made the R source code available via the internet in 1995. Because of its relatively easy-to-learn language, R was quickly extended with code and packages developed by its users (§ A.21). The rapid growth of R prompted the need for a group to guide its progress. This led, in 1997, to the establishment of the R-development core team, an international panel that modifies, troubleshoots, and manages source code. Because of its freeware status, the overall number of R-users is difficult to determine. Nonetheless, experts in the computer industry estimated that between 1 and 2 million individuals were actively using R in 20091. This number has undoubtedly increased as the number of R-packages has continued to increase exponentially (Fox 2009). Recent reports indicate that the internet traffic for R topics exceeds discussions for all other types of statistical software2. In addition, recent searches using Google scholar® (http://scholar.google.com/) show that the number of citations of R and its packages in scientific articles has increased dramatically since 2005 while citations for software giants SAS® and SPPS® have declined3,4. R is currently the most popular programming language in the world among data scientists5, and ranks 6th among Institute of Electrical and Electronics Engi1 http://bits.blogs.nytimes.com/2009/01/08/r-you-ready-for-r/ accessed 5/27/2016 http://r4stats.com/articles/popularity/ accessed 5/27/2016. 3 http://www.r-bloggers.com/rs-continued-growth-in-academia/ accessed 5/27/2016 4 For additional information on the historical and technical development of R see Hornick (2009) and Venebles et al. (2008). 5 http://blog.revolutionanalytics.com/2015/11/new-surveys-show-continued-popularity-of-r.html accessed 5/27/2016 2 Introduction to R 2 neers (IEEE) all-purpose programming languages6. A.1.2 The R language The R language is based on the older languages S, developed at Bell Laboratories (Becker and Chambers 1978, Becker and Chambers 1981, Becker et al. 1988), Scheme, developed initially at MIT (Sussman and Steele 1975, Steele 1978), and Lisp, also developed at MIT (McCarthy et al. 1960). Ihaka and Gentleman (1996) used the name R both to acknowledge the influence of S (because R and S are juxtaposed in the alphabet), and to celebrate their personal efforts (because R was the first letter of their first names). S has since been adapted to run the commercial software S-Plus®. The R and S languages remain very similar, and code written for S can generally be run unaltered in R. The method of function implementation in R, however, remains more similar to Scheme. R differs from S in two important ways (Ihaka and Gentelmen 1996). First, the R-environment is given a fixed amount of memory at startup. This is in contrast to S-engines which adjust available memory to session needs. Among other things, this difference means more available pre-reserved computer memory, and fewer virtual pagination7 problems in R (Ihaka and Gentelmen 1996). It also makes R faster than S-Plus for many applications (Hornik 2009). Second, R variable locations are lexically scoped. In computer science, variables are storage areas with identifiers, and scope defines the context in which a variable name is recognized. So-called global variables are accessible in every scope (for instance, both inside and outside functions). In contrast, local variables exist only within the scope of a function. Formal parameters defined in R functions, including arguments, are (generally) local variables, whereas variables created outside of functions are global variables. Lexical scoping allows functions in R access to variables that were in effect when the function was defined in a session. S, however, does not allow such free variables. The characteristics of R functions and details concerning lexical scoping are further addressed in § A.20. A.1.3 R copyrights and licenses R is free, and as a result there are no warranties on its environment or packages. As its copyright framework R uses the GNU (an acronym for not Unix) General Public License (GPL). This allows users to share and change R and its functions. The associated legalese can read by typing RShowDoc("COPYING")in the R-console. Because its functions can be legally (and easily) recycled and altered we should always give credit to developers, package maintainers, or whoever wrote the R functions we are using. A.1.4 R and reliability The lack of an R warranty has frightened away some scientists. But be assured, with few exceptions R works as well or better than “top of the line” analytical commercial software. Indeed, SAS® and SPSS® have recently made R applications accessible from within their products (Fox 2009). For specialized or advanced statistical techniques R often outperforms other alternatives because of its diverse array of continually updated applications. 6 http://spectrum.ieee.org/computing/software/the-2015-top-ten-programming-languages accessed 5/27/2016 (currently, the big five are Java, C, C++, Python, and C#) 7 Virtual pagination is a memory management scheme that allows a computer to store and retrieve data from secondary storage for use in main memory. Introduction to R 3 The computing engine and packages that come with a conventional R download (see § A.21) meet or exceed U.S. federal analytical standards for clinical trial research (R Foundation for Statistical Computing 2008). In addition, core algorithms used in R are based on seminal and trusted functions. For instance, R random number generators include some of the most venerated and thoroughly tested functions in computer history (Chambers 2008). Similarly, the LAPACK algorithms (Anderson et al. 1999), used by R for linear algebra, are among the world’s most stable and best-tested software (Chambers 2008). A.1.5 Installation To install R, first go to the website http://www.r-project.org/. On this page specify which platform you are using (Figure A.1, step 1). R can currently be used on Unix-like, Windows and Mac operating systems. This appendix has been created specifically for Windows users of R. In almost every case, however, it will also be applicable to other platforms. Once an operating system has been selected one can click on the “base” link to download the precompiled base binaries if R currently exists on your computer. If R has not been previously installed on your computer click on "Install R for the first time" (Figure A.1, step 2). You will delivered to a window containing a link to download the most recent version of R. Click on the "Download" link (Figure A.1, step 3). Figure A.1 Method for installing R for the 1st time. Two new versions of R are generally released each year, one in April and one in October. New packages (§ A.21), and newer versions of older packages are released on a continual basis at the discretion of their developers. Archived versions of R and its packages are also available from http://www.r-project.org/. Basics 4 A.2 Basics Upon opening R the first two things that will appear in the console of the Graphical User Interface (R-GUI) are the license disclaimers and the command line prompt, i.e., > (Figure A.2). The prompt indicates that R is ready for a command. All commands must begin at >. R user commands are colored red, and output, errors and warnings are colored blue. We can exit R at any time by typing q() in the console, closing the GUI window (non-Unix only), or by selecting “exit” from the pulldown File menu (non-Unix only). Figure A.2 The R console. R version 2.15.1, "Roasted Marshmallows" A.2.1 First operations As an introduction we can use R as a calculator. Type 2 + 2 and press Enter. 2 + 2 [1] 4 > The [1] means, “this is the first requested element”. In this case there is just one requested element, 4, the solution to 2 + 2. If the output elements cannot be held on a single console line, then R would begin the second line of output with the element number comprising the first element of the new line. For instance, the command rnorm(20) will take 20 random samples from a standard normal distribution (see Ch. 3 in the Foundational and Applied Statistics text). We have: rnorm(20) [1] 0.73704627 0.06572694 -0.19401659 1.55767302 -0.66656940 -0.63375586 [7] -0.38926816 0.46596203 -0.35382023 0.72729659 0.42944759 -0.50559415 [13] 0.95743654 0.49844963 0.01833043 -0.29257820 -0.56753070 0.25374833 [19] -0.27808670 -0.83199069 > The reappearance of the command line prompt indicates that R is ready for another command. Basics 5 Multiple commands can be entered on a single line, separated by semicolons. Note, however, that this is often considered poor programming style, as it may make your code more difficult to understand by a third party. 2 + 2; 3 + 2 A new command can be entered at the prompt or by separating commands with a semicolon. [1] 4 [1] 5 > R commands are generally insensitive to spaces. This allows the use of spaces to make code more legible. The command 2 + 2 is simply easier to read and debug than 2+2. A.2.2 Use your scroll keys As with many other command line environments, the scroll keys (Figure A.3) provide an important shortcut in R. Instead of editing a line of code by tediously mouse-searching for an earlier command to copy, paste and then modify it, you can simply scroll back through your earlier work using the upper scroll key, i.e., ↑. Accordingly, scrolling down using ↓ will allow you to move forward through earlier commands. Figure A.3 Direction keys on a keyboard. A.2.3 Note to self: # R will not recognize commands preceded by #. As a result this is a good way for us to leave messages to ourselves. # Note at beginning of line 2 + 2 [1] 4 2 + # Note in middle of line + 2 [1] 4 In the "best" code writing style it is recommended that one place a space after # before beginning a comment, and to insert two spaces following code before placing # in the middle of a line. This convention is followed Basics 6 above. A.2.4 Unfinished commands R will be unable to move on to a new task when a command line is unfinished. For example, type 2 + and press Enter. We note that a continuation prompt, by default + , is now in the place the command prompt should be. R is telling us the command is unfinished. We can get back to the command prompt by finishing the function, clicking Misc>Stop current computation or Misc>Stop all computations (non-Unix only), typing Ctrl-C (Unix only), or by hitting the Esc key (Windows only). A.2.5 Basic R options To enhance our experience, we can adjust the appearance of the R-console and customize options that affect function output. These include the characteristics of the graphics devices, the width of print output in the Rconsole, and the number of print lines and print digits. Changes to some of these parameters can be made by going to Edit>GUI Preferences in the R toolbar. Many other parameters can be changed using the options function. To see all alterable options one can type: options() To change default options one would simply define the desired change within parenthesis following the options function name. For instance, to see the default number of digits in R output I would type: options("digits") $digits [1] 7 To change the default number of digits in output from 7 to 5, in the current session I would type: options(digits = 5) To establish user defaults for all future sessions one will need to change the R-console file located (currently) in the etc directory of the version of R you are using. This can either be done manually, or by using Edit>GUI Preferences>Save in the R toolbar By default the R working directory is set to be the home directory of the workstation. The command getwd() shows the current file path for the working directory. getwd() [1] "C:/Users/User/Documents" The working directory can be changed with the command setwd(filepath), where filepath is the location Basics 7 of the directory, or by using pulldown menus, i.e., File>Change dir (non-Unix only). An .Rprofile file exists in your Windows R/R-version/etc directory. R will silently run commands in the file upon opening. By customizing the .Rprofile file one can set options, install frequently loaded non-default packages (§ A.21), define your favorite package repository, and create aliases and defaults for frequently used functions. Here is the content of my current .Rprofile file. options(repos = structure(c("http://ftp.osuosl.org/pub/cran/"))) .First <- function(){ library(asbio) cat("\nWelcome to R Ken! ", date(), "\n") } .Last <- function(){ cat("\nGoodbye Ken", date(), "\n") } The options(repos = structure(c("http://ftp.osuosl.org/pub/cran/"))) command defines my favorite R-repository. The function .First( ) will be run at the start of the R session and .Last( ) will be run at the end of the session. As we go through this primer it will become clear that these functions force R to say hello and to load the package asbio when it opens, and to say goodbye when it closes (although the farewell will only be seen when running R from a command line interface). The .Rprofile file in the /etc directory is the so-called .Rprofile.site file. Additional .Rprofile files can be placed in the working and user directories. R will check for these and run them after running the .Rprofile. site file. One can create .Rprofile files, or any type of R extension file using the function file.create. For instance, the code: file.create("defaults.Rprofile") places an empty, editable, .Rprofile file called defaults in the working directory. A.2.6 Saving and loading your work As noted earlier, an R session is allocated with a fixed amount of memory that is managed in an on-the-fly manner. An unfortunate consequence of this is that if R crashes all unsaved information from the work session will be lost. Thus, session work should be saved often. Note that R will not give a warning if you are writing over another file from the R console with the same name! The old file will simply be replaced. To save the history (i.e., all of the commands that have been used in a session) one can use File>Save History (non-Unix only) or the function savehistory. For instance, to save the session history to the working directory under the name history1 I could type: Basics 8 savehistory(file = "history1.Rhistroy") We can view the code in this file from any ascii text editor. To load the history from a previous session one can use File>Load History (non-Unix only) or the function loadhistroy. For instance, to load history1 I would type: loadhistory(file = "history1.Rhistroy") Importantly, the functions savehistory() and loadhistory() are not currently supported for Macs. To save all of the objects (see § A.5) available in the current R-session one can use File>Save Workspace (nonUnix only), or simply type save.image() These procedures save session objects to the working directory as a nameless file with an .RData extension. This file will be opened, silently, with the inception of the next R-session, and cause objects used or created in the previous session to be available. Indeed, R will automatically execute all .RData files in the working directory for use in a session. .RData files can also be loaded using File>Load Workspace (non-Unix only). One can also save the .RData objects to a specific directory location and use a specific file name using File>Save Workspace, or with the file argument in save.image, or with the flexible function save (see § A.12). R data file formats, including .rda, and .RData, (extensions for R data files), and .R (a format for R scripts), can be read into R using the function load. Users new to a command line environment will be reassured by typing: load(file.choose()) This will allow one to browse interactively for files using dialog boxes. Detailed procedures for importing and exporting (saving) data with a row and column format, and an explicit delimiter (e.g. .csv files) are described in § A.11 and A.12, respectively. To save R code as executable source file, one can save the code in the R-script editor (go to File>New script, or simply type fix()) or within some other R-compatible text editor (e.g., ESS, TinnR, etc.) as an .R extension file. R scripts can be called and executed using the function source. For instance to go looking interactively for source code to execute I could type: source(file.choose()) or go to File>Source R code. Getting help 9 A.3 Getting help in R There is no single perfect source for information/documentation for all aspects of R, for all levels of useRs. Detailed information concerning base operations and package development are described at the R website http:// www.r-project.org/, but is intended for those familiar with both Unix/Linux systems, and command based formats. They may not be especially helpful to new R users used to point-and-click formats, and with no familiarity with programming. A.3.1 help A comprehensive help system is built into R. The system can be accessed question mark (?) and help functions. For instance, if I wanted to know more about the plot function I could type: ?plot or help(plot) Documentation for functions will include a list of arguments for functions, and a description of variables for datasets, and other pertinent information. Quality of documentation will generally be excellent for functions from packages in the default R download, but will vary from package to package otherwise. A list of arguments for a function, and their default values, can (often) be obtained using the function formals. formals(plot) $x $y $... For help and information regarding programming metacharacters used in R (for instance $, ?, [[, &, or ^) I would enclose the characters with quotes. For example, to find out more information about the logical operator & I could type help("&") or ? "&". Placing two question marks in front of a word or function will cause R to search for help files concerning with respect to all libraries in a workstation. For instance type: ??lm or, alternatively help.search(lm) for a huge number of help files on linear model functions identified through fuzzy matching. Help for particular R-questions can often be found online using the search engine at http://search.r-project.org/. This link is Getting help 10 provided in the Help pulldown menu in the R console (non-Unix only). The function demo allows one access to examples developers have worked out (complete with R code) for a particular function or topic. For instance, type: demo(graphics) for a brief demonstration of R graphics or demo(persp)for a demonstration of 3D perspective plots. Typing: demo(Hershey) will provide a demonstration of available of modifiable symbols from the Hershey family of fonts (see Hershey 1967; A.8.5). Typing: demo() lists all of the demos available in the loaded libraries for a particular workstation. The function example usually provides less involved demonstrations from the man directories (short for user manual) in packages, along with code requirements. For instance, type example(plotmath) for a demonstration of mathematical graphics complete with code. R packages often contain vignettes. These are short documents that (generally) describe the theory underlying algorithms and how to use them. Vignettes can be accessed with the function vignette. To view all available vignettes for packages attached for a current work session type: vignette(all = FALSE) To view all vignettes for all installed packages type: vignette(all = TRUE) To view all vignettes for the installed package asbio: vignette(package = "asbio") To see the vignette simpson in package asbio I would load asbio, then type: vignette(simpson) The function browseVignettes() provides an HTML-browser that allows interactive vignette searches. A.3.2 Manuals and additional information Because the Foundational and Applied Statistics for Biologists using R textbook is primarily about statistics and not R, this appendix is not intended to be an exhaustive operators manual. General guidance for R-programming can be found in Chambers (2008), or more recently Matloff (2011). In addition to Foundational and Getting help 11 Applied Statistics, good resources for R in an applied statistical context include Venables and Ripley (2002), Crawley (2007), Adler (2010) and Fox (2011). The R website http://www.r-project.org/ provides access to the R journal, and a large number of helpful (and free) vignettes and manuals. R also has an a large and engaged electronic community that actively considers user programming questions, see www.stackoverflow.com/questions/ tagged/r, or http://statsexchange.com/questions/tagged/r, or even www.twitter.com/search/rstats. Expressions and Assignments 12 A.4 Expressions and assignments All entries in R are either expressions or assignments. If a command is an expression it will be evaluated, printed and discarded. Examples include: 2 + 2. Conversely, an assignment evaluates an expression, and assigns the output to an R-object (§ A.5), but does not automatically print the results. To convert an expression to an assignment we use the assignment operator, <-, which represents an arrow. The assignment operator can go on either side of an object. For instance, if I type: x <- 2 + 2 or 2 + 2 <- x then the sum 2 + 2 will be assigned to the object x. To print the result (to see x) I simply type: x [1] 4 or print(x) [1] 4 Note that we could have also typed x = 2 + 2 with the same assignment results. However in this document we will continue to use the arrow operator for assignments, and save = for specifying arguments. A.4.1 Naming objects When assigning names to R-objects we should try to keep the names simple, and avoid names that already represent important definitions and functions. These include: TRUE, FALSE, NULL, NA, NaN, and Inf. In addition, we cannot have names: • beginning with a numeric value, • containing spaces, colons, and semicolons, • containing mathematical operators (e.g., *, +, -, ^, /, =), • containing important R metacharacters (e.g., @, #, ?, !, %, &, |). Names should also be self explanatory. Thus, for a object containing 20 random observations from a normal distribution, the name rN20 is far superior to the, albeit easily-typed, name a. Finally, with assignment commands we should also remember that, like most software developed under Unix/Linux, R is case sensitive. That is, each of the following 24 combinations will be recognized as distinct: name, Name, nAme, naMe, namE, NAme, nAMe, naME, NaMe, nAmE, NamE, naME, NAMe, nAME, NaME, NAmE, NAME. Expressions and Assignments A.4.2 Combining objects with c 13 To define a collection of numbers (or other data, or objects) as a single entity, one can use the important R function c, which means "combine." For instance, to define the numbers 23, 34, and 10 as the object x, I would type: x <- c(23, 34, 10) We could then do something like: x + 7 [1] 30 41 17 Objects 14 A.5 R objects and R classes Before proceeding further it is important to make two observations. • First, everything created or loaded in R is an object. This means, in the sense of object oriented programming (OOP), that they possess certain traits that describe them8. • Second, every R-object belongs to at least one class. This requirement means that R-objects can be classified in a way that allows them to be recognized and used correctly by both users and other objects. Object classes allow for custom utility functions, e.g., plot, print, summary, to be created for particular classes. These can be called by simply using generic function names e.g., plot, print, summary (Section A.20.11). Among other things, this means fewer function names to memorize. For the object x below: x <- 2 + 2 we have the following class: class(x) [1] "numeric" Thus, the object x belongs to class numeric. Objects in class numeric can be evaluated mathematically. Objects can also be distinguished by their storage mode or type (the way R caches them in its primary memory). For x we have: typeof(x) [1] "double" Meaning that x is stored with 64 bit double precision. Object type names and class names are often identical. Many R-objects will also have attributes (characteristics particular to the object or object class). Typing: attributes(x) NULL indicates that x does not have additional attributes. However, using coercion (§ A.17) we can define x as an object of class matrix (a collection of data in a row and column format): x <- as.matrix(x) Now x has the attribute dim (i.e., dimension). 8 OOP languages include R, C#, C++, Objective-C, Smalltalk, Java, Perl, Python and PHP. C is not considered an OOP language. Objects attributes(x) $dim [1] 1 1 Now x is a one-celled matrix. It has one row and one column. Amazingly, classes and attributes allow R to simultaneously store and distinguish objects with the same name9. For instance: mean <- mean(c(1, 2, 3)) mean [1] 2 mean(c(1, 2, 3)) [1] 2 In general it is not advisable to name objects after frequently used functions. Nonetheless, the function mean is distinguishable from the new user-created object mean because these objects have different identifiable characteristics. A.5.1 Listing objects By typing: ls() or objects() one can see a list of the R-objects created or loaded in a particular R session. Currently, we have: ls() [1] "mean" "x" 9 Note, however, naming conflicts with global variables often causes issues with package development. The code: mean <- NA; rm(mean) will remove the offending user created object mean above, leaving behind the global variable. 