Spatial modelling using a new class of nonstationary covariance functions

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.

Spatial models with explanatory variables in the dependence structure

“…The non-stationary model were thought to solve some of the topographical challenges, but also this model smooths too much: at all stations with high predictions the actual observed values are even higher (see Figure 8). Also Paciorek and Schervish (2006) experienced that even though the results seem more reasonable with a non-stationary model, the predictive performance did not change much compared to a stationary model when analysing meteorological data.…”

“…Another popular approach has been kernel convolution methods (e.g. Higdon et al, 1999;Fuentes, 2002;Paciorek and Schervish, 2006;Reich et al, 2011;Kleiber and Nychka, 2012). Also a two stage approach with a discrete GRF controlling the variance in another discrete GRF have been used (Yue and Speckman, 2010).…”

“…Often advanced Markov Chain Monte Carlo schemes have to be used (e.g. Yue and Speckman, 2010;Schmidt et al, 2011;Reich et al, 2011), or step-wise procedures are used (e.g Paciorek and Schervish, 2006;Kleiber and Nychka, 2012).…”

On the diffusion equation and its application to isotropic and anisotropic correlation modelling in variational assimilation

“…Paciorek and Schervish (2006) describe a family of anisotropic and inhomogeneous correlation functions that generalize the standard isotropic and homogeneous Gaussian and Matérn family. These correlation functions have the form…”

A Study on Along-Track and Cross-Track Noise of Altimetry Data by Maximum Likelihood: Mars Orbiter Laser Altimetry (Mola) Example

“…This attribute is associated with non-stationarity of the spatial process and non-stationary kernels are developed to handle this problem in data modeling (Darbeheshti and Featherstone, 2009;Paciorek and Schervish, 2006). These kernels help in the modeling of the physical field in places where, due to different factors, different covariance parameters are required for different spatial directions.…”