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## Ilya Polyak

Print publication date: 1996

Print ISBN-13: 9780195099997

Published to Oxford Scholarship Online: November 2020

DOI: 10.1093/oso/9780195099997.001.0001

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# Random Processes and Fields

Chapter:
3 (p.110) Random Processes and Fields
Source:
Computational Statistics in Climatology
Publisher:
Oxford University Press
DOI:10.1093/oso/9780195099997.003.0005

In this chapter, the nonparametric methods of estimating the spectra and correlation functions of stationary processes and homogeneous fields are considered. It is assumed that the principal concepts and definitions of the corresponding theory are known (see Anderson, 1971; Box and Jenkins, 1976; Jenkins and Watts, 1968; Kendall and Stuart, 1967; Loeve, 1960; Parzen, 1966; Yaglom, 1986); therefore, only questions connected with the construction of numerical algorithms are studied. The basic results ranged from univariate process to multidimensional field are presented in Tables 3.1 and 3.2. These formulas make it possible to compare and trace the formal character of developing estimation procedures when the dimensionality is increasing. The schemes in these tables, as well as the formulas in the previous chapters, can be used for software development without any rearrangement. In part, this approach presents the application of the methods of Chapters 1 and 2 in evaluating random function characteristics. Of course, the final identification of the algorithm parameters (for example, the spectral window widths) can be made only through trial and error and by taking into account the character of the problem under study, that is, the physical properties of the processes and fields observed. The last section of this chapter presents results of the application of these methods to the analysis of some climatological fields. Here the basic results of the univariate spectral analysis are briefly discussed in order to develop algorithms for a multidimensional case by analogous reasoning. The complete description of the estimation procedures of the spectral and correlation analysis for univariate stationary process can be found, for example, in Jenkins and Watts, 1968.

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