Self-Organizing Maps

By: Teuvo Kohonen

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Paperback | 1 January 2001 | Edition Number 3

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530 Pages

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Revised

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24.13 x 15.88 x 2.54

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Since the second edition of this book came out in early 1997, the number of scientific papers published on the Self-Organizing Map (SOM) has increased from about 1500 to some 4000. Also, two special workshops dedicated to the SOM have been organized, not to mention numerous SOM sessions in neural- network conferences. In view of this growing interest it was felt desirable to make extensive revisions to this book. They are of the following nature. Statistical pattern analysis has now been approached more carefully than earlier. A more detailed discussion of the eigenvectors and eigenvalues of symmetric matrices, which are the type usually encountered in statistics, has been included in Sect. 1.1.3: also, new probabilistic concepts, such as factor analysis, have been discussed in Sect. 1.3.1. A survey of projection methods (Sect. 1.3.2) has been added, in order to relate the SOM to classical paradigms. Vector Quantization is now discussed in one main section, and derivation of the pointdensity of the codebook vectors using the calculus of variations has been added, in order to familiarize the reader with this otherwise com- plicated statistical analysis. It was also felt that the discussion of the neural-modeling philosophy should include a broader perspective of the main issues. A historical review in Sect. 2.2, and the general philosophy in Sects. 2.3, 2.5 and 2.14 are now expected to especially help newcomers to orient themselves better amongst the profusion of contemporary neural models.

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Non-Fiction Computing & I.T.Computer Science Artificial Intelligence Neural Networks & Fuzzy Systems Engineering & Technology Electronics & Communications Engineering Communications Engineering & Telecommunications Science Physics Applied Physics Biophysics

