Spatial statistics and stochastic partial differential equations: A mechanistic viewpoint - Archive ouverte HAL Access content directly
Journal Articles Spatial Statistics Year : 2022

Spatial statistics and stochastic partial differential equations: A mechanistic viewpoint

(1) , (1) , (1)
1

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 Levy 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 terms are introduced. The physical meaning of the latter and possible applications are discussed. Returning to the microscopic 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.

Dates and versions

hal-03843565 , version 1 (08-11-2022)

Identifiers

Cite

Lionel Roques, Denis Allard, Samuel Soubeyrand. Spatial statistics and stochastic partial differential equations: A mechanistic viewpoint. Spatial Statistics, 2022, 50, pp.100591. ⟨10.1016/j.spasta.2022.100591⟩. ⟨hal-03843565⟩

Collections

INRAE MATHNUM
0 View
0 Download

Altmetric

Share

Gmail Facebook Twitter LinkedIn More