Spatial statistics and stochastic partial differential equations: a mechanistic viewpoint
2021
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.
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