Variogram Matrix Functions for Vector Random Fields with Second-Order Increments
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Variogram Matrix Functions for Vector Random Fields with Second-Order Increments Juan Du · Chunsheng Ma
Received: 25 November 2010 / Accepted: 27 November 2011 / Published online: 23 December 2011 © International Association for Mathematical Geosciences 2011
Abstract The variogram matrix function is an important measure for the dependence of a vector random field with second-order increments, and is a useful tool for linear predication or cokriging. This paper proposes an efficient approach to construct variogram matrix functions, based on three ingredients: a univariate variogram, a conditionally negative definite matrix, and a Bernstein function, and derives three classes of variogram matrix functions for vector elliptically contoured random fields. Moreover, various dependence structures among components can be derived through appropriate mixture procedures demonstrated in this paper. We also obtain covariance matrix functions for second-order vector random fields through the Schoenberg–Lévy kernels. Keywords Bernstein function · Conditionally negative definite matrix · Covariance matrix · Elliptically contoured random field · Gaussian random field · Schoenberg-Lévy kernel · Variogram matrix 1 Introduction There is a great demand for analyzing multivariate measurements observed across space and over time, due to an increasing wealth of multivariate spatial or spatiotemJ. Du () Department of Statistics, Kansas State University, Manhattan, KS 66506-0802, USA e-mail: [email protected] C. Ma Department of Mathematics, Statistics, and Physics, Wichita State University, Wichita, KS 67260-0033, USA e-mail: [email protected] C. Ma School of Economics, Wuhan University of Technology, Wuhan 430070, Hubei, China
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Math Geosci (2012) 44:411–425
poral data; see Mardia (1988), Brown et al. (1994), Haas (1998), Zidek et al. (2000), Zhu et al. (2005), Calder (2007), Sain and Cressie (2007), Zhang (2007), Huang et al. (2009), and Sain et al. (2011), among others. This type of multivariate data is often treated as the realizations of a vector (multivariate) random field. For example, there are many stations in the world which record weather data hourly or daily, with components such as temperature, wind speed, wind direction, precipitation, and so on. There is a need to model such data for understanding how climate is varying in a given region, for capturing present climate variability and extreme events, and for assessing the impacts of climate change on our daily life, our society, and our environment. Although there are a variety of models for univariate processes, the available models for multivariate problems are limited due to the number of variables or components involved. Practical demands require us to develop covariance or variogram matrix functions for vector random fields with second-order moments or with second-order increments, so that practitioners have suitable models to fit data. Consider an m-variate random field {Z(x) = (Z1 (x), . . . , Zm (x)) , x ∈ D}, which is a family of real-valued random vecto
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