15 16 Mathematical expressions A.6 Mathematical operations A.6.1 Mathematical functions Usual (and unusual) mathematical operators are easily specified in R. Below are a few examples; x represents either a user-specified scalar (single number) or a vector (collection of numbers). +addition -subtraction *multiplication /division ^exponentiation %%modulo (find the remainder in division) Inf∞ -Inf -∞ piπ = 3.141593… exp(1)e = 2.718282… exp(x)ex sqrt(x) x log(x)loge(x) log(x, 10)log10(x) factorial(x)x! n x choose(n, x) gamma(x) Γ (x) cos(x)cosine (x) sin(x)sin(x) tan(x)tangent(x) acos(x) arccosin(x) asin(x)arcsin(x) atan(x)arctan(x) abs(x)absolute value of x round(x, digits) rounds x to the digits given in digits floor(x) rounds x down to closest whole number D(expression(y), x) dy δy or dx δx ceiling(x)rounds x up to closest whole number a integrate(function(x), a, b) ∫ f ( x)dx b 17 Mathematical expressions min(x)minimum value in x max(x)maximum value in x ∑ prod(x) ∏ sum(x) n x i =1 i n x i =1 i cumsum(x)cumulative sum of elements in x cumprod(x)cumulative product of elements in x As with functions from all programming languages R functions generally require a user to specify arguments (in parentheses) following the function name. For instance, sqrt and factorial each require one argument, a call to data. Thus, to solve 1/ 22! , I type: 1/sqrt(factorial(22)) [1] 2.982749e-11 and to solve Γ ( 3 ) 23π , I type: gamma((23 * pi)^(1/3)) [1] 7.410959 The log function can also compute logarithms to a specified base by supplying the base in an optional second argument called base. For instance, to solve the operation: log10 (3) + log 3 (5) , one would simply type: log(3, 10) + log(5, 3) [1] 1.942095 By default the function log computes natural logarithms, i.e., log(exp(1)) [1] 1 That is, by default base = Euler’s constant = e = 2.718282... . Arguments can be specified by the order that they occur in the list of arguments in the function code, or by calling the argument by name. In the code above I know that the first argument in log is a call to data, and the second argument is the base. I may not, however, remember the argument order in a function, or may wish to only change certain arguments from a large allotment. In this case it is better to specify the argument by calling its name and defining its value with an equals sign. log(x = 3, base = 10) + log(x = 5, base = 3) [1] 1.942095 18 Mathematical expressions Trigonometry R assumes that the inputs for the trigonometric functions are in radians. Of course to get degrees from radians we can use Deg = Rad ×180 / π , or conversely Rad = Deg × π /180 . For instance, to find cos(45º) I first convert 45 degrees to radians. cos(45 * pi/180) [1] 0.7071068 This is correct because cos(45º) = 2 / 2 = 0.7071068 . sqrt(2)/2 [1] 0.7071068 Derivatives The function D finds symbolic and numerical derivatives of simple expressions. It requires two arguments, a mathematical function specified as an expression [i.e., a list (§ A.9) that can be evaluated with the function eval], and the denominator in the difference quotient. Here is an example of how expression and eval are used: eval(expression(2 + 2)) [1] 4 Of course we wouldn’t bother to use expression and eval in this simple application. || Example A.1 To demonstrate expression and D (the latter calls eval) we will evaluate the following derivatives: d 0.5 x 4 , dx d log( x), and dx d cos(2 x). dx D(expression(0.5 * x^4), "x") 0.5 * (4 * x^3) D(expression(log(x)),"x") 1/x D(expression(cos(2 * x)),"x") -(sin(2 * x) * 2) 19 Mathematical expressions Symbolic solutions to partial derivatives are also possible. For instance, consider We have: δ 3 x 2 + 4 xy . δx D(expression(3 * x^2 + (4 * x * y)), "x") 3 * (2 * x) + 4 * y Integration Integration requires specifying an integrand, in the form of a function. A function is defined by a function named function. Arguments will be contained in a set of parentheses following the call to function. The function contents follow, generally delineated by curly brackets. Thus, we have the following form for a function called "example" with n arguments: example <- function(argument1, argument2,...., argumentn){function contents} Recall that the entity <- is called the command line operator (see § A.4). Functions are explained in detail in (§ A.20). The R function for integration, integrate, requires, as its 1st argument a user-defined function describing the integrand. The integrand function, must have as its 1st argument the name of the variable to be integrated. The 2nd and 3rd arguments for integrate will be the lower and upper bounds for definite integration. || Example A.2 To solve for: 3 ∫ 3x dx 2 2 we can use the following code: f <- function(x){3 * x^2} integrate(f, 2, 3) 19 with absolute error < 2.1e-13 A large number of other mathematical operations can be handled by R including multivariate integration (package cubature, Exercise 3), solving systems of differential equations (package deSolve, || Example A.11, Exercise 16), linear algebra (Exercise 4), and solving systems of linear equations (function solve). Linear algebra and 20 Mathematical expressions associated R functions are discussed in the Appendix for mathematics review in the Foundational and Applied Statistics for Biologists textbook. Some mathematical operators will ordinarily be used with a collection of data, i.e., a vector, not a single datum. These include min, max, sum and prod, along with most statistical operators. x <- c(23, 34, 10) prod(x) [1] 7820 A.6.2 Statistical operators R, of course, contains a huge number of functions for statistical point and interval estimates. Below is an extremely abbreviated list of statistical functions and their meaning. Some of the listed functions require the library asbio. The object x represents a user-specified vector (collection of numbers). mean(x) Arithmetic mean of x mean(x, trim = 0.2) Arithmetic mean of x with 20% trimming G.mean(x) Geometric mean of x H.mean(x) Harmonic mean of x median(x) Median of x HL.mean(x) Pseudomedian of x huber.mu(x) Huber M-estimator of location for x Mode(x) Mode(s) of x var(x) Variance of x sd(x) Standard deviation of x IQR(x) Interquartile range of x mad(x) Median absolute deviation of x quantile(x) Quantiles of x at the 0th, 25th, 50th, 75th and 100th percentiles quantile(x, probs = 0.75) Quantile of x at the 75th percentile skew(x) Skewness of x kurt(x) Kurtosis of x range(x) Range of x length(x) Number of observations in x length(x[is.na(x)]) Number of missing observations in x ci.mu.z(x, conf = 0.95) 95% confidence interval for μ, σ2 known ci.mu.t(x, conf = 0.95) 95% confidence interval for μ, σ2 unknown ci.median(x, conf = 0.95) 95% confidence interval for the true median 21 Mathematical expressions ci.sigma(x, conf = 0.95) 95% confidence interval for σ2 ci.p(x, conf = 0.95) 95% confidence interval for the binomial parameter π cov(x, y) Covariance of x and y cor(x, y) Pearson correlation of x and y For instance, consider the dataset x below: x <- c(1, 2, 3, 2, 1) The mean is mean(x) [1] 1.8 and the median is: median(x) [1] 2 The formulae and underlying theory for point estimators such the mean, variance, and median are dealt with in Chapter 4 in the Foundational and Applied Statistics text. Interval estimators are addressed in Ch. 5. Datasets 22 A.7 Datasets in R packages At this point it would be helpful to have a dataset to play with. The command: data(package = "base") lets us see all of the datasets available in the R base package while the command: data() lets us see all the datasets available within all the packages loaded onto a particular computer workstation. We will learn how to enter our own data into R shortly. || Example A.3 -- A brief R session The dataset Loblolly in the base package describes the height and age of loblolly pine trees (Pinus taeda), a commercially important species in the southeastern United States. To make the dataset available, we type: data(Loblolly) We can then type: Loblolly to take a look at the entire dataset. Loblolly height age Seed 1 4.51 3 301 15 10.89 5 301 29 28.72 10 301 43 41.74 15 301 57 52.70 20 301 71 60.92 25 301 …………………………………………… 70 49.12 20 331 84 59.49 25 331 We can type head(Loblolly) To view just the first few lines of data: head(Loblolly) height age Seed 1 4.51 3 301 15 10.89 5 301 29 28.72 10 301 43 41.74 15 301 57 52.70 20 301 71 60.92 25 301 Because the name Loblolly is rather difficult to type, we can use the assignment operator to call the dataset something else. To assign Loblolly the name lp I simply type: lp <- Loblolly Note that Loblolly still exists, lp is just a copy. Datasets 23 Typing: summary(lp) provides a simple statistical summary of lp, i.e. the minimum response, 1st quartile, median (or 2nd quartile), mean, 3rd quartile and maximum response. These statistics (excluding the mean) are known as the 5 number summary. summary(lp) height Min. : 3.46 1st Qu.:10.47 Median :34.00 Mean :32.36 3rd Qu.:51.36 Max. :64.10 age Min. : 3.0 1st Qu.: 5.0 Median :12.5 Mean :13.0 3rd Qu.:20.0 Max. :25.0 Seed 329 : 6 327 : 6 325 : 6 307 : 6 331 : 6 311 : 6 (Other):48 Notice that the third column “Seed” has a different sort of summary. R recognizes this column (correctly) as containing categorical data; i.e., six seed type categories and an “other” category (that includes still other seed types). The sixes are counts of experimental units within a particular seed type. A.7.1 R and spreadsheets A data editor is provided by typing fix(x), when x is a dataframe or matrix (see § A.9 for information on these object classes). For instance, the command: fix(lp) will open the Loblolly pine dataset in the R data editor (Figure A.4). When x is a function or character string, then fix(x) opens a script editor containing x. The data editor has limited flexibility compared to software whose interface is a worksheet, and whose primary purpose is data entry and manipulation, e.g., Microsoft Excel®. Note, however, that it is possible to access R from Excel® (and vice versa) by installing the RExcel package (see Heiberger and Nuewirth 2009). Changes made to an object using fix will only be maintained for the current work session. They will not permanently alter objects brought in remotely to a session. Datasets 24 Figure A.4 The first ten observations from the Lobolly dataset in an editable spreadsheet. A.7.2 attach and detach The function attach will let R recognize the column names of a dataframe (more on what a dataframe is later). For instance, typing: attach(lp) will let R recognize the individual columns in lp (a dataframe) by name. The function detach is the programming inverse of attach. For instance, typing: detach(Loblolly) does not remove Loblolly, but will prevent R from recognizing the variables in Loblolly by name. A.7.3 with A safer alternative to attach is the function with. Using with eliminates concerns about multiple variables with the same name becoming mixed up in functions. This is because the variable names for a dataframe specified in with will not be permanently attached in an R-session. For instance, try: detatch(lp) with(lp, age + height) Datasets 25 A.7.4 remove and rm Typing: remove(Loblolly) or rm(Loblolly) will remove Loblolly completely from a work session. Happily, the object Loblolly (a dataset in the package datasets) cannot actually be destroyed/deleted using rm or remove (or any other commands) at the R-console. A.7.5 Cleaning up By now we can see that the R-console can quickly become cluttered and confusing. To remove clutter on the console (without actually getting rid of any of the objects in a session) press Ctrl+L or using the Edit pulldown menu to click on “Clear console”10 (non-Unix only). To remove all objects in the current session, including saved objects brought in from the working directory, one can type: rm(list = ls(all = TRUE)) Of course, one should never include this or other equivalent lines of code in a distributed function, since it would cause users to delete their work. 10 Many other shortcuts are available including Ctrl+U which deletes all the text from the current line, and Ctrl+K which deletes text from the current character to the end of the line. Conventional shortcuts are also valid, e.g. Ctrl+C = copy and Ctrl+V = paste. For guidance go to Help>Console (non-Linux ony). 26 Graphics A.8 R graphics A strong feature of R is its capacity to create nice graphs with tremendous user flexibility. The conventional workhorse of R-graphics is the function plot. A.8.1 plot The function plot allows representation of objects defined with respect to Cartesian coordinates. It has only two required arguments. • x defines the x-coordinate values. while • y defines the y-coordinate values. Important optional arguments include the following: • pch, specifies the symbol type(s), i.e. the plotting character(s) to be used, • col, defines the color(s) to be used with the symbols, • cex, defines the size (character expansion) of the plot symbols and text, • xlab and ylab allow the user to specify the x and y-axis labels, • type allows the user to define the type of graph to be drawn. Possible types are "p" for points (the default), "l" for lines, "b" for both, "c" for the lines part alone of "b", "o" for overplotted, "h" for ‘histogram’ like vertical lines, "s" for stair steps, and "n" for no plotting. Note that the command box() will also place a box around the plotting region. We can see some symbol and color alternatives for R-plots in Figure A.5. 20 plot(1:20, 1:20, pch = 1:20, col = 1:20, ylab ="Symbol number", xlab = "Color number", cex = 1.5) ● ● 10 5 Symbol number 15 ● 5 10 15 20 Color number Figure A.5 Some symbol and color plotting possibilities. 27 Graphics In the code above the x and y coordinates are both sequences of numbers from 1 to 20 obtained from the command 1:20. I varied symbol colors and plotting characters (col and pch respectively) using 1:20 as well. The combination col = 1 and pch = 1 results in a black open circle, whereas the combination col = 20, pch = 20 results in a blue filled circle. Note that we need to enclose the axis names in quotations for R to recognize them as text. Symbol numbers 21-26 allow background color specification using the argument bg. Many other symbol types are also possible including thousands of unicode options. An enormous number of color choices for plots are possible and these can be specified in at least six different ways. First, we can specify colors with integers as I did in Figure A.5. Additional varieties can be obtained by drawing numbers colors()[number] (Figure A.6). e <- expand.grid(1:20, 1:32) plot(e[,1], e[,2], bg = colors()[1:640], pch = 22, cex = 2.5, xaxt = "n", yaxt = "n", xlab = "", ylab = "") text(e[,1], e[,2], 1:640, cex = .4) 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Figure A.6 Color choices from colors(). Second, we can specify colors using actual words, e.g. "white", "red", "yellow". For a visual display of essentially all the available named colors in R type: example(colors). 28 Graphics Third, we can define colors by requesting red green and blue color intensities with the function rgb. Usable light intensities can be made to vary individually from 0 to 255 (i.e., within an 8 bit format). Fourth, we can specify colors using the function hcl which controls hues, chroma, and luminescence and transparency. Fifth, we can define colors using hexadecimal codes11, e.g., blue = #0000FF. Finally we can specify colors using entirely different palettes. Figure A.7 shows six pie plots. Each pie plot uses a different color palette. Each pie slice from each pie represents a distinct segment of the palette. Fig.A.7() # See Appendix code Rainbow colors 4 Gray colors 3 5 4 3 5 2 2 6 1 6 1 7 12 7 12 8 8 11 9 10 Heat colors 4 10 Colors using hcl arguments 1 3 5 11 9 4 3 5 2 2 6 1 6 1 7 12 7 12 8 8 11 9 10 Topographic colors 4 10 Colors using hcl arguments 2 3 5 11 9 4 3 5 2 2 6 1 6 1 7 12 7 12 8 11 9 8 10 11 9 10 Figure A.7 Use of color palettes in R. Note that the numbers do not correspond to actual color type designations. A large number of additional palettes (including color-blind-safe palettes) can be obtained using the R-package RColorBrewer (more information on auxiliary packages is provided in § A.21) . display.brewer.all(n = 7, colorblindFriendly = T) 11 A data coding system that uses 16 symbols: the numbers 1-9, and the letters A-F. Hexadecimals are primarily used to provide a more intuitive representation of binary-coded values (see §A.18). 29 Graphics YlOrRd YlOrBr YlGnBu YlGn Reds RdPu Purples PuRd PuBuGn PuBu OrRd Oranges Greys Greens GnBu BuPu BuGn Blues Set2 Paired Dark2 RdYlBu RdBu PuOr PRGn PiYG BrBG Figure A.8 RColorBrewer color-blind-safe seven category palettes. Top palettes are so-called "sequential" palettes, middle palettes are "qualitative", and bottom palettes are "divergent". Here are the hexadecimal names for the "Set2" palette chunks in Figure A.8. brewer.pal(7,"Set2") [1] "#66C2A5" "#FC8D62" "#8DA0CB" "#E78AC3" "#A6D854" "#FFD92F" "#E5C494" 30 Graphics || Example A.4 – Further Exploration of the Loblolly Dataset. Let’s visualize the relationship of the age and height of Loblolly pines using a scatterplot (Figure A.9). 40 30 10 20 height 50 60 with(lp, plot(age, height)) 5 10 15 20 25 age Figure A.9 Scatterplot of height and age from the loblolly pine dataset. Not surprisingly, there appears to be a strong positive correlation between these variables. Now let’s fit a simple linear regression for loblolly pine height as a function of age. A regression line will show the best possible linear fit for a response variable as a function of an quantitative explanatory variable. The R function for a linear model is lm. It encompasses and allows a huge number of statistical procedures, including regression (see Chs. 9, 10, and 11 in the Foundational and Applied Statistics text). We have: ha.lm <- lm(height ~ age, data = lp) The tilde lets R know that I want height to be a function of age. We note that the function lm has a built-in version of with which is specified with the argument data. Objects of class lm have their own summary function. This can be called (in this case) by simply typing: summary(ha.lm) Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -1.31240 age 2.59052 0.62183 0.04094 -2.111 63.272 0.0379 * <2e-16 *** 31 Graphics The output shows us the Y-intercept, -1.31240, and slope, 2.59052, of the regression line. The abline function allows the plotting of a line over an existing plot. The first two arguments for abline are the Y-intercept and slope (Figure A.10). with(lp, plot(age,height, pch=2, col=3)) 40 30 10 20 height 50 60 abline(-1.312396, 2.590523) 5 10 15 20 25 age Figure A.10 Scatterplot of height and age with a regression line overlaid. A.8.2 Graphics devices Graphics in R are created within graphics devices that vary with respect to storage modes, display modes, available typefaces and other characteristics. In the current R-windows download, six graphics devices will be available including windows, pdf, postscript, and X11. The X11 device is a windows graphics system for bitmap displays, and is commonly used in Unix-alike operating systems. Six other devices will also exist, although they may return a warning if R was not compiled to use them upon installation. These are cairo_pdf, svg, png, jpeg, bmp, and tiff. Multiple devices (currently up to 63) may exist simultaneously in an R work session, although there will only be one active device. To find the current (active) graphics device’s flavor we can type dev.cur(). I get: dev.cur() windows 2 Graphics 32 R tells me there are two devices open. The current device is a windows device. The second device is the socalled "null device". The null device is always open but only serves as a placeholder. Any attempt to use it will open a new device in R. Occasionally, on purpose or by accident, all graphics devices (except the null device) may become turned off. A new active graphics device can be created at any time by typing: dev.new() The active device can be chaged using the function dev.set(). A.8.3 par Parameters for a graphics device (which may contain several plots) can be accessed and modified using the function par. Below are important arguments for par. Some of these can also be specified as arguments in plot, with different results. • bg gives the background color for the graphical device. When used in plot it gives the background color of plotting symbols. • bty is the box-type to be drawn around the plots. If bty is one of "o" (the default), "l", "7", "c", "u", or "]" the resulting box resembles the corresponding upper case letter. The value "n" suppresses the box. • fg gives the foreground color. • font is an integer that specifies the font typeface. 1 corresponds to regular text (the default), 2 to bold face, 3 to italic and 4 to bold italic • las is the style of axis labels: 0 always parallel to the axis (default), 1 always horizontal, 2 always perpendicular to the axis, 3 always vertical. • mar will have the form c(bottom, left, top, right) and gives the number of lines of margin to be specified on the four sides of the plot. The default is c(5, 4, 4, 2) + 0.1. • mfrow will have the form c(number of rows, number of columns) that indicates the number and position of plots in a graphical layout. • oma specifies the outer margins of a graphical device, given multiple plots, using a vector using a matrix of the form: c(bottom, left, top, right. • usr will have the form c(x1, x2, y1, y2) giving the extremes of the user coordinates of the plotting region. When setting graphical parameters, it is good practice to revert back to the original parameter values. Assume that I want to background of the graphics device to be black. To set this I would type: old.par <- par(no.readonly = TRUE) # save default, for resetting... par(bg = "black") # change background parameter To return to the default settings for background I would type: par(old.par) Defaults will also be reset by closing the graphics device containing the customized parameters, or by opening a new device. For instance, using dev.new(). Other fundamental properties of the default graphics device, such as device height, width and pointsize, can be adjusted using the dev.new function. For instance, to create a graphical device 9 inches wide, and 4 Graphics 33 inches high, I would type: dev.new(width = 9, height = 4) A.8.4 Exporting graphics To export R graphics, one can often copy snapshots to a Windows clipboard using pull down menus on the windows graphical device. These can then be pasted into Windows programs (e.g., word processors) as bitmaps (a spatially mapped array of bits) or metafiles, a generic term for a file format that can store multiple types of (generally graphical) data. To create the best possible graphs, however, one should save device output using postcript (ps), portable document format (pdf), or other universal high resolution graphical formats. This can be done directly by: 1. Using pulldown menus on a windows or X11 graphics device itself (currently only metafile and postscript saves are available). 2. Cicking on the extent of a windows or X11 graphics device and then going to File>Save as on the R-GUI pulldown menus (obviously, non-Linux only). 3. Using save arguments included in graphical device functions, i.e., pdf, tiff, jpeg, bmp, tiff, postscript, and win.metafile (= metafile). The third alternative allows fine scale control with respect to resolution (dpi), figure size, and other characteristics. For instance, to save a graphics device image as a pdf under the file name example.pdf in the working directory I would type: pdf(file ="example.pdf") I would then make the plot, for instance plot(1:10) The plot will not be shown because the graphical device is engaged with pdf. As a final step I close the device. dev.off() The graphics file will now be contained in your working directory. If the file argument is unspecified pdf will save a file called Rlot.pdf. Creating high resolution images in R: By default, BMP, JPEG, PNG and TIFF graphics devices have a width and height of 480 pixels, and a "large" point size (1/72 inch), This results in a rather coarse (72 ppi) image resolution. Changing the res (resolution) argument in a graphical device function without changing the pointsize, or height and width arguments will generally result in unusable figures. Because 500 ≈ 72 x 6, one can generate a > 400 ppi TIFF called fig1.tiff by typing: tiff("fig1.tiff", res = 72 * 6, height = 480 * 6, width = 480 * 6) plot(1:10) dev.off() 34 Graphics With respect to graphical formats, documentation in the grDevices library states: "The PNG format is lossless (data compression without loss of information) and is best for line diagrams and blocks of color. The JPEG format is lossy (data compression in which unnecessary information is discarded), but may be useful for image plots, for example. The BMP format is standard on Windows, and supported by most viewers elsewhere. TIFF is a meta-format: the default format written by the function tiff (compression = none) is lossless and stores RGB values uncompressed—such files are widely accepted, which is their main virtue over PNG. So-called scalar vector graphics (SVGs) can be created with the functions svg, cairo_pdf and cairo_ps. All three apply the cairographics application programming interface (API). These devices will recognize a large number of symbols and fonts not available for document and image generation in the default setting of the windows postscript and pdf devices. A.8.5 Typeface families Font typefaces can be changed using a number of graphical functions, including par, via the argument family. The general typeface families: "serif", "mono", and "sans", and the Hershey family of fonts (type ?Hershey for more information) are transferable across all graphics devices employed in R. To change the font in a graphical device from the default sans serif (similar to Arial) to serif (similar to Times New Roman) one could type par(family = "serif"). To use a Courier-type monospace font one would use par(family = "mono"). Many other typeface families are possible, although they may not be transportable to all graphical devices and graphical storage formats. In the code below I bring in a large number of conventional font families using a function from the book website. These typefaces (and many others) will typically be available on Windows platform machines, although not all will be supported by non-windows graphics devices. The result can be seen in Figure A.4. source(url("http://www2.cose.isu.edu/~ahoken/book/win_fonts.R")) tiff(file = "fonts.tiff", res = 600, width = 7.7, height = 7.7, units = "in") x <- rep(c(2.8, 6.4, 9.6), each = 33) y <- rep(seq(10, 0.25, by = -.2965), 3) font.type <- paste(rep("f", length(fonts)), 1:length(fonts), sep = "") par(mar = c(0.1,0.1,0.1,0.1), cex = 1.1) plot(0:10,type="n", xaxt= "n", yaxt = "n", xlab = "", ylab = "") for(i in 1:length(fonts)){ text(x[i],y[i], labels=fonts[i] , family = font.type[i]) } dev.off() Graphics 35 The figure displays examples of text from ninety-nine Windows typefaces. To save myself from typing an inordinate amount of code, I use a for loop to place the fonts one at a time in the graphics