Continuous logistic Gaussian random measure fields for spatial distributional modelling

TL;DR

This paper introduces Spatial Logistic Gaussian Process models for non-parametric spatial density estimation, analyzing their properties, regularity, and practical implementation with applications to temperature data.

Contribution

It provides a theoretical framework for SLGPs as random measures, extends regularity concepts, and proposes a scalable implementation using Random Fourier Features.

Findings

SLGP models exhibit joint Gaussianity of log-increments.

Sufficient conditions for mean-square continuity are established.

The method is successfully applied to temperature distribution data.

Abstract

We study Spatial Logistic Gaussian Process (SLGP) models for non-parametric estimation of probability density fields using scattered samples of heterogeneous sizes. SLGPs are examined from the perspective of random measures and their densities, investigating the relationships between SLGPs and underlying processes. Our inquiries are motivated by SLGP's abilities in delivering probabilistic predictions of conditional distributions at candidate points, allowing conditional simulations of probability densities, and jointly predicting multiple functionals of target distributions. We demonstrate that SLGP models exhibit joint Gaussianity of their log-increments, enabling us to establish theoretical results regarding spatial regularity. Additionally, we extend the notion of mean-square continuity to random measure fields and establish sufficient conditions on covariance kernels underlying SLGPs to ensure these models enjoy such regularity properties. Finally, we propose an implementation using Random Fourier Features and showcase its applicability on synthetic examples and on temperature distributions at meteorological stations.