Pavlov A.V., Stepashko V.S, Kondrashova N.V.
EFFECTIVE METHODS OF MODELS SELF-ORGANIZATION
Contents
Foreword 3
Acknowledgments 3
Introduction 4
Chapter 1.
Analysis of tasks, approaches, methods and algorithms for construction of models based on
observational data
1.1 Typical tasks, approaches and methods of computer heuristic modeling 8
1.1.1 The Task of Structural-Parametric Identification 10
1.1.2 Approaches to solving the problem of forecasting 13
1.1.3 Group Method of Data Handling (GMDH) as a part of mathematical methods 15
1.2 GMDH as an inductive method of heuristic self-organization of models 17
1.2.1 General Formulation of the Problem of the Inductive Method of Self-Organization of
Models 18
1.2.2 Tasks of the inductive method of model self-organization 19
1.2.3 Basic Principles of GMDH 19
1.2.4 The task of finding a model that corresponds to the global extremum of the quality
functional 20
1.2.5 The problem of partitioning a set of observations 21
1.3 Basic algorithms of GMDH 22
1.3.1 Sorting GMDH algorithms 23
1.3.2 Iterative GMDH algorithms 24
1.4 Convergence of iterative GMDH algorithms 25
1.5 Classical iterative algorithms 26
1.6 Modern iterative algorithms based on GMDH 30
1.7 Discussion 32
Chapter 2.
The Problem of Sample Partition in Heuristic Modeling
2.1 Optimal Sample Partition as a Result of Experiment Design 34
2.2 Relationship between optimal partitioning and partitioning by ‘variance’ 35
2.3 Quasi-optimal partitioning in the case of passive sampling 36
2.3.1 Criteria for finding a quasi-optimal partition 37
2.4 Matching the sample splitting criterion and the external complement criterion in model
selection by GMDH 39
2.4.1 Sample partitioning for model selection based on regularity criterion 39
2.4.2 Model selection by unbiasedness criterion for a quasi-optimal sample partition 40
2.4.3 Study of two types of norms of discrepancy between information matrices 41
2.5 Research of different ways for generating sample splitting options 43
2.5.1 Results of numerical experiments comparing quasi-optimal with traditional partitions of
GMDH algorithms 45
2.5.2 Study of the possibility of increasing the performance of the algorithm for searching for a
quasi-optimal partition of a sample 47
2.5.3 General conclusions based on the results of the numerical study 49
2.6 Using sample partitioning criteria to evaluate the transformation of geometric figures 49
2.7 Study of the influence of sample partitions on the structure of approximating and
extrapolating GMDH models 57
2.7.1 Approximating models 57
2.7.2 Extrapolation models 59
2.8. Discussion 62
Chapter 3.
Relaxation Iterative Algorithms of GMDH
3.1 Multilayered simplified GMDH algorithm 63
3.1.1 Description of a multi-layered simplified algorithm 63
3.1.2 Computational complexity of MSA 67
3.1.3 A way to find the best multiplicative mononomial 68
3.2 Generalized relaxation iterative algorithm (GRIA) 70
3.2.1 Model construction process 71
3.2.2 Algorithm architecture 73
3.2.3 Method for finding a multiplicative mononomial 73
3.2.4 Methods for estimating parameters and calculating selection criteria 74
3.2.5 Computational complexity of GRIA 77
3.3 Proof of GRIA convergence 78
3.4 Comparison of GRIA and MSA 81
3.4.1 Theoretical comparative analysis of the speed of GRIA and MSA 81
3.4.2 Comparison of the performance of GRIA and MSA software implementations 82
3.5. Discussion 83
Chapter 4.
Investigation of the Properties of Iterative Algorithms by Computer Experiments
4.1 Investigation of the Convergence Property of Algorithms 85
4.1.1 Methods of conducting computer experiments 87
4.1.2 Study of the GRIA convergence rate 88
4.2 Study of the Identifying Property of Algorithms 105
4.2.1 Study of the Impact of Data Characteristics 105
4.2.2 Studyof the Influence of Algorithm Parameters 107
4.3 Study of the Noise-Immunity Property of Modeling 108
4.4 Study of the Parameter of Freedom of Choice 112
4.5 Discussion 114
Chapter 5.
Solving practical modeling problems
5.1 Space weather forecasting 117
5.1.1 Description of initial data 117
5.1.2 Data pre-processing 118
5.1.3 Statement of the modeling problem 119
5.1.4 Problem Solution 120
5.1.5 Modeling results 121
5.1.6 Analysis of modeling results 122
5.1.7 Conclusions on the task of space weather forecasting 126
5.2 Evaluation of the effectiveness of medications 126
5.2.1 Problem Statement 127
5.2.2 Problem solution 128
5.2.3 Modeling results 131
5.2.4 Analysis of modeling results 132
5.2.5 Conclusions on the task of assessing drugs effecacy 133
5.3 Differential diagnosis of haemostasis pathologies 133
5.3.1 Problem statement 133
5.3.2 Problem solving concept 134
5.3.3 Solving the problem 135
5.3.4 Modeling results 136
5.3.5 Modeling analysis 138
5.3.6 Conclusions on the task of differential diagnosis 139
5.4 Discussion 139
Conclusion 140
Appendix A 142
Additional computer experiments 142
References 149