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Spatial modelling using a new class of nonstationary covariance functions
Author(s) -
Paciorek Christopher J.,
Schervish Mark J.
Publication year - 2006
Publication title -
environmetrics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.68
H-Index - 58
eISSN - 1099-095X
pISSN - 1180-4009
DOI - 10.1002/env.785
Subject(s) - covariance function , covariance , kriging , gaussian process , matérn covariance function , smoothing , bayesian probability , rational quadratic covariance function , mathematics , gaussian , differentiable function , covariance intersection , computer science , econometrics , statistics , physics , mathematical analysis , quantum mechanics
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 behaviour 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. Copyright © 2006 John Wiley & Sons, Ltd.