Chunsheng Ma

Families of spatio-temporal stationary covariance models

Abstract This paper provides simple methods for constructing new families of spatio-temporal stationary covariance models from purely spatial (or purely temporal) stationary covariance models. As the application of the methods developed, we introduce spatio-temporal stationary covariance models with the Gaussian and related spatial margins, and develop the Heine family and the Whittle–Matern family of spatio-temporal stationary covariance models.

Spatio-Temporal Covariance Functions Generated by Mixtures

Spatio-temporal covariance functions are introduced in this paper by using two approaches: (1) positive power mixture of purely spatial and purely temporal covariances, and (2) scale mixture of purely spatial and purely temporal covariances. Various parametric and nonparametric families of nonseparable spatio-temporal covariance functions are obtained with appropriate selections of the mixing function and covariances being mixed.

Linear combinations of space-time covariance functions and variograms

The difference or a linear combination of two space-time covariance functions (or variograms) is not necessarily a valid covariance function (or variogram). In general, there seems no simple condition that makes a linear combination permissible, unless its coefficients are non-negative. The permissibility is investigated in this paper for the linear combination of two spatial or spatio-temporal covariance functions (or variograms) isotropic in space so that we obtain flexible classes of spatial or spatio-temporal covariance functions with various properties such as long-range dependence and having different signs.

Convex orders for linear combinations of random variables

Abstract Linear combinations ∑i=1kbiXi and ∑i=1kaiXi of random variables X1,…,Xk are ordered in the sense of the decreasing convex order and the Laplace order, where (b1,…,bk) is majorized by (a1,…,ak), when the underlying random variables are independent but possibly nonidentically distributed, and the joint density is arrangement increasing, respectively. Finite mixture distributions ∑i=1kaiFXi(x) and ∑i=1kbiFXi(x) are compared in the sense of the usual stochastic order, the convex order and higher-order stochastic dominance. The comparison between ∑i=1kIbiXi and ∑i=1kIaiXi is also studied for binary random variables I a i ,I b i (i=1,…,k) . Some applications in economics and reliability are described.

Vector Random Fields with Second-Order Moments or Second-Order Increments

This article is concerned with vector (multivariate, or multidimensional) random fields with second-order moments or second-order increments. Two crucial questions for such a random field are what kind of the square matrix function can be employed as its covariance matrix or variogram matrix, and, in particular, what type of the functions can be employed as its cross covariances or cross variograms. We attempt to explore the relationships between the direct covariance and the cross covariance in a covariance matrix and the relationships between the direct variogram and the cross variogram in a variogram matrix. Necessary and sufficient conditions are obtained for a given square matrix function to be the covariance matrix or variogram matrix of a vector Gaussian or elliptically contoured random field, and some parametric or nonparametric examples are given for stationary and nonstationary cases in a temporal, spatial, or spatio-temporal domain.

Covariance Matrices for Second-Order Vector Random Fields in Space and Time

This paper deals with vector (or multivariate) random fields in space and/or time with second-order moments, for which a framework is needed for specifying not only the properties of each component but also the possible cross relationships among the components. We derive basic properties of the covariance matrix function of the vector random field and propose three approaches to construct covariance matrix functions for Gaussian or non-Gaussian random fields. The first approach is to take derivatives of a univariate covariance function, the second one is to work on the univariate random field whose index domain is in a higher dimension and the third one is based on the scale mixture of separable spatio-temporal covariance matrix functions. To illustrate these methods, many parametric or semiparametric examples are formulated.

Student's t vector random fields with power-law and log-law decaying direct and cross covariances.

This article deals with the Students t vector random field, which is formulated as a scale mixture of Gaussian vector random fields, and whose finite-dimensional distributions decay in power-law and have heavy tails. There are two classes of Students t vector random fields, one with second-order moments, and the other without a second-order moment. A Cauchy vector random field is an example of Students t vector random fields without a first-order moment, and is also an example of Stable vector random fields. A second-order Students t vector random field allows for any given correlation structure, just as a Gaussian vector random field does. We propose four types of covariance matrix structures for second-order Students t vector random fields, which decay in power-law or log-law.

Isotropic Variogram Matrix Functions on Spheres

This paper is concerned with vector random fields on spheres with second-order increments, which are intrinsically stationary and mean square continuous and have isotropic variogram matrix functions. A characterization of the continuous and isotropic variogram matrix function on a sphere is derived, in terms of an infinite sum of the products of positive definite matrices and ultraspherical polynomials. It is valid for Gaussian or elliptically contoured vector random fields, but may not be valid for other non-Gaussian vector random fields on spheres such as a χ2, log-Gaussian, or skew-Gaussian vector random field. Some parametric variogram matrix models are derived on spheres via different constructional approaches. A simulation study is conducted to illustrate the implementation of the proposed model in estimation and cokriging, whose performance is compared with that using the linear model of coregionalization.

Hyperbolic Vector Random Fields with Hyperbolic Direct and Cross Covariance Functions

This article introduces the hyperbolic vector random field whose finite-dimensional distributions are the generalized hyperbolic one, which is formulated as a scale mixture of Gaussian random fields and is thus an elliptically contoured (or spherically invariant) random field. Such a vector random field may or may not have second-order moments, while a second-order one is characterized by its mean function and its covariance matrix function, just as in a Gaussian case. Some covariance matrix structures of hyperbolic type are constructed in this paper for second-order hyperbolic vector random fields.

Spherically Invariant Vector Random Fields in Space and Time

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.