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Multivariate space-time data are increasingly available in various scientificdisciplines. When analyzing these data, one of the key issues is to describethe multivariate space-time dependencies. Under the Gaussian framework, oneneeds to propose relevant models for multivariate space-time covariancefunctions, i.e. matrix-valued mappings with the additional requirement ofnon-negative definiteness. We propose a flexible parametric class ofcross-covariance functions for multivariate space-time Gaussian random fields.Space-time components belong to the (univariate) Gneiting class of space-timecovariance functions, with Mat\'ern or Cauchy covariance functions in thespatial margins. The smoothness and scale parameters can be different for eachvariable. We provide sufficient conditions for positive definiteness. Asimulation study shows that the parameters of this model can be efficientlyestimated using weighted pairwise likelihood, which belongs to the class ofcomposite likelihood methods. We then illustrate the model on a French datasetof weather variables.

A Flexible Class of Non-separable Cross-Covariance Functions for Multivariate Space-Time Data