Mathworks NEURAL NETWORK TOOLBOX 6 user guide

Neural Network Toolbox™ 6

User’s Guide

Howard Demuth
Mark Beale
Martin Hagan

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Neural Network Toolbox™ User’s Guide

© COPYRIGHT 1992–2010 by The MathWorks, Inc.

The software described in this document is furnished under a license agreement. The software may be used
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FEDERAL ACQUISITION: This provision applies to all acquisitions of the Program and Documentation by,
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Revision History

June 1992 First printing
April 1993 Second printing
January 1997 Third printing
July 1997 Fourth printing
January 1998 Fifth printing Revised for Version 3 (Release 11)
September 2000 Sixth printing Revised for Version 4 (Release 12)
June 2001 Seventh printing Minor revisions (Release 12.1)
July 2002 Online only Minor revisions (Release 13)
January 2003 Online only Minor revisions (Release 13SP1)
June 2004 Online only Revised for Version 4.0.3 (Release 14)
October 2004 Online only Revised for Version 4.0.4 (Release 14SP1)
October 2004 Eighth printing Revised for Version 4.0.4
March 2005 Online only Revised for Version 4.0.5 (Release 14SP2)
March 2006 Online only Revised for Version 5.0 (Release 2006a)
September 2006 Ninth printing Minor revisions (Release 2006b)
March 2007 Online only Minor revisions (Release 2007a)
September 2007 Online only Revised for Version 5.1 (Release 2007b)
March 2008 Online only Revised for Version 6.0 (Release 2008a)
October 2008 Online only Revised for Version 6.0.1 (Release 2008b)
March 2009 Online only Revised for Version 6.0.2 (Release 2009a)
September 2009 Online only Revised for Version 6.0.3 (Release 2009b)
March 2010 Online only Revised for Version 6.0.4 (Release 2010a)

Acknowledgments

The authors would like to thank the following people:

Joe Hicklin of The MathWorks™ for getting Howard into neural network

research years ago at the University of Idaho, for encouraging Howard and
Mark to write the toolbox, for providing crucial help in getting the first toolbox
Version 1.0 out the door, for continuing to help with the toolbox in many ways,
and for being such a good friend.

Roy Lurie of The MathWorks for his continued enthusiasm for the possibilities

for Neural Network Toolbox™ software.

Mary Ann Freeman for general support and for her leadership of a great team of
people we enjoy working with.

Rakesh Kumar for cheerfully providing technical and practical help,
encouragement, ideas and always going the extra mile for us.

Alan LaFleur for facilitating our documentation work.

Tara Scott and Stephen Vanreusel for help with testing.

Orlando De Jesús of Oklahoma State University for his excellent work in

developing and programming the dynamic training algorithms described in
Chapter 6, “Dynamic Networks,” and in programming the neural network
controllers described in Chapter 7, “Control Systems.”

Martin Hagan, Howard Demuth, and Mark Beale for permission to include

various problems, demonstrations, and other material from Neural Network
Design, January, 1996.

Neural Network Toolbox™ Design Book

The developers of the Neural Network Toolbox™ software have written a
textbook, Neural Network Design (Hagan, Demuth, and Beale, ISBN
0-9717321-0-8). The book presents the theory of neural networks, discusses
their design and application, and makes considerable use of the MATLAB
environment and Neural Network Toolbox software. Demonstration programs
from the book are used in various chapters of this user’s guide. (You can find
all the book demonstration programs in the Neural Network Toolbox software
by typing

This book can be obtained from John Stovall at [phone], or by e-mail at

[e-mail].

The Neural Network Design textbook includes:

• An Instructor’s Manual for those who adopt the book for a class

• Transparency Masters for class use

If you are teaching a class and want an Instructor’s Manual (with solutions to
the book exercises), contact John Stovall at [phone], or by e-mail at

[e-mail].

To look at sample chapters of the book and to obtain Transparency Masters, go
directly to the Neural Network Design page at

http://hagan.okstate.edu/nnd.html

From this link, you can obtain sample book chapters in PDF format and you
can download the Transparency Masters by clicking Transparency Masters
(3.6MB).

nnd.)

®

You can get the Transparency Masters in PowerPoint or PDF format.

Getting Started

1

Product Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-2

Using the Documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-3

Applications for Neural Network Toolbox™ Software . . . . 1-4

Applications in This Toolbox . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-4

Business Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-4

Fitting a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-7

Defining a Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-7

Using Command-Line Functions . . . . . . . . . . . . . . . . . . . . . . . . 1-7

Using the Neural Network Fitting Tool GUI . . . . . . . . . . . . . . 1-13

Recognizing Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-24

Defining a Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-24

Using Command-Line Functions . . . . . . . . . . . . . . . . . . . . . . . 1-25

Using the Neural Network Pattern

Recognition Tool GUI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Contents

1-31

Clustering Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-42

Defining a Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-42

Using Command-Line Functions . . . . . . . . . . . . . . . . . . . . . . . 1-43

Using the Neural Network Clustering Tool GUI . . . . . . . . . . . 1-47

Neuron Model and Network Architectures

2

Neuron Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-2

Simple Neuron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-2

Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-3

Neuron with Vector Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-5

i

Network Architectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-8

A Layer of Neurons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-8

Multiple Layers of Neurons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-10

Input and Output Processing Functions . . . . . . . . . . . . . . . . . . 2-12

Data Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-14

Simulation with Concurrent Inputs in a Static Network . . . . 2-14

Simulation with Sequential Inputs in a Dynamic Network . . 2-15
Simulation with Concurrent Inputs in a Dynamic Network . . 2-17

Training Styles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-20

Incremental Training (of Adaptive and Other Networks) . . . . 2-20

Batch Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-22

Training Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-25

Perceptrons

3

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-2

Important Perceptron Functions . . . . . . . . . . . . . . . . . . . . . . . . . 3-2

ii Contents

Neuron Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-3

Perceptron Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-5

Creating a Perceptron (newp) . . . . . . . . . . . . . . . . . . . . . . . . . . 3-6

Simulation (sim) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-7

Initialization (init) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-8

Learning Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-11

Perceptron Learning Rule (learnp) . . . . . . . . . . . . . . . . . . . . 3-12

Training (train) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-15

Limitations and Cautions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-21

Outliers and the Normalized Perceptron Rule . . . . . . . . . . . . . 3-21

Graphical User Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-23

Introduction to the GUI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-23

Create a Perceptron Network (nntool) . . . . . . . . . . . . . . . . . . . 3-23

Train the Perceptron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-27

Export Perceptron Results to the Workspace . . . . . . . . . . . . . . 3-29

Clear the Network/Data Window . . . . . . . . . . . . . . . . . . . . . . . 3-30

Importing from the Command Line . . . . . . . . . . . . . . . . . . . . . 3-30

Save a Variable to a File and Load It Later . . . . . . . . . . . . . . . 3-31

Linear Filters

4

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-2

Neuron Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-3

Network Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-4

Creating a Linear Neuron (newlin) . . . . . . . . . . . . . . . . . . . . . . . 4-4

Least Mean Square Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-8

Linear System Design (newlind) . . . . . . . . . . . . . . . . . . . . . . . . 4-9

Linear Networks with Delays . . . . . . . . . . . . . . . . . . . . . . . . . 4-10

Tapped Delay Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-10

Linear Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-10

LMS Algorithm (learnwh) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-13

Linear Classification (train) . . . . . . . . . . . . . . . . . . . . . . . . . . 4-15

Limitations and Cautions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-18

Overdetermined Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-18

Underdetermined Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-18

iii

Linearly Dependent Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-18

Too Large a Learning Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-19

Backpropagation

5

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-2

Solving a Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-4

Improving Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-6

Under the Hood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-6

Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-8

Feedforward Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-10

Simulation (sim) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-14

Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-15

Backpropagation Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-15

iv Contents

Faster Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-19

Variable Learning Rate (traingda, traingdx) . . . . . . . . . . . . . . 5-19

Resilient Backpropagation (trainrp) . . . . . . . . . . . . . . . . . . . . . 5-21

Conjugate Gradient Algorithms . . . . . . . . . . . . . . . . . . . . . . . . 5-22

Line Search Routines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-26

Quasi-Newton Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-29

Levenberg-Marquardt (trainlm) . . . . . . . . . . . . . . . . . . . . . . . . 5-30

Reduced Memory Levenberg-Marquardt (trainlm) . . . . . . . . . 5-32

Speed and Memory Comparison . . . . . . . . . . . . . . . . . . . . . . . 5-34

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-50

Improving Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-52

Early Stopping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-53

Index Data Division (divideind) . . . . . . . . . . . . . . . . . . . . . . . . 5-54

Random Data Division (dividerand) . . . . . . . . . . . . . . . . . . . . . 5-54

Block Data Division (divideblock) . . . . . . . . . . . . . . . . . . . . . . . 5-54

Interleaved Data Division (divideint) . . . . . . . . . . . . . . . . . . . . 5-55

Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-55

Summary and Discussion of Early Stopping

and Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Preprocessing and Postprocessing . . . . . . . . . . . . . . . . . . . . . 5-61

Min and Max (mapminmax) . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-62

Mean and Stand. Dev. (mapstd) . . . . . . . . . . . . . . . . . . . . . . . . 5-63

Principal Component Analysis (processpca) . . . . . . . . . . . . . . . 5-64

Processing Unknown Inputs (fixunknowns) . . . . . . . . . . . . . . . 5-65

Representing Unknown or Don’t Care Targets . . . . . . . . . . . . 5-66

Posttraining Analysis (postreg) . . . . . . . . . . . . . . . . . . . . . . . . . 5-66

Sample Training Session . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-68

Limitations and Cautions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-71

Dynamic Networks

6

5-58

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-2

Examples of Dynamic Networks . . . . . . . . . . . . . . . . . . . . . . . . . 6-2

Applications of Dynamic Networks . . . . . . . . . . . . . . . . . . . . . . . 6-7

Dynamic Network Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-8

Dynamic Network Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-9

Focused Time-Delay Neural Network (newfftd) . . . . . . . . . 6-11

Distributed Time-Delay Neural Network (newdtdnn) . . . . 6-15

NARX Network (newnarx, newnarxsp, sp2narx) . . . . . . . . 6-18

Layer-Recurrent Network (newlrn) . . . . . . . . . . . . . . . . . . . . 6-24

v

Control Systems

7

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-2

NN Predictive Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-4

System Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-4

Predictive Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-5

Using the NN Predictive Controller Block . . . . . . . . . . . . . . . . . 7-6

NARMA-L2 (Feedback Linearization) Control . . . . . . . . . . 7-14

Identification of the NARMA-L2 Model . . . . . . . . . . . . . . . . . . 7-14

NARMA-L2 Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-16

Using the NARMA-L2 Controller Block . . . . . . . . . . . . . . . . . . 7-18

Model Reference Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-23

Using the Model Reference Controller Block . . . . . . . . . . . . . . 7-25

Importing and Exporting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-31

Importing and Exporting Networks . . . . . . . . . . . . . . . . . . . . . 7-31

Importing and Exporting Training Data . . . . . . . . . . . . . . . . . 7-35

vi Contents

Radial Basis Networks

8

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-2

Important Radial Basis Functions . . . . . . . . . . . . . . . . . . . . . . . 8-2

Radial Basis Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-3

Neuron Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-3

Network Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-4

Exact Design (newrbe) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-5

More Efficient Design (newrb) . . . . . . . . . . . . . . . . . . . . . . . . . . 8-7

Demonstrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-8

Probabilistic Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . . 8-9

Network Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-9

Design (newpnn) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-10

Generalized Regression Networks . . . . . . . . . . . . . . . . . . . . . 8-12

Network Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-12

Design (newgrnn) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-14

Self-Organizing and Learning

Vector Quantization Nets

9

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-2

Important Self-Organizing and LVQ Functions . . . . . . . . . . . . . 9-2

Competitive Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-3

Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-3

Creating a Competitive Neural Network (newc) . . . . . . . . . . . . 9-4

Kohonen Learning Rule (learnk) . . . . . . . . . . . . . . . . . . . . . . . . . 9-5

Bias Learning Rule (learncon) . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-5

Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-6

Graphical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-7

Self-Organizing Feature Maps . . . . . . . . . . . . . . . . . . . . . . . . . . 9-9

Topologies (gridtop, hextop, randtop) . . . . . . . . . . . . . . . . . . . . 9-10

Distance Functions (dist, linkdist, mandist, boxdist) . . . . . . . 9-14

Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-17

Creating a Self-Organizing MAP Neural Network (newsom) . 9-18

Training (learnsomb) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-19

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-22

Learning Vector Quantization Networks . . . . . . . . . . . . . . . 9-35

Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-35

Creating an LVQ Network (newlvq) . . . . . . . . . . . . . . . . . . . . . 9-36

LVQ1 Learning Rule (learnlv1) . . . . . . . . . . . . . . . . . . . . . . . . . 9-39

Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-40

Supplemental LVQ2.1 Learning Rule (learnlv2) . . . . . . . . . . . 9-42

vii

10

Adaptive Filters and Adaptive Training

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-2

Important Adaptive Functions . . . . . . . . . . . . . . . . . . . . . . . . . 10-2

Linear Neuron Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-3

Adaptive Linear Network Architecture . . . . . . . . . . . . . . . . 10-4

Single ADALINE (newlin) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-4

Least Mean Square Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-7

LMS Algorithm (learnwh) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-8

Adaptive Filtering (adapt) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-9

Tapped Delay Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-9

Adaptive Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-9

Adaptive Filter Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-10

Prediction Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-13

Noise Cancellation Example . . . . . . . . . . . . . . . . . . . . . . . . . . 10-14

Multiple Neuron Adaptive Filters . . . . . . . . . . . . . . . . . . . . . . 10-16

viii Contents

Applications

11

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-2

Application Scripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-2

Applin1: Linear Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-3

Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-3

Network Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-4

Network Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-4

Thoughts and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-6

Applin2: Adaptive Prediction . . . . . . . . . . . . . . . . . . . . . . . . . 11-7

Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-7

12

Network Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-8

Network Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-8

Network Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-8

Thoughts and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-10

Appelm1: Amplitude Detection . . . . . . . . . . . . . . . . . . . . . . . 11-11

Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-11

Network Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-11

Network Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-12

Network Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-12

Network Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-13

Improving Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-14

Appcr1: Character Recognition . . . . . . . . . . . . . . . . . . . . . . . 11-15

Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-15

Neural Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-16

System Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-18

Advanced Topics

13

Custom Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-2

Custom Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-2

Network Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-3

Network Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-12

Additional Toolbox Functions . . . . . . . . . . . . . . . . . . . . . . . . 12-15

Custom Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-16

Historical Networks

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13-2

ix

14

Important Recurrent Network Functions . . . . . . . . . . . . . . . . . 13-2

Elman Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13-3

Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13-3

Creating an Elman Network (newelm) . . . . . . . . . . . . . . . . . . . 13-4

Training an Elman Network . . . . . . . . . . . . . . . . . . . . . . . . . . . 13-5

Hopfield Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13-8

Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13-8

Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13-8

Design (newhop) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13-10

Network Object Reference

Network Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-2

Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-2

Subobject Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-5

Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-7

Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-9

Weight and Bias Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-10

Other . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-12

x Contents

Subobject Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-13

Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-13

Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-15

Outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-20

Biases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-21

Input Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-22

Layer Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-24

15

Function Reference

Analysis Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-3

Distance Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-4

Graphical Interface Functions . . . . . . . . . . . . . . . . . . . . . . . . 15-5

Layer Initialization Functions . . . . . . . . . . . . . . . . . . . . . . . . 15-6

Learning Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-7

Line Search Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-8

Net Input Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-9

Network Initialization Function . . . . . . . . . . . . . . . . . . . . . . 15-10

Network Use Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-11

New Networks Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-12

Performance Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-13

Plotting Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-14

Processing Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-15

Simulink® Support Function . . . . . . . . . . . . . . . . . . . . . . . . . 15-16

Topology Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-17

Training Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-18

Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-19

Utility Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-20

xi

16

A

Vector Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-21

Weight and Bias Initialization Functions . . . . . . . . . . . . . . 15-22

Weight Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-23

Transfer Function Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-24

Functions — Alphabetical List

Mathematical Notation

Mathematical Notation for Equations and Figures . . . . . . . A-2

Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-2

Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-2

Weight Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-2

Bias Elements and Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-2

Time and Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-2

Layer Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-3

Figure and Equation Examples . . . . . . . . . . . . . . . . . . . . . . . . . . A-3

xii Contents

Mathematics and Code Equivalents . . . . . . . . . . . . . . . . . . . . . A-4

Demonstrations and Applications

B

Tables of Demonstrations and Applications . . . . . . . . . . . . . B-2

Chapter 2, “Neuron Model and Network Architectures” . . . . . . B-2

Chapter 3, “Perceptrons” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-2

Chapter 4, “Linear Filters” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-3

C

Chapter 5, “Backpropagation” . . . . . . . . . . . . . . . . . . . . . . . . . . . B-3

Chapter 8, “Radial Basis Networks” . . . . . . . . . . . . . . . . . . . . . B-4

Chapter 9, “Self-Organizing and Learning

Vector Quantization Nets” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 10, “Adaptive Filters and Adaptive Training” . . . . . . . B-4

Chapter 11, “Applications” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-5

Chapter 13, “Historical Networks” . . . . . . . . . . . . . . . . . . . . . . . B-5

B-4

Blocks for the Simulink® Environment

Blockset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-2

Transfer Function Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-2

Net Input Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-3

Weight Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-3

Processing Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-4

Block Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-5

Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-5

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-7

D

Code Notes

Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-2

Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-3

Utility Function Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-4

Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-6

Code Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-7

Argument Checking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-8

xiii

E

Bibliography

Glossary

Index

xiv Contents

Getting Started

Product Overview (p. 1-2)

Using the Documentation (p. 1-3)

Applications for Neural Network Toolbox™ Software (p. 1-4)

Fitting a Function (p. 1-7)

Recognizing Patterns (p. 1-24)

Clustering Data (p. 1-42)

1

1 Getting Started

Neural Network
including connections
(called weights)
between neurons

Input Output

Target

Adjust
weights

Compare

Product Overview

Neural networks are composed of simple elements operating in parallel. These
elements are inspired by biological nervous systems. As in nature, the
connections between elements largely determine the network function. You
can train a neural network to perform a particular function by adjusting the
values of the connections (weights) between elements.