device (see Section A.20.6). Note that I use the function tiff to create a high resolution .tiff graphical file. Importantly, the imported typefaces will now be available for graphics functions using the windows graphical device. To see the available windows fonts one can type: windowsFonts() Similarly, one can see the available fonts for postscript and pdf graphics devices using: names(pdfFonts()) 36 Graphics Figure A.4 Examples of font families that can be used in R graphics. A.8.6 text, lines, points, paste The functions text, lines and points can be used to place text, lines and points in a plot, respectively. For all three functions, the first two arguments concern the x and y coordinates for the plotted items, while other arguments concern characteristics of the items. For instance, to plot the text "example" at plot coordinates x = 0, y =1, with a character expansion that was two times the default, I would type: text(x = 0, y = 1, "example", cex = 2) 37 Graphics The function paste can be used to concatenate elements from text strings in plots or output. For instance, try: a <- c("a", "b", "c") b <- c("d", "e", "f") paste(a, b) [1] "a d" "b e" "c f" To plot a dashed line between the points (0, 2) and (1, 3), I would type: lines(x = c(0, 1), y = c(2, 3), lty = 2) or points(x = c(0, 1), y = c(2, 3), lty = 2, type = "l") To place a red inverted triangle at the point (0, 1), I would type: points(x = 0, y = 1, pch = 6, col = 2) Geometric shapes can be drawn using a number of functions including rect (which draws rectangles) and polygon (which draws shapes based on user-supplied vertices). A.8.7 plotmath R has useful functions for the plotting of mathematical expressions. These include the Greek letters, mathematical operators, italicization, and sub- and super-scripts. These are generally called as an expression in the text argument in the functions text or mtext. For example, to paste the formula for the sample variance n ∑ (x − x ) i =1 i n −1 2 , into a plot at coordinates (0, 1) I would type: varexp <- expression(over(sum(paste("(",italic(x[i] - bar(x)),")"^2), italic(i)==1, italic(n)),(italic(n) - 1))) text(0, 1, varexp) Complete coverage of plotmath mathematical expressions would be unwieldy to summarize here. For more information type ?plotmath. A.8.8 axis The function axis can be used to create new axes on a plot or to customize axis characteristics. Its first argument (side) specifies the side of the plot that the new axis will occupy 1=bottom, 2=left, 3=top, 4=right. Other arguments include a vector of axis labels (argument labels), and the locations of labels Graphics 38 (argument at). For instance, to create right hand axis I would type: axis(4) A.8.9 mtext To place text in the margin of a plot we can use the function mtext. For its first argument the function requires the character string to be written into the plot. The 2nd argument defines the plot margin to be written on: 1=bottom, 2=left, 3=top, 4=right. For instance, to place the text "Axis 2" on the right hand axis I would type: mtext("Axis 2", 4) || Example A.5 --A complex multiplot example Consider a final comprehensive graphical example. The object C.isotope in the library asbio is a dataset describing δ14C [the ratio of carbon 14 to carbon 13 (14C is unstable, while 13C is a stable isotope) compared to a standard ratio], and the atmosphere carbon concentrations over time at La Jolla California. We will create a figure with four subplots (Figure A.5). • It will have dimensions 8" x 7". • The outer margins (in number of lines) will be bottom = 0.1, left = 0.1, top = 0, right = 0. • The inner margins (for each plot) will be bottom = 4, left = 4.4, top = 2, right = 2. The plot margins will be light gray. We can specify gray gradations with the function gray. We will use gray(0.97). • The first plot will show δ14C as a function of date. The plotting area will be dark gray, i.e., colors() [181]. Points will be white circles with a black border. • The second plot will be a line plot of atmospheric carbon as a function of date. It will have a light green plotting area: colors()[363]. • The third plot will be a scatterplot of δ14C as a function of atmospheric carbon. Points will be yellow circles with a black border. The plotting area will be light red: colors()[580]. • The fourth plot will show the sample variance of atmospheric carbon in the time series. It will have a custom axis with the labels: a, b, c, and d. It will also have a horizontal axis label inserted with mtext. library(asbio) # loads the library asbio data(C.isotope) # dataset in asbio dev.new(height = 8, width = 7) op <- par(mfrow = c(2, 2), oma = c(0.1, 0.1, 0, 0), mar = c(4, 4.4, 2, 2), bg = gray(.97)) Graphics #-------------------------------- plot 1 -------------------------------# with(C.isotope, plot(Decimal.date, D14C, xlab = "Date", ylab = expression(paste(delta^14,"C (per mille)")), type = "n")) rect(par("usr")[1], par("usr")[3], par("usr")[2], par("usr")[4], col = colors()[181]) with(C.isotope, points(Decimal.date, D14C, pch = 21, bg = "white")) #-------------------------------- plot 2 -------------------------------# with(C.isotope, plot(Decimal.date, CO2, xlab = "Date", ylab = expression(paste(CO[2]," (ppm)")), type = "n")) rect(par("usr")[1], par("usr")[3], par("usr")[2], par("usr")[4], col = colors()[363]) with(C.isotope, points(Decimal.date, CO2, type = "l")) #-------------------------------- plot 3 -------------------------------# with(C.isotope, plot(CO2, D14C, xlab = expression(paste(CO[2], "(ppm)")), ylab = expression(paste(delta^14,"C (per mille)")), type = "n")) rect(par("usr")[1], par("usr")[3], par("usr")[2], par("usr")[4], col = colors()[580]) with(C.isotope, points(CO2, D14C, pch = 21, bg = "yellow")) #-------------------------------- plot 4 -------------------------------# plot(1:10, 1:10, xlab = "", ylab = "", xaxt = "n", yaxt = "n", type = "n") rect(par("usr")[1], par("usr")[3], par("usr")[2], par("usr")[4], col = "white") text(5.5, 5.5, expression(paste(over(sum(paste("(",italic(x[i] bar(x)),")"^2), italic(i)==1, italic(n)),(italic(n) - 1)), " = 78.4")), cex = 1.5) axis(side = 1, at = c(2, 4, 6, 8), labels = c("a", "b", "c", "d")) mtext(side = 1, expression(paste("Variance of ", CO[2], " concentration")), line = 3) par(op) 39 40 2000 360 370 380 2005 350 40 1995 CO2 (ppm) ● ● ● ● ● ●● ● ●● ● ● ● ●● ●●● ● ● ●● ● ● ● ●● ● ●●● ● ●● ● ● ●● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ●● ●●● ●● ● ● ●● ● ● ●● ● ● ●●● ● ●●●● ● ● ● ●● ●● ● ● ●●● ● ● ● ● ● ● ● ● ●●● ●● ● ●● ● ●● ● ● ●●● ● ●● ●● ● ●● ●● ● ● ●● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● 80 100 ● ● ● ●● ● ●● ● ● ● ●● ● ●●● ● ●● ●●●● ● ● ● ●● ●● ● ●● ●●● ● ● ● ● ●● ● ● ● ● ●● ●● ●●● ● ● ● ● ● ● ● ● ● ●● ●● ●●● ● ● ●● ●● ● ● ● ●●● ●● ●● ● ●●● ● ●●●●● ● ●● ●● 60 δ14C (per mille) 120 140 Graphics 1995 2000 120 140 Date ● Date ● ● ● ● ● ● ● ● ● 40 60 80 100 ● ● ● δ14C (per mille) 2005 350 ●● ●● ●● ● ● ● ● ●● ●●● ● ● ● ● ●● ● ● ● ● ● ●● ●● ● ●● ● ● ● ●● ● ● ● ●● ●●●●● ● ●● ● ● ●● ●● ● ●● ● ●● ● ●● ●● ● ● ●● ● ● ● ● ●● ● ●● ●● ● ● ● ●● ● ●● ●● ● ● ●● ●● ●●● ● ● ●●●● ● ● ● ● ●● ● ●●● ● ● ●● ● ● ●● ● ● ● ● ●●●●● ● ●● ● ● ●● ● ● ● ●● ● ● ● ● ●● ● ● ● ●● ●● ●● ● ● ● ● ● ● ● ● ●● ●● ● ● ●● ●●● ●● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ●●● ● ●● ● ● ● ●● ● ● ● ●● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ●●● ● ● ●● ● ● ● ● ● 360 370 CO2(ppm) 380 n ∑ (x i − x)2 i =1 (n − 1) a b c = 78.4 d Variance of CO2 concentration Figure A.5 Figure resulting from code in || Example A.5 41 Graphics A.8.10 R data visualization packages There are a large number auxiliary packages (see § A.21) specifically for graphics. Several of these, including the popular packages lattice and ggplot2, depend on the lattice graphical system (Sarkar 2008), which uses graphical devices and arguments that are non-analogous to those of plot, discussed so far. A detailed description of these packages is beyond the scope of this document, although a coding example (from lattice) is given below (Figure A.6). plot(levelplot(volcano, col.regions = heat.colors, xlab = "x", ylab = "y"), split = c(1, 1, 1, 3), more = TRUE, panel.width = list(x = 6,units = "inches")) plot(contourplot(volcano, cuts = 20, label = FALSE, xlab = "x", ylab = "y", col = "green"), split = c(1, 2, 1, 3), more = TRUE, panel.width = list(x = 6,units = "inches")) y plot(wireframe(volcano, panel.aspect = 0.7, zoom = 1, lwd = 0.01, xlab = "x", ylab = "y", zlab = "z"), split = c(1, 3, 1, 3), more = FALSE, panel.width = list(x = 6,units = "inches")) 60 200 50 180 40 160 30 140 20 120 10 100 20 40 60 80 x 60 50 y 40 30 20 10 20 40 60 80 x z y x Figure A.6 Maunga Whau (Mt Eden), a volcanos in the Auckland volcanic field, on a 10m by 10m grid. Data structures 42 A.9 Data structures There are five primary data structures in R. We will use each of them repeatedly. Vectors In R a vector is collection of data with order and length, but no dimension. This is very different than the matrix algebra definition. In this case a row vector with n elements has dimension 1 x n (1 row and n columns), while a column vector has dimension n x 1. We can create vectors with the function c. Recall that c means combine. x <- c(1, 2, 3) is.vector(x) [1] TRUE length(x) [1] 3 dim(x) NULL dim(as.matrix(x)) [1] 3 1 The function as.matrix coerces the vector x to have a matrix structure with dimension 3 x 1. Thus, in R a matrix (see below) has dimension, but a vector does not. Elements in vectors must have a single data storage mode: e.g.,"integer", "double", "complex", "character" . That is, a vector cannot contain both numeric and categorical data. When an operation is simultaneously applied to two unequal length vectors, R will generate a warning and automatically recycle elements of the shorter vector, beginning with its first element, until it is has the same length as the longer one. For instance: c(1, 2, 3) + c(1, 0, 4, 5, 13) [1] 2 2 7 6 15 Warning message: In c(1, 2, 3) + c(1, 0, 4, 5, 13) : longer object length is not a multiple of shorter object length In this case the result of the addition of the two vectors is 1 + 1, 2 + 0, 3 + 4, 1 + 5, and 3 + 13. This is because the first two elements in the first object are recycled in the addition. R users should obviously be aware of this issue. Data structures 43 Matrices Matrices are two-dimensional (row and column) data structures whose elements are all comprised of a single type of data: quantitative, categorical, or ordinal. The function matrix can be used to create matrices. a <- c(1, 2, 3, 2, 3, 4) matrix(ncol = 2, nrow = 3, data = a) [,1] [,2] [1,] 1 2 [2,] 2 3 [3,] 3 4 Arrays Arrays are one, two (matrix), or three or more dimensional data structures whose elements contain a single type of data. The function array can be used to create arrays. Below we create a 2 x 2 x 2 array using data using the object a from above. a <- c(1, 2, 3, 4, 5, 6, 7, 8) array(a, c(2, 2, 2)) , , 1 [,1] [,2] [1,] 1 3 [2,] 2 4 , , 2 [,1] [,2] [1,] 5 7 [2,] 6 8 The first argument in array defines the data, while the second argument is a vector that defines both the number of dimensions (this will be the length of the vector) and the number of levels in each dimension (numbers in dimension elements). The function above took the first half of observations in the object a and put them in the first level of component 1. It put the other four observations in the second level of component 1. The four elements in each component are arranged into 2 x 2 matrices. Arrays are useful for containing results made up of multiple matrices or dataframes (see below). For instance, a Markov Chain Monte Carlo (MCMC) analysis of the dataset cuckoo requires six parameters, each of which will be described with multiple Markov chains. Below I specify the creation of two chains, each comprised of three steps. The output is an array. Data structures 44 data(cuckoo); mcmc.norm.hier(cuckoo, 3, 2) , , Chain# 1 theta1 theta2 theta3 mu s.sq tau.sq 1 23.25000 23.85000 23.85000 23.65000 1.7612736 0.0551980 2 23.71469 23.29092 23.33157 23.33031 1.3925030 2.8243915 3 23.37465 22.92141 22.55939 22.73779 0.8880371 3.9218969 , , Chain# 2 theta1 theta2 theta3 mu s.sq 1 23.85000 23.05000 23.05000 23.31667 0.7906798 2 23.13697 23.26426 22.78453 24.19186 0.8380785 tau.sq 1.473440 4.067955 3 23.17203 23.25774 22.59440 22.41249 0.5772941 46.341983 Dataframes Dataframes are two-dimensional data structures whose elements can be comprised of different types of data (although all data in a single column must be of the same type, e.g., quantitative or categorical). Each column and row in a dataframe may be given an identifying label. Columns can then be called by name using the functions attach or with, or the expression $. The function data.frame can be used to create dataframes. data <- data.frame(numeric = c(1, 2, 3), non.numeric = c("a", "b", "c")) data numeric non.numeric 1 1 a 2 2 b 3 3 c Here we access the column in data called non.numeric. data$non.numeric [1] a b c Levels: a b c Lists Lists are often used to contain associated objects. Like dataframes, they need not contain a single type of data (e.g. categorical or quantitative). Unlike dataframes lists can simultaneously include strings (i.e., units of charac- Data structures 45 ter variables), matrices, dataframes and even other lists. In addition, lists do not require a row-column structure, and list components need not be the same length. For instance, scalars and matrices can both be components in a list. Recall that the R-object we produced earlier, ha.lm, from the creation of a linear model, was a list. Like dataframes, objects in lists can be called using the expression $. The function list can be used to create lists. data <- list(a = c(1, 2, 3), b = "this.is.a.list") data $a [,1] [1,] 1 [2,] 2 [3,] 3 $b [1] "this.is.a.list" Lists elements can also be identified using double square brackets. Below is the first list component of the list data. data[[1]] [1] 1 2 3 || Example A.6 -- Downs syndrome data Let’s create a dataframe with three numeric columns using data in Table A.1. This is part of a dataset for Down’s syndrome collected in British Columbia by the British Columbia Health Surveillance Registry (Geyer 1991). Table A.1 British Columbia Down’s syndrome data. Mothers age 17 20.5 21.5 29.5 30.5 38.5 39.5 40.5 44.5 45.5 47 Number of births 13555 22005 23896 15685 13954 4834 3961 2952 596 327 249 We will give this data subset the name Downs. Number of Down’s syndrome cases 16 22 16 9 12 15 30 31 22 11 7 Data structures 46 Downs <- data.frame(Age = c(17, 20.5, 21.5, 29.5, 30.5, 38.5, 39.5, 40.5, 44.5, 45.5, 47), Births = c(13555, 22005, 23896, 15685, 13954, 4834, 3961, 2952, 596, 327, 249), Cases = c(16, 22, 16, 9, 12, 15, 30, 31, 22, 11, 7)) Downs Age Births Cases 1 17.0 13555 16 3 21.5 23896 16 2 4 5 6 7 8 9 20.5 29.5 22005 30.5 15685 22 9 12 3961 30 38.5 13954 39.5 40.5 44.5 10 45.5 11 47.0 4834 2952 596 327 249 15 31 22 11 7 Columns in Downs can be called without attaching the dataframe by using the $ expression. Downs$Age [1] 17.0 20.5 21.5 29.5 30.5 38.5 39.5 40.5 44.5 45.5 47.0 We can also assemble items from Downs into a list. For instance: Dlist <- list(age = Downs$Age, births = Downs$Births, message = "Data from the British Columbia health surveillance registry") Dlist $age [1] 17.0 20.5 21.5 29.5 30.5 38.5 39.5 40.5 44.5 45.5 47.0 $births [1] 13555 22005 23896 15685 13954 4834 3961 2952 596 327 $message [1] "Data from the British Columbia health surveillance registry" 249 Command line data entry 47 A.10 Data entry at the command line A question of obvious importance is: "how do I get my data into R?" The answer is: "two ways." First, one can enter data “by hand” at the command line. Second, one can read in data files. We concentrate on command line entry in this section. Data import is described in § A.11. A.10.1 scan, cbind, rbind As we know data can be combined into a single entity with the function c. a <- c(1, 2, 3); b <- c(2, 3, 4) To create an R-object containing character strings, e.g., a categorical variable, we will need to place quotation marks around entries. x <- c("low", "med", "high") x [1] "low" "med" "high" Command line data entry is made easier with the function scan (which can also be used for file import) because a prompt is given for each data point, and separators are created by the function itself. For instance: a <- scan() 1: 1 2 3 4: Read 3 items The function will be terminated by a blank line or an end of file (EOF) signal. These will be Ctrl+D in Unix and Ctrl+Z in Windows. We can use cbind to combine columns, cbind(a, b) a b [1,] 1 2 [2,] 2 3 [3,] 3 4 while rbind lets us combine rows. rbind(a, b) [,1] [,2] [,3] a 1 2 3 b 2 3 4 Command line data entry 48 We see that the objects a and b can be interpreted either as columns or rows. A.10.2 Facilitating command line entry: seq and rep R has a number of functions that can speed up command line data entry. For instance, what if I want to create a sequence from 1990 to 2008? I would type: seq(1990, 1998), or seq(1990 : 1998), or simply 1990 : 2008 [1] 1990 1991 1992 1993 1994 1995 1996 1997 1998 The first two arguments in seq are the start and end of the sequence (unless a sequence is specified in the first argument). The third argument specifies the increment between items in the sequence. For example, if I wanted the vector 1990, 1992, 1994, 1996, 1998, I could simply type: seq(1990, 1998, 2) [1] 1990 1992 1994 1996 1998 One can easily create a vector with repeated values using the function rep. For example, to repeat the sequence 1991, 1992, 1993, 1994 ten times. I could type: rep(c(1991, 1992, 1993, 1994), 10) [1] 1991 1992 1993 1994 1991 1992 1993 1994 1991 1992 1993 1994 1991 1992 1993 [16] 1994 1991 1992 1993 1994 1991 1992 1993 1994 1991 1992 1993 1994 1991 1992 [31] 1993 1994 1991 1992 1993 1994 1991 1992 1993 1994 The first argument, c(1991, 1992, 1993, 1994), defines the thing we want to repeat. The second argument, 10, specifies the number of repetitions. Using these foundations we can create extremely complex sequences. For instance, to compose a sequence in which 1991, 1992, and 1993 were each repeated twice, and then to repeat that sequence three times we have: rep(1991 : 1993, each = 2, times = 3) [1] 1991 1991 1992 1992 1993 1993 1991 1991 1992 1992 1993 1993 1991 1991 1992 [16] 1992 1993 1993 Data import 49 A.11 Importing data into R While it is possible to enter data into R at the command line (§ A.10) this will normally be inadvisable except for small datasets. In general it will be much easier to import data. R can import data from many different kinds of formats including .txt, and .csv (comma separated) files, and files with space, tab, and carriage return datum separators. I generally organize my datasets using Excel or some other spreadsheet program (although R can handle much larger datasets than these platforms; § A.11.5), then save them as .csv files. I then import the .csv files into R using the read.table, read.csv, or scan functions. As noted in the function load can be used to import data files in rda data formats, or other R objects. A.11.1 read.table In the code below I use read.table to import a data file called veg.csv which is located within a series of nested directories in my C drive. Missing data in the file are indicated with periods. read.table("C:/Users/User/Documents/veg.csv", sep = ",", header = TRUE, row.names = 1, na.strings = ".") The first three arguments from read.table are very important. • The first argument, file, refers to the location and name of the file. Above, I specify: file = "C:/Users/ User/Documents/veg.csv". • The second argument, header, refers to the dataset column names. If we specify header = TRUE this indicates that the first row of dataset are column names. By default header = FALSE, and the function gives column names as "V" (for variable) followed by the column number. • The third argument, sep, refers to the type of data separator used. Comma separated files use commas to distinguish data entries. Thus I specify sep = "," above. Other common separators include tabs, specified as "\t" and spaces, specified as " ". Other useful read.table arguments include row.names and na.strings: • By specifying row.names = 1 I indicate that the first column of data contains row names. • The argument na.strings = "." indicates that missing values in the imported dataset are designated with periods. R uses the term NA to indicate “not available”. Blank fields are considered by R to be missing values and are given NA entries upon import. Note that R locates files using forward slashes rather than backslashes. To use backslashes one must double them; a legacy of R’s development under Unix/Linux. Consider the file above described above read.table("C:\\R.data\\veg.csv", sep = ",", header = T, row.names = 1, na.strings = ".") Data can also be read directly from the working directory. For instance: read.table("veg.csv", sep = ",", row.names = 1, header = T) Data import 50 A.11.2 read.csv The function read.csv has the same arguments as read.table with the exception that data separators are assumed to be commas, precluding the necessity of the sep argument. Thus, for the example above we would have: read.csv("veg.csv", header = TRUE, row.names = 1, na.strings = ".") A.11.3 scan The function scan can read in data from an essentially unlimited number of formats, and is extremely flexible with respect to character fields and storage modes of numeric data In addition to arguments used by read.table, scan has the arguments: • what which describes the storage mode of data e.g., logical, integer, etc., or if what is a list, components of variables including column names (see below), and • dec which describes the decimal point character (European scientists and journals often use commas). Assume that veg.csv has column of species names, called species, that will serve as row names, and 3 columns of numeric data, named site1, site2, and site3. We would read the data in with scan using: scan("veg.csv", what = list(species = "", site1 = 0, site2 = 0, site3 = 0), na.strings = ".") The empty string species = "" in the list comprising the argument what, indicates that species contains character data. Stating that the remaining variables equal 0, or any other number, indicates that they contain numeric data. A.11.4 Easy imports: use of file.choose() Possibly the easiest way to import data is to use read.csv, read.table, or scan with file.choose function as the file argument. For instance, by typing: read.csv(file.choose()) we can now browse for .csv files to open. Other arguments (e.g., header, row.names) will need to be used, when appropriate, to import the file correctly. A.11.5 Additional comments It is generally recommended that datasets imported and used by R be smaller than 25% of the physical memory Data import 51 of the computer. For instance, they should use less than 3GB on a 32-bit operating system. Note that this still equates to a roughly 13,700 x 8000 element data array. R can handle extremely large datasets, i.e. > 10GB, and 10 > 1.2 x 10 rows. In this case specific R packages can be used to aid in efficient data handling. Parallel “cluster” computing and workstation modifications may allow even greater efficiency. The actual upper physical limit for an R dataframe is 2 x 1031-1 elements. Note that this exceeds the latest limits for Excel worksheets by 21 orders of magnitude (Excel 2010 worksheets can handle approximately 1.7 x 1010 cell elements). R allows interfacing with a number relational database storage platforms. These include open source entities that express queries in SQL (Structured Query Language). For more information see Chambers (2008, pg. 178) and Adler (2010, pg. 157). Data export 52 A.12 Exporting data from R It is easy to export data from R. The functions write.table and write.csv let one write output for a large number of formats. For example, the commands below will write the matrix test to the working directory as a .csv file. test <- matrix(nrow = 2, ncol = 2, data = c(2, 1, 3, 4)) write.csv(test, "test.csv", sep = ",") • The first argument in write.table is the object I wish to export, i.e. test. • The second argument tells R where to export the object, and the storage name of the object. • The third argument, sep, tells R what kind of separator to use to distinguish data entries. The function write.csv facilitates the creation of .csv spreadsheets and assumes sep = ",". Several other optional arguments in write.table are important. • • • col.names = TRUE indicates that the first row of data are column names. row.names = 1 indicates that the first column of data are row names. na = "NA" indicates that missing values are specified in the data matrix with "NA". We can also use the write.table command to “copy” data to a clipboard. For instance: write.table(test, "clipboard", sep = "\t", col.names = NA) Now we can go to Excel® or Word® or some other program and paste the information using toolbars or Crtl+V. One can save a function, dataframe, or data matrix as a binary .rda or .RData file using the save function. For instance: save(test,file = "test.RData") saves test.RData to the working directory. Subsetting components 53 A.13 Subsetting matrix, dataframe and array components R allows us to easily specify particular subsets of dataframe, matrix or array using subset brackets, i.e. [ ]. Gaining skills with subsets will greatly enhance one’s ability to manipulate datasets in R. A dataframe or matrix name followed by brackets with a comma preceding a number inside the brackets, indicates a column number, i.e. [,column number]. For instance, the command Downs[,1] specifies column 1 in the dataframe Downs from || Example A.6. Downs[,1] [1] 17.0 20.5 21.5 29.5 30.5 38.5 39.5 40.5 44.5 45.5 47.0 A dataframe or matrix name followed by brackets with a comma following a number inside the brackets, indicates a row number, i.e., [row number,]. For instance, the command Downs[1,] specifies row 1 in Downs. Downs[1,] Age Births Cases 1 17 13555 16 Brackets without commas can be used to subset individual elements in a data matrix. It should be noted that by default R reads datasets by column. For instance, the command Downs[16] indicates the sixteenth element in the Downs dataset, which also happens to be fifth element in column 2. Downs <- as.matrix(Downs) Downs[16] [1] 13954 Note that in the operation above I convert the Downs dataframe to a matrix. This will force the subset (element 16) to work correctly. For more information on coercing R-objects see § A.17. The command Downs[1, 1] or Downs[c(1, 1)] specifies element 1 in column 1. Downs[1, 1] Age 17 The command Downs[c(5:8),]specifies rows 5 through 8 in Downs. Subsetting components 54 Downs[c(5:8),] Age Births Cases 5 30.5 13954 12 6 38.5 4834 15 7 39.5 3961 30 8 40.5 2952 31 The command Downs[-c(5:8),] specifies Downs without rows 5 through 8 Downs[-c(5:8),] Age Births Cases 1 17.0 13555 16 2 20.5 22005 22 3 21.5 23896 16 4 29.5 15685 9 9 44.5 596 22 10 45.5 327 11 11 47.0 249 7 Rows, columns, and layers from arrays can also be accessed using subset brackets. For instance, to obtain all the row and columns from the second layer of the array obj I would type: obj[, , 2] Note: when a subset results in zero observations for a level in a categorical variable, then it will be useful to use droplevels() to remove the empty levels. This is because R will consider these levels to still be an implicit part of the subset data, potentially complicating or preventing analyses. Matrix operations 55 A.14 Operations on matrices and dataframes Operators can be applied individually to every row or column of a matrix using a number of time saving methods. Actual R-applications for linear algebra (e.g. matrix multiplication, matrix inverses, eigenanalysis and matrix decompositions) are described in the mathematical Appendix in the Foundational and Applied Statistics textbook. || Example A.7 As a simple example we will plot Down’s syndrome cases per live birth from the Downs dataset. To do this I would simply divide the Cases column by the Births column and plot the result as a function of Age (Figure A.7). To accomplish this using variable (column) names we first coerce (§ A.17) Downs back into a dataframe. Downs<-as.data.frame(Downs) #coerce back to dataframe 0.02 0.00 0.01 Cases/Births 0.03 with(Downs, plot(Age, Cases/Births)) 20 25 30 35 40 45 Age Figure A.7 The rate of Down’s syndrome births as a function of mother’s age. Converting back to a data frame allows the column names in Downs to be recognizable. We can see that the rate of Down’s syndrome increases dramatically in older mothers. Matrix operations 56 A.14.1 apply Operations can be performed quickly on matrices with the function apply. The function requires three arguments. • In the first argument, X, we specify a matrix, array, or dataframe to be analyzed. • In the second argument, MARGIN, we specify whether rows or columns are to be analyzed (1 indicates rows, 2 indicates columns, while c(1, 2) indicates rows and columns). • In the third argument, FUN, we specify a function to be applied to the margins of the object in the first argument. max.val <- apply(Downs, 2, max) max.val Age Births Cases 47 23896 31 In the operation above I created an object called max.val that contains the maximum value from each column of Downs. Similarly the command below creates a vector made up of the minimum responses at each row. min.val <- apply(Downs, 1, min) min.val [1] 16.0 20.5 16.0 9.0 12.0 15.0 30.0 31.0 22.0 11.0 7.0 We can use the apply command to apply any statistical function (i.e. mean, sd, median etc.) to all the rows and/or columns of a matrix. means <- apply(Downs, 2, mean) means Age Births Cases 34.04545 9274.00000 17.36364 Several summary statistical functions exist for matrices that can be used in the place of apply. These include rowMeans and colMeans which give the sample means of specified rows and columns, respectively, and rowSums and colSums which give the sums of specified rows and columns, respectively. For instance: colMeans(Downs) Age Births 34.04545 9274.00000 Cases 17.36364 Matrix operations 57 A.14.2 tapply Imagine that we have a categorical variable in the Downs dataset with two factor levels (categories). The first factor level is associated with the first 6 experimental units, while the second level is associated with the last 5 experimental units. That is, Categories <- factor(c(rep(1, 6), rep(2, 5))) cbind(Downs, Categories) Age Births Cases Categories 1 17.0 13555 16 1 2 20.5 22005 22 1 3 21.5 23896 16 1 4 29.5 15685 9 1 5 30.5 13954 12 1 6 38.5 4834 15 1 7 39.5 3961 30 2 8 40.5 2952 31 2 9 44.5 596 22 2 10 45.5 327 11 2 11 47.0 249 7 2 The mixture of categorical and quantitative variables is allowed because Downs is a dataframe. We can easily summarize our data with respect to the categories in Categories by using the function tapply. Like apply, tapply requires three arguments. • The first argument, X, specifies which vector to evaluate. • The second argument, INDEX, will be vector of categories that can be used to subset X. • The third argument, FUN, describes the function to be applied to X for each level in INDEX. tapply(X = Downs[,2], INDEX = Categories, FUN = mean) 1 2 15654.83 1617.00 A.14.3 outer Another important function for matrix operations is the function outer. The function lets us create an array that contains all possible combinations of two vectors or arrays with respect to a particular function. For example, suppose I wished to find the mean of all possible pairs of observations from a vector. I would type the following commands: Matrix operations x <o <o [,1] [1,] [2,] [3,] [4,] [5,] 58 c(1, 2, 3, 5, 4) outer(x, x, "+")/2 [,2] [,3] [,4] [,5] 1.0 1.5 2.0 3.0 1.5 2.0 2.5 3.5 2.0 2.5 3.0 4.0 3.0 3.5 4.0 5.0 2.5 3.0 3.5 4.5 2.5 3.0 3.5 4.5 4.0 The upper and lower triangles of the matrix o contain the pairwise means while the diagonal contains the means of the objects with themselves. That is, the diagonal contains the original data in x. A.14.4 lower.tri, upper.tri and diag We can use the commands lower.tri, upper.tri and diag to examine the upper triangle, lower triangle, and diagonal parts of a matrix. For instance, o[upper.tri(o)] [1] 1.5 2.0 2.5 3.0 3.5 4.0 2.5 3.0 3.5 4.5 o[lower.tri(o)] [1] 1.5 2.0 3.0 2.5 2.5 3.5 3.0 4.0 3.5 4.5 diag(o) [1] 1 2 3 5 4 Note that I use square brackets to subset the data in o. A.14.5 stack and unstack When manipulating matrices and dataframes it is often useful to stack and unstack columns. These operations are handled with the functions stack and unstack. Consider the 4 x 4 dataframe below. s <- data.frame(matrix(nrow=4, ncol=4, rnorm(16))) s [,1] [,2] [,3] [,4] [1,] -0.08879353 0.5075496 0.8077366 0.6685438 [2,] -0.84689275 -1.1091304 -0.1530534 -1.7209322 [3,] 2.33278611 1.1339444 -0.3599148 0.2705621 [4,] 0.06433390 0.2264004 0.2190580 0.3001240 Matrix operations 59 The command rnorm(16) generates 16 random values from a standard normal distribution (Ch. 3). Say that we wish to assign distinct treatments to the columns and stack them. Applying stack we have: st <- stack(s) st values ind 1 -0.04941294 X1 2 -1.37260663 X1 3 2.21704986 X1 4 0.89460072 X1 5 0.25489102 X2 6 0.81068275 X2 7 1.04691132 X2 8 -0.83447166 X2 9 2.72308681 X3 10 0.48653472 X3 11 0.12804772 X3 12 0.89494172 X3 13 -0.54162277 X4 14 -1.11219031 X4 15 -1.60632784 X4 16 0.32180719 X4 Unstacking we have: unstack(st) X1 X2 1 -0.04941294 0.2548910 2 -1.37260663 0.8106827 3 2.21704986 1.0469113 4 0.89460072 -0.8344717 X3 X4 2.7230868 -0.5416228 0.4865347 -1.1121903 0.1280477 -1.6063278 0.8949417 0.3218072 Logical commands 60 A.15 Logical commands Computer languages like R that can dichotomously classify true and false statements are called logical or Boolean. R uses the following logical operators: • • • • • • • • > “greater than” >= “greater than or equal to” < “less than” <= “less than or equal to” == “equal to” != “not equal to” & “and” | “or” In R logical queries, comparisons, or commands will return the Boolean categories TRUE and FALSE. || Example A.8 We will demonstrate the use of logical commands with the Downs dataset. attach(Downs) Age >= 30 [1] FALSE FALSE FALSE FALSE TRUE TRUE TRUE TRUE TRUE TRUE TRUE Age != 30.5 [1] TRUE TRUE TRUE TRUE FALSE TRUE TRUE TRUE TRUE TRUE TRUE Age != 30.5&Age < 40 [1] TRUE TRUE TRUE TRUE FALSE TRUE TRUE FALSE FALSE FALSE FALSE Age < 30.5|Age == 47 [1] TRUE TRUE TRUE TRUE FALSE FALSE FALSE FALSE FALSE FALSE TRUE We can subset data with logical commands by using square brackets. For instance the following code requests age data in the Downs dataset less than 30.5, or equal to 47. Age[Age < 30.5|Age == 47] [1] 17.0 20.5 21.5 29.5 47.0 Using the function subset we have: subset(Age, Age < 30.5|Age == 47) [1] 17.0 20.5 21.5 29.5 47.0 Logical commands 61 R allows users to use T and F in the place of TRUE and FALSE, although this may result in problems if you have variables named T or F. A.15.1 ifelse A number of functions can be used in combination with a logical argument to evaluate a vector and provide outcomes if the argument is true or false. One example is the ifelse function. It requires three arguments • The first argument, test, gives the logical test to be evaluated. • The second argument, yes, provides the output if the test is true. • The third argument, no, provides the output if the test is false. For instance: ifelse(Age < 25, "Young", "Not so young") [1] "Young" "Young" "Young" "Not so young" "Not so young" [6] "Not so young" "Not so young" "Not so young" "Not so young" "Not so young" [11] "Not so young" A.15.2 if, else, any, and all A more generalized approach to providing a condition and then defining the consequences uses the functions if and else. For instance: if(any(Age < 25))"Young" else "Not so Young" [1] "Young" The any function looks through the vector Age to see if any of the elements meet the specified criterion. Conversely, the function all will look through a vector or vectors to see if all of the specified elements meet the specified criterion. if(all(Age < 25))"Young" else "Not so Young" [1] "Not so Young" Functions for data management 62 A.16 Simple functions for data management An attractive attribute of R is its capacity to efficiently manage large, complex datasets. In this section I list a few functions and approaches useful for data management. A.16.1 replace We can replace elements in a vector with the function replace. The function requires three arguments. • The first argument, x, specifies the vector to be analyzed. • The second argument, list, tells R which elements need to be replaced. A logical argument can be used here as a replacement index. • The third argument, values, tells R what these elements need to be replaced with. For instance: replace(Age, Age < 25, "R is Cool") [1] "R is Cool" "R is Cool" "R is Cool" "29.5" [7] "39.5" "40.5" "44.5" "45.5" "30.5" "47" "38.5" Recall that a vector is not allowed to contain both quantitative and categorical data. As a result R made all of the output from replace categorical. A.16.2 which The function which.can be used with logical commands to subset data. For instance: which(Age > 30) [1] 5 6 7 8 9 10 11 The result tells us which elements in Age were in agreement with the logical argument Age>30. To find which element in Age is closest to 32 I type: which(abs(Age - 32) == min(abs(Age - 32))) [1] 5 According to R, the 5th element in the Age is closest to 32. We can see that this is true. abs(Age - 32) [1] 15.0 11.5 10.5 2.5 1.5 6.5 7.5 8.5 12.5 13.5 15.0 Three other functions related to which are sort, rank, and match. Functions for data management 63 A.16.3 sort The function sort sorts alphanumeric data from a vector into an ascending order. sort(Age) [1] 17.0 20.5 21.5 29.5 30.5 38.5 39.5 40.5 44.5 45.5 47.0 sort(c("a", "d", "c", "Inf")) [1] "a" "c" "d" "Inf" Data can be sorted in a descending order by specifying decreasing = T. sort(Age, decreasing = T) [1] 47.0 45.5 44.5 40.5 39.5 38.5 30.5 29.5 21.5 20.5 17.0 A.16.4 rank The function rank gives the ascending alphanumeric rank of elements in a vector. Ties are given the average of their ranks. This operation is important to rank-based permutation analyses (see Ch. 6 in the Foundational and Applied Statistics text). x <- c(1, 2, 3, 4, 4, 3) rank(x) [1] 1.0 2.0 3.5 5.5 5.5 3.5 A.16.5 order The function order is more difficult to understand than sort and rank. It alphanumerically sorts a vector and returns the original element order overlaid on the sorted vector. This allows us to sort a vector, matrix or dataframe into an ascending or descending order, based on one or several vectors. For instance: x <- c(1, 3, 2, 4, 0.5) o <- order(x) o [1] 5 1 3 2 4 The 5th element in x has the smallest value, thus the number five is placed first in vector o. The 1st element in x has the next smallest value, thus the number one is placed second in o. The 3rd element is the next smallest, and so on. We see here how we would utilize the vector o. Functions for data management 64 x[o] [1] 0.5 1.0 2.0 3.0 4.0 Of course we get the same answer as if we were to use sort(x). sort(x) [1] 0.5 1.0 2.0 3.0 4.0 However, the uses of order extend beyond sorting single vectors. Consider the following dataset describing the percent cover of plant species, in which species are signified with four letter (genus, species) codes: field.data <- data.frame(code = c("ACMI", "ELSC", "CAEL", "TACE"), cover = c(12, 13, 14, 11)) field.data code cover 1 ACMI 12 2 ELSC 13 3 CAEL 14 4 TACE 11 What if I want to sort the dataframe alphabetically by species codes? I would type: field.data[order(field.data[,1]),] code cover 1 ACMI 12 3 CAEL 14 2 ELSC 13 4 TACE 11 Similarly, to order by species cover I would type: field.data[order(field.data[,2]),] code cover 4 TACE 11 1 ACMI 12 2 ELSC 13 3 CAEL 14 A.16.6 unique Functions for data management 65 To find unique values in dataset (and eliminate unwanted repeats) we can use the function unique. Here is list of species codes from a bird survey on islands in Southeast Alaska. Notice that here are a large number of repeats. AK.bird "GRSC", "BAEA", "TUSW", <- c("GLGU", "MEGU", "DOCO", "PAJA", "COLO", "BUFF", "COGO", "WHSC", "TUSW", "GRTE", "REME", "BLOY", "REPH", "SEPL", "LESA", "ROSA", "WESA", "WISN", "SHOW", "GLGU", "MEGU", "PAJA", "DOCO", "GRSC", "GRTE", "BUFF", "MADU", "REME", "SEPL", "REPH","ROSA", "LESA", "COSN", "BAEA", "ROHA") Using unique we have: unique(AK.bird) [1] "GLGU" "MEGU" "DOCO" "PAJA" "COLO" "BUFF" "COGO" "WHSC" "TUSW" "GRSC" "GRTE" [12] "REME" "BLOY" "REPH" "SEPL" "LESA" "ROSA" "WESA" "WISN" "BAEA" "SHOW" "MADU" [23] "COSN" "ROHA" A.16.7 match Given two vectors, the function match finds where objects in the second vector appear in the elements of the first vector. For instance: x <- c(6, 5, 4, 3, 2, 7) y <- c(2, 1, 4, 3, 5, 6) match(y, x) [1] 5 NA 3 4 2 1 The number 2 (the 1st element in y) is the 5th element of x, thus the number 5 is put 1st in the vector m created from match. The number 1 (the 2nd element of y) does not occur in x (it is NA). The number 4 is the 3rd element of y and x, thus number 3 is placed in the third element of m, and so on. The value of this function may seem unclear at first, but consider a scenario where I want to convert field data with species codes into a dataset containing species names. Consider the following species list (which includes species not in the field data from § A.16.5). species.list <- data.frame(code = c("ACMI", "ASFO", "ELSC", "ERRY", "CAEL", "CAPA", "TACE"), names = c("Achillea millefolium", "Aster foliaceus", "Elymus scribneri", "Erigeron rydbergii", "Carex elynoides", "Carex paysonis", "Taraxacum ceratophorum")) Functions for data management 66 species.list code names 1 ACMI Achillea millefolium 2 ASFO Aster foliaceus 3 ELSC Elymus scribneri 4 ERRY Erigeron rydbergii 5 CAEL Carex elynoides 6 CAPA Carex paysonis 7 TACE Taraxacum ceratophorum Here I give the correct species names to the field codes using the match function. m <- match(field.data[,1], species.list[,1]) field.data[,1] <- species.list[,2][m] field.data code cover 1 Achillea millefolium 12 2 Elymus scribneri 13 3 Carex elynoides 14 4 Taraxacum ceratophorum 11 A.16.8 which and %in% We can use the commands %in% and which together to achieve the same results as match. Under the current example we have: m <- which(species.list[,1] %in% field.data[,1]) field.data[,1] <- species.list[,2][m] field.data code cover 1 Achillea millefolium 12 2 Elymus scribneri 13 3 Carex elynoides 14 4 Taraxacum ceratophorum 11 A.16.9 strsplit and strtrim Elements within a text string can be rearranged, identified or extracted using a number of functions. The function strsplit splits a character string into substrings based on user defined criteria. It contains two important arguments. Functions for data management 67 • The first argument, x, specifies the character string to be analyzed. • The second argument, split, is a character criterion that is used for splitting. Letting this argument equal NULL (§ A.19) results in spaces being placed between every character in the string. noquote(strsplit("Achillea millefolium", NULL)) [[1]] A c h i l l e a m i l l e f o l i u m To split the string whenever "l" occurs, I have: noquote(strsplit("Achillea millefolium", "l")) [[1]] Achi ea mi efo ium The function noquote() removes quotes when printing. The function strtrim is useful for extracting characters from vectors. For instance, for the species codes in the plant character vector below, the first capital letter indicates whether the species are flowering plants (anthophytes) or mosses (bryophytes). Assume I want to create a new categorical variable distinguishing anthophytes from bryophytes by extracting the first letter. This is defined by specifying 1 in the second strtrim argument, width. plant <- c("A_CAAT", "B_CASP", "A_SARI") strtrim(plant, 1) [1] "A" "B" "A" A.16.10 Complex pattern matching: gsub, grep and metacharacters The functions grep and gsub are specifically designed for pattern matching and replacement within a character vector comprised of multiple text strings. Consider the following character object made up of six strings: sample <- c("amy", "joe", "fred", "mike", "betty") If for some reason we wanted to convert every occurrence of a lower-case ‘m’ to an upper case ‘M’ we could use: gsub("m", "M", sample) [1] "aMy" "joe" "fred" "Mike" "betty" Here the function grep tells us whih elements in sample contain the letter ‘m’. grep("m", sample) [1] 1 4 A metacharacter is a keyboard character that has a special (non-literal) meaning to a computer program. In R these include the symbols: \ | () [ { ^ $ * + and ?. We can use metacharacters in conjunction with Functions for data management 68 gsub and grep for the purpose of complex pattern matching and replacement. In using this approach we can also call the Perl Compatible Regular Expressions (PCRE) library, which has been incorporated into R. The PCRE library implements the computer language Perl. As an example of how metacharacters can be used by the PCRE library we will tell R to capitalize the first letter of each element in the object sample. gsub("(\\w)(\\w*)", "\\U\\1\\L\\2", sample, perl=TRUE) [1] "Amy" "Joe" "Fred" "Mike" "Betty" In the first argument, the string (\\w)(\\w*) indicates that I want the function to consider the first letter in each element of sample, and all later letters, separately. In the second argument, the string \\U\\1\\L\\2 indicates that I want the first letter of each element in sample to be replaced with an upper case version of itself, and that I want the other letters to be lower case. We can also make all the letters in sample upper-case. Here I ask R to give me the elements in sample which are four or more letters long. grep("[[:alnum:]]{4,}", sample, value = T) [1] "fred" "mike" "betty" The string [[:alnum:]] indicates all alphanumeric characters. I am combing this expression with the metacharacter expression {n,}. This combination tells R to report all elements that have four or more alphanumeric units. The argument value = T tells R to provide the actual contents of the elements in sample, and not the element indices. For more information type ?grep. Testing and coercing 69 A.17 Testing and coercing There are a number of functions that are designed to test whether an R object has particular characteristics, or to coerce an R object to have a desired class. For instance, the function is.numeric tests whether an object is numeric while the function as.numeric coerces an object to be of class numeric. x<-c("a", "b", 4) is.numeric(x) [1] FALSE as.numeric(x) [1] NA NA 4 Warning message: NAs introduced by coercion We note that in as.numeric(x) the non-numeric parts of x are discarded. Testing and coercing functions exist for all important R-classes. These include: vector, matrix, dataframe, array, list, factor, numeric, function, dist, double, character and many others. Coercion of data to categorical and ordinal classes can be managed using the functions factor and ordered respectively. For instance: x <- c(1, 2, 1, 2, 1, 2) is.numeric(x) [1] TRUE #quantitative y <- factor(x) #categorical y [1] 1 2 1 2 1 2 Levels: 1 2 z <- ordered(c(1, 2, 3, 4, 5)) z [1] 1 2 3 4 5 Levels: 1 < 2 < 3 < 4 < 5 #ordinal All numeric objects in R are stored in a double-precision format, meaning that the object occupies two adjacent locations in computer memory. a <- 1 is.double(a) [1] TRUE Objects coerced to be integers will be stored as double precision, although one of the storage locations will not be used. As a result integers are not conventional double precision data. This is explained in greater detail in Testing and coercing the next section. a <- 1 is.double(as.integer(a)) [1] FALSE 70 Binary numbers, bits and bytes 71 A.18 Binary numbers, bits and bytes To acquire a deeper understanding of the workings of R we must gain some familiarity with basic principles of computer science. Computers are designed around bits and bytes. A bit is a binary piece of information, i.e. a 0 or a 1, while a byte consists of a collection of eight bits12. This convention occurs because conventional computer systems use electronic circuits that exist in only one of two states, on or off. In most programs on most workstations the results of computations are stored as 32 bits (4 bytes) or as 64 bits (8 bytes) of information. Double precision storage (used by R) requires 64 bits. This allows expression and storage of numerical quantities between ca. 5.0 × 10-324 and ca. 1.8 × 10308, and a precision of at least 15 - 17 significant digits (see below). We can see that the current upper numerical limit of R (ver 2.15.1) is somewhere between: 1.8 * 10^307 [1] 1.8e+307 and 1.8 * 10^308 [1] Inf And that the (non-negative) lower limits are between 5.0 * 10^-323 [1] 4.940656e-323 and 5.0 * 10^-324 [1] 0 With a single bit we can describe only 21 = 2 distinct digital objects. These are, an object represented by a 0 and an object represented by a 1. It follows that 22 = 4 distinct objects can be described with two bits, 23 = 8 objects can be described with three bits, and so on13. We count to ten in binary using: 0 = 0, 1 = 1, 10 = 2, 11 = 3, 100 = 4, 101 = 5, 110 = 6, 111 = 7, 1000 = 8, 1001 = 9, 1010 = 10. Thus, we require four bits to count to ten. Note that the binary sequences for all positive non-zero integers start with one. 12 According the IEEE, a computer hard drive with 1 gigabyte (1 billion bytes) of memory will have 1 × 1011 bytes = 8 × 1011 bits of memory. This occurs because of the conversion: 1 terabyte = 1000 Gb , 1 Gb = 1000 Mb, 1Mb = 1000 Kb, 1Kb = 1000 bytes, 1 bytes = 8 bits. A second convention (used frequently to describe RAM) defines byte units in multiples of 1024. In this case 1 Gb = 10243 bytes = 8(1,073,741,824) bits. 13 For instance, older remotely sensed images often contained eight bit (one byte or octet) variables. The color red was expressed in such an image as a number between 0 and 255 (i.e. red could have 28 = 256 distinct values). Given that the colors blue and green were also eight bit, there would be 2563 = 16,777,216 color possibilities (combinations) for any pixel in the image. Binary numbers, bits and bytes 72 1) For the smallest integer, given a particular number of bits, the leftmost placeholder will be a one, and all other bits will be zeros (e.g., 100 = 4). 2) For the next largest integer, a one is placed in the rightmost placeholder occupied by a zero in the previous step (e.g., 101 = 5). 3) For the next largest integer the one inserted in the previous step is moved one placeholder to the left if that bit is zero (e.g. 110 = 6). If the bit to the left is non-zero, then go to step 2 (e.g., 111 = 7). If all bits are ones following (or preceding) step 2, then add a bit, and begin again at step 1 (e.g., 1000 = 8). The addition of a binary digit represents an increasing power of 2. As a result we say that the rightmost digit in a set of binary digits represents 20, the next represents 21, then 22, and so on. This can be defined with the equation k αβ ,(A.1) where α is quantity known as the significand, that describes the number of significant digits followed by a modifying number, β (usually 2), and k is called (appropriately) the exponent. The number of bits in the significand determines the precision of a binary expression, while the exponent determines minimum and maximum possible number. In a 64 bit double precision format 1 bit is allocated to the sign of the stored item, 53 bits are assigned to the significand, and 11 bits are given to the exponent. Equation A.1 actually represents a dot product, i.e., it is the sum of the piecewise multiplication of two vectors. For instance, to find the decimal number version of a single binary bit (consisting of only the right-most digit) we multiply the digit value by the power of two it represents. Thus, as we noted above, 1× 20 = 1 1 , and 1× 20 = . Accordingly, to find the decimal version of a set of binary values we take the sum of the products of the binary digits and the powers of base 2 that they represent. For instance, the binary number 010101 equals: (0 × 25 ) + (1× 24 ) + (0 × 23 ) + (1× 22 ) + (0 × 21 ) + (1× 20 ) = 0 + 16 + 0 + 4 + 0 + 1 = 21 . The function bin2dec in asbio does the calculation for us. bin2dec(010101) [1] 21 A.18.1 Floating point arithmetic Depicting undefinable real numbers (e.g. an irrational number), and certain rational fractional numbers in binary, requires binary approximation (Goldberg 1991). One approach is to express numeric values with non-zero fractional parts using floating point arithmetic. This framework is used by all conventional software (although its mechanisms are easily revealed in R). As with binary decimal numbers, binary fractional numbers are expressed with respect to a decimal, and the number of digits will (often) be dictated by the significand. Given 13 bits we have the following binary translations for decimal numbers: 1/1 = 1, 1/2 = 0.1, 1/3 = 0.01010101…, 1/4 = 0.01, 1/5 = 0.00110011..., 1/6 = 0.0010101…, 1/7 = 0.001001..., 1/8 = 0.001, 1/9 = 0.000111000111... , 1/10 = 0.000110011.... Binary numbers, bits and bytes 73 To obtain decimal fractions from binary fractions we multiply the bits by decreasing inverse powers of base two, starting at 0 (excluding 2-0), and find the sum. For example, back-calculating the decimal value 1/4 from the binary value 0.01 we have: (0 × 20) + (0 × 2-1) + (1 × 2-2) = 0 + 0 + 0.25 = 0.25. bin2dec(0.01) [1] 0.25 Floating point arithmetic may give unexpected results. For instance: options(digits = 20) 1/10 [1] 0.10000000000000000555 Due to the character of the binary floating point 1/10 = 0.1 can only be approximated by a binary number. The approximation is very close (out to the 18th significant digit), but not exactly equal to 0.1. Many other fractional quantities cannot be expressed exactly. Indeed, real (terminal) binary fractions will only exist if the only prime number is in the decimal fraction denominator, and it is 2 (although exception handling allows terminality for many fractions). As other examples, consider 1/3 [1] 0.3333333333333333148296 sqrt(2)^2 == 2 [1] FALSE and even, 12.3 - 49.2 [1] -36.900000000000005684 Obviously, such “rounding errors” may result in user-function failures14. 