Continuous logistic Gaussian random measure fields for spatial distributional modelling

TL;DR

This work develops Spatial Logistic Gaussian Processes (SLGPs) for non-parametric density-field estimation indexed by a spatial domain and accommodating heterogeneous sampling. By re-framing logistic Gaussian processes as random measures and analyzing their increments, the authors establish connections between SLGPs, LRPMFs, and Gaussian increment structures, along with sufficient conditions on covariance kernels that guarantee almost-sure and mean-power continuity. A scalable implementation using Random Fourier Features and Matérn kernels is proposed, enabling MAP and MCMC inference to produce probabilistic density-field predictions and uncertainty quantification, demonstrated on synthetic fields and a meteorological temperature dataset from Switzerland. The results enable probabilistic predictions of conditional distributions at candidate points and joint functionals with uncertainty, offering a flexible framework for distributional spatial modelling with practical applicability to real-world environmental data. Overall, the paper advances both the theoretical understanding of SLGPs as random measure fields and their practical applicability through an efficient, uncertainty-aware computational approach.

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.

Figures (11)

• Figure 1: One example of probability distribution field: The daily mean temperatures in Switzerland indexed by latitude, longitude and altitude. We display the histogram of the available data for three meteorological stations (365 replications for each location).
• Figure 2: Summarising the nature and relationships between the mathematical objects considered in this section
• Figure 3: Visualising $\mathbb{E}\left[ \mathfrak{D} \left( \ifstrempty{0} { \Xi }{ \ifstrempty{} { \Xi_{0} }{ \Xi_{0}\left( \right) } }, \ifstrempty{\mathbf{x}'} { \Xi }{ \ifstrempty{} { \Xi_{\mathbf{x}'} }{ \Xi_{\mathbf{x}'}\left( \right) } }\right)^\gamma \right]$ (plain lines) and the theoretical bound (dotted lines) for both kernels, all four dissimilarities and $\gamma \in \{0.5, 1, 2\}$.
• Figure 4: Representation of the four density fields used as reference: probability density functions over slices at some prescribed $\mathbf{x} \in [0, 1]$.
• Figure 5: Results for the first reference field (lowest spatial regularity): Showing the MAP estimator for varying sample sizes, with samples scattered across space.

Theorems & Definitions (34)

• Theorem 1: \ref{['mainA:th:Expected_quadratic_cty']}
• Definition 2.1: Spatial logistic density transformation
• Definition 2.2: Logistic Random Probability Measure Field - LRPMF
• Definition 2.3: Spatial Logistic Process - SLP
• Remark
• Proposition 2.1: Condition for the indistinguishability of LRPMF and SLPs
• Proof
• Remark : Indistinguishability compared to others notions of coincidence between RPMF
• Lemma 1
• Proof