Typically, neural networks are adjusted, or trained, so that a particular input
leads to a specific target output. The next figure illustrates such a situation.
There, the network is adjusted, based on a comparison of the output and the
target, until the network output matches the target. Typically, many such
input/target pairs are needed to train a network.

Neural networks have been trained to perform complex functions in various
fields, including pattern recognition, identification, classification, speech,
vision, and control systems.

Neural networks can also be trained to solve problems that are difficult for
conventional computers or human beings. The toolbox emphasizes the use of
neural network paradigms that build up to—or are themselves used in—
engineering, financial, and other practical applications.

The next sections explain how to use three graphical tools for training neural
networks to solve problems in function fitting, pattern recognition, and
clustering.

1-2

Using the Documentation

The neuron model and the architecture of a neural network describe how a
network transforms its input into an output. This transformation can be
viewed as a computation.

This first chapter gives you an overview of the Neural Network Toolbox™
product and introduces you to the following tasks:

• Training a neural network to fit a function

• Training a neural network to recognize patterns

• Training a neural network to cluster data

These next two chapters explain the computations that are done and pave the
way for an understanding of training methods for the networks. You should
read them before advancing to later topics:

• Chapter 2, “Neuron Model and Network Architectures,” presents the
fundamentals of the neuron model, the architectures of neural networks. It
also discusses the notation used in this toolbox.

• Chapter 3, “Perceptrons,” explains how to create and train simple networks.
It also introduces a graphical user interface (GUI) that you can use to solve
problems without a lot of coding.

Using the Documentation

1-3

1 Getting Started

Applications for Neural Network Toolbox™ Software

Applications in This Toolbox

Chapter 7, “Control Systems” describes three practical neural network control
system applications, including neural network model predictive control, model
reference adaptive control, and a feedback linearization controller.

Chapter 11, “Applications” describes other neural network applications.

Business Applications

The 1988 DARPA Neural Network Study [DARP88] lists various neural
network applications, beginning in about 1984 with the adaptive channel
equalizer. This device, which is an outstanding commercial success, is a single­neuron network used in long-distance telephone systems to stabilize voice
signals. The DARPA report goes on to list other commercial applications,
including a small word recognizer, a process monitor, a sonar classifier, and a
risk analysis system.

Neural networks have been applied in many other fields since the DARPA
report was written, as described in the next table.

1-4

Industry Business Applications

Aerospace High-performance aircraft autopilot, flight path

simulation, aircraft control systems, autopilot
enhancements, aircraft component simulation,
and aircraft component fault detection

Automotive Automobile automatic guidance system, and

warranty activity analysis

Banking Check and other document reading and credit

application evaluation

Applications for Neural Network Toolbox™ Software

Industry Business Applications

Defense Weapon steering, target tracking, object

discrimination, facial recognition, new kinds of
sensors, sonar, radar and image signal processing
including data compression, feature extraction
and noise suppression, and signal/image
identification

Electronics Code sequence prediction, integrated circuit chip

layout, process control, chip failure analysis,
machine vision, voice synthesis, and nonlinear
modeling

Entertainment Animation, special effects, and market forecasting

Financial Real estate appraisal, loan advising, mortgage

screening, corporate bond rating, credit-line use
analysis, credit card activity tracking, portfolio
trading program, corporate financial analysis,
and currency price prediction

Industrial Prediction of industrial processes, such as the

output gases of furnaces, replacing complex and
costly equipment used for this purpose in the past

Insurance Policy application evaluation and product

optimization

Manufacturing Manufacturing process control, product design

and analysis, process and machine diagnosis,
real-time particle identification, visual quality
inspection systems, beer testing, welding quality
analysis, paper quality prediction, computer-chip
quality analysis, analysis of grinding operations,
chemical product design analysis, machine
maintenance analysis, project bidding, planning
and management, and dynamic modeling of
chemical process system

1-5

1 Getting Started

Industry Business Applications

Medical Breast cancer cell analysis, EEG and ECG

analysis, prosthesis design, optimization of
transplant times, hospital expense reduction,
hospital quality improvement, and
emergency-room test advisement

Oil and gas Exploration

Robotics Trajectory control, forklift robot, manipulator

controllers, and vision systems

Speech Speech recognition, speech compression, vowel

classification, and text-to-speech synthesis

Securities Market analysis, automatic bond rating, and

stock trading advisory systems

Telecommunications Image and data compression, automated

information services, real-time translation of
spoken language, and customer payment
processing systems

1-6

Transportation Truck brake diagnosis systems, vehicle

scheduling, and routing systems

Fitting a Function

Neural networks are good at fitting functions and recognizing patterns. In fact,
there is proof that a fairly simple neural network can fit any practical function.

Suppose, for instance, that you have data from a housing application
[HaRu78]. You want to design a network that can predict the value of a house
(in $1000s), given 13 pieces of geographical and real estate information. You
have a total of 506 example homes for which you have those 13 items of data
and their associated market values.

You can solve this problem in three ways:

• Use a command-line function, as described in “Using Command-Line
Functions” on page 1-7.

• Use a graphical user interface,
Network Fitting Tool GUI” on page 1-13.

• Use

Defining a Problem

To define a fitting problem for the toolbox, arrange a set of Q input vectors as
columns in a matrix. Then, arrange another set of Q target vectors (the correct
output vectors for each of the input vectors) into a second matrix. For example,
you can define the fitting problem for a Boolean AND gate with four sets of
two-element input vectors and one-element targets as follows:

Fitting a Function

nftool, as described in “Using the Neural

nntool, as described in “Graphical User Interface” on page 3-23.

inputs = [0 1 0 1; 0 0 1 1];
targets = [0 0 0 1];

The next section demonstrates how to train a network from the command line,
after you have defined the problem. This example uses the housing data set
provided with the toolbox.

Using Command-Line Functions

1 Load the data, consisting of input vectors and target vectors, as follows:

load house_dataset

2 Create a network. For this example, you use a feed-forward network with

the default tan-sigmoid transfer function in the hidden layer and linear

1-7

1 Getting Started

transfer function in the output layer. This structure is useful for function
approximation (or regression) problems. Use 20 neurons (somewhat
arbitrary) in one hidden layer. The network has one output neuron, because
there is only one target value associated with each input vector.

net = newfit(houseInputs,houseTargets,20);

Note More neurons require more computation, but they allow the network to
solve more complicated problems. More layers require more computation, but
their use might result in the network solving complex problems more
efficiently.

3 Train the network. The network uses the default Levenberg-Marquardt

algorithm for training. The application randomly divides input vectors and
target vectors into three sets as follows:

- 60% are used for training.

- 20% are used to validate that the network is generalizing and to stop

training before overfitting.

- The last 20% are used as a completely independent test of network

generalization.

1-8

To train the network, enter:

net=train(net,houseInputs,houseTargets);

During training, the following training window opens. This window displays
training progress and allows you to interrupt training at any point by
clicking

Stop Training.

Fitting a Function

This example used the train function. All the input vectors to the network
appear at once in a batch. Alternatively, you can present the input vectors
one at a time using the

adapt function. “Training Styles” on page 2-20

describes the two training approaches.

This training stopped when the validation error increased for six iterations,
which occurred at iteration 23. If you click

Performance in the training

window, a plot of the training errors, validation errors, and test errors
appears, as shown in the following figure. In this example, the result is
reasonable because of the following considerations:

1-9

1 Getting Started

- The final mean-square error is small.

- The test set error and the validation set error have similar characteristics.

- No significant overfitting has occurred by iteration 17 (where the best

validation performance occurs).

1-10

4 Perform some analysis of the network response. If you click Regression in

the training window, you can perform a linear regression between the
network outputs and the corresponding targets.

The following figure shows the results.

Fitting a Function

The output tracks the targets very well for training, testing, and validation,
and the R-value is over 0.95 for the total response. If even more accurate
results were required, you could try any of these approaches:

• Reset the initial network weights and biases to new values with

init and

train again.

• Increase the number of hidden neurons.

• Increase the number of training vectors.

1-11

1 Getting Started

• Increase the number of input values, if more relevant information is
available.

• Try a different training algorithm (see “Speed and Memory Comparison” on
page 5-34).

In this case, the network response is satisfactory, and you can now use
put the network to use on new inputs.

To get more experience in command-line operations, try some of these tasks:

• During training, open a plot window (such as the regression plot), and watch
it animate.

