Bezruchko / Smirnov Extracting Knowledge From Time Series

An Introduction to Nonlinear Empirical Modeling
1. Auflage 2010
ISBN: 978-3-642-12601-7
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 410 Seiten

Reihe: Springer Series in Synergetics

Mathematical modelling is ubiquitous. Almost every book in exact science touches on mathematical models of a certain class of phenomena, on more or less speci?c approaches to construction and investigation of models, on their applications, etc. As many textbooks with similar titles, Part I of our book is devoted to general qu- tions of modelling. Part II re?ects our professional interests as physicists who spent much time to investigations in the ?eld of non-linear dynamics and mathematical modelling from discrete sequences of experimental measurements (time series). The latter direction of research is known for a long time as 'system identi?cation' in the framework of mathematical statistics and automatic control theory. It has its roots in the problem of approximating experimental data points on a plane with a smooth curve. Currently, researchers aim at the description of complex behaviour (irregular, chaotic, non-stationary and noise-corrupted signals which are typical of real-world objects and phenomena) with relatively simple non-linear differential or difference model equations rather than with cumbersome explicit functions of time. In the second half of the twentieth century, it has become clear that such equations of a s- ?ciently low order can exhibit non-trivial solutions that promise suf?ciently simple modelling of complex processes; according to the concepts of non-linear dynamics, chaotic regimes can be demonstrated already by a third-order non-linear ordinary differential equation, while complex behaviour in a linear model can be induced either by random in?uence (noise) or by a very high order of equations.

Autoren/Hrsg.

Bezruchko, Boris P.

Smirnov, Dmitry A.

