ContentsXVIII
11.4.5 Clusterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
11.4.6 Traveling Salesman Problem
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
11.4.7 Dynamic System Modelling
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
11.4.8 Conclusion
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
11.5 Typical Input Signal of Multilayer Neural Networks
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
Literature
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
12 Analysis of Closed-Loop Multilayer Neural Networks
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
12.1 Problem Statement for the Synthesis of the Multilayer Neural Networks
Adjusted in the Closed Cycle
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
12.2 Investigation of the Neuron Under the Multi-Modal Distribution
of the Input Signal
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
12.2.1 One-Dimensional Case – Search Adjustment Algorithm
. . . . . . . . . . . . . . . . . 224
12.2.2 Multidimensional Case – Analytical Adjustment Algorithm
. . . . . . . . . . . . . 226
12.3 Investigation of Dynamics for the Neural Networks of Particular Form
for the Non-Stationary Pattern Recognition
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
12.4 Dynamics of the Three-Layer Neural Network in the Learning Mode
. . . . . . . . . . . 235
12.5 Investigation of the Particular Neural Network
with Backward Connections
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
12.6 Dynamics of One-Layer Neural Networks in the Learning Mode
. . . . . . . . . . . . . . . . 242
12.6.1 Neural Network with the Search
of the Distribution Mode Centers f(x)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
12.6.2 Neural Network with N
*
Output Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
12.6.3 Neuron with K
p
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
12.7 Two-Layer Neural Network in the Self-Learning Mode
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
12.8 About Some Engineering Methods for the Selection of Matrix Parameters
in the Multilayer Neural Network Closed Cycle Adjustment Algorithms
. . . . . . . . 257
12.9 Design of the Multilayer Neural Network for the Matrix Inversion Problem
. . . 258
12.10 Design of the Multilayer Neural Network for the Number Transformation
from the Binary System into the Decimal One
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
12.11 Investigation of the Multilayer Neural Network
under the Arbitrary Teacher Qualification
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
12.12 Analytical Methods of Investigations of the Neural Network
Closed Cycle Adjustment
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
Literature
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
13 Synthesis of Multilayer Neural Networks with Flexible Structure
. . . . . . . . . . . . . . 273
13.1 Sequential Learning Algorithm for the First Neuron Layer
of the Multilayer Neural Network
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
13.2 Learning Algorithm for the First Neuron Layer of the Multilayer
Neural Network Using the Method of Random Search of Local
and Global Function Extrema
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
13.3 Analysis of Algorithm Convergence under the Hyperplane
Number Increase
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
13.4 Learning Algorithms for the Second Layer Neurons
of the Two-Layer Neural Network
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283