• Plot from the command line with functions such as

plotregression, plottrainstate and plotperform. (For more information

on using these functions, see their reference pages.)

plotfit,

sim to

1-12

Using the Neural Network Fitting Tool GUI

1 Open the Neural Network Fitting Tool with this command:

nftool

Fitting a Function

1-13

1 Getting Started

2 Click Next to proceed.

1-14

3 Click Load Example Data Set in the Select Data window. The Fitting Data

Set Chooser window opens.

Note You use the Inputs and Targets options in the Select Data window
when you need to load data from the MATLAB® workspace.

Fitting a Function

4 Select Simple Fitting Problem, and click Import. This brings you back to

the Select Data window.

1-15

1 Getting Started

5 Click Next to display the Validate and Test Data window, shown in the

following figure.

The validation and test data sets are each set to 15% of the original data.

1-16

6 Click Next.

Fitting a Function

The number of hidden neurons is set to

20. You can change this value in

another run if you want. You might want to change this number if the
network does not perform as well as you expect.

1-17

1 Getting Started

7 Click Next.

1-18

8 Click Train.

This time the training continued for the maximum of 1000 iterations.

Fitting a Function

9 Under Plots, click Regression.

For this simple fitting problem, the fit is almost perfect for training, testing,
and validation data.

1-19

1 Getting Started

1-20

These plots are the regression plots for the output with respect to training,
validation, and test data.

10 View the network response. For single-input/single-output problems, like

this simple fitting problem, under the

Plots pane, click Fit.

Fitting a Function

The blue symbols represent training data, the green symbols represent
validation data, and the red symbols represent testing data. For this
problem and this network, the network outputs match the targets for all
three data sets.

11 Click Next in the Neural Network Fitting Tool to evaluate the network.

1-21

1 Getting Started

1-22

At this point, you can test the network against new data.

If you are dissatisfied with the network’s performance on the original or new
data, you can take any of the following steps:

- Train it again.

- Increase the number of neurons.

- Get a larger training data set.

12 If you are satisfied with the network performance, click Next.

13 Use the buttons on this screen to save your results.

Fitting a Function

- You have the network saved as

net1 in the workspace. You can perform

additional tests on it or put it to work on new inputs, using the
function.

- You can also click

Generate M-File to create an M-file that can be used to

reproduce all of the previous steps from the command line. Creating an
M-file can be helpful if you want to learn how to use the command-line
functionality of the toolbox to customize the training process.

14 When you have saved your results, click Finish.

sim

1-23

1 Getting Started

Recognizing Patterns

In addition to function fitting, neural networks are also good at recognizing
patterns.

For example, suppose you want to classify a tumor as benign or malignant,
based on uniformity of cell size, clump thickness, mitosis, etc. [MuAh94]. You
have 699 example cases for which you have 9 items of data and the correct
classification as benign or malignant.

As with function fitting, there are three ways to solve this problem:

• Use a command-line solution, as described in “Using Command-Line

• Use the

• Use

Defining a Problem

To define a pattern recognition problem, arrange a set of Q input vectors as
columns in a matrix. Then arrange another set of Q target vectors so that they
indicate the classes to which the input vectors are assigned. There are two
approaches to creating the target vectors.

Functions” on page 1-43.

nprtool GUI, as described in “Using the Neural Network Clustering

Tool GUI” on page 1-47.

nntool, as described in “Graphical User Interface” on page 3-23.

1-24

One approach can be used when there are only two classes; you set each scalar
target value to either 1 or 0, indicating which class the corresponding input
belongs to. For instance, you can define the exclusive-or classification problem
as follows:

inputs = [0 1 0 1; 0 0 1 1];
targets = [0 1 0 1];

Alternately, target vectors can have N elements, where for each target vector,
one element is 1 and the others are 0. This defines a problem where inputs are
to be classified into N different classes. For example, the following lines show
how to define a classification problem that divides the corners of a 5-by-5-by-5
cube into three classes:

• The origin (the first input vector) in one class

• The corner farthest from the origin (the last input vector) in a second class

• All other points in a third class

Recognizing Patterns

inputs = [0 0 0 0 5 5 5 5; 0 0 5 5 0 0 5 5; 0 5 0 5 0 5 0 5];
targets = [1 0 0 0 0 0 0 0; 0 1 1 1 1 1 1 0; 0 0 0 0 0 0 0 1];

Classification problems involving only two classes can be represented using
either format. The targets can consist of either scalar 1/0 elements or
two-element vectors, with one element being 1 and the other element being 0.

The next section demonstrates how to train a network from the command line,
after you have defined the problem.

Using Command-Line Functions

1 Use the cancer data set as an example. This data set consists of 699

nine-element input vectors and two-element target vectors.

Load the tumor classification data as follows:

load cancer_dataset

2 Create a network. For this example, you use a pattern recognition network,

which is a feed-forward network with tan-sigmoid transfer functions in both
the hidden layer and the output layer. As in the function-fitting example,
use 20 neurons in one hidden layer:

- The network has two output neurons, because there are two categories

associated with each input vector.

- Each output neuron represents a category.

- When an input vector of the appropriate category is applied to the

network, the corresponding neuron should produce a 1, and the other
neurons should output a 0.

To create a network, enter this command:

net = newpr(cancerInputs,cancerTargets,20);

3 Train the network. The pattern recognition network uses the default Scaled

Conjugate Gradient algorithm for training. The application randomly
divides the input vectors and target vectors into three sets:

- 60% are used for training.

- 20% are used to validate that the network is generalizing and to stop

training before overfitting.

1-25

1 Getting Started

- The last 20% are used as a completely independent test of network

generalization.

To train the network, enter this command:

net=train(net,cancerInputs,cancerTargets);

During training, as in function fitting, the training window opens. This
window displays training progress. To interrupt training at any point, click

Stop Training.

1-26

Recognizing Patterns

This example uses the train function. It presents all the input vectors to the
network at once in a batch. Alternatively, you can present the input vectors
one at a time using the

adapt function. “Training Styles” on page 2-20

describes the two training approaches.

This training stopped when the validation error increased for six iterations,
which occurred at iteration 15.

4 To find the validation error, click Performance in the training window. A

plot of the training errors, validation errors, and test errors appears, as

1-27

1 Getting Started

shown in the following figure. The best validation performance occurred at
iteration 9, and the network at this iteration is returned.

1-28

5 To analyze the network response, click Confusion in the training window.

A display of the confusion matrix appears that shows various types of errors
that occurred for the final trained network.

The next figure shows the results.

Recognizing Patterns

The diagonal cells in each table show the number of cases that were correctly
classified, and the off-diagonal cells show the misclassified cases. The blue cell
in the bottom right shows the total percent of correctly classified cases (in
green) and the total percent of misclassified cases (in red). The results for all
three data sets (training, validation, and testing) show very good recognition.
If you needed even more accurate results, you could try any of the following
approaches:

1-29

1 Getting Started

• Reset the initial network weights and biases to new values with init and
train again.

• Increase the number of hidden neurons.

• Increase the number of training vectors.

• Increase the number of input values, if more relevant information is

available.

• Try a different training algorithm (see “Speed and Memory Comparison” on
page 5-34).

In this case, the network response is satisfactory, and you can now use
put the network to use on new inputs.

To get more experience in command-line operations, here are some tasks you
can try:

• During training, open a plot window (such as the confusion plot), and watch
it animate.

• Plot from the command line with functions such as

plottrainstate, and plotperform. (For more information on using these

functions, see their reference pages.)

plotconfusion, plotroc,

sim to

1-30

Recognizing Patterns

Using the Neural Network Pattern Recognition Tool GUI

1 Open the Neural Network Pattern Recognition Tool window with this

command:

nprtool

1-31

1 Getting Started

2 Click Next to proceed. The Select Data window opens.

1-32

3 Click Load Example Data Set. The Pattern Recognition Data Set Chooser

window opens.

Recognizing Patterns

4 In this window, select Simple Classes, and click Import. You return to the

Select Data window.

1-33

1 Getting Started

5 Click Next to continue to the Validate and Test Data window, shown in the

following figure.

Validation and test data sets are each set to 15% of the original data.

1-34

6 Click Next.

Recognizing Patterns

The number of hidden neurons is set to

20. You can change this in another

run if you want. You might want to change this number if the network does
not perform as well as you expect.

1-35

1 Getting Started

7 Click Next.

1-36

8 Click Train.

Recognizing Patterns

The training continues for 55 iterations.

9 Under the Plots pane, click Confusion in the Neural Network Pattern

Recognition Tool.

The next figure shows the confusion matrices for training, testing, and
validation, and the three kinds of data combined. The network's outputs are
almost perfect, as you can see by the high numbers of correct responses in

1-37

1 Getting Started

the green squares and the low numbers of incorrect responses in the red
squares. The lower right blue squares illustrate the overall accuracies.

1-38

10 Plot the Receiver Operating Characteristic (ROC) curve. Under the Plots

pane, click Receiver Operating Characteristic in the Neural Network
Pattern Recognition Tool.

