Bayesian estimation of semi-parametric non-stationary spatial covariance structures

Abstract

We use the Sampson and Guttorp approach to model the non!stationary correlation function r"x\ x?# of a Gaussian spatial process through a bijective space deformation\ f\ so that in the deformed space the spatial correlation function can be considered isotropic\ namely r"x\ x?# r"= f "x#−f "x?#=#\ where r belongs to a known parametric family[ Given the locations in the deformed space of a number of geographic sites at which data are available\ we smoothly extrapolate the deformation to the whole region of interest[ Using a Bayesian framework\ we estimate jointly these locations\ as well as the parameters of the correlation function and the variance parameters[ The advantage of our Bayesian approach is that it allows us to obtain measures of uncertainty of all these parameters[ As the parameter space is of a very high dimension\ we implement an MCMC method for obtaining samples from the posterior distributions of interest[ We demonstrate our method through a simulation study\ and show an application to a real data set[ Copyright Þ 1990 John Wiley + Sons\ Ltd[ KEY WORDS] thin!plate splines^Markov Chain Monte Carlo^Gaussian spatial processes 0[ INTRODUCTIONThe estimation of heterogeneous spatial covariance is an important problem in environmetrics and geostatistics[ A general\ semi!parametric approach to this\ based on repeated observations from a network of monitoring stations\ was introduced by Sampson and Guttorp "0881# and developed further in a sequence of papers "Guttorp and Sampson\ 0883^Meiring et al[\ 0886\ 0887^Perrin and Meiring\ 0888#[ One of the advantages with the SampsonÐGuttorp approach is that it enables estimation of the covariance between spatial locations where no observations have been made\ without relying on an assumption of homogeneity or stationarity of the underlying spaceÐtime random _eld[ A weakness of the current methodology is its reliance on somewhat adCorrespondence to] D[ Damian\

tgp: AnRPackage for Bayesian Nonstationary, Semiparametric Nonlinear Regression and Design by Treed Gaussian Process Models

“…The addition of axis-aligned treed partitioning provides a divide-and-conquer mechanism that can not only reduce the computational burden relative to the base GP model, but can also facilitate the efficient modeling of nonstationarity and heteroskedasticity in the data. This is in stark contrast to other recent approaches to nonstationary spatial models (e.g., via deformations (Damian, Sampson, and Guttorp 2001;Schmidt and O'Hagan 2003), or process convolutions (Higdon, Swall, and Kern 1999;Fuentes and Smith 2001;Paciorek 2003)) which can require orders of magnitude more effort relative to stationary GPs.…”

A simple non-separable, non-stationary spatiotemporal model for ozone

“…For many spatiotemporal processes, there is little reason to expect the spatial stationarity of the covariance function. Spatially varying anisotropy can sometimes be modelled through deformations of the geographic coordinates system (Sampson and Guttorp 1992;Damian et al 2000), where the non-stationary spatial correlation function of a spatiotemporal process is a function of the Euclidean distances between site locations in a bijective transformation of the geographic coordinate system. After an appropriate adjustment has been made to the coordinate system, the correlation structure may be considered isotropic (i.e.…”

“…A further complication that frequently arises is that of anisotropy, which can be removed by deformation analysis (Sampson and Guttorp 1992;Damian et al 2000). Deformation may be applied to Z * (t, x), when the spatiotemporal model considered is (3), or to Z 2 (t, x), when model (4) is assumed.…”

Distribution of the maximum in air pollution fields

“…However, for our purposes, the important thing is that the mean can be computed at any spatial point at any time point. The underlying process ξ(x, t) is assumed to have a Sampson-Guttorp nonstationary spatial covariance, see Damian et al (2001). This means that in the geographic coordinates the covariance is nonstationary but if the geographic plane is deformed using a function f , say, then, in the deformed geography, the covariance is approximately isotropic.…”