Inhaltsverzeichnis

1;Springer Complexity;1
2;Preface;5
3;Introduction;8
4;Contents;13
5;Some Abbreviations and Notations;19
6;Part I Models and Forecast;20
7;Chapter
1 The Concept of Model. What is Remarkable in Mathematical Models ;21
7.1;1.1 What is Called ``Model'' and ``Modelling'';21
7.2;1.2 Science, Scientific Knowledge, Systematisation of Scientific Models;24
7.3;1.3 Delusion and Intuition: Rescue via Mathematics;28
7.4;1.4 How Many Models for a Single Object Can Exist?;32
7.5;1.5 How the Models are Born;34
7.6;1.6 Structural Scheme of Mathematical Modelling Procedure;35
7.7;1.7 Conclusions from Historical Practice of Modelling: Indicative Destiny of Mechanics Models;37
7.8;References;41
8;Chapter
2 Two Approaches to Modelling and Forecast ;42
8.1;2.1 Basic Concepts and Peculiarities of Dynamical Modelling;43
8.1.1;2.1.1 Definition of Dynamical System;43
8.1.2;2.1.2 Non-rigorous Example: Variables and Parameters;45
8.1.3;2.1.3 Phase Space. Conservative and Dissipative Systems. Attractors, Multistability, Basins of Attraction;48
8.1.4;2.1.4 Characteristics of Attractors;52
8.1.5;2.1.5 Parameter Space, Bifurcations, Combined Spaces, Bifurcation Diagrams;57
8.2;2.2 Foundations to Claim a Process ``Random'';59
8.2.1;2.2.1 Set-Theoretic Approach;59
8.2.2;2.2.2 Signs of Randomness Traditional for Physicists;69
8.2.3;2.2.3 Algorithmic Approach;70
8.2.4;2.2.4 Randomness as Unpredictability;71
8.3;2.3 Conception of Partial Determinancy;72
8.4;2.4 Lyapunov Exponents and Limits of Predictability;73
8.4.1;2.4.1 Practical Prediction Time Estimator;73
8.4.2;2.4.2 Predictability and Lyapunov Exponent: The Case of Infinitesimal Perturbations;74
8.5;2.5 Scale of Consideration Influences Classification of a Process (Complex Deterministic Dynamics Versus Randomness);78
8.6;2.6 ``Coin Flip'' Example;81
8.7;References;85
9;Chapter
3 Dynamical (Deterministic) Models of Evolution;87
9.1;3.1 Terminology;87
9.1.1;3.1.1 Operator, Map, Equation, Evolution Operator;87
9.1.2;3.1.2 Functions, Continuous and Discrete time;88
9.1.3;3.1.3 Discrete Map, Iterate;89
9.1.4;3.1.4 Flows and Cascades, Poincare Section and Poincare Map;89
9.1.5;3.1.5 Illustrative Example;89
9.2;3.2 Systematisation of Model Equations;91
9.3;3.3 Explicit Functional Dependencies;95
9.4;3.4 Linearity and Non-linearity;97
9.4.1;3.4.1 Linearity and Non-linearity of Functions and Equations;97
9.4.2;3.4.2 The Nature of Non-linearity;98
9.4.3;3.4.3 Illustration with Pendulums;99
9.5;3.5 Models in the form of Ordinary Differential Equations;101
9.5.1;3.5.1 Kinds of Solutions;101
9.5.2;3.5.2 Oscillators, a Popular Class of Model Equations;104
9.5.3;3.5.3 ``Standard form'' of Ordinary Differential Equations;108
9.6;3.6 Models in the Form of Discrete Maps;109
9.6.1;3.6.1 Introduction;109
9.6.2;3.6.2 Exemplary Non-linear Maps;110
9.6.3;3.6.3 Role of Discrete Models;115
9.7;3.7 Models of Spatially Extended Systems;121
9.7.1;3.7.1 Coupled Map Lattices;121
9.7.2;3.7.2 Cellular Automata;126
9.7.3;3.7.3 Networks with Complex Topology;128
9.7.4;3.7.4 Delay Differential Equations;129
9.7.5;3.7.5 Partial Differential Equations;130
9.8;3.8 Artificial Neural Networks;131
9.8.1;3.8.1 Standard Formal Neuron;132
9.8.2;3.8.2 Architecture and Classification of Neural Networks;134
9.8.3;3.8.3 Basic Properties and Problems;135
9.8.4;3.8.4 Learning;136
9.9;References;137
10;Chapter
4 Stochastic Models of Evolution ;142
10.1;4.1 Elements of the Theory of Random Processes;142
10.1.1;4.1.1 Concept of Random Process;142
10.1.2;4.1.2 Characteristics of Random Process;144
10.1.3;4.1.3 Stationarity and Ergodicity of Random Processes;145
10.1.4;4.1.4 Statistical Estimates of Random Process Characteristics;146
10.2;4.2 Basic Models of Random Processes;146
10.3;4.3 Evolutionary Equations for Probability Distribution Laws;149
10.4;4.4 Autoregression and Moving Average Processes;150
10.5;4.5 Stochastic Differential Equations and White Noise;153
10.5.1;4.5.1 The Concept of Stochastic Differential Equation;153
10.5.2;4.5.2 Numerical Integration of Stochastic DifferentialEquations;156
10.5.3;4.5.3 Constructive Role of Noise;158
10.6;References;161
11;Part II Modelling from Time Series;163
12;Chapter
5 Problem Posing in Modelling from Data Series ;164
12.1;5.1 Scheme of Model Construction Procedure;164
12.2;5.2 Systematisation in Respect of A Priori Information;166
12.3;5.3 Specific Features of Empirical Modelling Problems;167
12.3.1;5.3.1 Direct and Inverse Problems;167
12.3.2;5.3.2 Well-posed and Ill-posed Problems;168
12.3.3;5.3.3 Ill-conditioned Problems;170
12.4;References;170
13;Chapter
6 Data Series as a Source for Modelling ;172
13.1;6.1 Observable and Model Quantities;172
13.1.1;6.1.1 Observations and Measurements;172
13.1.2;6.1.2 How to Increase or Reduce a Number of Characterising Quantities;176
13.2;6.2 Analogue-to-Digital Converters;177
