Article

Spherically Invariant Vector Random Fields in Space and Time

IEEE Transactions on Signal Processing (Impact Factor: 2.79). 01/2012; 59(12):5921 - 5929. DOI: 10.1109/TSP.2011.2166391
Source: IEEE Xplore

ABSTRACT

This paper is concerned with spherically invariant or elliptically contoured vector random fields in space and/or time, which are formulated as scale mixtures of vector Gaussian random fields. While a spherically invariant vector random field may or may not have second-order moments, a spherically invariant second-order vector random field is determined by its mean and covariance matrix functions, just like the Gaussian one. This paper explores basic properties of spherically invariant second-order vector random fields, and proposes an efficient approach to develop covariance matrix functions for such vector random fields.

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Available from: Chunsheng ma, Dec 03, 2015
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    • "may not exist a non-Gaussian, such as a χ 2 (Ma, 2011c), log-Gaussian, skew-Gaussian, or K-distributed random field with C(x 1 , x 2 ) as its covariance matrix function. The logistic vector random field developed here belongs to the family of elliptically contoured (spherically invariant) vector random fields (Du and Ma, 2011; Ma, 2011a), and thus allows for any given correlation structure. "
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    ABSTRACT: The logistic vector random field is introduced in this paper as a scale mixture of Gaussian vector random fields, and is thus a particular elliptically contoured (spherically invariant) vector random field. Such a logistic vector random field is characterized by its mean function and covariance matrix function just as in the case of Gaussian vector random fields, and so it is flexible enough to allow for any possible mean structure or covariance matrix structure. We also derive covariance matrix functions whose direct and cross covariances are of the logistic type.
    Full-text · Article · Feb 2015 · Journal of Statistical Planning and Inference
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    • "The covariance function of the log-Gaussian scalar random field has a specific format as well (Matheron, 1989). Nevertheless, (1) is not only a necessary condition but also a sufficient condition for an isotropic and symmetric matrix function to be the covariance matrix function of a Gaussian or second-order elliptically contoured (spherically invariant) vector random field (Yao, 2003; Du and Ma, 2011; Ma, 2011a), to which or whose covariance matrix function we pay most our attention in what Downloaded by [Kansas State University Libraries], [Juan Du] at 15:19 06 May 2014 follows. Examples of elliptically contoured vector random fields include Gaussian, Student's t, stable, logistic, hyperbolic, Mittag-Leffler, Laplace, and Linnik ones; see R∅islien and Omre (2006), Du and Ma (2011), Du et al. (2012), Ma (2011a, 2013), among others. "
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    ABSTRACT: An isotropic scalar or vector random field is a second-order random field in [Inline formula] (d 2), whose covariance function or direct/cross covariance functions are isotropic. While isotropic scalar random fields have been well developed and widely used in various sciences and industries, the theory of isotropic vector random fields needs to be investigated for applications. The objective of this article is to study properties of covariance matrix functions associated with vector random fields in [Inline formula] which are stationary, isotropic, and mean square continuous, and derives the characterizations of the covariance matrix function of the Gaussian or second-order elliptically contoured vector random field in [Inline formula]. In particular, integral or spectral representations for isotropic and continuous covariance matrix functions are derived.
    Full-text · Article · May 2014 · Communication in Statistics- Theory and Methods
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    • "It may be interpreted as the instantaneous intensity of an m-variate complex elliptically contoured (or spherically invariant) random field. For properties and applications of real elliptically contoured (or spherically invariant) random fields see Yao (2003), Du and Ma (2011), and Ma (2011b,c), among others. First and secondorder moments of K -distributed vector random fields are derived in Section 3.2, and finite-dimensional probability laws are given in Section 3.3. "
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    ABSTRACT: This paper introduces two types of second-order vector random fields or stochastic processes whose marginals are K-distributed, through certain mixture procedures. The first type is formulated as an independent product of a Gamma random variable and a χ2χ2 vector random field, with an arbitrary spatial, temporal, or spatio-temporal index domain. The second type is formed as an independent product of a Gamma process and a χ2χ2 vector random field, with the index domain limited on the nonnegative part of the real line. We derive the mean and covariance matrix functions of these K-distributed vector random fields, as well as the corresponding finite-dimensional Laplace transformations.
    Full-text · Article · Apr 2013 · Statistics [?] Probability Letters
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