SelfOrganizing MapsSince the second edition of this book came out in early 1997, the number of scientific papers published on the SelfOrganizing 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 point density 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 neuralmodeling 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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Contents
1 Mathematical Preliminaries  1 
11 Mathematical Concepts and Notations  2 
112 Matrix Notations  8 
113 Eigenvectors and Eigenvalues of Matrices  11 
114 Further Properties of Matrices  13 
115 On Matrix Differential Calculus  15 
12 Distance Measures for Patterns  17 
122 Measures of Similarity and Distance Between Symbol Strings  21 
572 The Batch Map for Strings  206 
The SOM of Phonemic Transcriptions  207 
59 EvolutionaryLearning SOM  211 
592 SelfOrganization According to a Fitness Function  212 
510 Supervised SOM  215 
511 The AdaptiveSubspace SOM ASSOM  216 
5112 Relation Between Invariant Features and Linear Subspaces  218 
5113 The ASSOM Algorithm  222 
123 Averages Over Nonvectorial Variables  28 
13 Statistical Pattern Analysis  29 
132 Projection Methods  34 
133 Supervised Classification  39 
134 Unsupervised Classification  44 
14 The Subspace Methods of Classification  46 
142 Adaptation of a Model Subspace to Input Subspace  49 
143 The Learning Subspace Method LSM  53 
59  
152 Derivation of the VQ Algorithm  60 
153 Point Density in VQ  62 
16 Dynamically Expanding Context  64 
161 Setting Up the Problem  65 
162 Automatic Determination of ContextIndependent Productions  66 
163 Conflict Bit  67 
164 Construction of Memory for the ContextDependent Productions  68 
166 Estimation Procedure for Unsuccessful Searches  69 
2 Neural Modeling  71 
22 A History of Some Main Ideas in Neural Modeling  72 
23 Issues on Artificial Intelligence  75 
24 On the Complexity of Biological Nervous Systems  76 
25 What the Brain Circuits Are Not  78 
26 Relation Between Biological and Artificial Neural Networks  79 
27 What Functions of the Brain Are Usually Modeled?  81 
29 Transformation Relaxation and Decoder  82 
210 Categories of ANNs  85 
211 A Simple Nonlinear Dynamic Model of the Neuron  87 
212 Three Phases of Development of Neural Models  89 
213 Learning Laws  91 
2132 The RiccatiType Learning Law  92 
2133 The PCAType Learning Law  95 
214 Some Really Hard Problems  96 
215 Brain Maps  99 
3 The Basic SOM  105 
31 A Qualitative Introduction to the SOM  106 
32 The Original Incremental SOM Algorithm  109 
33 The DotProduct SOM  115 
34 Other Preliminary Demonstrations of TopologyPreserving Mappings  116 
342 Demonstrations of Ordering of Responses in the Output Space  120 
35 Basic Mathematical Approaches to SelfOrganization  127 
351 OneDimensional Case  128 
352 Constructive Proof of Ordering of Another OneDimensional SOM  132 
36 The Batch Map  138 
37 Initialization of the SOM Algorithms  142 
38 On the Optimal LearningRate Factor  143 
39 Effect of the Form of the Neighborhood Function  145 
310 Does the SOM Algorithm Ensue from a Distortion Measure?  146 
311 An Attempt to Optimize the SOM  148 
312 Point Density of the Model Vectors  152 
3122 Numerical Check of Point Densities in a Finite OneDimensional SOM  153 
313 Practical Advice for the Construction of Good Maps  159 
314 Examples of Data Analyses Implemented by the SOM  161 
Poverty Map  165 
316 Interpretation of the SOM Mapping  166 
3162 Contribution of a Variable to Cluster Structures  169 
317 Speedup of SOM Computation  170 
3172 Increasing the Number of Units in the SOM  172 
3173 Smoothing  175 
3174 Combination of Smoothing Lattice Growing and SOM Algorithm  176 
4 Physiological Interpretation of SOM  177 
42 Two Different Lateral Control Mechanisms  178 
421 The WTA Function Based on Lateral Activity Control  179 
422 Lateral Control of Plasticity  184 
43 Learning Equation  185 
45 Recapitulation of the Features of the Physiological SOM Model  188 
461 Magnification  189 
5 Variants of SOM  191 
52 Adaptive Tensorial Weights  194 
53 TreeStructured SOM in Searching  197 
