Multidimensional Stationary Time Series: Dimension Reduction and Prediction

Description

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. Cover Half Title Title Page Copyright Page Dedication Contents Foreword Preface List of Figures Symbols 1. Harmonic analysis of stationary time series 1.1. Introduction 1.2. Covariance function and spectral representation 1.3. Spectral representation of multidimensional stationary time series 1.4. Constructions of stationary time series 1.4.1. Construction 1 1.4.2. Construction 2 1.4.3. Construction 3 1.4.4. Construction 4 1.4.4.1. Discrete Fourier Transform 1.4.4.2. The construction 1.5. Estimating parameters of stationary time series 1.5.1. Estimation of the mean 1.5.2. Estimation of the covariances 1.5.3. Periodograms 1.6. Summary 2. ARMA, regular, and singular time series in 1D 2.1. Introduction 2.2. Time invariant linear ltering 2.3. Moving Average processes 2.4. Autoregressive processes 2.5. Autoregressive moving average processes 2.6. Wold decomposition in 1D 2.7. Spectral form of the Wold decomposition 2.8. Factorization of rational and smooth densities 2.8.1. Rational spectral density 2.8.2. Smooth spectral density 2.9. Classi cation of stationary time series in 1D 2.10. Examples for singular time series 2.10.1. Type (0) singular time series 2.10.2. Type (1) singular time series 2.10.3. Type (2) singular time series 2.11. Summary 3. Linear system theory, state space models 3.1. Introduction 3.2. Restricted input/output map 3.3. Reachability and observability 3.4. Power series and extended input/output maps 3.5. Realizations 3.6. Stochastic linear systems 3.6.1. Stability 3.6.2. Prediction, miniphase condition, and covariance 3.7. Summary 4. Multidimensional time series 4.1. Introduction 4.2. Linear transformations, subordinated processes 4.3. Stationary time series of constant rank 4.4. Multidimensional Wold decomposition 4.4.1. Decomposition with an orthonormal process 4.4.2. Decomposition with innovations 4.5. Regular and singular time series 4.5.1. Full rank processes 4.5.2. Generic regular processes 4.5.3. Classification of non-regular multidimensional time series 4.6. Low rank approximation 4.6.1. Approximation of time series of constant rank 4.6.2. Approximation of regular time series 4.7. Rational spectral densities 4.7.1. Smith–McMillan form 4.7.2. Spectral factors of a rational spectral density matrix 4.8. Multidimensional ARMA (VARMA) processes 4.8.1. Equivalence of di erent approaches 4.8.2. Yule–Walker equations 4.8.3. Prediction, miniphase condition, and approximation by VMA processes 4.9. Summary 5 Dimension reduction and prediction in the time and frequency domain 5.1. Introduction 5.2. 1D prediction in the time domain 5.2.1. One-step ahead prediction based on finitely many past values 5.2.2. Innovations 5.2.3. Prediction based on the infinite past 5.3. Multidimensional prediction 5.3.1. One-step ahead prediction based on finitely many past values 5.3.2. Multidimensional innovations 5.4. Spectra of spectra 5.4.1. Bounds for the eigenvalues of Cn 5.4.2. Principal component transformation as discrete Fourier transformation 5.5. Kálmán's filtering 5.6. Dynamic principal component and factor analysis 5.6.1. Time domain approach via innovations 5.6.2. Frequency domain approach 5.6.3. Best low-rank approximation in the frequency domain, and low-dimensional approximation in the time domain 5.6.4. Dynamic factor analysis 5.6.5. General Dynamic Factor Model 5.7. Summary A. Tools from complex analysis A.1. Holomorphic (or analytic) functions A.2. Harmonic functions A.3. Hardy spaces A.3.1. First approach A.3.2. Second approach B. Matrix decompositions and special matrices C. Best prediction in Hilbert spaces D. Tools from algebra Bibliography Index