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Abstract:
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For multivariate spatial processes, common cross-covariance functions do not exploit graphical models to ensure process-level conditional independence among the variables. This is undesirable, especially for highly multivariate settings, where popular cross-covariance functions such as the multivariate Matérn suffer from a "curse of dimensionality" as the number of parameters and floating point operations scale up in quadratic and cubic order, respectively, in the number of variables. We propose a class of multivariate "graphical Gaussian Processes" using a general construction called "stitching" that crafts cross-covariance functions from graphs and ensure process-level conditional independence among variables. For the Matérn family, stitching yields a multivariate GP whose univariate components are Matérn, and conforms to process-level conditional independence as specified by the graph. For highly multivariate settings and decomposable graphical models, stitching offers massive computational gains and parameter dimension reduction. We demonstrate our approach to jointly model highly multivariate spatial data using simulation examples and an application to air-pollution modelling.
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