Recognizing Patterns

The colored lines in each axis represent the ROC curves for each of the four
categories of this simple test problem. The

ROC curve is a plot of the true

positive rate (sensitivity) versus the false positive rate (1 - specificity) as the
threshold is varied. A perfect test would show points in the upper-left corner,
with 100% sensitivity and 100% specificity. For this simple problem, the
network performs almost perfectly.

1-39

1 Getting Started

11 In the Neural Network Pattern Recognition Tool, click Next to evaluate the

network.

1-40

At this point, you can test the network against new data.

If you are dissatisfied with the network’s performance on the original or new
data, you can train it again, increase the number of neurons, or perhaps get
a larger training data set.

Recognizing Patterns

12 When you are satisfied with the network performance, click Next.

13 Use the buttons on this screen to save your results.

- You now have the network saved as

net1 in the workspace. You can

perform additional tests on it or put it to work on new inputs using the
function.

- If you click

Generate M-File, the tool creates an M-file, with commands

that recreate the steps that you have just performed from the command
line. Generating an M-file is a good way to learn how to use the
command-line operations of the Neural Network Toolbox™ software.

14 When you have saved your results, click Finish.

sim

1-41

1 Getting Started

Clustering Data

Clustering data is another excellent application for neural networks. This
process involves grouping data by similarity. For example, you might perform:

• Market segmentation by grouping people according to their buying patterns

• Data mining by partitioning data into related subsets

• Bioinformatic analysis by grouping genes with related expression patterns

Suppose that you want to cluster flower types according to petal length, petal
width, sepal length, and sepal width [MuAh94]. You have 150 example cases
for which you have these four measurements.

As with function fitting and pattern recognition, there are three ways to solve
this problem:

• Use a command-line solution, as described in “Using Command-Line
Functions” on page 1-43.

• Use the nctool GUI, as described in “Using the Neural Network Clustering
Tool GUI” on page 1-47.

• Use

nntool, as described in “Graphical User Interface” on page 3-23.

1-42

Defining a Problem

To define a clustering problem, simply arrange Q input vectors to be clustered
as columns in an input matrix. For instance, you might want to cluster this set
of 10 two-element vectors:

inputs = [7 0 6 2 6 5 6 1 0 1; 6 2 5 0 7 5 5 1 2 2]

The next section demonstrates how to train a network from the command line,
after you have defined the problem.

Clustering Data

Using Command-Line Functions

1 Use the flower data set as an example. The iris data set consists of 150

four-element input vectors.

Load the data as follows:

load iris_dataset

This data set consists of input vectors and target vectors. However, you only
need the input vectors for clustering.

2 Create a network. For this example, you use a self-organizing map (SOM).

This network has one layer, with the neurons organized in a grid. (For more
information, see “Self-Organizing Feature Maps” on page 9-9.) When
creating the network, you specify the number of rows and columns in the
grid:

net = newsom(irisInputs,[6,6]);

3 Train the network. The SOM network uses the default batch SOM algorithm

for training.

net=train(net,irisInputs);

4 During training, the training window opens and displays the training

progress. To interrupt training at any point, click

Stop Training.

1-43

1 Getting Started

1-44

5 For SOM training, the weight vector associated with each neuron moves to

become the center of a cluster of input vectors. In addition, neurons that are
adjacent to each other in the topology should also move close to each other
in the input space. The default topology is hexagonal; to view it, click

Topology

from the network training window.

SOM

Clustering Data

In this figure, each of the hexagons represents a neuron. The grid is 6-by-6,
so there are a total of 36 neurons in this network. There are four elements
in each input vector, so the input space is four-dimensional. The weight
vectors (cluster centers) fall within this space.

Because this SOM has a two-dimensional topology, you can visualize in two
dimensions the relationships among the four-dimensional cluster centers.
One visualization tool for the SOM is the
the

U-matrix).

6 To view the U-matrix, click SOM Neighbor Distances in the training

weight distance matrix (also called

window.

1-45

1 Getting Started

1-46

In this figure, the blue hexagons represent the neurons. The red lines
connect neighboring neurons. The colors in the regions containing the red
lines indicate the distances between neurons. The darker colors represent
larger distances, and the lighter colors represent smaller distances.

A band of dark segments crosses from the lower-center region to the
upper-right region. The SOM network appears to have clustered the flowers
into two distinct groups.

To get more experience in command-line operations, try some of these tasks:

• During training, open a plot window (such as the SOM weight position plot)
and watch it animate

• Plot from the command line with functions such as

plotsomnd, plotsomplanes, plotsompos, and plotsomtop. (For more

information on using these functions, see their reference pages.)

plotsomhits, plotsomnc,

Clustering Data

Using the Neural Network Clustering Tool GUI

1 Open the Neural Network Clustering Tool window with this command:

nctool

1-47

1 Getting Started

2 Click Next. The Select Data window appears.

1-48

Clustering Data

3 Click Load Example Data Set. The Clustering Data Set Chooser window

appears.

4 In this window, select Simple Clusters, and click Import. You return to the

Select Data window.

1-49

1 Getting Started

5 Click Next to continue to the Network Size window, shown in the following

figure.

The size of the two-dimensional map is set to
side of a two-dimensional grid. The total number of neurons is 100. You can
change this number in another run if you want.

10. This map represents one

1-50

6 Click Next. The Train Network window appears.

Clustering Data

1-51

1 Getting Started

7 Click Train

1-52

The training runs for the maximum number of epochs, which is 200.

Clustering Data

8 Investigate some of the visualization tools for the SOM. Under the Plots

pane, click SOM Sample Hits.

This figure shows how many of the training data are associated with each of
the neurons (cluster centers). The topology is a 10-by-10 grid, so there are
100 neurons. The maximum number of hits associated with any neuron is

22. Thus, there are 22 input vectors in that cluster.

9 You can also visualize the SOM by displaying weight places (also referred to

as

component planes). Click SOM Weight Planes in the Neural Network

Clustering Tool.

1-53

1 Getting Started

This figure shows a weight plane for each element of the input vector (two,
in this case). They are visualizations of the weights that connect each input
to each of the neurons. (Darker colors represent larger weights.) If the
connection patterns of two inputs were very similar, you can assume that
the inputs are highly correlated. In this case, input 1 has connections that
are very different than those of input 2.

1-54

10 In the Neural Network Clustering Tool, click Next to evaluate the network.

Clustering Data

At this point you can test the network against new data.

If you are dissatisfied with the network’s performance on the original or new
data, you can increase the number of neurons, or perhaps get a larger
training data set.

11 When you are satisfied with the network performance, click Next.

1-55

1 Getting Started

12 Use the buttons on this screen to save your results.

1-56

• You now have the network saved as
additional tests on it, or put it to work on new inputs, using the function

• If you click
recreate the steps that you have just performed from the command line.
Generating an M-file is a good way to learn how to use the command-line
operations of the Neural Network Toolbox™ software.

13 When you have saved your results, click Finish.

Generate M-File, the tool creates an M-file, with commands that

net1 in the workspace. You can perform

sim.

Neuron Model and Network Architectures

Neuron Model (p. 2-2)

Network Architectures (p. 2-8)

Data Structures (p. 2-14)

Training Styles (p. 2-20)

2

2 Neuron Model and Network Architectures

Input - Title -

- Exp -

an

p

w

f

Neuron without bias

a = f (wp )

Input - Title -

- Exp -

an

p

f

Neuron with bias

a = f (wp + b)

b

1

w

Neuron Model

Simple Neuron

A neuron with a single scalar input and no bias appears on the left below.

The scalar input p is transmitted through a connection that multiplies its
strength by the scalar weight w to form the product wp, again a scalar. Here
the weighted input wp is the only argument of the transfer function f, which
produces the scalar output
can view the bias as simply being added to the product wp as shown by the
summing junction or as shifting the function f to the left by an amount b. The
bias is much like a weight, except that it has a constant input of 1.

a. The neuron on the right has a scalar bias, b. You

2-2

The transfer function net input n, again a scalar, is the sum of the weighted
input wp and the bias b. This sum is the argument of the transfer function f.
(Chapter 8, “Radial Basis Networks,” discusses a different way to form the net
input n.) Here f is a transfer function, typically a step function or a sigmoid
function, that takes the argument n and produces the output a. Examples of
various transfer functions are in “Transfer Functions” on page 2-3. Note that
w and b are both adjustable scalar parameters of the neuron. The central idea
of neural n e t works is th a t such param e ters can b e adjuste d s o that the n etwork
exhibits some desired or interesting behavior. Thus, you can train the network
to do a particular job by adjusting the weight or bias parameters, or perhaps
the network itself will adjust these parameters to achieve some desired end.

All the neurons in the Neural Network Toolbox™ software have provision for
a bias, and a bias is used in many of the examples and is assumed in most of
this toolbox. However, you can omit a bias in a neuron if you want.