54 Different Definitions of the Neighborhood  198 
55 Neighborhoods in the Signal Space  200 
56 Dynamical Elements Added to the SOM  204 
57 The SOM for Symbol Strings  205 
5114 Derivation of the ASSOM Algorithm by Stochastic Approximation  226 
5115 ASSOM Experiments  228 
512 FeedbackControlled AdaptiveSubspace SOM FASSOM  242 
6 Learning Vector Quantization  245 
62 The LVQ1  246 
63 The OptimizedLearningRate LVQ1 OLVQ1  250 
64 The BatchLVQ1  251 
65 The BatchLVQ1 for Symbol Strings  252 
67 The LVQ3  253 
68 Differences Between LVQ1 LVQ2 and LVQ3  254 
610 The HypermapType LVQ  256 
611 The LVQSOM  261 
7 Applications  263 
71 Preprocessing of Optic Patterns  264 
711 Blurring  265 
712 Expansion in Terms of Global Features  266 
714 Expansion in Terms of Local Features Wavelets  267 
72 Acoustic Preprocessing  268 
73 Process and Machine Monitoring  269 
732 Analysis of Large Systems  270 
74 Diagnosis of Speech Voicing  274 
76 Texture Analysis  280 
77 Contextual Maps  281 
771 Artifically Generated Clauses  283 
772 Natural Text  285 
78 Organization of Large Document Files  286 
782 Construction of Very Large WEBSOM Maps by the Projection Method  292 
783 The WEBSOM of All Electronic Patent Abstracts  296 
79 RobotArm Control  299 
792 Another Simple RobotArm Control  303 
710 Telecommunications  304 
7102 Channel Equalization in the Adaptive QAM  305 
7103 ErrorTolerant Transmission of Images by a Pair of SOMs  306 
711 The SOM as an Estimator  308 
7112 Asymmetric Heteroassociative Mapping  309 
8 Software Tools for SOM  311 
82 Desirable Auxiliary Features  313 
83 SOM Program Packages  315 
832 SOM Toolbox  317 
833 Nenet Neural Networks Tool  318 
84 Examples of the Use of SOIV_PAK  319 
842 Description of the Programs in SOM_PAK  322 
843 A Typical Training Sequence  326 
85 NeuralNetworks Software with the SOM Option  327 
9 Hardware for SOM  329 
92 Fast Digital Classifier Circuits  332 
93 SIMD Implementation of SOM  337 
94 Transputer Implementation of SOM  339 
95 SystolicArray Implementation of SOM  341 
96 The COKOS Chip  342 
98 NBISOM_25 Chip  344 
10 An Overview of SOM Literature  347 
102 Early Works on Competitive Learning  348 
103 Status of the Mathematical Analyses  349 
1032 Alternative Topological Mappings  350 
1034 Functional Variants  351 
1035 Theory of the Basic SOM  352 
104 The Learning Vector Quantization  358 
1052 Optical Character and Script Reading  360 
1054 Acoustic and Musical Studies  361 
1055 Signal Processing and Radar Measurements  362 
1058 Process Control  363 
1059 Robotics  364 
10512 Chemistry  365 
10514 Neurophysiological Research  366 
10516 Linguistic and AI Problems  367 
10517 Mathematical and Other Theoretical Problems  368 
106 Applications of LVQ  369 
107 Survey of SOM and LVQ Implementations  370 
11 Glossary of Neural Terms  373 
References  403 
487  
Common terms and phrases
accuracy adaptive algorithm analysis applications approximation array Artificial Neural Networks ASSOM assumed asymptotic basic basis vectors Batch Map brain cells classification clustering codebook vectors components computing convergence corresponding cortex defined denoted density function described dimensionality discussed in Sect distance distribution dot product dynamic elements equation Euclidean feature filters input data input samples input vector labels lattice learning learningrate factor Levenshtein distance linear linear subspace map units mathematical matrix memory method model vectors neighborhood function neighboring neuron nodes nonlinear operation optimal orthogonal output parameters pattern recognition phoneme point density probability density function problem projection quantization error random reference vectors relating respectively scalar SelfOrganizing Map sequence signal space similar speech recognition statistical step stochastic stochastic approximation strings subset subspace supervised learning symbol synaptic tion topology transformation twodimensional values variables vector quantization visual Voronoi tessellation wavelets whereby winner zero