14 Alternatives to binary floating point arithmetic that address this problem exist, but are rarely implemented because: 1) they are less efficient, and 2) currently no IEEE standards have been specified. In order of increasing precision and decreasing efficiency alternative systems include Limited-Precision Decimal, Arbitrary-Precision Decimal, and Symbolic Calculation systems. Binary numbers, bits and bytes 74 A.18.2 Binary characters Characterscan also be expressed in binary. The American Standard Code for Information Interchange (ACSCII) consists of 128 characters and requires eight bits (one byte). Likewise, Unicode has 65,536 characters (the first 128 are the ASCII characters), and requires 16 bits (two bytes)15. R uses the Unicode system of characters. We can observe the process of binary character assignment using the functions as.raw, rawToChar, and rawToBits. Here is a list of the fist 128 Unicode characters (i.e. the ASCII characters) rawToChar(as.raw(1:128), multiple [1] "\001" "\002" "\003" "\004" [12] "\f" "\r" "\016" "\017" "\026" [23] "\027" "\030" "\031" "\032" [34] "\"" "#" "$" "%" [45] "-" "." "/" "0" [56] "8" "9" ":" ";" [67] "C" "D" "E" "F" [78] "N" "O" "P" "Q" [89] "Y" "Z" "[" "\\" [100] "d" "e" "f" "g" [111] "o" "p" "q" "r" [122] "z" "{" "|" "}" = TRUE) "\005" "\006" "\a" "\b" "\t" "\n" "\020" "\021" "\022" "\023" "\024" "\025" "\033" "&" "1" "<" "G" "R" "]" "h" "s" "~" "\034" "’" "2" "=" "H" "S" "^" "i" "t" "\177" "\035" "(" "3" ">" "I" "T" "_" "j" "u" "€" "\036" ")" "4" "?" "J" "U" "`" "k" "v" "\037" "*" "5" "@" "K" "V" "a" "l" "w" " " "+" "6" "A" "L" "W" "b" "m" "x" "\v" "!" "," "7" "B" "M" "X" "c" "n" "y" Note that the exclamation point is character number 33. Its 16 bit binary code is: rawToBits(as.raw(33)) [1] 01 00 00 00 00 01 00 00 This is the number of characters that can be defined with 16 bits. The current number of Unicode characters, however, is actually much larger than 65,536. The 16-bit Unicode Transformation Format (UTF-16) handles these additional characters by using two sixteen bit units. 15 NA NaN and NULL 75 A.19 NA, NaN, and NULL R identifies missing values (empty cells) as NA, which indicates “not available”. Hence the R function to identify a missing value is: is.na. For example: x <- c(2, 3, 1, 2, NA, 3, 2) is.na(x) [1] FALSE FALSE FALSE FALSE TRUE FALSE FALSE To say “identify all parts of a vector except missing values”, we set a logical test to be true when values are not missing. The unary (single argument) operator for "not" in R is !. Thus, the correct command is: !is.na( ) For example: !is.na(x) [1] TRUE TRUE TRUE TRUE FALSE TRUE TRUE There are a number of functions to get rid of missing values. These include na.omit. na.omit(x) [1] 2 3 1 2 3 2 attr(, "na.action") [1] 5 attr(, "class") [1] "omit" We see that R omitted the missing observation and then told us which observation was omitted. Functions in R are often set up to handle missing data. In many cases, however, it will be desirable to have a complete set of observations for a dataframe, matrix, or vector. Consider the following dataframe which contains plant percent cover for four plant species which are identified with codes. NA NaN and NULL field.data <- data.frame(code=c("ACMI", "ELSC", "CAEL", "CAPA", "TACE"), cover = c(12, 13, 14, NA, 11)) field.data code cover 1 ACMI 12 2 ELSC 13 3 CAEL 14 4 CAPA NA 5 TACE 11 I can check for completeness of the data, i.e. the appearance of missing data for any experimental unit (row) using the complete.cases function. complete.cases(field.data) [1] TRUE TRUE TRUE FALSE TRUE I can omit rows with missing data using na.omit. na.omit(field.data) code cover 1 ACMI 12 2 ELSC 13 3 CAEL 14 5 TACE 11 One shouldn’t worry too much about memorizing a million R functions. Instead you should develop a basic vocabulary of functions that you will use frequently. For instance, instead of using na.omit the example above, I could have gotten the same result using brackets and logical commands. field.data[!is.na(field.data[,2]),] code cover 1 ACMI 12 2 ELSC 13 3 CAEL 14 5 TACE 11 The final comma in the command above tells R that I want to go through the field.data dataframe row by row and eliminate rows with missing values. The designation NaN is associated with the current conventions of the IEEE 754-2008 (IEEE is an acronym for Institute of Electrical and Electronics Engineers, and pronounced "I triple E") arithmetic used by R. It means “not a number”. Mathematical operations which produce NaN include: 76 NA NaN and NULL 77 0/0 [1] NaN Inf-Inf [1] NaN sin(Inf) [1] NaN Occasionally one may wish to specify that an R-object is NULL, meaning that it is absent. A NULL object can be included as an argument in a function without requiring that it have a particular value or meaning. As with NA the NULL specification is easy: x <- NULL It should be emphasized that R-objects or elements within objects that are NA or NULL cannot be identified with the logical commands == or !=. For instance: x <- NA; y <- NULL x == NA [1] NA y == NULL logical(0) Instead one should use is.na (as above) or is.null to identify NA or NULL elements or objects. is.na(x) [1] TRUE is.null(y) [1] TRUE Writing functions 78 A.20 Writing functions Perhaps the most useful thing about R is its capacity to implement user-defined functions. At some point you may even wish to compile your functions into an R-package. No more will be said on such matters here. Interested readers are directed to the reference for R-package construction “Writing R Extensions”, available at http:// cran.r-project.org/doc/manuals/R-exts.html. Programs like RWinEdt (an R package plugin for WinEdt), Tinn-R (a recursive acronym for Tinn is not Notepad, http://www.sciviews.org/Tinn-R), ESS (Emacs Speaks Statistics, http://ess.r-project.org) and RStudio 16 (http://rstudio.org) have been developed primarily to facilitate function writing in R17. Text editors from these programs provide syntax highlighting of the R (and S) code and (generally) highlighting for other programming languages, e.g. C/C++, Java, Perl, Pascal, Visual Basic, and Fortran. In addition, some editors, including RWinEdt, Tinn-R, ESS and RStudio allow direct interaction with an R-console running on the same computer, or can open R and generate an R-console. R-scripts, accessed with File>New script (non-Unix only) or with Ctrl+F+N on the command line prompt, provide another excellent way to edit R-code. A single line of code (or selected lines of code) can be sent to the R-console from an R-script interface using the shortcut Ctrl+R. The entire contents of a script can be selected using Ctrl+A, and then sent to the R-console using Ctrl+R (Figure A.8). Figure A.8 Script editor in R In all the text editors described above, functions or groups of functions can be saved as .R files. These files can then be read as source code directly in R or opened as scripts in the script editor. It is strongly recommended that word processing programs (e.g., MS Word) not be used to create R scripts and functions as the resulting 16 The attributes of RStudio greatly exceed simple function writing (see § A.20.12) Other frequently used text editors with some support for R include, but are not limited to, NppToR in Notepad++ (http://sourceforge.net/projects/npptor/) , Tinn-R (http://www.sciviews.org/Tinn-R/), Bluefish (http://bluefish.openoffice.nl/index.html), Crimson Editor (http://www.crimsoneditor.com/), ConTEXT http://www.contexteditor.org/), Eclipse (http://www.eclipse.org/eclipse/), ESS (http://ess.r-project.org/) , Vim (http://www.vim.org/), Geany (http://www.geany. org/), jEdit (http://www.jedit.org/), Kate (http://kate-editor.org/), TextMate (http://macromates.com/), gedit (http://projects.gnome.org/gedit/), SciTE (http://www.scintilla.org/SciTE.html), and RWinEdt (http://cran.r-project.org/web/packages/RWinEdt/). All websites accessed 4/10/2012. 17 Writing functions 79 code may contain hidden formatting characteristics that may affect its implementation in R. A.20.1 An introductory example At the risk of sounding repetitive, we specify a function using the function function. The arguments for function are completely user-defined and will be the argument list for one’s personalized function. For instance, in the excerpt example <- function(x)the function example contains a single argument named x (naming conventions for arguments must follow the conventions defined in A.4.1). The code for the function itself follows the call to function, and the argument list, generally delimited by braces. This results in the format: function.name <- function(argument1, argument2,....argumentn){function contents} Consider the function Exp.growth below that solves the function N0λt, where N0= initial number of individuals, λ = the rate of increase, and t = the number of time intervals or generations. Exp.growth <- function(N.0, lambda,t){ Nt <- N.0 * lambda^t Nt } The function requires three arguments: N.0, lambda, and t. The first line of code following the arguments solves the function N0λt utilizes the argument definitions and stores the solution as N.t. The last line gives the object I want returned by the function, N.t. Note that if I didn’t specify some "return value", nothing would be returned by the function. If a function has multiple return objects, then one can place them in single suitable container like a list. To increase clarity one should place the first curly bracket on same line as the arguments, and place last curly bracket on its own line. This convention is followed above. Readability can also be improved by use of spaces. Note that I have inserted two spaces to begin lines containing related operations. This distinguishes these lines from the first (argument) line and the end (return) line. Note also that spaces are placed after commas, and before and after operators, including the assignment operator. Once I have read the function Exp.growth into R (by scanning, reading, typing, or pasting it), I will be able to run it by typing the function name and specifying arguments. Here we run function for N0= 100, λ = 0.8, and t = 10 Exp.growth(N.0 = 100, lambda = 0.8, t = 10) [1] 10.73742 The population size is decreasing because λ is less than one. Writing functions 80 A.20.2 Global variables versus local variables It is important to note that the arguments and objects created in a function are local variables. That is, they only exist within the confines of the function. We can use Exp.growth to demonstrate that the variable N.t, which was defined in the function, is local. Nt Error: object ‘Nt’ not found Global variables are (generally) defined outside of functions, and thus can be called within a function as arguments or other sorts of objects. Lexical scoping allows R to distinguish global and local variables. Languages like S that don’t have lexical scoping would not “know about” the variables N.0, lamba, and t unless they were defined outside of the function, i.e., N.0 <- 100; lambda <- 0.8; t <-3 Conversely, R allows one to define the free variables in the text of the function, e.g., Exp.growth(100, 0.8,3) [1] 51.2 Global variables can be assigned within functions using the so-called superassignment operator, <<-. Exp.growth <- function(N.0, lambda,t){ Nt <<- N.0 * lambda^t Nt } Exp.growth(N.0 = 100, lambda = 0.8, t = 10) [1] 10.73742 Nt [1] 10.73742 rm(Nt) Nt Error: object ‘Nt’ not found || Example A.9 --Summary statistics: variations on a theme The simple function shown below centers and scales (standardizes) outcomes in the Downs dataset (i.e., each element in the data matrix is subtracted from its column mean, and divided by its column standard deviation). The function itself is defined and called within the function apply. Recall that the third argument in apply must be an existing or user-defined function, e.g., mean. We note that braces are not necessary if the "block" of function code only requires a single line, and this line is placed on the Writing functions same line as the function call. stan <- (apply(Downs, 2, function(x){(x - mean(x))/sd(x)})) stan Age Births Cases [1,] -1.5797549 0.4841299 -0.1698524 [2,] -1.2553785 1.4397237 0.5774983 [3,] -1.1626996 1.6535732 -0.1698524 [4,] -0.4212680 0.7250074 -1.0417617 [5,] -0.3285890 0.5292520 -0.6680863 [6,] 0.4128426 -0.5021109 -0.2944109 [7,] 0.5055216 -0.6008367 1.5739660 [8,] 0.5982005 -0.7149425 1.6985244 [9,] 0.9689163 -0.9813779 0.5774983 [10,] 1.0615953 -1.0117986 -0.7926447 [11,] 1.2006137 -1.0206195 -1.2908786 Below I create a function called stats that will simultaneously calculate a large number of summary statistics. stats <- function(x, digits = 5){ ds <- data.frame(statistics = round(c(length(x), min(x), max(x), mean(x), median(x), sd(x), var(x) ,IQR(x), sd(x)/sqrt(length(x)), kurt(x), skew(x)), digits)) rownames(ds) <- c("n", "min", "max", "mean", "median", "sd", "var", "IQR", "SE", "kurtosis", "skew") return(ds) } The function contains two arguments. A call to a vector of data (x), and the number of significant digits (digits). Because I have given digits the default value 5, only the first argument needs to specified by a user. Following the arguments, the first three lines of the function create a dataframe called ds. It has one column called "statistics" that will contain eleven statistical summaries of x. The summaries are rounded to the number of digits specified in digits. The next lines of code define the row names of ds. These are the names of the statistics calculated by the function. The last command, return(ds), returns ds. Note that the lines beginning median(x), digits and "SE" are not indented because they are con- 81 Writing functions 82 tinuations of the previous line. Here are summary stats for the Age column in Downs. stats(Downs$Age) statistics n 11.00000 min 17.00000 max 47.00000 mean 34.04545 median 38.50000 sd 10.78994 var 116.42273 IQR 17.00000 SE 3.25329 kurtosis -1.41039 skew -0.40977 The lone argument for stats, x, must be a vector of data. The function, however, can easily be extended to matrices and dataframes. For instance, lets now apply stats as the last argument of the apply function, to summarize the stan dataframe we created above. apply(stan, 2, stats) $Age statistics n 11.00000 min -1.57975 max 1.20061 mean 0.00000 median 0.41284 sd 1.00000 var 1.00000 IQR 1.57554 SE 0.30151 kurtosis -1.41039 skew -0.40977 $Births statistics n 11.00000 min -1.02062 max 1.65357 mean 0.00000 median -0.50211 sd 1.00000 var 1.00000 IQR 1.47529 SE 0.30151 kurtosis -1.28862 skew 0.53721 $Cases statistics n 11.00000 min -1.29088 max 1.69852 mean 0.00000 median -0.16985 sd 1.00000 var 1.00000 IQR 1.30786 SE 0.30151 kurtosis -0.63921 skew 0.61985 The output above is a list. The attributes of the list are the column names of Downs. As the result of standardization, three variables now have mean zero, and unit standard deviations and variances. The function stats will work on any R object with a numerical class. For instance, try it on the first two columns of lp, the loblolly pine dataset. Writing functions 83 A.20.3 uniroot Parameter estimation in statistics often requires (at least implicitly) function optimization. However parameter estimation functions may not have a closed form, because an entire dataset needs to be considered with respect to proposed estimate values. A useful function in these situations is uniroot which searches an interval for the zero root of a function. For instance, many location estimators (those which estimate "central" or "typical" values, e.g., the true mean) will be the zero root the function: n ∑ ( xi − µˆ ) , (A.2) i =1 where xi is the ith observation from a dataset, and µ̂ is an estimator that uses the data to estimate a true location value. We will use uniroot to find a location estimate that provides a zero root for this function. To do this we must first create a function defining Eq. A.2. f <- function(x, data) sum(data - x) We will evaluate the loblolly tree height data. data <- lp[,1] Here we apply uniroot: uniroot(f(data), c(min(data), max(data)))$root [1] 32.3644 This value is identical to the sample mean. Indeed, the sample mean will be always be the zero root of Eq. A.2. Normally the difference of the data points and the location estimate is squared. Minimizing this function is called the process of ordinary least squares. A.20.4 switch A useful command in function writing is switch. It evaluates and switches among user-designated alternatives, included as a function argument. The function below switches between five different estimators of location (the typical or central value from a sample). These are the sample mean, a trimmed mean (using 10% trimming), the geometric mean, the median, and Huber’s M-estimator (see Chapter 4 in the Foundational and Applied Statistics text). location <- function(x, estimator){ switch(estimator, mean = mean(x), # arithmetic mean trim = mean(x, trim=0.1), # trimmed mean κ = 0.1 geo = exp(mean(log(x))), # geometric mean med = median(x), # median huber = huber.mu(x), # Huber M-estimator, likelihood maximizing method stop("Estimator not included")) } Writing functions 84 Here are some data. x <- c(2, 1, 4, 5.6, 7, 6.7, 3, 4.5, 10, 12, 2, 5, 6) I can run the function using any of the four location estimators. location(x, "mean") [1] 5.292308 location(x, "geo") [1] 4.357524 location(x, "huber") [1] 4.959546 A.20.5 Triple dot ... argument A particularly handy item for writing wrapper functions (functions which embed other functions) is the triple dot ( ...) argument. Imagine you wish to create a wrapper for the function plot that allows you to automatically display the results from a simple linear regression. We might create the following: reg.plot <- function(x, y, ...){ plot(x, y,...) lm.temp <- lm(y ~ x) coef <- lm.temp$coefficients abline(coef[1], coef[2]) } The triple dots in the arguments of reg.plot, and the associated triple dots we included in plot let us specify any additional argument from plot as an argument in reg.plot. For instance we could include specification for X and Y axis labels in reg.plot with the plot arguments xlab and ylab. For instance: with(Loblolly, reg.plot(age, height, xlab = "age", ylab = "height")) || Example A.10 –Function writing: quantifying biological diversity In this extended exercise we examine functions for summarizing a community ecology dataset. Alpha diversity measures the degree of evenness and richness within individual plots in a dataset. Low levels of monodominance and high richness result in high alpha diversity. A large number of alpha diversity indices have been utilized by ecologists (Magurran 1988). The most widely used are Simpson’s index (D1, Simpson 1949) and the Shannon-Weiner index (H′, MacArthur and MacArthur 1961). Writing functions 85 D1 = 1 − ∑ i pi2 H ' = ∑ pi ln pi i where pi is the proportional abundance of the ith species. (A.3) (A.4) Simpson’s index (D1) has a straightforward interpretation. It is the probability of reaching into a plot and simultaneously pulling out two different species. The Shannon Weiner index (H´) does not allow straightforward interpretation, although its values conventionally fall between 1.5 and 3.5, and rarely surpass 4.5 (Magurran 1988, Margalef 1972). D1 and is sensitive to abundance changes in common species, while H´ is sensitive to abundance changes in rare species (Gurevitch et al. 2006). D1 and H´ will both increase with increasing diversity (i.e., increased richness and evenness). Table A.2 Part of a community dataset (six species and five sites) from a Scandinavian Pinus sylvestris forest (Väre et al. 1995). Responses are percent ground cover. Empetrum nigrum Vaccinium myrtillus Vaccinium vitis-idaea Pinus sylvestris Vaccinium uliginosum Betula pubescens Site 1 11.13 0 17.8 0.07 1.6 0 Site 2 8.92 2.42 10.28 0.12 0 0 Site 3 6.45 0 14.13 0.07 0.47 0 Site 4 9.3 0 8.5 0.03 0 0 Site 5 3.47 0.25 20.5 0.25 0 0 To read the data into R we can use the data.frame function, as we have done before, or the matrix function (since the variables are all quantitative). Using the matrix command we specify how many rows and columns we want and then simply enter the data by rows or columns. ps.data <- matrix(nrow = 5,ncol = 6,data = c(11.13, 0, 17.8, 0.07, 1.6, 0, 8.92, 2.42, 10.28, 0.12, 0, 0, 6.45, 0, 14.13, 0.07, 0.47, 0, 9.3, 0, 8.5, 0.03, 0, 0, 3.47, 0.25, 20.5, 0.25, 0, 0), byrow = TRUE) We can give the matrix the correct row and column names. colnames(ps.data)<-c("Empetrum nigrum", "Vaccinium myrtillus", "Vaccinium vitis-idaea", "Pinus sylvestris", "Vaccinium uliginosum", "Betula pubescens"); rownames(ps.data)<-c("Site 1", "Site 2", "Site 3", "Site 4", "Site 5") ps.data Writing functions Site Site Site Site Site 1 2 3 4 5 Site Site Site Site Site 1 2 3 4 5 86 Empetrum nigrum Vaccinium myrtillus Vaccinium vitis-idaea 11.13 0.00 17.80 8.92 2.42 10.28 6.45 0.00 14.13 9.30 0.00 8.50 3.47 0.25 20.50 Pinus sylvestris Vaccinium uliginosum Betula pubescens 0.07 1.60 0 0.12 0.00 0 0.07 0.47 0 0.03 0.00 0 0.25 0.00 0 Next, let’s create a function that can calculate Simpson’s Index (Eq. 2.1). First we need to calculate the relative proportion, pi, of each species at each site. We can use the command apply again for this. p.i <- apply(ps.data, 1, function(x){x/sum(x)}) As with my first example in this section I have inserted a function where we would normally tell apply what to do to the rows or columns of the matrix. In this case I have told the function (within the function apply) to divide each element within a particular row by the sum of its respective row. p.i Site 1 Empetrum nigrum 0.363725490 Vaccinium myrtillus 0.000000000 Vaccinium vitis-idaea 0.581699346 Pinus sylvestris 0.002287582 Vaccinium uliginosum 0.052287582 Betula pubescens 0.000000000 Site 5 Empetrum nigrum 0.14180629 Vaccinium myrtillus 0.01021659 Vaccinium vitis-idaea 0.83776052 Pinus sylvestris 0.01021659 Vaccinium uliginosum 0.00000000 Betula pubescens 0.00000000 Site 2 0.410303588 0.111315547 0.472861086 0.005519779 0.000000000 0.000000000 Site 3 0.305397727 0.000000000 0.669034091 0.003314394 0.022253788 0.000000000 Site 4 0.521592821 0.000000000 0.476724621 0.001682557 0.000000000 0.000000000 Next I need to: 1) square the proportions, 2) take the sum within each row (each site), and 3) subtract this sum from one. D <- 1 - apply(p.i^2, 2, sum) It looks like we only really need two lines of code to calculate Simpson’s index. We can write the function as: Writing functions Simp.index <- function(x){ p.i <- apply(x, 1, function(x){x/sum(x)}) 1-apply(p.i^2, 2, sum) } To run Simp.index for the Pinus sylvestris data, ps.data, I read or type the function into R and then type: Simp.index(ps.data) Site 1 Site 2 Site 3 Site 4 Site 5 0.5265904 0.5956317 0.4586194 0.5006717 0.2778395 Next we will create a function for the Shannon Weiner index. This measure also utilizes pi; as a result it will be similar to the Simpson’s index function. SW.index <- function(x){ p.i <- apply(x, 1, function(x){x/sum(x)}) h <- apply(p.i, 1, function(x){log(x) * x}) -1 * apply(h, 1, function(x){sum(x[!is.na(x)])}) } Note that we run into a problem calculating H’ when any pi = 0 because ln(0) is undefined. As a result zeroes in the data will cause the function to fail. To account to for this I told R to add pi elements only when they are not undefined. This is in the line: h <- -1 * apply(h, 1, function(x){sum(x[!is.na(x)])}) The line works because an undefined (NaN) value is regarded as NA. Here we run the function on ps.data. SW.index(ps.data) Site 1 Site 2 Site 3 Site 4 Site 5 0.8512311 0.9927535 0.7347475 0.7034007 0.5189526 || Example A.11 Advanced mathematical applications -- solving systems of ODEs Biologists often need to solve systems of dependent differential equations in models involving species interactions (e.g., competition or predation). For instance the Lotka-Volterra competition model has the form: 87 Writing functions 88 dN1 / dt = rmax1 N1 ( K1 − N1 − α12 N 2 ) K1 ( K − N 2 − α 21 N1 ) dN 2 / dt = rmax 2 N 2 2 K2 (A.5) where t = time, rmax1 = the maximum per capita rate of increase for species 1 under ideal conditions, rmax2 = the maximum per capita rate of increase for species 2 under ideal conditions, N1 = the number of individuals from species 1, N2 = the number of individuals from species 2, K1 = the carrying capacity for species 1, i.e., the maximum population size of that species that the environment can support. K2 = the carrying capacity for species 2, α12 = the competitive effect of species 2 on the growth rate of species 1, and α21 = the competitive effect of species 1 on the growth rate of species 2. We first bring in the package deSolve (see Section A.21) which contains function for solving ordinary differential equations (ODEs)18. install.packages("deSolve") library(deSolve) We then define starting values for N1 and N2 and model parameters xstart <- c(N1 = 10, N2 = 10) pars <- list(r1 = 0.5, r2 = 0.4, K1 = 400, K2 = 300, a2.1 = 0.4, a1.2 = 1.1) We then specify the Lotkka-Volterra equations as a function. We will include the argument time = time even though it is not used in the function. This is required by ODE evaluators from deSolve. LV <- function(time=time, xstart=xstart, pars=pars){ N1 <- xstart[1] N2 <- xstart[2] with(as.list(pars),{ dn1 <- r1 * N1 * ((K1 - N1 - (a1.2 * N2))/K1) dn2 <- r2 * N2 * ((K2 - N2 - (a2.1 * N1))/K2) res <- list(c(dn1, dn2)) }) } We will run the Lotka-Volterra function with the function rk4 from library deSolve. The rk4 function solves simultaneous differential equations using classical Runge-Kutta 4th order integration (Butcher As opposed to partial differential equations 18 Writing functions 89 1987). The method of Euler (the simplest method to find approximate solutions to first order equations) can be specified with the function euler. The arguments for rk4, in order, are the initial population numbers from species 1 and 2, the time frames to be considered, the function to be evaluated, and the parameter values. out <- as.data.frame(rk4(xstart, time = 1:200, LV, pars)) The object out contains the number of individuals in species 1 and 2 for time frames 1-200 (Figure A.9). 