Extracting Knowledge From Time Series (eBook)

An Introduction to Nonlinear Empirical Modeling

Boris P. Bezruchko, Dmitry A. Smirnov (Autoren)

eBook Download: PDF

2010 | 2010
XXII, 410 Seiten
Springer Berlin (Verlag)
978-3-642-12601-7 (ISBN)

Infos

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.

Inhalt

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

Produkt-Details

• Erscheint lt. Verlag: 3.9.2010
• Reihe/Serie: Springer Series in Synergetics
• Reihe/Serie: Springer Series in Synergetics
• Zusatzinfo: XXII, 410 p. 162 illus.
• Verlagsort: Berlin
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Astronomie / Astrophysik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Theoretische Physik
• Themenwelt: Technik
• Themenwelt: Wirtschaft
• Schlagworte: chaotic signals • model equations • Modeling • modeling and forecast • nonlinear dynamical systems • Quantitative Finance • Sets • stochastic model • stochastic models • Time Series Analysis
• ISBN-10: 3-642-12601-4 / 3642126014
• ISBN-13: 978-3-642-12601-7 / 9783642126017