# E. Castillo, A. Cobo, J.M. Gutiérrez, and E. Pruneda.

Kluwer International Publishers

Some Mathematica programs (notebooks for earlier Mathematica versions: v2.2 and v3.0) implementing the various algorithms and methodologies presented in this book. For a full understanding of these programs some knowledge of Mathematica is needed.

Some Java Applets implementing the various algorithms and methodologies presented in this book are also available.

PREFACE ix
I NEURAL NETWORKS 1
 1 Introduction to Neural Networks
5
 1.1 Introduction
5
 1.2 Inspiration from Neuroscience
6
 1.3 Components of Neural Networks
7
 1.4 Learning
13
 1.5 Validation
14
 1.6 The Hopfield Neural Network
16
 1.7 Feed Forward Networks: Perceptrons
21
 1.8 Multi-layer Perceptrons
29
 1.9 Feed Forward Neural Network Examples
33
 1.10 Competitive Neural Networks
41
 Exercises
43
II FUNCTIONAL NETWORKS 47
 2 Introduction to Functional Networks
51
 2.1 Introduction
51
 2.2 Motivating Functional Networks
52
 2.3 Elements of a Functional Network
56
 2.4 Differences Between Neural and Functional Networks
59
 2.5 Working With Functional Networks
61
 2.6 An Introductory Example
62
 Exercises
68
 3 Functional Equations
71
 3.1 Introduction
71
 3.2 A First Definition and Some Examples of Functional Equations
72
 3.3 Some Motivating Examples of Functional Equations
73
 3.4 Some General Methods for Solving Functional Equations
84
 3.5 Some Functional Equations in Functions of a Single Variable
89
 3.6 Some Functional Equations in Functions of Several Variables
92
 Exercises
95
 4 Some Functional Network Models
97
 4.1 Introduction
97
 4.2 The Uniqueness Model
97
 4.3 The Generalized Associativity Model
104
 4.4 The Separable Model
107
 4.5 The Generalized Bisymmetry Model
111
 4.6 Serial Functional Model
113
 4.7 Independent Multiple Output Models
116
 4.8 Dependent Multiple Output Network I
117
 4.9 Dependent Multiple Output Network II
120
 4.10 One-Layer Functional Networks
124
 Exercises
130
 5 Model Selection
133
 5.1 Introduction
133
 5.2 The Minimum Description Length Principle
134
 5.3 Encoding Integer and Real Numbers
135
 5.4 Selection of Models
137
 5.5 Application to Functional Networks
138
 Exercises
145
III APPLICATIONS 147
 6 Applications to Time Series
151
 6.1 Introduction
151
 6.2 Univariate Box-Jenkins Time Series Models
152
 6.3 Functional Networks and Univariate Models
157
 6.4 Applications to Box-Jenkins Models
158
 6.5 Applications to Economic Problems
160
 6.6 Applications to Chaotic Series
166
 6.7 Applications to Noise Reduction and Information Masking
179
 6.8 Multivariate Box-Jenkins Time Series Models
184
 Exercises
188
 7 Applications to Differential Equations
195
 7.1 Introduction
195
 7.2 The Equivalence Problem
198
 7.3 Approximations Using Functional Networks
206
 7.4 Example of Application: The Beam Problem
212
 Exercises
219
221
 8.1 Introduction
221
 8.2 Surfaces in Implicit Form
222
 8.3 Surfaces in Explicit Form
228
 8.4 Surfaces in Parametric Form
233
 8.5 Generalizations
236
 Exercises
237
 9 Applications to Regression
239
 9.1 Introduction
239
 9.2 Linear Regression Model
239
 9.3 Non-Linear Regression
251
 9.4 Functional Networks and Regression Models
252
 Exercises
254
IV COMPUTER PROGRAMS 259
 10 Mathematica Programs
263
 10.1 The Associative Model Program
263
 10.2 The Uniqueness Model Program
266
 10.3 The Separable Model Program
271
 10.4 The Difference Equation Model Program
274
 10.5 Equivalence of Difference and Differential Equations
277
 10.6 The Iterator Model Program
278
 11 A Java Applet
283
 11.1 How to Use the Program
283
Notation 291
References 299
Index 305