150 100 50 Number of individuals 200 250 plot(out$time, out$N2, xlab = "Time", ylab = "Number of individuals", type = "l") lines(out$time, out$N1, type = "l", col = "red", lty = 2) legend("bottomright", lty = c(1, 2), legend = c("Species 2", "Species 1"), col = c(1, 2)) Species 2 Species 1 0 50 100 150 200 Time Figure A.9 Result of solutions from Lotka Volterra ODEs for t = {1, 2, ..., 200}. Species 1 and 2 coexist, but at levels appreciably below their carrying capacities as a result of interspecific competition. Functions can obviously be more complicated than these previous examples. Below is a function called radiation.heatl. Its purpose is to calculate northern hemisphere annual incident solar radiation (MJ cm-2 yr-1) and heatload given geographic and topographic data (McCune and Keough 2002). Heatload is a radiation index based on the idea that the highest amounts of radiation occur on southwest facing slopes, because they receive late afternoon sun. The function has three arguments: slope (measured in degrees), aspect, and latitude. Writing functions 90 radiation.heatl <- function(slope, aspect, lat){ asp.wrap.rad <- sapply(aspect, function(x){(-1 * abs(x - 180)) + 180}) asp.wrap.hl <- sapply(aspect, function(x){abs(180 - abs(x - 225))}) rad.lat <- (lat/180) * pi rad.asp <- sapply(asp.wrap.rad, function(x){(x/180) * pi}) hl.asp <- sapply(asp.wrap.hl, function(x){(x/180) * pi}) rad.slope <- sapply(slope, function(x){(x/180) * pi}) rad <- matrix(nrow = length(slope), ncol = 1) hl <- matrix(nrow = length(slope), ncol = 1) for(i in 1 : length(slope)){ rad[i] <- 0.339 + 0.808 * (cos(rad.lat) * cos(rad.slope[i])) - 0.196 * (sin(rad. lat) * sin(rad.slope[i]))-0.482 * (cos(rad.asp[i]) * sin(rad.slope[i])) hl[i] <- 0.339+0.808 * (cos(rad.lat) * cos(rad.slope[i])) - 0.196 * (sin(rad.lat) * sin(rad.slope[i]))-0.482 * (cos(hl.asp[i]) * sin(rad.slope[i])) } list(radiation = rad, heat.load = hl) } After reading the function into R I can find the annual incident radiation is for a site with a slope of 20º, with a northeast (30º) aspect, that is located at 40 degrees N latitude, by typing: radiation.heatl(20, 30, 40) $radiation [,1] [1,] 0.7347784 $heat.load [,1] [1,] 0.7183095 This site receives 0.7347784 mega joules of radiation per cm2 per year, and has a heat load index of 0.7183095. A.20.6 Looping A major component of the radiation.heatl function is a for loop. It starts with the line: for(i in 1 : length(slope)){, and ends with a closing brace, }. Loop functions exist in some form in virtually all programming languages. The call to a for loop in R is made using the command for. An R for loop requires specification of three entities, in parentheses, following for. These are: 1) an index variable, e.g., i, 2) the statement in, and 3) a sequence that the index variable refers to as loops commence. The code defining the loop follows, generally (if the loop requires multiple lines) delineated by curly brackets. Writing functions 91 In parallel to function writing it is good style to place the first curly bracket on the same line as the call to for, and to place the last curly bracket on its own line. For clarity, spaces should also be used to distinguish the loop contents. Thus, we have the basic form: for(i in seq){ loop contents } In the loop the ith element of something (e.g., matrix column, vector entry, etc.) is defined (or replaced) as the for sequence commences. The replacement/definition process takes place in the "loop contents". || Example A.12 One application for a loop is to make functions with scalar input arguments amenable to vector, matrix or dataframe inputs. For example, what if I wanted to apply the radiation.heatl function (which requires scalar inputs) to a three columned dataframe with columns containing slope, aspect, and latitude information, respectively? For instance: x <- data.frame(slope = c(10, 12, 15, 20, 3), aspect = c(148, 110, 0, 30, 130), latitude = c(40, 50, 20, 25, 45)) The first step in many loop applications is defining an object to contain the results. Because I will obtain a list with two objects from radiation.heatl, I will create two one column matrices to contain my results, one for radiation (I will call this rad.res), and one for heatload (I will call this heat.res). Because I have five observations, each matrix will have five elements. rad.res <- matrix(ncol = 1, nrow = 5) heat.res <- matrix(ncol = 1, nrow = 5) Next I create the for loop. for(i in 1:5){ rad.res[i] <- radiation.heatl(x[i,(1)],x[i,(2)],x[i,(3)])$radiation heat.res[i] <- radiation.heatl(x[i,(1)],x[i,(2)],x[i,(3)])$heat.load } The function loops around on itself letting i = 1 during the first loop, i = 2, during the second loop, up to i = 5 on the final loop. Here are the results. Writing functions rad.res [1,] [2,] [3,] [4,] [5,] [,1] 0.9976634 0.8500810 0.9302991 0.8560357 0.9185208 heat.res [1,] [2,] [3,] [4,] [5,] [,1] 0.9455112 0.7734540 0.9668378 0.8395667 0.9001073 Look at the function and try to really I understand what I did, using the information from earlier in this Appendix. || Example A.13 Another application of a loop is to iteratively build on the results of the previous step in the loop. Consider the following function that counts the number of even entries in a vector of integers. evencount <- function(x){ res <- 0 for(i in 1 : length(x)){ if(x[i]%%2 == 0) res <- res + 1 } res } Recall that %% is the modulus operator in R. That is, it finds the remainder in division. By definition the remainder of any even integer divided by two will be zero. At each loop iteration the function adds one to the number in res if the current integer in the loop is even (has remainder zero if divided by two). evencount(c(1,2,3)) [1] 1 evencount(c(1,2,3,4,10)) [1] 3 92 Writing functions 93 || Example A.14 A third loop application is the summarization of data with respect to levels in a categorical variable. Consider the categorical extension to the Downs dataset discussed in section A.14.2. We have: Categories <- factor(c(rep(1, 6), rep(2, 5))) cbind(Downs, Categories) What if we wished to statistically summarize the variables in Downs (Age, Births, and Cases) simultaneously for each level in the categorical variable Categories? One solution is a for loop. We first create an empty list to hold the result: result <- list() The components in the list will be dataframes of summary statistcs. These will be assembled in a for loop that steps through the levels in Categories for(i in levels(Categories)){ temp <- Downs[Categories==i,] result[[i]] <- as.data.frame(apply(temp,2,stats)) names(result[[i]]) <- c("Age","Births","Cases") } Instead of a numeric sequence, the loop will walk through the levels of Categories. This is specified with the code: for(i in levels(Categories)). Note that in the first line following the for specification, the Downs dataset is subset by levels in categories. The data are then summarized with the function stats that we created in Example 7. Column names for the summary dataframes are given in the last line of the loop. Here is the loop result: Writing functions result $`1` Age Births n 6.00000 6.000000e+00 min 17.00000 4.834000e+03 max 38.50000 2.389600e+04 mean 26.25000 1.565483e+04 median 25.50000 1.481950e+04 sd 7.99844 6.821069e+03 var 63.97500 4.652698e+07 IQR 9.50000 6.770250e+03 SE 3.26535 2.784690e+03 kurtosis -0.79755 2.097100e-01 skew 0.50457 -4.399500e-01 94 Cases 6.00000 9.00000 22.00000 15.00000 15.50000 4.38178 19.20000 3.25000 1.78885 0.92773 0.36372 $`2` Age Births Cases n 5.00000 5.00000 5.00000 min 39.50000 249.00000 7.00000 max 47.00000 3961.00000 31.00000 mean 43.40000 1617.00000 20.20000 median 44.50000 596.00000 22.00000 sd 3.24808 1721.52011 10.89495 var 10.55000 2963631.50000 118.70000 IQR 5.00000 2625.00000 19.00000 SE 1.45258 769.88720 4.87237 kurtosis -2.48635 -2.20448 -2.64479 skew -0.32174 0.77477 -0.28092 A.20.7 Looping without for Looping in R is also possible using the style of the language C which conventionally uses the functions while, repeat, and break. || Example A.15 Consider an example in which 2 is added to a base number until the updated number becomes greater then or equal to 10: We have: i <- 1 while (i < 10)i <- i + 2 i [1] 11 Writing functions 95 Or, alternatively i <- 1 while(TRUE){ i <- i + 2 if (i > 9) break } i [1] 11 Here the variable i took on values 1, 3, 5, 7, 9, and 11 as the loop commenced. When i equalled 11 the condition for continuation of the loop failed and the loop was halted. A.20.8 Calling and receiving code from other languages Code from C, C++, Fortran, MATLAB, Python and other languages can be linked to R at the command prompt. Indeed, most of the code for R is written in C. R can also be called from a number of different languages including C, Java, or Python (Lang 2005). For instance, the R package RCytoscape allows cross communication between the popular Java-driven software molecular network package Cytoscape and the R-console. We have already learned that Perl applications can be used to identify and manipulate R objects (see § A.16.10). R is an interpreted language, as a result functions written in R must be tacitly translated into binary before they can be executed. This process is in contrast to compiled languages, e.g., C, Fortran, and Java. For these languages a compiler (a translation program) is used to transform the source code into a target "object" language, which is generally binary. The product of the compilation is called an executable file (Figure A.10). Compiler (translator program) (e.g., javac, R CMD SHLIB) Suitable environment (e.g., program in a workstation) Object code Source code Outcome (executable, binary) (e.g., .exe, .dll) (e.g. .java, .c, .f ) Compilation Steps in the translation of source code Execution Figure A.10 Illustration of translation steps in compiled languages. Raw Fortran source code is generally saved as an entity called an .f file, C source code is saved as an .c file, and Java source code is saved as a .java file. Windows executable files, complied from source code, will generally have an .exe or .dll extension. R contains both Fortran and C compilers. Note, however, that the compilers will only work for Fortran code written as a subroutine and void C code. As a result neither will return a value directly. To write and compile Fortran and C in Windows one can save files containing the source code in the bin directory of the current version of R (where the compilers are located), and then invoke the shell command: R CMD Writing functions 96 SHLIB from a workstation command prompt followed by the name of the file. The R CMD algorithms located in the R bin directory are an important group of functions that are called directly from the command prompt shell (not interactively from the R-console). They include R CMD check, which checks user designed packages for errors, and R CMD batch which creates batch R output files (e.g. several .pdf files, dataframes or graphs). Calling Fortran and C for looping Because its functions are not compiled, R requires a large amount of system time for iterative procedures like loops. As a result, it may be expedient to call compiled code, originally written in Fortran or C, for complex looping with large datasets. Below is a simple example (that could easily be handled without a for loop) that compares R, Fortran, and C looping. Familiarity with C and Fortran is assumed. || Example A.16 Here is a Fortran subroutine to convert a vector of Fahrenheit measurements to degrees Celsius. subroutine ftoc(n, x) integer n double precision x(n) integer i do 100 i = 1, n x(i) = (x(i)-32)*(5./9.) 100 continue end After typing up the code in a text editor we will save it in the R bin directory as the file FtoC.f. We will then compile it by opening up an operating system command line prompt, going to the bin directory (if you are using the 64 bit R, be sure to specify the 64 bit version before the 32 bit version in your Windows system search path), and typing: R CMD SHLIB FtoC.f This shell command will create a compiled routine (in a Windows operating system this will be a .dll file, in Unix/Linux it will be an .so file). Here is an analogous C function for converting ⁰F to ⁰C: Writing functions void ftocc(int *nin, double *x) { int n = nin[0]; int i; } for (i=0; i<n; i++) x[i] = (x[i] - 32) * 5 / 9; We would save the code as the file ftocc.c file in the bin directory and run the shell command: R CMD SHLIB ftocc.c will create a compiled routine in the form of a .dll file. Here is an R-wrapper that can call either the Fortran subroutine, call = "f", or the C function, call = "c". F2C <- function(x, call = "f"){ n <- length(x) if(call == "f"){ dyn.load("C:/Program Files/R/R-3.0.2/bin/x64/ftoc.dll") out <- .Fortran("ftoc", n = as.integer(n), x = as.double(x)) } if(call == "c"){ dyn.load("C:/Program Files/R/R-3.0.2/bin/x64/ftocc.dll", nin = n, x) out <- .C("ftocc", n = as.integer(n), x = as.double(x)) } out } Let’s convert 1,000,000 randomly generated Fahrenheit temperatures to Celsius. x <- runif(1000000, 0, 100) Here is a loop using R alone. F2CR<-function(x){ res <- 1 : length(x) for(i in 1 : length(x)){ res[i] <- (x[i] - 32) * (5/9) } res } 97 Writing functions 98 Let’s compare the run times. system.time(F2CR(x)) user system elapsed 2.93 0.00 2.95 system.time(F2C(x, call = "f")) user system elapsed 0 0 0 system.time(F2C(x, call = "c")) user system elapsed 0.03 0.00 0.03 The Fortran and C loops gave the same transformation results (not shown here), but ran much faster. The Fortran subroutine was particularly fast. This is despite the fact that this language greatly predates C (and certainly R). A.20.9 Functions with animation Animation can be in R used to illustrate a wide range of processes (Xie 2011, Xie and Chang 2008). Functions with animation are generally based on for loops with some method of slowing the loop; usually the function Sys.sleep(). Slowing the loop allows examination of the iterative steps defined in the creation of a plot. || Example A.17 Consider the following function: R.zoom<-function(){ # requires R >= 2.15.1 for(i in seq(30, 1800, 30)/10){ dev.hold() plot(1:10, 1:10, type = "n", xaxt = "n", yaxt = "n", xlab = "", ylab = "") text(5, 5, "R", cex = i, srt = i) dev.flush() Sys.sleep(.1) #Loop stopped for 0.1 seconds at each iteration } } Using functions from package animation (Xie and Chang 2008) one can save an R animation as a media file for inclusion in other formats, e.g., webpages, powerpoint presentations, interactive pdf documents, etc. For instance the code below creates an adobe flash (.swf) file of Rzoom to a temporary directory. Writing functions 99 install.packages("animation") library(animation) saveSWF(R.zoom(), swf.name = "Rzoom.swf", interval = 0.1, nmax = 30, ani.dev = "pdf", ani.type = "pdf", ani.height = 6, ani.width = 6) The function saveSWF requires installation of SWF Tools, and a path in the Windows environment to the tools. The result, embedded in an .flv file, can be shown by clicking on the flashplayer link below. A.20.10 Building GUIs GUIs (Graphical User Interfaces) are a mixed bag. A person who is terrified of command line entry will be reassured by their point and click familiarity. However much user flexibility will be lost for controlling functions. In addition, a GUI implementing too many arguments may quickly become visually confusing, and hiding code inside a GUI "black box" is contrary to the "mission statement" of R. Command line entry is valuable for two other reasons. First, one can scroll through earlier commands (using the up arrow), and easily repeat or modify earlier steps. Second, command line entry provides an exact record of everything you have done during a particular Rsession. Despite these caveats the package asbio relies on interactive GUIs for pedagogic demonstrations. Type: library(asbio) book.menu() Devices like those in asbio can be created in a number of programming languages including Java and Perl. I used the language tcltk (pronounced "tickle tee kay") to create asbio GUIs, because tcltk GUIs are easy to build, and an interpreter for tcltk is included with a conventional install of R. tcltk GUIs An extensive description of the tcltk language and its capacities in R are beyond the scope of this book. I include simple examples here (with little code explanation) merely to show the reader that such devices can be easily constructed in R. || Example A.18 Consider a GUI whose only purpose is to destroy itself (Figure A.11). Writing functions 100 require(tcltk) tt <- tktoplevel() DM.but <- tkbutton(tt, text = "Exit", command = function() tkdestroy(tt)) tkgrid(DM.but) Figure A.11 An extremely simple radio button GUI. Click "Exit" (in R) to destroy. • • • The function ttoplevel defines the "toplevel" widget (graphical elements that allow a user to interactively change a function), allowing it to be manipulated. The tkbutton function creates the "Exit" button. The first argument is name of the widget that the button is to be placed upon. The text argument defines the text on the button. The command argument defines the function that the button initiates. In this case it is tkdestroy, which closes open tcltk widgets. The function tkgrid places the button on the widget. We can also use GUIs to run other R-functions. For instance the GUI below (Figure A.12) calls the function R.zoom from § A.20.9. tt <- tktoplevel() but <- tkbutton(tt, text = "Zoom", command = R.zoom) tkgrid(but) Figure A.12 A GUI for calling the function R.zoom. GUIs can contain a number of widgets, including sliders. || Example A.19 Below is code that defines the function plot.me. The function simply plots a large blue dot. Below it is code for a GUI slider. The dot’s vertical location is defined by the slider (Figure A.13). Writing functions 101 plot.me <- function(){ y <- evalq(tclvalue(SliderValue))# Evaluate the expression plot(1,as.numeric(y),xlab = "", ylab = "%", xaxt = "n", ylim = c(0,100), cex = 4, col = 4, pch = 19) } 0 20 40 % 60 80 100 slider.env <<- new.env() tt <- tktoplevel() SliderValue <- tclVar("50") SliderValueLabel <- tklabel(tt,text = as.character(tclvalue(SliderValue))) tkgrid(tklabel(tt, text = "Slider Value : "), SliderValueLabel, tklabel(tt, text = "%")) tkconfigure(SliderValueLabel, textvariable = SliderValue) slider <- tkscale(tt, from=100, to=0, showvalue=F, variable = SliderValue, resolution=1, orient = "vertical", command = substitute(plot.me())) tkgrid(slider) Figure A.13 A plotted symbol along with the slider that alters its position. The superassignment operator <<- used above initiates a search through the function environment for an existing definition of the variable being assigned. If the variable is found then its value is redefined, otherwise assignment takes place with respect to the global environment and will be translated in all functions. Other code specifications include: Writing functions • • • • • • 102 SliderValue <- tclVar("50")which creates a variable that can be used interactively, and defines the starting value of the slider to be 50. SliderValueLabel <- tklabel(tt,... which causes the value of the slider to be formatted for projection tkgrid(tklabel(tt, text = "Slider Value : "),... which places the slider values,along with text "Slider Value" and "%", above the slider. tkconfigure(SliderValueLabel, textvariable = SliderValue) which tells R that the slider output to be written will come from the variable SliderValue. slider <- tkscale(tt, from=100, to=0,... which defines the range of the slider along with other slider characteristics including the name of the function (plot.me) that the slider runs. tkgrid(slider)which places the slider on the GUI. For more information on GUIs see the tcltk manual included with a conventional installation of R (but written for a non-R language framework). Amazingly, one can call and utilize R from a local online server. Further, using shiny apps (http://shiny.rstudio. com/), an implementation of Rstudio, one can control R using sliders and widgets. Using such applications R is run unseen and a user need know nothing about the R language (see https://ahoken.shinyapps.io/seeNorm/). A.20.11 Functions with class and method This is an advanced topic for package developers, but may be of interest to those who wish to embed particular methods for summarizing a family of functions. It turns out that we are not limited to the pre-existing classes in R (e.g., numeric, factor, etc.). Instead, we can create user-defined classes with associated methods for plotting, printing and summarization. Object classes allow one to customize generic functions, e.g., plot, print, summary, for particular classes. These functions are called with the generic function names e.g., plot, print, summary (Section A.20.9). Here are functions on my workstation that will be called with plot, depending on the class of the object that is being plotted. methods(plot) [1] plot.acf* [5] plot.dendrite* [9] plot.ecdf [13] plot.hclust* [17] plot.lm* [21] plot.ppr* [25] plot.shingle* [29] plot.table* [33] plot.TukeyHSD* plot.data.frame* plot.dendrogram* plot.factor* plot.histogram* plot.medpolish* plot.prcomp* plot.spec* plot.trellis* plot.decomposed.ts* plot.density* plot.formula* plot.HoltWinters* plot.mlm* plot.princomp* plot.stepfun plot.ts plot.default plot.deSolve* plot.function plot.isoreg* plot.pairw plot.profile.nls* plot.stl* plot.tskernel* For example, let x be an object of class pairw. If I type plot(x), then the command plot.pairw(x) is run. The package asbio contains a number of functions for summary statistics including the function G.mean below Writing functions 103 which calculates the geometric mean (see Ch. 4 in Foundational and Applied Statistics textbook). It might be desirable to create a class called as.stat to print such summary statistics in a consistent way. G.mean <- function(x){ x <- na.omit(x) res <- list(stat = (prod(x))^(1/length(x)), name = "Geometric Mean") class(res) <- "as.stat" res } print.as.stat<-function(x, ...){ cat("\n") cat(x$name, "\n") cat(x$stat, "\n") invisible(x) } x<-c(2, 1, 4, 5.6, 7, 6.7, 3, 4.5, 10, 12, 2, 5, 6) G.mean(x) Geometric Mean 4.357524 We have gotten rid of some of the clunky default R output (for instance the [1] is gone). G.mean calls the function print.as.stat to print its results. Within print.as.stat the function cat concatenates and prints G.mean output, the argument "\n" means “put a line break in the output”, and the term invisible lets additional items be in print.as.stat that are not printed, but can still be called. Additional methods can also be made for plotting and other summaries for objects of class as.stat. The code for internal functions will not be directly accessible by typing the function name, but can be accessed with the double colon metacharacter (::) For instance a user of as.stat can see the internal function print.as.stat (if it exists) by typing the package name followed by ::print.as.stat. The triple colon operator (:::) allows access to so-called "unexported" objects in a package. It should be noted that the implementation here describes the R S3 system for classes and methods. The socalled S4 system, described in Chambers (2008) requires a much stricter set of conditions for user-classes19. A.20.12 Documenting functions and workflow A concern with R is tracking the characteristics of your functions and workflow. I generally use Rstudio or a text editor like TinnR to create .R script files that can be loaded into R, or pasted piecemeal into the R-console. 