Tony O'Hagan - Academic pages - Abstracts

## Considering Covariates
in the Covariance Structure of Spatial Processes

Alexandra M. Schmidt, Peter Guttorp, and Anthony O'Hagan.

*Universidade Federal do Rio de Janeiro, Brazil*;
*University of Washington, USA and
Norwegian Computing Center, Oslo, Norway*; and
*University of Sheffield, UK*

**Publication details: **
*Environmentrics* **22**, 487-500, 2011.

### Abstract

In spatial statistics one usually assumes that observations are partial
realizations of a stochastic process {Y(x); x in R^C}, where commonly C = 2,
and the components of the location vector x are geographical coordinates.
Frequently, it is assumed that Y(.) follows a Gaussian process (GP) with
stationary covariance structure. In this setting the usual aim is to make spatial
interpolation to unobserved locations of interest, based on observed values at
monitored locations. This interpolation is heavily based on the specification
of the mean and covariance structure of the GP. In environmental problems
the assumption of stationary covariance structures is commonly violated due
to local influences in the covariance structure of the process.

We propose models which relax the assumption of stationary GP by accounting
for covariate information in the covariance structure of the process.
Usually at each location x, covariates related to Y(.) are also observed. We
initially propose the use of covariates to allow the latent space model of
Sampson & Guttorp to be of dimension C > 2. Then we discuss a particular case of
the latent space model by using a representation projected down from C
dimensions to 2 in order to model the 2D correlation structure better. Inference
is performed under the Bayesian paradigm, and Markov chain Monte Carlo
methods are used to obtain samples from the resultant posterior distributions
under each model. As illustration of the proposed models, we analyze solar
radiation in British Columbia, and mean temperature in Colorado.

**Keywords: **Anisotropy; Deformation; Manifold; Non-stationarity; Projection.

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Updated: 6 May 2011
Maintained by: Tony O'Hagan