Spatial modelling using a new class of nonstationary covariance functions

By Christopher J. Paciorek, Mark J. Schervish. Environmetrics, 2006

533 citations recorded by the source corpus.

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.

Subfields: Artificial Intelligence, Environmental Engineering, Global and Planetary Change

View Paper: https://www.semanticscholar.org/paper/847561c52be4d400760a6d67cc4478e2c16850dc

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Citations in the corpus

Second-order non-stationary modeling approaches for univariate geostatistical data (2017, methodology): “Fouedjio et al. (2016) show that the closed-form nonstationary covariance functions with locally varying geometric anisotropy introduced by Paciorek and Schervish (2006) and Stein (2005) can be derived from a generalized convolution model.”
Space–time modelling of precipitation by using a hidden Markov model and censored Gaussian distributions (2009, methodology): “The dimensionless factor κ2.x, y/ (the ratio of the geometric and arithmetic means of x2 and y2) is bounded above by 1, and the fact that Σ.s/ is positive definite follows from Paciorek and Schervish (2006), theorem 1, using the exponential correlation function with diagonal kernel matrix.”
Nonstationary covariance models for global data (2008, methodology): “Kernel convolution and its variants have been applied in several papers to create nonstationary covariance functions such as in Higdon (1998), Higdon, Swall and Kern (1999), Fuentes (2002) and Paciorek and Schervish (2006).”
Moving window kriging with geographically weighted variograms (2010, methodology): “1 which mainly operate within the classical geostatistics framework, but also more modern nonstationary predictors such as the kernel-based (KS‐StK) models of Fuentes (2001), Fuentes et al. (2003), the (different) kernel-based models of Higdon et al. (1999), Paciorek and Schervish (2006) , and (pos”
Some topics in convolution-based spatial modeling (2007, unknown): “Paciorek and Schervish (2006) extend these nonstationary process-convolution models proposed by Higdon et al. (1998).”
Non-Stationary Dependence Structures for Spatial Extremes (2014, methodology): “In the present paper, we focus on the kernel convolution approach advocated by Paciorek and Schervish (2006). The latter have proposed a very general construction of non-stationary correlation functions that are based on known isotropic correlation models.”
Estimation of a non-stationary model for annual precipitation in southern Norway using replicates of the spatial field (2014, methodology): “Others have also reported only small improvements in predictive performance changing from a stationary to a non-stationary model (Paciorek and Schervish, 2006; Reich et al., 2011).”
Considering covariates in the covariance structure of spatial processes (2011, methodology): “As outlined in (Paciorek and Schervish, 2006), a covariance structure that generalizes to anisotropic covariance functions and account for directionality is one that uses the Mahalanobis distance between locations.”
A class of Matérn-like covariance functions for smooth processes on a sphere (2015, methodology): “We used the nonstationary Matérn covariance model that allows for spatially varying local dependence through ν (Stein, 2005; Paciorek and Schervish, 2006; Kleiber and Nychka, 2012).”
Gaussian Processes for time-marked time-series data (2012, background): “While other non-parametric Gaussian Processes have been studied (Sampson and Guttorp, 1992; Nott and Dunsmuir, 2002; Schmidt and O’Hagan, 2003; Murray-Smith and Pearlmutter, 2003; Paciorek and Schervish, 2006, 2003), we consider in depth the particular case of causal GPs.”
An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach (2011, background): “…function and there is a vast literature on this subject (Sampson and Guttorp, 1992; Higdon, 1998; Hughes-Oliver et al., 1998; Cressie and Huang, 1999; Higdon et al., 1999; Fuentes, 2001; Gneiting, 2002; Stein, 2005; Paciorek and Schervish, 2006; Jun and Stein, 2008; Yue and Speckman, 2010).”
A full scale approximation of covariance functions for large spatial data sets (2012, methodology): “We adopted a non-stationary version of the Matérn covariance as used in Banerjee et al. (2008), section 5.1.2, to simulate the spatial random effects (see also Paciorek and Schervish (2006)).”
Modeling an augmented Lagrangian for improved blackbox constrained optimization (2014, methodology): “There are modern GP methods that accommodate nonstationarity (Schmidt and O’Hagan, 2003; Paciorek and Schervish, 2006) and even regime-changing behavior (Gramacy and Lee, 2008).”
Local Gaussian Process Approximation for Large Computer Experiments (2013, background): “…input space to one where stationarity reigns (e.g. Schmidt and O’Hagan, 2003, and references therein), and process convolution approaches that allow the kernels to vary smoothly in parameterization as an unknown function of their spatial location (e.g., Paciorek and Schervish, 2006, and therein).”
A Spatio-Temporal Model for Mean, Anomaly, and Trend Fields of North Atlantic Sea Surface Temperature (2009, methodology): “functions using convolutions (Paciorek and Schervish 2006).”