Spatial statistics and stochastic partial differential equations: a mechanistic viewpoint

TL;DR

This paper revisits the SPDE approach in spatial statistics from a mechanistic perspective, linking particle movement models to differential operators, and introduces new generalizations for modeling spatial dependencies and dynamics.

Contribution

It establishes a mechanistic interpretation of SPDEs using particle movement, connects Le9vy flights to PDEs, and proposes generalized models with applications to non-stationary and nonlinear spatial processes.

Findings

Link between Le9vy flights and Fractional Laplacian PDEs

Introduction of generalized SPDEs with dispersal, drift, and reaction terms

Connection of particle models to point processes and hierarchical estimation

Abstract

The Stochastic Partial Differential Equation (SPDE) approach, now commonly used in spatial statistics to construct Gaussian random fields, is revisited from a mechanistic perspective based on the movement of microscopic particles, thereby relating pseudo-differential operators to dispersal kernels. We first establish a connection between L\'evy flights and PDEs involving the Fractional Laplacian (FL) operator. The corresponding Fokker-Planck PDEs will serve as a basis to propose new generalisations by considering a general form of SPDE with terms accounting for dispersal, drift and reaction. We detail the difference between the FL operator (with or without linear reaction term) associated with a fat-tailed dispersal kernel and therefore describing long-distance dependencies, and the damped FL operator associated with a thin-tailed kernel, thus corresponding to short-distance dependencies. Then, SPDE-based random fields with non-stationary external spatially and temporally varying force are illustrated and nonlinear bistable reaction term are introduced. The physical meaning of the latter and possible applications are discussed. Returning to the particulate interpretation of the above-mentioned equations, we describe in a relatively simple case their links with point processes. We unravel the nature of the point processes they generate and show how such mechanistic models, associated to a probabilistic observation model, can be used in a hierarchical setting to estimate the parameters of the particle dynamics.