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.
Table of Contents
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 in the time and frequency domain. Appendices.
Marianna Bolla, DSc is professor in the Institute of Mathematics, Budapest University of Technology and Economics. She authored the book Spectral Clustering and Biclustering, Learning Large Graphs and Contingency Tables, Wiley (2013) and the article Factor Analysis, Dynamic in Wiley StatsRef: Statistics Reference Online (2017). She is coauthor of a Hungarian book on Multivariate Statistical Analysis and a textbook Theory of Statistical Inference; further, provides lectures on these topics at her home institution and in the Budapest Semesters in Mathematics program. Research interest: spectral clustering, graphical models, time series, application of spectral and block matrix techniques in multivariate regression and prediction, based on classical works of CR Rao.
Tamás Szabados, PhD is a retired associate professor in the Institute of Mathematics, Budapest University of Technology and Economics. He used to give lectures on stochastic analysis and probability theory in his home institute and on probability theory in the Budapest Semesters in Mathematics program as well. He is a coauthor of a Hungarian textbook (1983) on vector analysis. He holds master’s degrees in electrical engineering and applied mathematics and PhD in mathematics. Research interests: discrete approximations in stochastic calculus, theory of time series, and mathematical immunology.
" The book is a well-structured point of view of time series theory, contains many theorems along with proofs. In addition, the book presents the necessary lemmas, definitions, and remarks. It should be noted, that at the end of the book in the form of appendices you can find the material needed to understand the theory of time series – tools from linear algebra, matrix theory and complex analysis. So, the book "Multidimensional Stationary Time Series: Dimension Reduction and Prediction" by Marianna Bolla and Tamas Szabados is a very good guide for specialists in time series predictions and dimension reduction."
Taras Lukashiv, Ukraine, ISCB News, June 2022.