Neuron Model

a = hardlim(n)

Hard-Limit Transfer Function

-1

n

0

+1

a

As previously noted, the bias b is an adjustable (scalar) parameter of the
neuron. It is

not an input. However, the constant 1 that drives the bias is an

input and must be treated as such when you consider the linear dependence of
input vectors in Chapter 4, “Linear Filters.”

Transfer Functions

Many transfer functions are included the Neural Network Toolbox software.
Three of the most commonly used functions are shown below.

The hard-limit transfer function shown above limits the output of the neuron
to either 0, if the net input argument n is less than 0, or 1, if n is greater than
or equal to 0. This function is used in Chapter 3, “Perceptrons,” to create
neurons that make classification decisions.

The toolbox has a function,

hardlim, to realize the mathematical hard-limit

transfer function shown above. Try the following code:

n = -5:0.1:5;
plot(n,hardlim(n),'c+:');

It produces a plot of the function hardlim over the range -5 to +5.

All the mathematical transfer functions in the toolbox can be realized with a
function having the same name.

The following figure illustrates the linear transfer function.

2-3

2 Neuron Model and Network Architectures

n

0

-1

+1

a = purelin(n)

Linear Transfer Function

a

-1

n

0

+1

a

Log-Sigmoid Transfer Function

a = logsig(n)

Neurons of this type are used as linear approximators in Chapter 4, “Linear
Filters.”

The sigmoid transfer function shown below takes the input, which can have
any value between plus and minus infinity, and squashes the output into the
range 0 to 1.

2-4

This transfer function is commonly used in backpropagation networks, in part
because it is differentiable.

The symbol in the square to the right of each transfer function graph shown
above represents the associated transfer function. These icons replace the
general

f in the boxes of network diagrams to show the particular transfer

function being used.

For a complete listing of transfer functions and their icons, You can also specify
your own transfer functions.

You can experiment with a simple neuron and various transfer functions by
running the demonstration program

nnd2n1.

Neuron Model

p1, p2,... p

R

w

11,

, w

12,

, ... w

1 R,

Input

p

1

an

p

2

p

3

p

R

w

1, R

w

1,1

f

b

1

Where
R = number of

elements in
input vector

Neuron w Vector Input

a = f(Wp +b)

nw

11,p1w12,p2

... w

1 R,pR

b+++ +=

Neuron with Vector Input

A neuron with a single R-element input vector is shown below. Here the
individual element inputs

are multiplied by weights

and the weighted values are fed to the summing junction. Their sum is simply

Wp, the dot product of the (single row) matrix W and the vector p.

The neuron has a bias b, which is summed with the weighted inputs to form
the net input n. This sum, n, is the argument of the transfer function f.

This expression can, of course, be written in MATLAB

n = W*p + b

®

code as

However, you will seldom be writing code at this level, for such code is already
built into functions to define and simulate entire networks.

Abbreviated Notation

The figure of a single neuron shown above contains a lot of detail. When you
consider networks with many neurons, and perhaps layers of many neurons,
there is so much detail that the main thoughts tend to be lost. Thus, the

2-5

2 Neuron Model and Network Architectures

p a

1

n

W

b

R x 1

1

x R

1

x 1

1

x 1

1

x 1

Input

R

1

f

Where...
R = number of

elements in
input vector

Neuron

a = f(Wp +b)

authors have devised an abbreviated notation for an individual neuron. This
notation, which is used later in circuits of multiple neurons, is shown.

Here the input vector
The dimensions of

p is represented by the solid dark vertical bar at the left.

p are shown below the symbol p in the figure as Rx1. (Note

that a capital letter, such as R in the previous sentence, is used when referring
to the size of a vector.) Thus,
postmultiply the single-row, R-column matrix

p is a vector of R input elements. These inputs

W. As before, a constant 1 enters

the neuron as an input and is multiplied by a scalar bias b. The net input to the
transfer function f is n, the sum of the bias b and the product

Wp. This sum is

passed to the transfer function f to get the neuron’s output a, which in this case
is a scalar. Note that if there were more than one neuron, the network output
would be a vector.

A layer of a network is defined in the previous figure. A layer includes the
combination of the weights, the multiplication and summing operation (here
realized as a vector product
array of inputs, vector

Wp), the bias b, and the transfer function f. The

p, is not included in or called a layer.

Each time this abbreviated network notation is used, the sizes of the matrices
are shown just below their matrix variable names. This notation will allow you
to understand the architectures and follow the matrix mathematics associated
with them.

As discussed in “Transfer Functions” on page 2-3, when a specific transfer
function is to be used in a figure, the symbol for that transfer function replaces
the f shown above. Here are some examples.

2-6

Neuron Model

purelinhardlim logsig

You can experiment with a two-element neuron by running the demonstration
program

nnd2n2.

2-7

2 Neuron Model and Network Architectures

f

f

f

w

1,1

w

SR,

S

S

S

n

1

p

1

p

2

p

3

p

R

n

2

n

S

b

1

b

2

b

S

a

1

a

2

a

S

1

1

1

Inputs

Layer of Neurons

a=f(Wp+b)

Where

= number of

elements in
input vector

= number of

neurons in layer

R

S

Network Architectures

Two or more of the neurons shown earlier can be combined in a layer, and a
particular network could contain one or more such layers. First consider a
single layer of neurons.

A Layer of Neurons

A one-layer network with R input elements and S neurons follows.

2-8

In this network, each element of the input vector
input through the weight matrix

W. The ith neuron has a summer that gathers

p is connected to each neuron

its weighted inputs and bias to form its own scalar output n(i). The various n(i)
taken together form an S-element net input vector

the figure.

Note that it is common for the number of inputs to a layer to be different from
the number of neurons (i.e., R is not necessarily equal to S). A layer is not
constrained to have the number of its inputs equal to the number of its
neurons.

outputs form a column vector

a. The expression for a is shown at the bottom of

n. Finally, the neuron layer

Network Architectures

W

w

11,w12,

… w

1 R,

w

21,w22,

… w

2 R,

w

S 1,wS 2,

… w

SR,

=

a= f (Wp + b)

pa

1

n

W

b

R x 1

S

x R

S

x 1

S

x 1

Input

Layer of Neurons

RS

f

S x 1

R = number of

elements in
input vector

Where...

S = number of
neurons in layer 1

You can create a single (composite) layer of neurons having different transfer
functions simply by putting two of the networks shown earlier in parallel. Both
networks would have the same inputs, and each network would create some of
the outputs.

The input vector elements enter the network through the weight matrix

Note that the row indices on the elements of matrix

W indicate the destination

W.

neuron of the weight, and the column indices indicate which source is the input
for that weight. Thus, the indices in w

from the second input element

to the first (and only) neuron is w

say that the strength of the signal

1,2

1,2

.

The S neuron R input one-layer network also can be drawn in abbreviated
notation.

Here

p is an R length input vector, W is an SxR matrix, and a and b are S

length vectors. As defined previously, the neuron layer includes the weight
matrix, the multiplication operations, the bias vector
transfer function boxes.

Inputs and Layers

To describe networks having multiple layers, the notation must be extended.
Specifically, it needs to make a distinction between weight matrices that are

b, the summer, and the

2-9

2 Neuron Model and Network Architectures

pa

1

1

n

1

S

1

x R

S

1

x 1

S

1

x 1

S

1

x 1

Input

IW

1,1

b

1

Layer 1

S

1

f

1

R

a

1

= f1(IW

1,1

p +b1)

S

1

x 1

R

x 1

R = number of
elements in
input vector

S = number of
neurons in Layer 1

Where...

connected to inputs and weight matrices that are connected between layers. It
also needs to identify the source and destination for the weight matrices.

We will call weight matrices connected to inputs input weights; we will call
weight matrices coming from layer outputs layer weights. Further,
superscripts are used to identify the source (second index) and the destination
(first index) for the various weights and other elements of the network. To
illustrate, the one-layer multiple input network shown earlier is redrawn in
abbreviated form below.

As you can see, the weight matrix connected to the input vector
an input weight matrix (
destination 1 (first index). Elements of layer 1, such as its bias, net input, and
output have a superscript 1 to say that they are associated with the first layer.

“Multiple Layers of Neurons” uses layer weight (
weight (

IW) matrices.

Multiple Layers of Neurons

A network can have several layers. Each layer has a weight matrix W, a bias
vector
output vectors, etc., for each of these layers in the figures, the number of the
layer is appended as a superscript to the variable of interest. You can see the
use of this layer notation in the three-layer network shown below, and in the
equations at the bottom of the figure.