13.3;6.3 Time Series;179
13.3.1;6.3.1 Terms;179
13.3.2;6.3.2 Examples;180
13.4;6.4 Elements of Time Series Analysis;185
13.4.1;6.4.1 Visual Express Analysis;185
13.4.2;6.4.2 Spectral Analysis (Fourier and Wavelet Transform);188
13.4.3;6.4.3 Phase of Signal and Empirical Mode Decomposition;200
13.4.4;6.4.4 Stationarity Analysis;204
13.4.5;6.4.5 Interdependence Analysis;206
13.5;6.5 Experimental Example;208
13.6;References;210
14;Chapter
7 Restoration of Explicit Temporal Dependencies ;214
14.1;7.1 Parameter Estimation;214
14.1.1;7.1.1 Estimation Techniques;216
14.1.2;7.1.2 Comparison of Techniques;220
14.2;7.2 Approximation;225
14.2.1;7.2.1 Problem Formulation and Terms;225
14.2.2;7.2.2 Parameter Estimation;227
14.2.3;7.2.3 Model Size Selection, Overfitting and Ockham's Razor;228
14.2.4;7.2.4 Selecting the Class of Approximating Functions;233
14.3;7.3 Model Validation;235
14.3.1;7.3.1 Independence of Residuals;236
14.3.2;7.3.2 Normality of Residuals;236
14.4;7.4 Examples of Model Applications;238
14.4.1;7.4.1 Forecast;238
14.4.2;7.4.2 Numerical Differentiation;240
14.5;References;243
15;Chapter
8 Model Equations: Parameter Estimation ;245
15.1;8.1 Parameter Estimators and Their Accuracy;247
15.1.1;8.1.1 Dynamical Noise;247
15.1.2;8.1.2 Measurement Noise;248
15.2;8.2 Hidden Variables;251
15.2.1;8.2.1 Measurement Noise;252
15.2.2;8.2.2 Dynamical and Measurement Noise;256
15.3;8.3 What One Can Learn from Modelling Successes and Failures;260
15.3.1;8.3.1 An Example from Cell Biology;261
15.3.2;8.3.2 Concluding Remarks;264
15.4;References;264
16;Chapter
9 Model Equations: Restoration of Equivalent Characteristics ;267
16.1;9.1 Restoration Procedure and Peculiarities of the Problem;268
16.1.1;9.1.1 Discrete Maps;268
16.1.2;9.1.2 Ordinary Differential Equations;269
16.1.3;9.1.3 Stochastic Differential Equations;270
16.2;9.2 Model Structure Optimisation;272
16.3;9.3 Equivalent Characteristics for Two Real-World Oscillators;274
16.3.1;9.3.1 Physiological Oscillator;274
16.3.2;9.3.2 Electronic Oscillator;278
16.4;9.4 Specific Choice of Model Structure;280
16.4.1;9.4.1 Systems Under Regular External Driving;280
16.4.2;9.4.2 Time-Delay Systems;282
16.5;References;284
17;Chapter
10 Model Equations: ``Black Box'' Reconstruction ;286
17.1;10.1 Reconstruction of Phase Orbit;287
17.1.1;10.1.1 Takens' Theorems;288
17.1.2;10.1.2 Practical Reconstruction Algorithms;295
17.2;10.2 Multivariable Function Approximation;301
17.2.1;10.2.1 Model Maps;301
17.2.2;10.2.2 Model Differential Equations;310
17.3;10.3 Forecast with Various Models;311
17.3.1;10.3.1 Techniques Which Are not Based on Non-linear Dynamics Ideas;311
17.3.2;10.3.2 Iterative, Direct and Combined Predictors;312
17.3.3;10.3.3 Different Kinds of Model Maps;313
17.3.4;10.3.4 Model Maps Versus Model ODEs;314
17.4;10.4 Model Validation;315
17.5;References;316
18;Chapter
11 Practical Applications of Empirical Modelling ;320
18.1;11.1 Segmentation of Non-stationary Time Series;321
18.2;11.2 Confidential Information Transmission;323
18.3;11.3 Other Applications;325
18.4;References;328
19;Chapter
12 Identification of Directional Couplings ;330
19.1;12.1 Granger Causality;330
19.2;12.2 Phase Dynamics Modelling;333
19.3;12.3 Brain -- Limb Couplings in Parkinsonian Resting Tremor;337
19.4;12.4 Couplings Between Brain Areas in Epileptic Rats;340
19.5;12.5 El Niño -- Southern Oscillation and North Atlantic Oscillation;344
19.5.1;12.5.1 Phase Dynamics Modelling;344
19.5.2;12.5.2 Granger Causality Analysis;346
19.6;12.6 Causes of Global Warming;348
19.6.1;12.6.1 Univariate Models of the GST Variations;349
19.6.2;12.6.2 GST Models Including Solar Activity;352
19.6.3;12.6.3 GST Models Including Volcanic Activity;354
19.6.4;12.6.4 GST Models Including CO2 Concentration;354
19.7;References;355
20;Chapter
13 Outdoor Examples ;360
20.1;13.1 Coupled Electronic Generators;360
20.1.1;13.1.1 Object Description;360
20.1.2;13.1.2 Data Acquisition and Preliminary Processing;362
20.1.3;13.1.3 Selection of the Model Equation Structure;364
20.1.4;13.1.4 Model Fitting, Validation and Usage;366
20.2;13.2 Parkinsonian Tremor;374
20.2.1;13.2.1 Object Description;374
20.2.2;13.2.2 Data Acquisition and Preliminary Processing;375
20.2.3;13.2.3 Selection of the Model Equation Structure;378
20.2.4;13.2.4 Model Fitting, Validation and Usage;379
20.2.5;13.2.5 Validation of Time Delay Estimation;382
20.3;13.3 El-Niño/Southern Oscillation and Indian Monsoon;386
20.3.1;13.3.1 Object Description;386
20.3.2;13.3.2 Data Acquisition and Preliminary Processing;387
20.3.3;13.3.3 Selection of the Model Equation Structure;389
20.3.4;13.3.4 Model Fitting, Validation and Usage;390
20.4;13.4 Conclusions;397
20.5;References;397
21;Summary and Outlook;400
22;List of Mathematical Models;405
23;List of Real-World Examples;408
24;Index;409