Statistical Physics Energy Technology & Engineering Electrical Engineering

Mathematical Preliminaries	p. 1
Mathematical Concepts and Notations	p. 2
Vector Space Concepts	p. 2
Matrix Notations	p. 8
Eigenvectors and Eigenvalues of Matrices	p. 11
Further Properties of Matrices	p. 13
On Matrix Differential Calculus	p. 15
Distance Measures for Patterns	p. 17
Measures of Similarity and Distance in Vector Spaces	p. 17
Measures of Similarity and Distance Between Symbol Strings	p. 21
Averages Over Nonvectorial Variables	p. 28
Statistical Pattern Analysis	p. 29
Basic Probabilistic Concepts	p. 29
Projection Methods	p. 34
Supervised Classification	p. 39
Unsupervised Classification	p. 44
The Subspace Methods of Classification	p. 46
The Basic Subspace Method	p. 46
Adaptation of a Model Subspace to Input Subspace	p. 49
The Learning Subspace Method (LSM)	p. 53
Vector Quantization	p. 59
Definitions	p. 59
Derivation of the VQ Algorithm	p. 60
Point Density in VQ	p. 62
Dynamically Expanding Context	p. 64
Setting Up the Problem	p. 65
Automatic Determination of Context-Independent Productions	p. 66
Conflict Bit	p. 67
Construction of Memory for the Context-Dependent Productions	p. 68
The Algorithm for the Correction of New Strings	p. 68
Estimation Procedure for Unsuccessful Searches	p. 69
Practical Experiments	p. 69
Neural Modeling	p. 71
Models, Paradigms, and Methods	p. 71
A History of Some Main Ideas in Neural Modeling	p. 72
Issues on Artificial Intelligence	p. 75
On the Complexity of Biological Nervous Systems	p. 76
What the Brain Circuits Are Not	p. 78
Relation Between Biological and Artificial Neural Networks	p. 79
What Functions of the Brain Are Usually Modeled?	p. 81
When Do We Have to Use Neural Computing?	p. 81
Transformation, Relaxation, and Decoder	p. 82
Categories of ANNs	p. 85
A Simple Nonlinear Dynamic Model of the Neuron	p. 87
Three Phases of Development of Neural Models	p. 89
Learning Laws	p. 91
Hebb's Law	p. 91
The Riccati-Type Learning Law	p. 92
The PCA-Type Learning Law	p. 95
Some Really Hard Problems	p. 96
Brain Maps	p. 99
The Basic SOM	p. 105
A Qualitative Introduction to the SOM	p. 106
The Original Incremental SOM Algorithm	p. 109
The "Dot-Product SOM"	p. 115
Other Preliminary Demonstrations of Topology-Preserving Mappings	p. 116
Ordering of Reference Vectors in the Input Space	p. 116
Demonstrations of Ordering of Responses in the Output Space	p. 120
Basic Mathematical Approaches to Self-Organization	p. 127
One-Dimensional Case	p. 128
Constructive Proof of Ordering of Another One-Dimensional SOM	p. 132
The Batch Map	p. 138
Initialization of the SOM Algorithms	p. 142
On the "Optimal" Learning-Rate Factor	p. 143
Effect of the Form of the Neighborhood Function	p. 145
Does the SOM Algorithm Ensue from a Distortion Measure?	p. 146
An Attempt to Optimize the SOM	p. 148
Point Density of the Model Vectors	p. 152
Earlier Studies	p. 152
Numerical Check of Point Densities in a Finite One-Dimensional SOM	p. 153
Practical Advice for the Construction of Good Maps	p. 159
Examples of Data Analyses Implemented by the SOM	p. 161
Attribute Maps with Full Data Matrix	p. 161
Case Example of Attribute Maps Based on Incomplete Data Matrices (Missing Data): "Poverty Map"	p. 165
Using Gray Levels to Indicate Clusters in the SOM	p. 165
Interpretation of the SOM Mapping	p. 166
"Local Principal Components"	p. 166
Contribution of a Variable to Cluster Structures	p. 169
Speedup of SOM Computation	p. 170
Shortcut Winner Search	p. 170
Increasing the Number of Units in the SOM	p. 172
Smoothing	p. 175
Combination of Smoothing, Lattice Growing, and SOM Algorithm	p. 176
Physiological Interpretation of SOM	p. 177
Conditions for Abstract Feature Maps in the Brain	p. 177
Two Different Lateral Control Mechanisms	p. 178
The WTA Function, Based on Lateral Activity Control	p. 179
Lateral Control of Plasticity	p. 184
Learning Equation	p. 185
System Models of SOM and Their Simulations	p. 185
Recapitulation of the Features of the Physiological SOM Model	p. 188
Similarities Between the Brain Maps and Simulated Feature Maps	p. 188
Magnification	p. 189
Imperfect Maps	p. 189
Overlapping Maps	p. 189
Variants of SOM	p. 191
Overview of Ideas to Modify the Basic SOM	p. 191
Adaptive Tensorial Weights	p. 194
Tree-Structured SOM in Searching	p. 197
Different Definitions of the Neighborhood	p. 198