2-10

b, and an output vector a. To distinguish between the weight matrices,

1,1

IW

) having a source 1 (second index) and a

LW) matrices as well as input

p is labeled as

Network Architectures

iw

1,1

1,1

lw

2,1

1,1

lw

,

1,1

32

iw

1,1

1

SR,

lw

2,1

21

SS ,

lw

3,2

SS

32

,

S S S

S S S

S S S

n

1

1

n

1

2

n

1

3

p

1

p

2

p

3

p

R

n

2

1

n

2

2

n

2

3

n

1

S

1 n

2

S

2 n

3

S

3

b

1

1

b

1

2

b

1

3

b

2

1

b

2

2

b

2

3

b

1

S

1 b

2

S

2 b

3

S

3

a

1

1

a

1

2

a

1

3

a

2

1

a

2

2

a

2

3

a

1

S

1 a

2

S

2 a

3

S

3

111

111

111

Inputs Layer 1 Layer 2 Layer 3

f

2

f

2

f

2

f

3

f

3

f

3

f

1

f

1

f

1

aIWpb

1 1,1 1

=( +)f

1

aLWab

2 2,1 1 2

=( +)f

2

aLWab

3 3,2 2 3

=( +)f

3

a LWLWIWpbbb

3 3,2 2,1 1,1 1 2 3

= ( ( ( +)+)+)fff

321

The network shown above has R1 inputs, S1 neurons in the first layer, S
neurons in the second layer, etc. It is common for different layers to have
different numbers of neurons. A constant input 1 is fed to the bias for each
neuron.

Note that the outputs of each intermediate layer are the inputs to the following
layer. Thus layer 2 can be analyzed as a one-layer network with S
neurons, and an S

2

is

a

. Now that all the vectors and matrices of layer 2 have been identified, it
can be treated as a single-layer network on its own. This approach can be taken
with any layer of the network.

The layers of a multilayer network play different roles. A layer that produces
the network output is called an output layer. All other layers are called hidden
layers. The three-layer network shown earlier has one output layer (layer 3)
and two hidden layers (layer 1 and layer 2). Some authors refer to the inputs
as a fourth layer. This toolbox does not use that designation.

2xS1

weight matrix W2. The input to layer 2 is a1; the output

2

1

inputs, S2

2-11

2 Neuron Model and Network Architectures

pa1a

2

1 1

n

1

n

2

a

3 =

y

n

3

1

S

2 x S

1

S

2 x

1

S

2 x

1

S 2 x

1

S

3

x S

2

S

3 x

1

S

3

x 1

S

3

x 1

R x

1

S

1 x R

S

1 x

1

S 1 x

1

S

1 x

1

Input

IW

1,1

b

1

b

2

b

3

LW

2,1

LW

3,2

R S

3

S

1

S

2

f

2

f

3

Layer 1 Layer 2 Layer 3

a

1

=

f

1

(IW

1,1

p +b1)

a

2

=

f

2

(LW

2,1 a1

+b2) a3 =

f

3

(LW

3,2a2

+b3)

a

3

=

f

3

(LW

3,2

f

2

(LW

2,1

f

1

(IW

1,1p +b1

)+ b2)

+

b

3 =

y

f

1

The same three-layer network can also be drawn using abbreviated notation.

Multiple-layer networks are quite powerful. For instance, a network of two
layers, where the first layer is sigmoid and the second layer is linear, can be
trained to approximate any function (with a finite number of discontinuities)
arbitrarily well. This kind of two-layer network is used extensively in Chapter
5, “Backpropagation.”

Here it is assumed that the output of the third layer,
of interest, and this output is labeled as
output of multilayer networks.

3

a

, is the network output

y. This notation is used to specify the

Input and Output Processing Functions

Network inputs might have associated processing functions. Processing
functions transform user input data to a form that is easier or more efficient for
a network.

For instance,
interval [-1, 1]. This can speed up learning for many networks.

removeconstantrows removes the values for input elements that always have

2-12

the same value because these input elements are not providing any useful
information to the network. The third common processing function is

fixunknowns, which recodes unknown data (represented in the user’s data

with

NaN values) into a numerical form for the network. fixunknowns preserves

information about which values are known and which are unknown.

mapminmax transforms input data so that all values fall into the

Network Architectures

Similarly, network outputs can also have associated processing functions.
Output processing functions are used to transform user-provided target vectors
for network use. Then, network outputs are reverse-processed using the same
functions to produce output data with the same characteristics as the original
user-provided targets.

Both

mapminmax and removeconstantrows are often associated with network

outputs. However,
(represented by

fixunknowns is not. Unknown values in targets

NaN values) do not need to altered for network use.

Processing functions are described in more detail in “Preprocessing and
Postprocessing” in Chapter 5.

2-13

2 Neuron Model and Network Architectures

p

1

an

Inputs

b

p

2

w

1,2

w

1,1

1

a = purelin (Wp + b)

Linear Neuron

p

1

1
2

= ,

p

2

2
1

= ,

p

3

2
3

= ,

p

4

3
1

=

Data Structures

This section discusses how the format of input data structures affects the
simulation of networks. It starts with static networks, and then continues with
dynamic networks.

There are two basic types of input vectors: those that occur concurrently (at the
same time, or in no particular time sequence), and those that occur sequentially
in time. For concurrent vectors, the order is not important, and if there were a
number of networks running in parallel, you could present one input vector to
each of the networks. For sequential vectors, the order in which the vectors
appear is important.

Simulation with Concurrent Inputs in a Static Network

The simplest situation for simulating a network occurs when the network to be
simulated is static (has no feedback or delays). In this case, you need not be
concerned about whether or not the input vectors occur in a particular time
sequence, so you can treat the inputs as concurrent. In addition, the problem is
made even simpler by assuming that the network has only one input vector.
Use the following network as an example.

2-14

Suppose that the network simulation data set consists of Q = 4 concurrent
vectors:

Data Structures

W

12

=

b

0

=

Concurrent vectors are presented to the network as a single matrix:

P = [1 2 2 3; 2 1 3 1];

Suppose that this network typically outputs -100, 50 and 100. This is arbitrary
for this example. If you were solving a real problem, you would have actual
values.

T = [-100 50 50 100];

To set up this feedforward network, use the following command:

net = newlin(P,T);

For simplicity assign the weight matrix and bias to be

and

The commands for these assignments are

net.IW{1,1} = [1 2];
net.b{1} = 0;

You can now simulate the network:

A = sim(net,P)
A =
5 4 8 5

A single matrix of concurrent vectors is presented to the network, and the
network produces a single matrix of concurrent vectors as output. The result
would be the same if there were four networks operating in parallel and each
network received one of the input vectors and produced one of the outputs. The
ordering of the input vectors is not important, because they do not interact with
each other.

Simulation with Sequential Inputs in a Dynamic Network

When a network contains delays, the input to the network would normally be
a sequence of input vectors that occur in a certain time order. To illustrate this
case, the next figure shows a simple network that contains one delay.

2-15

2 Neuron Model and Network Architectures

a(t)n(t)

Inputs

w

1,1



D

w

1,2

Linear Neuron

p(t)

a(t) = w

p(t) + w

p(t - 1)

p11= , p22= , p33= , p44=

W

12

=

Suppose that the input sequence is

Sequential inputs are presented to the network as elements of a cell array:

P = {1 2 3 4};

2-16

Suppose you know that the typical output values include 10, 3 and -7. These
values are arbitrary for this example; if you were solving a real problem, you
would have real output values.

T = {10, 3, 7};

The following commands create this network:

net = newlin(P,T,[0 1]);
net.biasConnect = 0;

Assign the weight matrix to be

The command is

net.IW{1,1} = [1 2];

You can now simulate the network:

A = sim(net,P)
A =

Data Structures

p

1

1

= ,

p

2

2

= ,

p

3

3

= ,

p

4

4

=

[1] [4] [7] [10]

You input a cell array containing a sequence of inputs, and the network
produces a cell array containing a sequence of outputs. The order of the inputs
is important when they are presented as a sequence. In this case, the current
output is obtained by multiplying the current input by 1 and the preceding
input by 2 and summing the result. If you were to change the order of the
inputs, the numbers obtained in the output would change.

Simulation with Concurrent Inputs in a Dynamic Network

If you were to apply the same inputs as a set of concurrent inputs instead of a
sequence of inputs, you would obtain a completely different response.
(However, it is not clear why you would want to do this with a dynamic
network.) It would be as if each input were applied concurrently to a separate
parallel network. For the previous example, “Simulation with Sequential
Inputs in a Dynamic Network” on page 2-15, if you use a concurrent set of
inputs you have

which can be created with the following code:

P = [1 2 3 4];

When you simulate with concurrent inputs, you obtain

A = sim(net,P)
A =
1 2 3 4

The result is the same as if you had concurrently applied each one of the inputs
to a separate network and computed one output. Note that because you did not
assign any initial conditions to the network delays, they were assumed to be 0.
For this case the output is simply 1 times the input, because the weight that
multiplies the current input is 1.

