Blur-generated non-separable space–time models

TLDR

Statistical space–time modelling has traditionally focused on separable covariance functions, where the covariance is a product of purely temporal and purely spatial components. The authors highlight a physical dispersion model capable of representing phenomena such as the spread of an air pollutant. The model evolves by blurring the spatial field at each time step and adding a spatial random field, first defined at discrete times and then extended to continuous time, with Gaussian smoothing required under certain conditions. The model possesses a non‑separable covariance function, yields a consistent continuous formulation with interpretable parameters independent of sampling intervals, is well suited to realistic problems poorly fitted by separable models, and is generated by a previously studied stochastic differential equation.

Abstract

Summary Statistical space–time modelling has traditionally been concerned with separable covariance functions, meaning that the covariance function is a product of a purely temporal function and a purely spatial function. We draw attention to a physical dispersion model which could model phenomena such as the spread of an air pollutant. We show that this model has a non-separable covariance function. The model is well suited to a wide range of realistic problems which will be poorly fitted by separable models. The model operates successively in time: the spatial field at time t +1 is obtained by ‘blurring’ the field at time t and adding a spatial random field. The model is first introduced at discrete time steps, and the limit is taken as the length of the time steps goes to 0. This gives a consistent continuous model with parameters that are interpretable in continuous space and independent of sampling intervals. Under certain conditions the blurring must be a Gaussian smoothing kernel. We also show that the model is generated by a stochastic differential equation which has been studied by several researchers previously.