19 Google has an R style guide: http://google-styleguide.googlecode.com/svn/trunk/Rguide.xml that offers good advice for code writing. The webpage recommends S3 over S4 methods for most applications. Writing functions 104 However, even with use of notes using # it may be difficult to deduce what you have done, particularly given a an extended long separation from the work. One solution is to create .Rd document files describing functions, translate them into a LaTex style .tex documents , and use these to be build a pdf document. LaTex is a high quality typesetting freeware system that is the de facto standard for many publishers of scientific documents (see http://www.latex-project.org/). Consider the function below: myfunction <- function(text = "Hello world"){ print(text) } After typing the function into R type prompt("myfunction") This will create the skeleton of an .Rd documentation file for myfunction that will be placed in your working directory. The document can be completed using any text editor. Now... • Place the finished file in your R///bin directory, • Go to a command prompt and navigate to the directory using the MS-DOS command cd • In the command prompt type R CMD Rd2pdf myfunction.rd. This creates a .tex file that is read (in LaTex) to build a .pdf. The approach requires that both a LaTeX style repository (e.g., MiKTeX), and a Windows environmental path to this repository exist on your machine. A more sophisticated approach is to use a LaTeX text editor to create an .Rnw file that can be read with either Sweave or knitr. These programs can write beautiful equations and text (although the LaTeX language can seem daunting initially), distinguish R code from documentation, create graphs and tables from R code, and correctly place all of these into a .pdf document. Guidance for Sweave can be found at: http://www.stat.uni-muenchen.de/~leisch/Sweave/ Guidance for knitr can be found at: http://yihui.name/knitr/ Examples for some of my consulting work, documented in this way, can be found with the links below. Zero-inflated generalized linear models and elk Dall’s sheep population dynamics Autism neurological genetic networks Good LaTex editors include TeXnicCenter, which can be customized to call the R CMD Sweave script using the Define Output Profile dialog box in the Build pulldown menu. Writing functions || Example A.20 To create knitr documentation for an R work session we first install and load the knitr and xtabs packages. install.packages("knitr"); library(knitr) install.packages("xtable") We then write an empty .Rnw file to our working directory: file.create("stats.Rnw") LaTex documents require a preamble. We open stats.Rnw and type the following: \documentclass{article} \begin{document} \title{Stats summary of loblolly pine data} \maketitle We can, if we wish, now type in some R code. We indicate (to Knitr or Sweave) that characters are R code by beginning a code chunk with <<>>= and ending it with @. The following code provides a statistical summary of loblolly pine (\emph{Pinus taeda}) data. <<>>= data(Loblolly) stats <- function(x, digits = 5){ ds <- data.frame(statistics = round(c(length(x), min(x), max(x), mean(x), median(x), sd(x), var(x) ,IQR(x), sd(x)/sqrt(length(x)), kurt(x), skew(x)), digits)) rownames(ds) <- c("n", "min", "max", "mean", "median", "sd", "var", "IQR", "SE", "kurtosis", "skew") return(ds) } pine.height <- stats(Loblolly$height) @ We can use the xtable library to allow us to make an pretty table of this output. We load the package (and hide the code in the output document that we will create) by typing: 105 Writing functions Summary stats for tree hieghts are shown in Table 1\\ <<echo=FALSE>>= library("xtable") @ Two backslashes (i.e. \\) is a LaTex carriage return. We insert the table using: <<label = tab1, results = "asis", echo = FALSE>>= mat1 <- xtable(as.matrix(pine.height), caption = "Loblolly pine height summary", label = "tab:one") print(mat1, table.placement = "H", caption.placement = "top") @ The argument table.placement = "H" means: place the table exactly at the indicated location with respect to text. We can also make on-the-fly graphs: Figure 1 shows a histogram of pine ages \begin{center} \begin{figure}[H] <<out.width="0.8\\linewidth">>= hist(Loblolly$age) @ \caption{Distribution of pine ages.} \end{figure} \end{center} The argument center in begin centers the figure. The command out.width="0.8\\linewidth" forces the figure width to be 80% of the defined document line width. Figure dimensions will be held constant in this reduction. The quotations and backslashes in this code would be unnecessary in a nonknitr context. Lets end the document. \end{document} We now save the document. Assuming that stats.Rnw is still in the working directory we go back to 106 Writing functions R and type: knit2pdf("stats.Rnw") The document stats.pdf will now be in your working directory. The document is shown here. LaTex/ knitr code for the entire example can be found here. The function purl in knitr can be used to pull R code from a knit document and compile an .R file of the code. To create such a file for the current example I would type: purl("stats.Rnw") RStudio has a number of outstanding features that simplify writing documentation. Generating html R-documentation in RStudio requires no additional LaTex support. Generation of Sweave and knitr pdfs does require MikTex-alike installation although this process is greatly simplified in RStudio through the straightforward creation of .Rnw and .Rd files, among other features. Implementation of a simple documentation framework called R Markdown is also provided in RStudio, which allows generation of many output formats including html, pdf, and even MS Word. 107 Auxiliary packages 108 A.21 Auxiliary packages As we have already seen, an R package contains a set of related, functions, help files and data files that have been bundled together. Aside from the companion package to the Foundational and Applied Statistics text, asbio, there are a large number of packages that are useful to biologists and biometrists. Table A.3 lists a few. Table A.3 Example of packages of potential interest to biologists. Package Maintainer Topic(s) addressed by package: Current Citation/Author akima A. Gebhardt Interpolation of irregularly spaced data Akima (2012) car J. Fox A companion package to the book “an R and S companion to applied regression” Fox (2011) coin T. Hothorn Advanced applications of non-parametric analysis Hothorn et al. (2006, 2008) lme4 D. Bates Linear and non-linear mixed effect models Bates (2008) lmodel2 Jari Oksanen Model II regression Legendre (2013) MASS B. Ripley Wide variety of statistical topics. Venables and Ripley (2002) mgcv S. Wood Generalized Additive Models (GAMS) Wood (2011) nlme R-development core team Linear and non-linear mixed effect models Pinheiro et al. (2012) plotrix J. Lemon et al. Helpful graphical ideas Lemon (2006) spdep R. Bivand Spatial analysis Bivand et al. (2013) vegan J. Oksanen Multivariate analysis and ecological analysis Oksanen et al. (2013) The packages MASS (Modern Applied Statistics with S), nlme, and mgcv are included with the R base install along with a number of other valuable packages20. Other packages from Table A.3 are not included with the R base install, but can be downloaded from CRAN (Comprehensive R Archive Network). To do this, one can go to Packages>Install package(s) on the R-GUI toolbar, and choose a nearby CRAN mirror site to minimize download time (non-Unix only). Once a mirror site is selected, the packages available at the site will appear. One can simply click on the desired packages to install them. Packages can also be downloaded directly from the command line. To install vegan I would simply type: install.packages("vegan") Of course both of these approaches require that your computer has web access. If local web access is not available, libraries can be saved from the CRAN website as .zip files which can then be placed into the “library” folder within R. This folder will be installed onto your workstation with the installation of R. Once installed, information about the vegan package can be accessed by typing help(package= "vegan"). 20 Packages included in a conventional download of R are: base, boot, class, cluster, codetools, datasets, foreign, graphics, grDevices, grid, KernSmooth, lattice, MASS methods, mgcv, nlme, nnet, rpart, spatial, splines, stats, stats4, survival, tcltk, tools, and utils. Auxiliary packages 109 Given web access, newer versions of packages can be obtained using the command: update.packages() Note, however, that this will only give you the latest possible package applicable to the version of R that you are running. Acquiring the “very latest” version of a package may require downloading the latest version of R Any installed package can be loaded into a work session with the command library(). For instance, if it has been installed, vegan can be loaded by typing: library(vegan) Packages can also be loaded using Packages>Load packages in the R-GUI (non-Unix only). Typing search() lets one see all the packages presently loaded for a workstation. To remove vegan from a particular session I would type: detach(package:vegan) Aside from CRAN, there are currently two large repositories of R packages. First, the bioconductor project (http://www.bioconductor.org/packages/release/Software/html) contains a large number of packages for genomic analysis (466 as of Sept 2011) that are not found at CRAN. Packages can be downloaded from bioconductor using an R script called biocLite. To download the package rRytoscape from biocondctor I would type: source("http://bioconductor.org/biocLite.R") biocLite("RCytoscape") Second, R-forge (http://r-forge.r-project.org/) contains alpha releases of packages that have not yet been implemented into CRAN, and other miscellaneous code. Either of these sources can be specified from Packages>Select Repositories in the R-GUI (non-Unix only)21. 21 I do not encourage downloading all of the R packages that have been developed for analysis of biological data, much less trying out all of their functions. It is remarkable, however, to consider the range of topics these packages encompass. There are R packages for bioinformatics including packages devoted only to phylogenetics (e.g. packages ouch, picante, apTreeshape) or compiling amino acid sequences (package: seqinr). There are libraries that allow interpolation with geographic information system software (e.g. GRASS), packages specifically created for animal tracking data (adehabitat), functions for finding spatial patterns in cells (applications in package spatstat), and even algorithms for quantifying the existence of indicator species using count data from benthic environments (package bio.infer), or discerning the effect of fishing methods (package fishingmethods). Biological data and R 110 A.22 Exploration of biological data using R In concluding this appendix I will demonstrate some simple ways to graphically and numerically summarize biological datasets in R. The topics chosen reflect my interests in plant ecology and large scale ecology. || Example A.21 – Exploration of Data from Bryce Canyon National Park The brycesite dataset from library labdsv consists of environmental variables recorded at, or calculated for, each of 160 plots in Bryce Canyon National Park in Southern Utah. We will examine different ways of summarizing these data. install.packages("labdsv") library(labdsv) data(brycesite) Here are the names of the site environmental variables (columns) in the brycesite dataset: names(brycesite) [1] "annrad" "asp" [9] "pos" "quad" "av" "slope" "depth" "east" "elev" "grorad" "north" For an additional overview can get the first few lines of data using the function head. Here we apply it the first nine columns of brycesite. head(brycesite[,1:9]) annrad asp av 50001 241 30 1.00 50002 222 50 0.96 50003 231 50 0.96 50004 254 360 0.93 50005 232 300 0.48 50006 216 330 0.76 depth deep shallow shallow shallow shallow shallow east 388220.9 388477.2 388384.0 388307.5 389026.3 389297.6 elev grorad north pos 8720 162 4144784 ridge 8360 156 4147573 mid_slope 8560 159 4147347 mid_slope 8660 166 4146971 ridge 8480 159 4146846 up_slope 8560 155 4146853 mid_slope Let’s look at the distribution of the slope variable (Figure A.14). This variable describes slope (in degrees) of sites in the dataset. with(brycesite, hist(slope, xlab = "Slope (Degrees)", ylab = "Frequency of observations", main = "")) Biological data and R 100 50 0 Frequency of observations 150 111 0 10 20 30 40 50 60 70 Slope (Degrees) Figure A.14 Histogram of slope observations in the brycesite dataset. The distribution of slope is strongly right skewed (i.e. most slopes are gradual, and only few are extremely steep). Consideration of raw aspect values in analyses is problematic because the measurements are circular. As a result the values 1 and 360 are numerically 359 units apart, although they in fact only differ by one degree. One solution is to use the transformation [1 - cos(aspect° - 45)]/2. This index will have highest values on southwest slopes (at 225 degrees), and lowest values on northeast facing slopes (at 45 degrees). This acknowledges the fact that highest temperatures in the Northern Hemisphere occur on Southwest facing slopes because they receive ambient warming during the morning, coupled with late afternoon direct radiation. We have: asp.val <- (1-cos(((brycesite$asp-45)*pi)/180))/2 The distribution of aspect values in the Bryce Canyon dataset is shown in Figure A.15. hist(asp.val, ylab = "Frequency of observations", xlab = "Aspect value", main = "") Biological data and R 20 10 0 Frequency of observations 30 112 0.0 0.2 0.4 0.6 0.8 1.0 Aspect value Figure A.15 Histogram of observed aspect values from the brycesite dataset. We have a bimodal distribution. Specifically, there are a lot of northeast-facing and southwest-facing sites, and fewer northwest and southeast-facing sites. We can look at the relationship of two quantitative variables, annual solar radiation and aspect value, with a scatterplot (Figure A.16) with(brycesite, plot(asp.val, annrad, xlab = "Aspect value", ylab = "Annual radiation (Langleys)")) Biological data and R 260 240 220 200 180 Annual radiation (Langleys) 280 300 113 0.0 0.2 0.4 0.6 0.8 1.0 Aspect value Figure A.16 Scatterplot of aspect value versus annual radiation from the brycesite dataset. There is a strong positive association between aspect value and annual radiation (Figure A.16). Thus, highest levels of radiation tend to occur on southwestern slopes. This is what we would expect in the northern hemisphere. A Langley (Ly) is a measure of energy per unit area. To be precise one Ly = 1calorie m-2 min-1. In SI units 1Ly = 41840.00 J m-2. Soil depth (i.e. depth) and topographic position (pos) are both categorical variables in the brycesite dataset. We can use the table command to see the distribution of the two categories of soil depth. table(brycesite$depth) depth deep shallow 56 89 We can use table(depth,pos) to see the joint distribution of soil depth and topographic position. Biological data and R 114 with(brycesite, table(depth, pos)) pos depth bottom low_slope mid_slope ridge up_slope deep 13 14 18 5 6 shallow 6 14 31 12 26 It is often useful to distinguish points in scatterplots with respect to a categorical variable. For instance, the plot below shows radiation as a function of aspect value, and has the five topographic positions distinguished using both point color and shape. For clarity I also create a legend. We see that ridgetop sites have mostly northeastern aspect, and hence have lower radiation inputs (Figure A.17). with(brycesite, plot(asp.val, annrad, xlab = "Aspect value", ylab = "Annual radiation (Langleys)", col = as.numeric(pos), pch = as.numeric(pos))) 260 240 220 200 bottom low_slope mid_slope ridge up_slope 180 Annual radiation (Langleys) 280 300 legend("bottomright",legend = levels(brycesite$pos), pch = 1:5, col = 1:5) 0.0 0.2 0.4 0.6 0.8 1.0 Aspect value Figure A.17 Scatterplot of aspect value versus annual radiation with topographic positions indicated To assign colors and plotting characters appropriately I coerce the categorical topographic position vector, pos, to be numeric with as.numeric. The result is: Biological data and R as.numeric(brycesite$pos) [1] 4 3 3 4 5 3 3 5 3 3 2 2 3 4 3 3 3 1 2 2 2 5 [38] 1 2 4 4 3 3 3 3 4 3 5 3 3 3 2 5 3 5 3 3 5 5 [75] 3 2 2 3 3 2 4 3 4 2 5 3 3 2 2 3 5 5 3 5 5 3 [112] 1 2 3 5 1 5 3 3 3 3 1 3 2 2 5 2 1 2 2 1 2 1 [149] 2 4 1 5 5 5 3 2 2 1 5 4 115 4 4 3 1 4 3 3 1 3 3 3 1 5 5 5 1 4 2 5 1 3 3 3 2 5 3 3 1 3 5 3 1 5 2 3 4 3 2 5 5 2 5 1 5 5 2 2 5 5 2 4 4 4 3 1 5 1 3 2 2 Ones correspond to the first alphanumeric level in pos, bottom, whereas fives correspond to the last alphanumeric level, up_slope. The color and symbols assignments are made within the plot using: col = as.numeric(brycesite$pos) pch = as.numeric(brycesite$pos) Legends in R can be created using the function legend. The first argument(s) will be a specific X,Y position for the legend, or one of: "bottomright", "bottom", "bottomleft", "left", "topleft", "top", "topright", "right" or "center". The legend argument names the categories to be depicted. The function levels used in this argument lists the categories in a categorical variable alphanumerically. It may be useful to add additional axes to a scatterplot. In R this will involve laying one plot on top of another, by specifying par(new = TRUE), and defining axes = FALSE, and xlab and ylab = "" in the arguments of the second plot. Consider Figure A.18 in which both annual radiation and annual growing season radiation are plotted as a function of aspect value. op <- par(mar=c(5,4.5,1,4.5),cex=1.1) with(brycesite, plot(asp.val, annrad, xlab = "Aspect value", ylab = "Annual radiation (Langleys)")) par(new = TRUE) with(brycesite, plot(asp.val, grorad, pch = 19, axes = FALSE, xlab = "", ylab = "")) axis(4) mtext(side=4,"Growing season radiation (Langleys)",line = 3) legend("bottomright", pch=c(1,19), legend=c("Annual radiation", "Growing season radiation"),bty = "n") par(op) Biological data and R ● ●● ● ● ● ● ● ● ● 150 ● ● ● ● ● ● ● 145 220 170 ● ● ● 240 ● 155 260 ● ● ● ● 0.0 0.2 ● 0.4 Annual radiation Growing season radiation 0.6 0.8 135 180 140 200 Annual radiation (Langleys) ● ● Growing season radiation (Langleys) ● ● ● ● ● ● ●● ● ● ● ● ● ● ●●●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● 165 ●● ● ●● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 160 280 300 116 1.0 Aspect value Figure A.18 Plot of annual radiation and growing season radiation as a simulataneous function of aspect value. The line par(new = TRUE) tells R to not clean the graphical device before drawing a new plot. The argument axes = FALSE in the second plot, suppresses default plot plotting of axis units on the left and bottom axes. Boxplots or box and whisker plots and their variants are an excellent way to look at categorical treatment effects on a quantitative variable. I can easily create a boxplot of the slope variable with respect to shallow or topographic position categories (Figure A.19). The upper and lower hinges of a boxplot show the 1st and 3rd quartiles. The black stripe in the middle of each box shows the median. The whiskers extend to the most extreme data point which is no more than coef times the length of the box away from the box (by default coef = 1.5). Circle symbols are outlying observations (outside the whiskers). with(brycesite, plot(slope ~ pos, ylab = "Slope (Degrees)", xlab = "Topographic position")) Biological data and R 40 30 0 10 20 Slope (Degrees) 50 60 117 bottom low_slope mid_slope ridge up_slope Topographic position Figure A.19 Slope versus topographic position for the brycesite dataset. The function bplot in asbio creates bars whose heights show location measures (e.g. means, medians, etc.) along with user specified error bars. || Example A.22 –Taiga/tundra vegetation in Scandinavia To simultaneously examine environmental and plant species data we will use two datasets from the library vegan describing Scandinavian taiga/tundra. Vegetation data are contained in the dataset varespec while soil chemistry data for the same sites are contained in the dataset varechem. data(varespec) data(varechem) Let’s look at the distribution of the heath plant Vaccinium vitis-idaea (a common species in boreal forest understories) with respect to both pH and % soil nitrogen (Figure A.20). Note that we allow symbol sizes to change with the cover of V. vitis-idaea. with(varechem, plot(N, pH, xlab = "% soil N", pch = 16, cex = varespec$Vac. vit/100 * 15) Biological data and R 3.6 118 3.4 ● 3.2 pH ● ● ● 3.0 ● 2.8 ● ● ● ● 15 ● ●● ● ● ● ● 20 ● ● ● ● ● ● ● 25 30 % soil N Figure A.20 Cover of Vaccinium vitis-idaea with respect to pH and % soil nitrogen. Larger symbols indicate higher percent plant cover. Vaccinium vitis-idaea appears to prefer intermediate to low levels of soil N, and acidic soils. The negative association between soil N and pH is probably due to soil leaching, because H+ (and Al3+) cations are more strongly adsorbed by soil colloids than bases in poorly drained soils. A 3D plot of the same associations can be created using the scatterplot3d library (Figure A.21). install.packages("scatterplot3d"); library(scatterplot3d) Fig <- function(angle = 55){ s3d <- scatterplot3d(cbind(varechem$N, varechem$pH, varespec$Vaccviti), type="h", highlight.3d = TRUE, angle = angle, scale = .7, pch = 16, xlab = "N", ylab = "pH", zlab = express ion(paste(italic(Vaccinium), " ", italic(vitis-idaea), " % cover"))) lm1 <-lm(varespec$Vaccviti ~ varechem$N + varechem$pH) s3d$plane3d(lm1) } Fig() I define the figure to be a function to allow the angle of rotation for the 3D scatterplot to be easily changed. By stipulating highlight.3d = TRUE objects that are closer to us in the X plane are given warmer Biological data and R 119 colors. A regression “plane” is also overlaid on the graph. The fitted plane is produced from a linear model (the command lm). 25 ● ● ● 20 ● ● ● ● ● ● ● ● ● ● ● ● 10 3.6 ● ● 3.4 ● ● 3.2 ● ● 5 pH 15 ● ● 3.0 2.8 2.6 0 V acci ni um vi t i s − i d aea % cover ● 10 15 20 25 30 35 N Figure A.21 Cover of Vaccinium vitis-idaea in a 3D plot with respect to pH and % soil nitrogen. To clarify relationships among the variables may want to create an animation of the figure rotating in three dimensions (see Section A.20.7). rotate.Fig <- function(){ angle <- 0:360 for(i in 1:length(angle)){ dev.hold() Fig(angle = angle[i]) dev.flush() Sys.sleep(.1) } } library(animation) saveSWF(rotate.Fig(), swf.name = "rotate.swf", interval = 0.1, nmax = 30, ani.dev = "pdf", ani.type = "pdf", ani.height = 6, ani.width = 6) Biological data and R 120 The rotation, embedded in an .flv file, can be run by clicking on the flashplayer link below. We can also create hand-rotatable 3D figures under the rgl real-time rendering system (Figure A.22). expg <- expand.grid(varechem$pH, varechem$N) subs <- cbind(varechem$pH, varechem$N) tf <- (expg[,1] == subs[,1])&(expg[,2] == subs[,2]) y <- ifelse(tf == TRUE, varespec$Vaccviti, NA) surface <- data.frame(N = expg[,1], pH = expg[,2], vac.vit = y) library(car) scatter3d(vac.vit ~ N+pH, data = surface, surface = TRUE, fit = "linear", zlab = "N", xlab = "pH", ylab = "Vaccinium vitilus (% cover)") Figure A.22 Hand rotatable graphics object. Biological data and R || Example A.23 Greenhouse gasses Of great concern to both citizens and scientists are rising global levels of atmospheric greenhouse gasses. Atmospheric CO2 concentrations have increased approximately 40% since the start of the industrial revolution while more potent greenhouse gasses like CH4 and NO2 have increased 150% and 23% respectively (Mann and Kump 2009). We will take a detailed look at recent global patterns of CO2 emissions and human population numbers in this exercise. The US department of energy has data since 1980 detailing total CO2 emissions from the consumption and flaring of fossil fuels. In addition, midyear population data can be obtained, by country, from the US census bureau. CO2 and population data are available as world.co2 and world.pop respectively in asbio. We will first import these data. library(asbio); data(world.co2); data(world.pop) To make them easier to call, we will give the datasets shorter names. co2 <- world.co2 wp <- world.pop The CO2 data has two additional countries (columns) compared to the world population data: Belgium and Ghana. We will get rid of these columns and the “year” column in the CO2 dataset. co2.1 <- co2[,-c(1, 3, 8)] We will also want to look at the 2006 CO2 data by itself. It is in row 27. co2.2006 <- co2.1[27,] The names of some of the counties are too long to fit on the X-axis for the barplot we wish to create (Figure A.23). We can deal with this in at least four ways. First, we can increase the bottom margin (e.g. par(mar = c(6, 4, 2, 2)). Second, we can decrease the font-size of the names using the cex. names argument (the default for barplot names is cex.names =1). Third, we can make the country names perpendicular to the X-axis instead of parallel using the las argument. Lastly, we can simply make the country names shorter by changing the column names in the dataframe or matrix. For instance we could use some variant on: colnames(co2.1) <- c("Afgan.", "Brazil", "Canada", "China", "Finland", "Italy", "Japan", "Kenya", "Mexico", "S. Arabia", "UAE", "US", "Total") We use the column names from co2.1 as the column names for co2.2006 121 Biological data and R 122 colnames(co2.2006) <- colnames(co2.1) barplot(as.matrix(co2.2006),las = 3,ylab = expression(paste("2006 ", CO[2], " Emissions (metric tons x ", 10^6, ")"))) We add a horizontal grid to make levels among countries more discernible. 20000 15000 10000 Total US UAE S. Arabia Mexico Kenya Japan Italy Finland China Canada Brazil Afgan. 0 5000 2006 CO2 Emissions (metric tons x 106) 25000 grid(ny = 20, nx = 0) Figure A.23 2006 CO2 emissions, by country. Next we will create a stacked bar graph of carbon emissions from 1980-2006, with countries as bars, and each bar stacked by year (Figure A.24). We will leave out world totals to make patterns among countries easier to discern. By default R will impose its own coloring scheme for the stacked bars. We will create a twenty-seven step grayscale coloring scheme of our own by using the argument col = gray(seq(0:26)/27). Type ?gray for more information. We will also include a legend to describe the stacked bars. Type ?legend for more information. Biological data and R 123 20000 40000 60000 80000 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 US UAE S. Arabia Mexico Kenya Japan Italy Finland China Canada Brazil Afgan. 0 6 CO2 Emissions (metric tons x 10 ) 120000 barplot(as.matrix(co2.1[,1:12]), las = 3, ylab = expression(paste(CO[2], " Emissions (metric tons x ", 10^6, ")")), col = gray(seq(0:26)/27)) legend("topleft", fill = gray(seq(0,26,1)/27), legend = seq(1980, 2006, 1), cex = .5) Figure A.24 Stacked bars of the CO2 emissions data. By default barplot builds stacked bar plots. To create side-by-side barplots one would use the arguments beside = TRUE. I will make the figure wider to make it easier to view the large number of sideby-side bars (Figure A.25). Biological data and R 124 5000 4000 3000 2000 1000 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 US UAE S. Arabia Mexico Kenya Japan Italy Finland China Canada Brazil Afgan. 