Bivariate Gaussian random fields : models, simulation, and inference

Moreva, Olga

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URN: urn:nbn:de:bsz:180-madoc-453800
Document Type: Doctoral dissertation
Year of publication: 2018
Place of publication: Mannheim
University: Universität Mannheim
Evaluator: Schlather, Martin
Date of oral examination: 22 June 2018
Publication language: English
Institution: School of Business Informatics and Mathematics > Applied Stochastics (Schlather 2012-)
Subject: 510 Mathematics
Subject headings (SWD): Geostatistik , Kovarianzfunktion , Positiv-definite Funktion
Keywords (English): circulant embedding , cokriging , compactly supported covariance function , cut-off embedding , multivariate covariance function , multivariate Gaussian random field , multivariate geostatistics
Abstract: Spatial data with several components, such as observations of air temperature and pressure in a certain geographical region or the content of two metals in a geological deposit, require models which can capture the spatial dependence structure of individual components and the relationship between them. In a wealth of applications, multivariate Gaussian random fields are sensible models for multivariate spatial data and their second order structure specifies the marginal correlations and the cross-correlations between the components. In this thesis we focus on covariance models and simulation techniques for bivariate fields. In Chapter 2 we summarize some definitions and facts from univariate and multivariate Geostatistics which are essential for the subsequent chapters. Chapter 3 introduces two novel bivariate parametric covariance models, the powered exponential (or stable) covariance model and the generalized Cauchy covariance model. Both models allow for flexible smoothness, variance, scale, and cross-correlation parameters. The smoothness parameter is in (0, 1]. Additionally, the bivariate generalized Cauchy model allows for distinct long range parameters. The results are based on general sufficient conditions for the positive definiteness of 2×2-matrix valued functions. These conditions are easy to check, since they require only computing the derivatives of a bivariate covariance function and calculating an infimum of a function of one variable. We also show that the univariate spherical model can be generalized to the bivariate case with spherical marginal and cross-covariance functions only in a trivial way. Circulant embedding is a powerful algorithm for fast simulation of stationary Gaussian random fields on a rectangular grid in R^n , which works perfectly for compactly supported covariance functions. Cut-off circulant embedding techniques have been developed for univariate random fields for dimensions up to R^3 and rely on the modification of a covariance function outside the simulation window, such that the modified covariance function is compactly supported. In Chapter 4 we propose extensions of the cut-off approach for bivariate Gaussian random fields. In particular, we provide a method for simulating bivariate fields with a bivariate powered exponential covariance model and the full bivariate Matérn covariance model for certain sets of parameters. On the way we extend the cut-off circulant embedding method even for univariate models. In Chapter 5 we illustrate the use of the bivariate powered exponential model for a data example.

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