Multidimensional Stationary Time Series

Dimension Reduction and Prediction

Marianna Bolla, Tamás Szabados

• Publisher: Chapman and Hall/CRC
• Year: 2021
• Print ISBN:
  9780367619701
  9780367569327
• Edition: 1
• eBook ISBN:
  9781000392401
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

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.

1

Harmonic analysis of stationary time series

1.1 Introduction

A widely applicable notion of random fields is the one of multidimensional (or multivariate) time series $X_t=(X_t^1,\dots ,X_t^d)$, where $t\in ℝ$ is the time, $d\in \mathbb{N}$ is the dimension, and for each t fixed, each coordinate $X_t^j$ $(j=1,\dots ,d)$ is a complex valued random variable on the same probability space $(\mathrm{\Omega },\mathcal{F},\mathrm{\mathbb{P}})$. Sometimes, we investigate the special case when the random variables are real valued. Here we consider time series with discrete time and state space ℂ^d.

We concentrate on stationary time series, mainly in the wide sense, the behavior of which is irrespective of time shift. Moreover, the assumption of weak stationarity needs only the first and second moments and cross-moments (second-order processes). This allows us to use a huge machinery of analytical tools and the theory of Hilbert spaces. Fortunately, many time series in practi...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Contents
• Foreword
• Preface
• List of Figures
• Symbols
• 1 Harmonic analysis of stationary time series
• 2 ARMA, regular, and singular time series in 1D
• 3 Linear system theory, state space models
• 4 Multidimensional time series
• 5 Dimension reduction and prediction
• A Tools from complex analysis
• B Matrix decompositions and special matrices
• C Best prediction in Hilbert spaces
• D Tools from algebra
• Bibliography
• Index