In certain special cases, you might want to simulate the network response to
several different sequences at the same time. In this case, you would want to
present the network with a concurrent set of sequences. For example, suppose
you wanted to present the following two sequences to the network:

2-17

2 Neuron Model and Network Architectures

p

1

1()

1

,= p

1

2()

2

,= p

1

3()

3

,= p

1

4()4=

p

2

1()

4

,= p

2

2()

3

,= p

2

3()

2

,= p

2

4()1=

p11()p21()…pQ1(),,,[]p12()p22()…pQ2(),,,[]

·

… p1TS()p2TS()…pQTS(),,,[],,,{ }

First Sequence

Qth Sequence

The input P should be a cell array, where each element of the array contains
the two elements of the two sequences that occur at the same time:

P = {[1 4] [2 3] [3 2] [4 1]};

You can now simulate the network:

A = sim(net,P);

The resulting network output would be

A = {[1 4] [4 11] [7 8] [10 5]}

As you can see, the first column of each matrix makes up the output sequence
produced by the first input sequence, which was the one used in an earlier
example. The second column of each matrix makes up the output sequence
produced by the second input sequence. There is no interaction between the
two concurrent sequences. It is as if they were each applied to separate
networks running in parallel.

The following diagram shows the general format for the input

P to the sim

function when there are Q concurrent sequences of TS time steps. It covers all
cases where there is a single input vector. Each element of the cell array is a
matrix of concurrent vectors that correspond to the same point in time for each
sequence. If there are multiple input vectors, there will be multiple rows of
matrices in the cell array.

In this section, you apply sequential and concurrent inputs to dynamic
networks. In “Simulation with Concurrent Inputs in a Static Network” on
page 2-14, you applied concurrent inputs to static networks. It is also possible

2-18

Data Structures

to apply sequential inputs to static networks. It does not change the simulated
response of the network, but it can affect the way in which the network is
trained. This will become clear in “Training Styles” on page 2-20.

2-19

2 Neuron Model and Network Architectures

t 2p1p2+=

p

1

1
2

= ,

p

2

2
1

= ,

p

3

2
3

= ,

p

4

3
1

=

t

1

4

= ,

t

2

5

= ,

t

3

7

= ,

t

4

7

=

Training Styles

This section describes two different styles of training. In incremental training
the weights and biases of the network are updated each time an input is
presented to the network. In batch training the weights and biases are only
updated after all the inputs are presented.

Incremental Training (of Adaptive and Other Networks)

Incremental training can be applied to both static and dynamic networks,
although it is more commonly used with dynamic networks, such as adaptive
filters. This section demonstrates how incremental training is performed on
both static and dynamic networks.

Incremental Training with Static Networks

Consider again the static network used for the first example. You want to train
it incrementally, so that the weights and biases are updated after each input is
presented. In this case you use the function
are presented as sequences.

adapt, and the inputs and targets

2-20

Suppose you want to train the network to create the linear function:

Then for the previous inputs,

the targets would be

For incremental training, you present the inputs and targets as sequences:

P = {[1;2] [2;1] [2;3] [3;1]};
T = {4 5 7 7};

First, set up the network with zero initial weights and biases. Also, set the
initial learning rate to zero to show the effect of incremental training.

Training Styles

net = newlin(P,T,0,0);
net.IW{1,1} = [0 0];
net.b{1} = 0;

Recall from “Simulation with Concurrent Inputs in a Static Network” on
page 2-14 that, for a static network, the simulation of the network produces the
same outputs whether the inputs are presented as a matrix of concurrent
vectors or as a cell array of sequential vectors. However, this is not true when
training the network. When you use the

adapt function, if the inputs are

presented as a cell array of sequential vectors, then the weights are updated as
each input is presented (incremental mode). As shown in the next section, if the
inputs are presented as a matrix of concurrent vectors, then the weights are
updated only after all inputs are presented (batch mode).

You are now ready to train the network incrementally.

[net,a,e,pf] = adapt(net,P,T);

The network outputs remain zero, because the learning rate is zero, and the
weights are not updated. The errors are equal to the targets:

a = [0] [0] [0] [0]
e = [4] [5] [7] [7]

If you now set the learning rate to 0.1 you can see how the network is adjusted
as each input is presented:

net.inputWeights{1,1}.learnParam.lr=0.1;
net.biases{1,1}.learnParam.lr=0.1;
[net,a,e,pf] = adapt(net,P,T);
a = [0] [2] [6] [5.8]
e = [4] [3] [1] [1.2]

The first output is the same as it was with zero learning rate, because no
update is made until the first input is presented. The second output is different,
because the weights have been updated. The weights continue to be modified
as each error is computed. If the network is capable and the learning rate is set
correctly, the error is eventually driven to zero.

Incremental Training with Dynamic Networks

You can also train dynamic networks incrementally. In fact, this would be the
most common situation.

2-21

2 Neuron Model and Network Architectures

Here are the initial input Pi and the inputs P and targets T as elements of cell
arrays.

Pi = {1};
P = {2 3 4};
T = {3 5 7};

Create a linear network with one delay at the input, as used in a previous
example. Initialize the weights to zero and set the learning rate to 0.1.

net = newlin(P,T,[0 1],0.1);
net.IW{1,1} = [0 0];
net.biasConnect = 0;

You want to train the network to create the current output by summing the
current and the previous inputs. This is the same input sequence you used in
the previous example (using
term in the sequence as the initial condition for the delay. You can now
sequentially train the network using

[net,a,e,pf] = adapt(net,P,T,Pi);
a = [0] [2.4] [7.98]
e = [3] [2.6] [-0.98]

sim) with the exception that you assign the first

adapt.

2-22

The first output is zero, because the weights have not yet been updated. The
weights change at each subsequent time step.

Batch Training

Batch training, in which weights and biases are only updated after all the
inputs and targets are presented, can be applied to both static and dynamic
networks. Both types of networks are discussed in this section.

Batch Training with Static Networks

Batch training can be done using either adapt or train, although train is
generally the best option, because it typically has access to more efficient
training algorithms. Incremental training can only be done with
can only perform batch training.

For batch training of a static network with

adapt, the input vectors must be

placed in one matrix of concurrent vectors.

P = [1 2 2 3; 2 1 3 1];

adapt; train

Training Styles

T = [4 5 7 7];

Begin with the static network used in previous examples. The learning rate is
set to 0.1.

net = newlin(P,T,0,0.1);
net.IW{1,1} = [0 0];
net.b{1} = 0;

When you call adapt, it invokes trains (the default adaptation function for the
linear network) and
biases).

trains uses Widrow-Hoff learning.

[net,a,e,pf] = adapt(net,P,T);
a = 0 0 0 0
e = 4 5 7 7

learnwh (the default learning function for the weights and

Note that the outputs of the network are all zero, because the weights are not
updated until all the training set has been presented. If you display the
weights, you find

»net.IW{1,1}

ans = 4.9000 4.1000

»net.b{1}

ans =

2.3000

This is different from the result after one pass of adapt with incremental
updating.

Now perform the same batch training using
rule can be used in incremental or batch mode, it can be invoked by

train. (There are several algorithms that can only be used in batch mode (e.g.,

Levenberg-Marquardt), so these algorithms can only be invoked by

train. Because the Widrow-Hoff

adapt or

train.)

For this case, the input vectors can be in a matrix of concurrent vectors or in a
cell array of sequential vectors. Because the network is static and because

train always operates in batch mode, train converts any cell array of

sequential vectors to a matrix of concurrent vectors. Concurrent mode
operation is used whenever possible because it has a more efficient
implementation in MATLAB

P = [1 2 2 3; 2 1 3 1];
T = [4 5 7 7];

®

code:

2-23

2 Neuron Model and Network Architectures

The network is set up in the same way.

net = newlin(P,T,0,0.1);
net.IW{1,1} = [0 0];
net.b{1} = 0;

Now you are ready to train the network. Train it for only one epoch, because
you used only one pass of
network is
is

learnwh, so you should get the same results obtained using adapt in the

previous example, where the default adaptation function was

net.inputWeights{1,1}.learnParam.lr = 0.1;
net.biases{1}.learnParam.lr = 0.1;
net.trainParam.epochs = 1;
net = train(net,P,T);

If you display the weights after one epoch of training, you find

»net.IW{1,1}

»net.b{1}

2.3000

trainb, and the default learning function for the weights and biases

ans = 4.9000 4.1000

ans =

adapt. The default training function for the linear

trains.

2-24

This is the same result as the batch mode training in adapt. With static
networks, the

adapt function can implement incremental or batch training,

depending on the format of the input data. If the data is presented as a matrix
of concurrent vectors, batch training occurs. If the data is presented as a
sequence, incremental training occurs. This is not true for

train, which always

performs batch training, regardless of the format of the input.

Batch Training with Dynamic Networks

Training static networks is relatively straightforward. If you use train the
network is trained in batch mode and the inputs are converted to concurrent
vectors (columns of a matrix), even if they are originally passed as a sequence
(elements of a cell array). If you use
the method of training. If the inputs are passed as a sequence, then the
network is trained in incremental mode. If the inputs are passed as concurrent
vectors, then batch mode training is used.

With dynamic networks, batch mode training is typically done with
especially if only one training sequence exists. To illustrate this, consider again

adapt, the format of the input determines

train only,