0 CO2 Emissions (metric tons x 106) 6000 dev.new(height=5,width=8); op <- par(mar=c(5,4.5,2,1)) barplot(as.matrix(co2.1[,1:12]), las = 3, ylab = expression(paste(CO[2], " Emissions (metric tons x ", 10^6, ")")), col = gray(seq(0:26)/27), beside = T) legend("topleft", fill = gray(seq(0,26,1)/27), legend = seq(1980, 2006, 1), cex = .5) par(op) Figure A.25 Side by side bars of the CO2 emissions data. Next we will create a line plot with 13 lines (one for each country, and one for world totals) showing the carbon emission variability from 1980 to 2006 (Figure A.26). We will make the lines in the plot of different types. We will also include a legend to describe the lines. We can create the lines for the plot individually or create them simultaneously using the function matplot. Biological data and R 125 matplot(seq(1980,2006,1), co2.1, ylab = expression(paste(CO[2], " Emissions (metric tons x ", 10^6,")")),type = "l",xlab = "Year", col = c(gray(seq(0 : 11)/15), 1), lwd = c(rep(1,12),2), lty = c(seq(1:6),seq(1:6),1),ylim=c(1,100 000),log="y") legend("topleft",col=c(gray(seq(0 : 11)/15), 1),lty = c(seq(1:6),seq(1:6),1),lwd = c(rep(1,12), 2), legend = colnames(co2.1), cex=.7, ncol=4, bty = "n") Mexico S. Arabia UAE US Total 100 10000 Finland Italy Japan Kenya 1 CO2 Emissions (metric tons x 106) Afgan. Brazil Canada China 1980 1985 1990 1995 2000 2005 Year Figure A.26 Line plot of world CO2 emissions. We see that the US and China have the highest emissions, and that China’s emissions are increasing rapidly. Of additional interest, Afghanistan’s CO2 emissions peaked in the late 80’s, then collapsed as a result of war and political strife. Note that we log transform the Y-axis to allow better discrimination among countries. Finally, we utilize the census data to create a scatterplot of per capita emissions by country (Figure A.27) Biological data and R 126 wp.1 <- wp[,-c(1)]#get rid of the year column per.cap <- (10^6) * co2.1/wp.1 10 20 30 40 Afgan. Brazil Canada China Finland Italy Japan Kenya Mexico S. Arabia UAE US 0 Per capita emissions (metric tons) 50 matplot(seq(1980,2006,1), per.cap, type = "l", col = gray(seq(0 : 13)/15), xlab = "Year", ylab = "Per capita emissions (metric tons)", lty = seq(1:6)) legend("topleft", col = gray(seq(0 : 13)/15), lty = seq(1:6), legend = colnames(co2.1)[-13],cex = .8, bg = "white") 1980 1985 1990 1995 2000 2005 Year Figure A.27 Per capita CO2 emissions by country. The United Arab Emirates has, by far, the highest levels of per capita CO2 emissions. These examples present just a few R graphical ideas. For additional demonstrations go to my personal webpage https://sites.google.com/a/isu.edu/aho/ or the book website http://www2.cose.isu.edu/~ahoken/ book/. 127 Exercises Exercises 1. The following questions concern the history of and general characteristics of R. (a) Who were the creators of R? (b) When was R first introduced? (c) What languages is R derived from/most similar to? (d) What features distinguish R from other languages and statistical software? (e) What are the three operating systems R works with? 2. Perform the following operations. (a) Leave a note to yourself in the console. (b) Identify your working directory. (c) Create an object called x that contains the numeric entries 1, 2, and 3. (d) Examine x in the R console. (e) Make a copy of x called y. (f) List the current objects in your work station. (g) Save your history and open the file in a text editor. (h) Save your objects using save.image(). Close and reopen R. Do x and y exist? 3. Solve the following mathematical operations using lines of code in R. Show code. (a) 1 + 3/10 + 2 (b) (1 + 3)/10 + 2 (c) ((3 × 4)/23)2 (d) log2(31/2) (e) 3x3 + 3x2 + 2 where x ={0, 1.5, 4, 6, 8, 10} (f) 4(x + y) where x and y are: x = {0, 1.5, 4, 6, 8} and y = {-2, 0.5, 3, 5, 8}. (g) (h) d tan( x) + 2.3e3 x dx ∫ 12 3 ∫ ∞ (j) ∫ ∞ (k) ∫ ∞ (i) −∞ −∞ −∞ 24 x3 + ln( x)dx 2 1 − x2 e dx (i.e., find the area under a standard normal pdf). 2π 2 x − x2 e dx (i.e., find E(X) for a standard normal pdf). 2π 2 x 2 − x2 e dx (i.e., find E(X2) for a standard normal pdf). 2π (l) Find the arithmetic mean, variance, median, skew and kurtosis of the data x ={0, 1.5, 4, 6, 8, 10}. 4. Read the section on linear algebra in the mathematical appendix from the Foundational and Applied Stats textbook (Section A.5, pg 546). Let 2 −3 1 = A = , b . Find 1 0 5 (a) Ab 128 Exercises bA det(A) A-1 A’ (b) (c) (d) (e) (f) We can solve systems of linear equations using matrix algebra with the function Ax = b, and A-1b = x. In this notation A contains the coefficients from the linear equations (by row), b is a vector of solutions given in the individuals equations, and x is a vector of solutions sought in the system of models. Thus, for the linear equations: 2x + 4 y = −4 , x + 2y = 1 we have: 2 4 x −4 A= , x = and b = . 1 2 1 y Solve the system below using linear algebra with R: 3x + 2 y − z = 1 2x − 2 y + 4z = −2 . − x + 0.5 y − z =0 f (x, y ) = (x 2 + 6y 2)e (−x 2 − y2 Find the area under the function surface between x = 0 and 2 . −y 2) f(x, y) y ) ( x 2 + 6 y 2 )e − x 5. The function to the right22 has the form f ( x,= 2 and y = -1 and 2. That is, find: 2 0 −1 ( x 2 + 6 y 2 )e − x 2 − y2 dydx . x 2 ∫∫ The solution in R will require the function adaptIntegrate in package cubature. y 6. Distinguish expressions and assignments in R. 7. Consider the following variables: x<-c(1,2,2.5,3,4,3,5) 22 code for figure: x <- seq(-3, 3, by = .1); y <- x f <- function(x, y) {(x^2 + 6 * y^2) * exp(-x^2 - y^2)} z <- outer(x, y, f) persp(x, y, z, theta = 70, phi = 20, expand = 0.5, col = "lightblue", ltheta = 120, shade = 0.75, xlab = "x", ylab = "y", zlab = "f(x, y)") Exercises 129 y<-c(6,4.3,3,3.1,2,1.7,1) (a) Plot y as a function of x, i.e. make a plot with x on the X-axis and y on the Y-axis. (b) Make every point in the scatterplot a different color. (c) Make every point a different shape. (d) Create a legend describing all the shape and color combinations of all points one through seven (call them Point 1, Point 2, etc.). Type ?legend for more information. (e) Convert from a point to an overplotted line and point plot. Type ?plot to get more information. (f) Change the name of the X-axis to “Abscissa axis” and the name of the y-axis to “Ordinate axis”. As indicated, italicize the words “Abscissa” and “Ordinate” while leaving the word axis in a normal font. Type ?expression, ?paste and ?italic for more information. (g) Place the text "Y = -1.203X + 6.539" at coordinates x = 2, y = 2.5. Italicize as indicated. Type ?text for more information. (h) Place a line with a slope of -1.203 and an Y-intercept of 6.539 on the plot. Type ?abline for more information. 8. Create the follow data structures: (a) A matrix object with two rows and two columns with the numeric entries 1,2,3,4. (b) A dataframe object with two columns; one column containing the numeric entries 1,2,3,4, an one column containing the character entries "a","b","c","d". (c) A list containing the objects created in (a) and (b). 9. Load the C.isotope dataset from package asbio. (a) Create a graphical device holding three plots in a single row, i.e., the three plots will be side by side. (b) In the first plot, show δ14C as a function of time (decimal.date) using a line plot. Use appropriate axis labels. (c) In the second plot, show CO2 concentration as a function of time in a scatterplot. (d) In the third plot, show measurement precision (column four in the dataset) as a function of δ14C. 10. The speed of the earth rotating on its axis, E, is approximately 1700km/hr, or 1037 mph, at the equator. We can calculate the velocity of the rotation of the earth at any latitude with the equation, V = cos(latitude⁰) × E. The latitude of Pocatello, Idaho is 42.871 degrees. Use R to find the rotational speed of the earth at this latitude. Remember, the function cos() assumes inputs are in radians, not degrees. 11. Make a plot showing the relationship between the speed of the earth’s rotation and latitude. See question 10 above for more information. (a) Create a sequence of numbers (degrees) from 0 to 90. Give this vector a name. (b) Convert the vector in (a) from degrees to radians. Give this vector a name. (c) Calculate velocities for the vector in (b), give this vector a name. (d) Plot these velocities versus latitude, i.e. the vector from (a). Label axes appropriately. 12. Access the dataset cliff.sp in the library asbio which describes cliff vegetation from Yellowstone National Park. (a) Use the names function to find the names of the variables. (b) Show how one would access the first row of data using R. (c) Show how one would access the third column of data using R. 130 Exercises (d) Show how one would access the fourth element form the third column using R. 13. Given the following dataset provide solutions and R code. Q.13 <- data.frame(height.in = c(70, 76, 72, 73, 81, 66, 69, 75, 80, 81, 60, 64, 59, 61, 66, 63, 59, 58, 67, 59), weight.lbs = c(160, 185, 180, 186, 200, 156, 163, 178, 186, 189,140, 156, 136, 141, 158, 154, 135, 120, 145, 117), sex = c(rep("M", 10), rep("F", 10))) (a) Find the mean height of females (i.e. F) greater than or equal to 59 inches but less than 63 inches. (b) The mean weight of males who are 75 or 76 inches tall. (c) Classify heights equal to 58 or 59 as “small”. (d) Sort the dataset with respect to ascending values of weight using the function order. (e) Simultaneously find the mean of columns one and two (height and weight) with the apply command. 14. Enter data from the table below into an R dataframe. Use the functions seq and rep to help you (show your code for accomplishing this). Attach your code and resulting dataframe. Plant height (dm) 22.3 21 24.7 25 26.3 22 31 32 34 42 28.9 33.3 35.2 36.7 34.4 33.2 35 41 Water index (1-10) 1 2 3 4 5 6 7 8 9 10 1 2 NA 4 5 NA 7 8 Nutrient Treatment A A B B C C D D E E A A B B C C D D 43 44 9 10 E E (a) Find the mean and variance of plant heights for each nutrient treatment using tapply. (b) Use complete.cases to eliminate rows with missing data. (c) Create a plot showing plant height as a function of N treatment. Hint: type ?boxplot. Use appropriate axis labels. 15. Create .csv and .txt datasets and read them into R. 131 Exercises 16. The following questions concern binary data and floating point applications. (a) Find the binary representation of the number 32 by hand. (b) Find the digital numbers for the binary numbers 101101, 101010101, and 0.111 by hand. Verify your answer using bin2dec. (c) Use bin2dec to evaluate different binary representation of the decimal number 1/3. Use 0.0101, 0.01010101, and 0.01010101010101010101. Which provides the most precise representation of 1/3? Why? (d) Use R to find the Unicode binary representation of the letter "A". 17. Below is a mock dataset of plant height with respect to three treatments. height.data<-data.frame(height=c(20,30,40,40,40,20,15,10,15,20,50,35,40,50,60) ,treatment=rep(1:3,each=5,times=1)) (a) Create a function utilizing tapply that simultaneously calculates means, variances, standard deviations, minimums, maximums and medians with respect to a vector of categories. Use it to find these estimates simultaneously for all three treatments in height.data. (b) Use the function which to find which value in height.data is closest to 9. (c) Using a for loop, create a function that simultaneously calculates means, variances, standard deviations, minimums, maximums and medians for all three treatments. (d) Use the function system.time to compare the run time for the functions in (a) and (c). Describe your results. 18. Solve the systems of ODEs below for t = {1,2,...,20}. dx = ax + by dt dy = cx + dy dt Let a = 3, b = 4, c = 5, d = 6. Initial values for x and y can be anything but {0,0}. 19. Divide the values in the dataset in Q. 13 by their respective column sums by specifying an appropriate function in the 3rd (function) argument for apply. 20. Below is McIntosh’s index of diversity (McIntosh 1967): U= S ∑n i =1 2 i , where S is the total number of species and ni is the number of individuals in the ith species. Write a function to calculate this index. Run it on a sample from a site that contains five species represented by 5, 4, 5, 3, and 2 individuals respectively. 21. Below is the Satterthwaite formula for approximating degrees of freedom for the t distribution: 2 S x2 SY2 + nX nY , ν= 2 2 2 2 ( S X nX ) + ( SY nY ) nX − 1 nY − 1 Exercises 132 where SY2 is the sample variance for X, SY2 is the variance for the variable Y, nX is the sample size for variable X, and nY is the sample size for variable Y. Write a function for this equation that has the variables X and Y as arguments. Run it for x <- c(1,2,3,2,4,5) and y <- c(2,3,7,8,9,10,11). 22. Use a for loop to create a 100 x 100 element matrix of random numbers. Accomplish this by completing the following steps. (a) Create an empty matrix of the correct dimensions to hold the result, and give it a name. For instance, rand. (b) Use a for loop to create columns in rand. Use the function runif(100) to generate 100 random numbers (the contents of one column) based on a uniform probability distribution. Loop to create other columns. 23. More fun with for loops. Here are some classical computer science applications. (a) A sequence of Fibonacci numbers is based on the function: f (n=) f (n − 1) + f (n − 2) for n > 2 . = f (1) f= (2) 1 where n represents the nth step in the sequence. Using a for loop create the first 100 numbers in the sequence, i.e. find f(1) to f(100). As a check, the first 5 numbers in the sequence should be: 1, 1, 2, 3, 5. (b) An interesting chaotic recursive sequence has the function: f (n)= f (n − f (n − 1)) + f (n − f (n − 2)) for n > 2 . = f (1) f= (2) 1 Using a for loop create the first 100 numbers in the sequence, i.e. find f(1) to f(100). As a check, the first 5 numbers in the sequence should be: 1, 1, 2, 3, 3. 24. Create an R animation function using your name that changes font-size and color as the animation loop proceeds. 25. Create an .rd documentation file for the function in Q 24. Make a .pdf from the .rd file. 26. Create a tcltk GUI to run the animation function in Q. 24. 27. Load the goats data set in asbio. (a) Create a histogram for the variable NO3. (b) Create a scatterplot of NO3 as a function of feces. (c) Change symbol sizes in (b) to reflect the values in organic.matter. (d) Make a plot showing NO3 and organic.matter as a simultaneous function of feces by adding a second Y axis. (e) Create a 3D scatterplot depicting NO3 as a function of organic.matter and feces. 28. Write R documentation using Sweave, knitr or R Markdown describing the work done in Q. 27. This will be greatly facilitated through the use of RStudio. Exercises 133 29. Load the data set asthma in asbio. (a) Make an overplotted line scatterplot showing all FEV levels to be function of time. Vary symbols types to reflect patients, and vary line and symbol color to reflect the drug treatment. Make a legend (or two) to clarify the meaning of symbol types and colors. (b) In a barplot display the FEV1 levels of the 24 patients in the study in treatment "a". (c) Create a stacked barplot (or a beside barplot using beside = TRUE) to show how average FEV1 levels for each time frame differed for treatments a, b, and c. To do this you will want to create a 3 x 24 dimension matrix containing means for each patient (columns) at each treatment (rows). (d) Create a legend to describe (c). 30. Write R documentation using Sweave, knitr or R Markdown describing the work done in Q. 29. This will be greatly facilitated through the use of RStudio. 31. Perform a statistical and/or graphical analysis of your own data in R and document the workflow using weave, knitr or R Markdown. References 134 References Adler, J. 2010. R in a Nutshell. O’ Reilly Media, Inc. Sebastopol, CA. Akima, H., Gebhardt, A., and Petzoldt, T. 2012. Akima: Interpolation of irregularly spaced data. R package version 0.5-7. Anderson, E., et al. 1999. LAPACK User’s Guide, Third Edition. SIAM, Philadelphia. Becker, R. A., and Chambers, J. M. 1978. Design and Implementation of the ‘S’ System for Interactive Data Analysis. Proceedings of the International Computer Software and Applications Conference (COMPSAC). 78: 626-629. Becker, R. A., and Chambers, J. M. 1981, S: A Language and System for Data Analysis. Bell Laboratories, Computer Information Service. Murray Hill, New Jersey. Becker, R.A., Chambers, J. M., and Wilks, A. R. 1988. The new S language. Chapman and Hall. Bivand, R. S., et al. 2013. spdep: Spatial dependence: weighting schemes, statistics and models. R package version 0.5-56. Butcher, J. C. 1987. The numerical analysis of ordinary differential equations, Runge-Kutta and general linear methods, Wiley, Chichester and New York. Crawley, M. J. 2007. The R Book. Wiley. Fox, J., and Weisberg, S. 2011. An R Companion to Applied Regression, 2nd edition. Sage, Thousand Oaks CA. http://socserv.socsci.mcmaster.ca/jfox/Books/Companion. (Accessed 5/28/2013) Fox, J. 2009. Aspects of the social organization and trajectory of the R project. The R Journal Vol. 1/2: 5-13. Geyer, C. J. 1991. Constrained maximum likelihood exemplified by isotonic convex logistic regression. Journal of the American Statistical Association. 86: 717–724. Goldberg, D. 1991 What Every Computer Scientist Should Know About Floating-Point Arithmetic, Computing Surveys. Association for Computing Machinery, Inc. Gurevitch, J., Scheiner, S. M., and Fox, G. A. 2006. The Ecology of Plants. Sinauer. Heiberger, R. M., and Neuwirth, E. 2009. R Through Excel, A Spreadsheet Interface for Statistics, Data Analysis, and Graphics. Springer. Hershey, A. V. 1967. Calligraphy for Computers, Dahlgren, VA: U.S. Naval Weapons Laboratory, ASIN B0007EV KFI, OCLC 654265615, NWL Report No. 2101. NTIS AD662398 Hothorn, T., Hornik, K. van de Wiel, M. A., and Zeileis, A. 2006. A lego system for conditional inference. The American Statistician 60(3): 257-263. References 135 Hothorn, T., Hornik, K. van de Wiel, M. A., and Zeileis, A. 2008. Implementing a class of permutation tests: The coin package. Journal of Statistical Software 28(8), 1-23. Hornik, K. 2009. The R FAQ Version 3.0.2013-05-12. ISBN: 3-900051-08-9. http://CRAN.R-project.org/doc/FAQ/R-FAQ.html. (Accessed 5/27/013). Ihaka, R., and Gentleman, R. 1996. R: A language for data analysis and graphics, Journal of Computational and Graphical Statistics. 5: 299-314. Lang, D. T. 2005. Calling R from Java. http://www.nuiton.org/attachments/168/RFromJava.pdf. (Accessed 4/10/2012) Legendre, P. 2013. lmodel2: Model II regression. R package version 1.7-1. http://CRAN.R-project.org/package=lmodel2 (Accessed 5/27/2013) Lemon, J. 2006. Plotrix: a package in the red light district of R. R-News, 6(4): 8-12. MacArthur, R. H., and MacArthur, J. W. 1961. On bird species diversity. Ecology. 42: 594-598. Magurran, A. 1988. Ecological diversity and its measurement. Princeton University Press, Princeton NJ. Mann, M. E., and Kump, L. R. 2009. Dire Predictions: Understanding Global Warming. DK Press, London. Margalef, R. 1972. Homage to Evelyn Hutchinson. Why there is no upper limit to diversity. Trans. Connect. Acad. Arts Sci. 44: 211-235. Matloff, N. 20111. The Art of R Programming. No Starch Press. McCarthy, J., Brayton, R., Edwards, D., Fox, P., Hodes, L., Luckham, D., Maling, K., Park, D., et al. 1960. LISP I Programmers Manual. Boston, Massachusetts: Artificial Intelligence Group, M.I.T. Computation Center and Research Laboratory. McCune, B., and Keon, D. 2002. Equations for potential annual direct radiation and heat load. Journal of Vegetation Science,13: 603-606. McIntosh, R. P. 1967. An index of diversity and the relationship of certain concepts to diversity. Ecology 48: 392-404. Oksanen, J., et al. 2012. vegan: community ecology package. R package version 2.0-4. http://CRAN.R-project.org/package=vegan. (Accessed 5/28/2013) Pinheiro, J., Bates, D., DebRoy, S., Sarkar, D., and the R Development Core Team. 2012. nlme: linear and nonlinear mixed effects models. R package version 3.1-104. R Development Core Team. 2012. R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0, http://www.R-project.org. (Accessed References 136 5/28/2013) R Foundation for Statistical Computing. 2012. R: Regulatory compliance and validation issues a guidance document for the use of R in regulated clinical trial environments. http://www.r-project.org/doc/R-FDA.pdf. (Accessed 5/28/2013) Roberts, D. W. 2012. labdsv: ordination and multivariate analysis for ecology. R package version 1.5-0. http://ecology.msu.montana.edu/labdsv/R Sarkar, D. 2008. Lattice: Multivariate Data Visualization with R. Springer. Steele, G. L. Jr. 1978 Rabbit: A Compiler for Scheme. Masters Thesis. MIT AI Lab. AI Lab Technical Report AITR-474. Sussman, G. J., and Steele, G. L. Jr. 1975. Scheme: an interpreter for extended lambda calculus. MIT Artificial Intelligence Lab. AI Lab Memo AIM-349. Väre, H., Ohtonen, R., and Oksanen, J. 1995. Effects of reindeer grazing on understory vegetation in dry Pinus sylvestris forests. Journal of Vegetation Science. 6: 523–530. Venables, W. N., and Ripley, B. D. 2002. Modern Applied Statistics with S. Fourth Edition. Springer, New York. Wood, S. N. 2011. Fast stable restricted maximum likelihood and marginal likelihood estimation of semiparametric generalized linear models. Journal of the Royal Statistical Society (B) 73(1):3-36. 137 Index Index B Symbols barplot() bin2dec() binary bit bitmap Bluefish box() browseVignettes() Bryce Canyon National Park brycesite byte :: ::: ? ... [ ] # <<- $ + (addition) & (and) / (division) == (equal to) ^ (exponentiation) ~ (function of) > (greater than) >= (greater than or equal to) %in% <= (less than or equal to) * (multiplication) != (not equal to) | (or) - (subtraction) 103 103 9 84 53 5 80 44 16 60 16 60 16 30 60 60 66 60 16 60 60 16 A abline() abs() acos() akima all() alpha diversity animation any() apply() arguments, also see formals() array() arrays as.array() asbio as.character() as.dataframe() as.dist() as.double() as.factor() as.function() asin() as.list() as.matrix() as.numeric() assignments as.vector() atan() attach() attributes() axis() 31 16 16 108 61 84 98 61 56 17 43 43 69 20 69 69 69 69 69 69 16 69 69 69 12 69 16 24 14 37, 115 122 72 71 71 33 78 26 10 110 110 71 C c() C C# C++ cairo_pdf() cairo_ps() car cat() cbind() choose() class class() clipboard coercing coin colMeans() colnames() color palettes colors() colSums() command line prompt complete.cases() ConTEXT continuation prompt cos() CRAN (Comprehensive R Archive Network) Crimson Editor cuckoo 13, 42, 47 78 14 14, 78 34 34 108 103 47 16 14, 30, 69, 102 14 52 69 108 56 85 28 27 56 4 76 78 6 16 7, 108 78 43 D data() data.frame() deSolve detach() dev.cur() dev.new() D() (find derivative) diag() dim() dot product double() Down’s syndrome droplevels() 22 44 88 24 31 32 16 58 42 72 69 45 54 138 Index E Eclipse else() example() Excel exp() exponent (in floating point arithmetic) exporting data expressions 78 61 10 23 16 72 52 12 F factor() factorial() Fibonacci numbers file.choose() file.create() fix() fonts for() formals() Fortran forward slashes rather than backslashes function() 69 16 132 50 7 23 34 91 9 95 49 19, 78–79, 81 G gamma() Geany gedit getwd() ggplot2 global variables G.mean() GNU (not Unix) graphics devices grep() grid() gsub() GUIs (Graphical User Interfaces) 16 78 78 6 41 2, 80 20 2 31 67 122 67 99 H hcl() head() help() Hershey family of fonts hexadecimal HL.mean() H.mean() huber() 28 22, 110 9 10 28 20 20 20 I IEEE if() ifelse() importing data Inf integrate() 71, 76 61 61 49 16 16 invisible() IQR() is.array() is.character() is.dataframe() is.double() is.factor() is.function() is.list() is.matrix() is.na(x) is.null() is.numeric() is.vector() 103 20 69 69 69 69 69 69 69 69 75 77 69 42 J Java jEdit jpeg() 14, 78 78 33 K Kate Knitr kurt() 78 104 20 L Langley LAPACK algorithms large datasets LaTex lattice legend() length() lexical scoping library() lines() Lisp lists lm() lme4 lmodel2 load() loadhistroy() Loblolly loblolly pine trees (Pinus taeda) local variables log() logical commands looping lower.tri() ls() 113 3 51 104 41 125 20, 42 2, 80 109 36 2 44 30 108 108 8 8 22 22 2, 80 16 60 90 58 15 M MAD() Markov Chain Monte Carlo (MCMC analysis) MASS match() 20 43 108 65 139 Index MATLAB matrices McIntosh’s index of diversity mcmc.norm.hier() mean() median() metacharacter metafile methods() mgcv mode() mtext() 95 42 131 44 20 20 67 33 102 108 20 38 N NA NaN na.omit() nlme NppToR in Notepad NULL 75 76 76, 77, 81 108 78 77 O object oriented programming options() order() ordered() ordinary differential equations (ODEs) ordinary least squares 14 6 63 69 88 83 P par() Pascal pattern matching pdf() pdfFonts() Perl Perl Compatible Regular Expressions (PCRE) PHP Pinus sylvestris plot() plotmath plotrix points() polygon() postscript() purl() Python 32 78 66, 67 33 35 14, 68 68 14 85 26, 32, 36 37 108 36 37 33 107 14 Q q() quantile() 4 20 R R calling and receiving code from other languages cleaning up 95 25 copyrights datasets in development core team exiting getting help graphics history installation language Logical commands in looping in mathematical operations memory usage objects options popularity precision preferences reliability R journal rounding errors saving your work stop all computations stop current computation storage upper physical limit of website writing functions in range() rank() R bin R CMD R CMD batch R CMD check R CMD SHLIB RColorBrewer read.csv() read.table() rect() recycling elements replace() RExcel rgb() rownames() RStudio RWinEdt 2 22 1 4 9 26 1 3 2 60 90 16 2 14 6 2 71 6 2 11 73 7, 12 6 6 71 51 9, 11 78 20 63 96 96 96 96 96 28 49 49 37 42 62 23 28 85 78 78 S S S3 system for class and method SAS Satterthwaite formula save() savehistory() save.image() scan() Scheme SciTE 2 103 2 131 52 7 8 47, 50 2 78 140 Index scope script editor sd() setwd() Shannon-Weiner index shiny significand Simpson’s index sin() skew() Smalltalk spdep S-Plus SPSS SQL (Structured Query Language) sqrt() stack() strsplit() strtrim() summary() svg() Sweave switch() Sys.sleep() 2 23 20, 32 6 84 102 72 84 16 20 14 108 2 2 51 16 58 66 66 23 34 104 83 98 T table() tan() tapply() tcltk testing text() TextMate tiff() Tinn-R 113 16 57 99 69 36 78 33 78 U unary operator uniroot() unstack() update.packages() upper.tri() 75 83 58 109 58 V Vaccinium vitis-idaea var() varechem varespec vectors vegan vignette() Vim Visual Basic 117 20, 32 117 117 42 108 10 78 78 W windowsFonts() WinEdt 35 78 win.metafile() with() working directory world.co2 world.pop write.csv() write.table() 33 24 6 121 121 52 52

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