Tanya Apanasovich
Thomas Jefferson University 
Jefferson Medical College 
Department of Pharmacology and Experimental Therapeutics 
Division of Biostatistics 

TITLE: Modeling Cross-Covariance Functions for Multivariate Random
Fields

Multivariate data indexed by spatial coordinates have become
ubiquitous in a large number of applications, for instance in
environmental and climate sciences to name but a few. This has
prompted a renewed interest in multivariate random fields in recent
years.

Various approaches to construct valid cross-covariance models can be
found in the literature. I will briefly discuss the most common ones.
However, I will pay more attention to our recently proposed approach
(Apanasovich and Genton (2010)) based on latent dimensions. Next, I
will introduce a valid parametric family of cross-covariance functions
for multivariate spatial random fields where each component has a
covariance function from a well-celebrated Mat\'ern class (Apanasovich
et al (2011)). Unlike previous attempts, our model indeed allows for
various smoothnesses and rates of correlation decay for any number of
vector components. I will present the conditions on the parameter
space that result in valid models with varying degrees of complexity.
Practical implementation, including reparametrizations to reflect the
conditions on the parameter space will be  discussed.  The application
of the proposed multivariate Mat\'ern model will be illustrated on two
meteorological datasets: Temperature/pressure over the Pacific
Northwest (bivariate) and wind/temperature/pressure in Oklahoma
(trivariate).