Cressie, Noel; Burden, Sandy; Davis, Walter; Krivitsky, Pavel; Mokhtarian, Payam; Suesse, Thomas; and Zammit-Mangion, Andrew, Capturing multivariate spatial dependence: model, estimate, and then predict, National Institute for Applied Statistics Research Australia, University of Wollongong, Working Paper 02-15, 2015, 9.
We would like to thank Marc Genton and William Kleiber (hereafter, GK) for their informative review, "Cross-covariance functions for multivariate geostatistics" (forthcoming in Statistical Science), and the editor of Statistical Science for the opportunity to contribute to the discussion.
Physical processes rarely occur in isolation, rather they influence and interact with one another. Thus, there is great benefit in modelling potential dependence between both spatial locations and different processes. It is the interaction between these two dependencies that is the focus of GK.
We see the problem of ensuring that the matrix given in GK-(2) is nonnegative definite (nnd) as important, but we also see it as a means to an end. That "end" is solving the scientific problem of predicting a multivariate eld of, say, temperature and rainfall, based on noisy and spatially incomplete data from weather stations in a region of interest. There is also scientific interest in the behaviour of the measures of cross-spatial dependence (e.g., cross-covariance functions), but usually spatial prediction is the ultimate goal.