Premium
Blur‐generated non‐separable space–time models
Author(s) -
Brown Patrick E.,
Roberts Gareth O.,
Kåresen Kjetil F.,
Tonellato Stefano
Publication year - 2000
Publication title -
journal of the royal statistical society: series b (statistical methodology)
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 6.523
H-Index - 137
eISSN - 1467-9868
pISSN - 1369-7412
DOI - 10.1111/1467-9868.00269
Subject(s) - covariance function , covariance , separable space , smoothing , mathematics , limit (mathematics) , space time , gaussian , function (biology) , field (mathematics) , kernel (algebra) , range (aeronautics) , statistical physics , mathematical analysis , statistics , physics , discrete mathematics , materials science , quantum mechanics , chemical engineering , evolutionary biology , pure mathematics , composite material , biology , engineering
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.