Front Cover 1
Dynamic Stochastic Models from Empirical Data 4
Copyright Page 5
Contents 8
Preface 12
Acknowledgments 14
Notation and Symbols 16
CHAPTER I. INTRODUCTION TO THE CONSTRUCTION OF MODELS 18
1a. Nature and Goals of Modeling 19
1b. Description of Models 21
1c. Choice of a Model for the Given Data 24
1d. Validation 25
Notes 27
CHAPTER II. PRELIMINARY ANALYSIS OF STOCHASTIC DYNAMICAL SYSTEMS 28
Introduction 28
2a. Assumptions and Discussion 28
2b. Stationarity 30
2c. Invertibility 31
2d. Covariance Functions and Correlograms 32
2e. Spectral Analysis 37
2f. Prediction 40
2g. Prediction in Multiplicative Systems 44
2h. Prediction in Systems with Noisy Observations 46
2i. Rescaled Range–Lag Characteristic 48
2j. Fractional Noise Models 53
2k. Conclusions 56
Appendix 2.1. Characteristics of Fractional Noise Models 56
Problems 57
CHAPTER III. STRUCTURE OF UNIVARIATE MODELS 59
Introduction 59
3a. Types of Dynamic Stochastic Models 61
3b. Types of Empirical Time Series 73
3c. Causality 80
3d. Choice of Time Scale for Modeling 81
3e. Conclusions 82
Notes 83
Problems 83
CHAPTER IV. ESTIMABILITY IN SINGLE OUTPUT SYSTEMS 84
Introduction 84
4a. Estimability of Systems in Standard Form 85
4b. Estimability in Systems with Noisy Observations 92
4c. Estimability in Systems with AR Disturbances 94
4d. The Estimation Accuracy 95
4e. Conclusions 102
Appendix 4.1 103
Appendix 4.2. Evaluation of the Cramér–Rao Matrix Lower Bound in Single Output Systems 108
Problems 109
CHAPTER V. STRUCTURE AND ESTIMABILITY IN MULTIVARIATE SYSTEMS 110
Introduction 110
5a. Characterization 112
5b. The Triangular Canonical Forms 116
5c. Diagonal Canonical Forms 119
5d. Pseudocanonical Forms 124
5e. Discussion of the Three Canonical Forms 125
5f. Estimation Accuracy 126
5g. Conclusions 128
Appendix 5.1. Proofs of Theorems 129
Problems 137
CHAPTER VI. ESTIMATION IN AUTOREGRESSIVE PROCESSES 139
Introduction 139
6a. Maximum Likelihood Estimators 140
6b. Bayesian Estimators 150
6c. Quasi-Maximum Likelihood (QML) Estimators in Single Output Systems 154
6d. Computational Methods 156
6e. Combined Parameter Estimation and Prediction 159
6f. Systems with Slowly Varying Coefficients 164
6g. Robust Estimation in AR Models 166
6h. Conclusions 168
Appendix 6.1. Proofs of Theorems in Section 6a 168
Appendix 6.2. The Expressions for the Posterior Densities 172
Appendix 6.3. The Derivation of Computational Algorithms 173
Appendix 6.4. Evaluation of the Cramér–Rao Lower Bound in Multi- variate AR Systems 175
Problems 176
CHAPTER VII. PARAMETER ESTIMATION IN SYSTEMS WITH BOTH MOVING AVERAGE AND AUTOREGRESSIVE TERMS 177
Introduction 177
7a. Maximum Likelihood Estimators 178
7b. Numerical Methods for CML Estimation 182
7c. Limited Information Estimates 188
7d. Numerical Experiments with Estimation Methods 191
7e. Conclusions 196
Problems 196
CHAPTER VIII. CLASS SELECTION AND VALIDATION OF UNIVARIATE MODELS 197
Introduction 197
8a.The Nature of the Selection Problem 198
8b.The Different Methods of Class Selection 200
8c. Validation of Fitted Models 218
8d. Discussion of Selection and Validation 231
8e. Conclusions 233
Appendix 8.1. Mean Square Prediction Error of Redundant Models 233
Problems 234
CHAPTER IX. CLASS SELECTION AND VALIDATION OF MULTIVARIATE MODELS 236
Introduction 236
9a. Nature of the Selection Problem 237
9b. Causality and the Construction of Preliminary Models 239
9c. Direct Comparison of Multivariate Classes of Models 244
9d. Validation of Models 249
9e. Conclusions 251
Appendix 9.1. Geometry of Correlation and Regression 252
Notes 253
Problems 254
CHAPTER X. MODELING RIVER FLOWS 255
10a. The Need and Scope of Modeling 255
10b. Discussion of Data 256
10c. Models for Monthly Flows 265
10d. Modeling Daily Flow Data 271
10e. Models for Annual Flow Data 277
10f. Conclusions 298
Notes 298
CHAPTER XI. SOME ADDITIONAL CASE STUDIES IN MODEL BUILDING 300
Introduction 300
11a. Modeling Some Biological Populations 302
11b. Analysis of the Annual Sunspot Series 321
11c. The Sales Data of Company X: An Empirical Series with Both Growth and Systematic Oscillations 325
11d. The Time Series E2 : Role of Moving Average Terms 329
11e. Causal Connection between Increases in Rainfall and Increased Urbanization 331
11f. A Multivariate Model for Groundwater Levels and Precipitation 335
11g. Conclusions 341
References 342
Index 348