Neighborhoods in the Signal Space	p. 200
Dynamical Elements Added to the SOM	p. 204
The SOM for Symbol Strings	p. 205
Initialization of the SOM for Strings	p. 205
The Batch Map for Strings	p. 206
Tie-Break Rules	p. 206
A Simple Example: The SOM of Phonemic Transcriptions	p. 207
Operator Maps	p. 207
Evolutionary-Learning SOM	p. 211
Evolutionary-Learning Filters	p. 211
Self-Organization According to a Fitness Function	p. 212
Supervised SOM	p. 215
The Adaptive-Subspace SOM (ASSOM)	p. 216
The Problem of Invariant Features	p. 216
Relation Between Invariant Features and Linear Subspaces	p. 218
The ASSOM Algorithm	p. 222
Derivation of the ASSOM Algorithm by Stochastic Approximation	p. 226
ASSOM Experiments	p. 228
Feedback-Controlled Adaptive-Subspace SOM (FASSOM)	p. 242
Learning Vector Quantization	p. 245
Optimal Decision	p. 245
The LVQ1	p. 246
The Optimized-Learning-Rate LVQ1 (OLVQ1)	p. 250
The Batch-LVQ1	p. 251
The Batch-LVQ1 for Symbol Strings	p. 252
The LVQ2 (LVQ 2.1)	p. 252
The LVQ3	p. 253
Differences Between LVQ1, LVQ2 and LVQ3	p. 254
General Considerations	p. 254
The Hypermap-Type LVQ	p. 256
The "LVQ-SOM"	p. 261
Applications	p. 263
Preprocessing of Optic Patterns	p. 264
Blurring	p. 265
Expansion in Terms of Global Features	p. 266
Spectral Analysis	p. 266
Expansion in Terms of Local Features (Wavelets)	p. 267
Recapitulation of Features of Optic Patterns	p. 267
Acoustic Preprocessing	p. 268
Process and Machine Monitoring	p. 269
Selection of Input Variables and Their Scaling	p. 269
Analysis of Large Systems	p. 270
Diagnosis of Speech Voicing	p. 274
Transcription of Continuous Speech	p. 274
Texture Analysis	p. 280
Contextual Maps	p. 281
Artifically Generated Clauses	p. 283
Natural Text	p. 285
Organization of Large Document Files	p. 286
Statistical Models of Documents	p. 286
Construction of Very Large WEBSOM Maps by the Projection Method	p. 292
The WEBSOM of All Electronic Patent Abstracts	p. 296
Robot-Arm Control	p. 299
Simultaneous Learning of Input and Output Parameters	p. 299
Another Simple Robot-Arm Control	p. 303
Telecommunications	p. 304
Adaptive Detector for Quantized Signals	p. 304
Channel Equalization in the Adaptive QAM	p. 305
Error-Tolerant Transmission of Images by a Pair of SOMs	p. 306
The SOM as an Estimator	p. 308
Symmetric (Autoassociative) Mapping	p. 308
Asymmetric (Heteroassociative) Mapping	p. 309
Software Tools for SOM	p. 311
Necessary Requirements	p. 311
Desirable Auxiliary Features	p. 313
SOM Program Packages	p. 315
SOM_PAK	p. 315
SOM Toolbox	p. 317
Nenet (Neural Networks Tool)	p. 318
Viscovery SOMine	p. 318
Examples of the Use of SOMLPAK	p. 319
File Formats	p. 319
Description of the Programs in SOM_PAK	p. 322
A Typical Training Sequence	p. 326
Neural-Networks Software with the SOM Option	p. 327
Hardware for SOM	p. 329
An Analog Classifier Circuit	p. 329
Fast Digital Classifier Circuits	p. 332
SIMD Implementation of SOM	p. 337
Transputer Implementation of SOM	p. 339
Systolic-Array Implementation of SOM	p. 341
The COKOS Chip	p. 342
The TInMANN Chip	p. 342
NBISOM_25 Chip	p. 344
An Overview of SOM Literature	p. 347
Books and Review Articles	p. 347
Early Works on Competitive Learning	p. 348
Status of the Mathematical Analyses	p. 349
Zero-Order Topology (Classical VQ) Results	p. 349
Alternative Topological Mappings	p. 350
Alternative Architectures	p. 350
Functional Variants	p. 351
Theory of the Basic SOM	p. 352
The Learning Vector Quantization	p. 358
Diverse Applications of SOM	p. 358
Machine Vision and Image Analysis	p. 358
Optical Character and Script Reading	p. 360
Speech Analysis and Recognition	p. 360
Acoustic and Musical Studies	p. 361
Signal Processing and Radar Measurements	p. 362
Telecommunications	p. 362
Industrial and Other Real-World Measurements	p. 362
Process Control	p. 363
Robotics	p. 364
Electronic-Circuit Design	p. 364
Physics	p. 364
Chemistry	p. 365
Biomedical Applications Without Image Processing	p. 365
Neurophysiological Research	p. 366
Data Processing and Analysis	p. 366
Linguistic and AI Problems	p. 367
Mathematical and Other Theoretical Problems	p. 368
Applications of LVQ	p. 369
Survey of SOM and LVQ Implementations	p. 370
Glossary of "Neural" Terms	p. 373
References	p. 403
Index	p. 487
Table of Contents provided by Publisher. All Rights Reserved.

Self-Organizing Maps

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