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We investigate a class of models for non-parametric estimation of probabilitydensity fields based on scattered samples of heterogeneous sizes. Theconsidered SLGP models are Spatial extensions of Logistic Gaussian Processmodels and inherit some of their theoretical properties but also of theircomputational challenges. We introduce SLGPs from the perspective of randommeasures and their densities, and investigate links between properties of SLGPsand underlying processes. Our inquiries are motivated by SLGP's abilities todeliver probabilistic predictions of conditional distributions at candidatepoints, to allow (approximate) conditional simulations of probabilitydensities, and to jointly predict multiple functionals of target distributions.We show that SLGP models induced by continuous GPs can be characterized by thejoint Gaussianity of their log-increments and leverage this characterization toestablish theoretical results pertaining to spatial regularity. We extend thenotion of mean-square continuity to random measure fields and establishsufficient conditions on covariance kernels underlying SLGPs for associatedmodels to enjoy such regularity properties. From the practical side, we proposean implementation relying on Random Fourier Features and demonstrate itsapplicability on synthetic examples and on temperature distributions atmeteorological stations, including probabilistic predictions of densities atleft-out stations.

Continuous logistic Gaussian random measure fields for spatial distributional modelling