Multidimensional Stationary Time Series : Dimension Reduction and Prediction

This book gives a brief survey of the theory of multidimensional (multivariate), weakly stationary time series, with emphasis on dimension reduction and prediction. Understanding the covered material requires a certain mathematical maturity, a degree of knowledge in probability theory, linear algebra, and also in real, complex and functional analysis. For this, the cited literature and the Appendix contain all necessary material. The main tools of the book include harmonic analysis, some abstract algebra, and state space methods: linear time-invariant filters, factorization of rational spectral densities, and methods that reduce the rank of the spectral density matrix. \* Serves to find analogies between classical results (Cramer, Wold, Kolmogorov, Wiener, Kálmán, Rozanov) and up-to-date methods for dimension reduction in multidimensional time series. \* Provides a unified treatment for time and frequency domain inferences by using machinery of complex and harmonic analysis, spectral and Smith--McMillan decompositions. Establishes analogies between the time and frequency domain notions and calculations. \* Discusses the Wold's decomposition and the Kolmogorov's classification together, by distinguishing between different types of singularities. Understanding the remote past helps us to characterize the ideal situation where there is a regular part at present. Examples and constructions are also given. \* Establishes a common outline structure for the state space models, prediction, and innovation algorithms with unified notions and principles, which is applicable to real-life high frequency time series. It is an ideal companion for graduate students studying the theory of multivariate time series and researchers working in this field. This book gives a brief survey of the theory of multidimensional (multivariate), weakly stationary time series, with emphasis on dimension reduction and prediction. Cover 1 Half Title 2 Title Page 4 Copyright Page 5 Dedication 6 Contents 8 Foreword 12 Preface 14 List of Figures 18 Symbols 20 1. Harmonic analysis of stationary time series 24 1.1. Introduction 24 1.2. Covariance function and spectral representation 24 1.3. Spectral representation of multidimensional stationary time series 32 1.4. Constructions of stationary time series 41 1.4.1. Construction 1 41 1.4.2. Construction 2 43 1.4.3. Construction 3 45 1.4.4. Construction 4 46 1.4.4.1. Discrete Fourier Transform 46 1.4.4.2. The construction 48 1.5. Estimating parameters of stationary time series 49 1.5.1. Estimation of the mean 49 1.5.2. Estimation of the covariances 53 1.5.3. Periodograms 56 1.6. Summary 58 2. ARMA, regular, and singular time series in 1D 62 2.1. Introduction 62 2.2. Time invariant linear ltering 63 2.3. Moving Average processes 65 2.4. Autoregressive processes 69 2.5. Autoregressive moving average processes 76 2.6. Wold decomposition in 1D 81 2.7. Spectral form of the Wold decomposition 83 2.8. Factorization of rational and smooth densities 88 2.8.1. Rational spectral density 88 2.8.2. Smooth spectral density 89 2.9. Classi cation of stationary time series in 1D 90 2.10. Examples for singular time series 98 2.10.1. Type (0) singular time series 98 2.10.2. Type (1) singular time series 101 2.10.3. Type (2) singular time series 101 2.11. Summary 104 3. Linear system theory, state space models 110 3.1. Introduction 110 3.2. Restricted input/output map 110 3.3. Reachability and observability 112 3.4. Power series and extended input/output maps 113 3.5. Realizations 119 3.6. Stochastic linear systems 126 3.6.1. Stability 126 3.6.2. Prediction, miniphase condition, and covariance 128 3.7. Summary 132 4. Multidimensional time series 136 4.1. Introduction 136 4.2. Linear transformations, subordinated processes 136 4.3. Stationary time series of constant rank 139 4.4. Multidimensional Wold decomposition 143 4.4.1. Decomposition with an orthonormal process 143 4.4.2. Decomposition with innovations 145 4.5. Regular and singular time series 147 4.5.1. Full rank processes 149 4.5.2. Generic regular processes 158 4.5.3. Classification of non-regular multidimensional time series 163 4.6. Low rank approximation 164 4.6.1. Approximation of time series of constant rank 165 4.6.2. Approximation of regular time series 169 4.7. Rational spectral densities 170 4.7.1. Smith–McMillan form 171 4.7.2. Spectral factors of a rational spectral density matrix 173 4.8. Multidimensional ARMA (VARMA) processes 174 4.8.1. Equivalence of di erent approaches 174 4.8.2. Yule–Walker equations 181 4.8.3. Prediction, miniphase condition, and approximation by VMA processes 184 4.9. Summary 186 5 Dimension reduction and prediction in the time and frequency domain 192 5.1. Introduction 192 5.2. 1D prediction in the time domain 193 5.2.1. One-step ahead prediction based on finitely many past values 193 5.2.2. Innovations 196 5.2.3. Prediction based on the infinite past 198 5.3. Multidimensional prediction 201 5.3.1. One-step ahead prediction based on finitely many past values 201 5.3.2. Multidimensional innovations 203 5.4. Spectra of spectra 206 5.4.1. Bounds for the eigenvalues of Cn 212 5.4.2. Principal component transformation as discrete Fourier transformation 213 5.5. Kálmán's filtering 214 5.6. Dynamic principal component and factor analysis 222 5.6.1. Time domain approach via innovations 222 5.6.2. Frequency domain approach 224 5.6.3. Best low-rank approximation in the frequency domain, and low-dimensional approximation in the time domain 225 5.6.4. Dynamic factor analysis 227 5.6.5. General Dynamic Factor Model 230 5.7. Summary 231 A. Tools from complex analysis 238 A.1. Holomorphic (or analytic) functions 238 A.2. Harmonic functions 242 A.3. Hardy spaces 245 A.3.1. First approach 245 A.3.2. Second approach 248 B. Matrix decompositions and special matrices 250 C. Best prediction in Hilbert spaces 264 D. Tools from algebra 272 Bibliography 288 Index 292 discrete,time;,state-space discrete time,state-space