Spatial modelling using a new class of nonstationary covariance functions

Christopher J Paciorek,Mark J Schervish

doi:10.1002/env.785

Abstract

We introduce a new class of nonstationary covariance functions for spatial modelling. Nonstationary covariance functions allow the model to adapt to spatial surfaces whose variability changes with location. The class includes a nonstationary version of the Matérn stationary covariance, in which the differentiability of the spatial surface is controlled by a parameter, freeing one from fixing the differentiability in advance. The class allows one to knit together local covariance parameters into a valid global nonstationary covariance, regardless of how the local covariance structure is estimated. We employ this new nonstationary covariance in a fully Bayesian model in which the unknown spatial process has a Gaussian process (GP) prior distribution with a nonstationary covariance function from the class. We model the nonstationary structure in a computationally efficient way that creates nearly stationary local behavior and for which stationarity is a special case. We also suggest non-Bayesian approaches to nonstationary kriging.To assess the method, we use real climate data to compare the Bayesian nonstationary GP model with a Bayesian stationary GP model, various standard spatial smoothing approaches, and nonstationary models that can adapt to function heterogeneity. The GP models outperform the competitors, but while the nonstationary GP gives qualitatively more sensible results, it shows little advantage over the stationary GP on held-out data, illustrating the difficulty in fitting complicated spatial data.

Highlights

One focus of spatial statistics research has been spatial smoothing - estimating a smooth spatial process from noisy observations or smoothing over small-scale variability
The eigendecomposition approach extends more readily to higher dimensions, which may be of interest for spatial data in three dimensions and more general nonparametric regression problems (Paciorek and Schervish 2004)
With either approach it appears very difficult to estimate a precipitation surface in western Colorado based on such a small number of weather stations

Summary

Introduction

One focus of spatial statistics research has been spatial smoothing - estimating a smooth spatial process from noisy observations or smoothing over small-scale variability. Two of the most prominent approaches have been kriging and thin plate splines The covariance function, C(·, ·; θ), of the GP is parameterized by θ and determines the covariance between any two locations This model underlies the standard kriging approach, in which C(·; θ) is a stationary covariance, a function only of Euclidean distance (or possibly a more general Mahalanobis distance) between any two locations. 66) after integrating the unknown process values at the observation locations out of the model. While various approaches to kriging and thin plate spline models have been used successfully for spatial process estimation, they have the weakness of being global models, in which the variability of the estimated process is the same throughout the domain because θ applies to the entire domain

Methods

Results

Conclusion