Non-Stationary Spatial Modeling

By D. Higdon, Jenise L. Swall, John C. Kern. 2022

363 citations recorded by the source corpus.

Standard geostatistical models assume stationarity and rely on a variogram model to account for the spatial dependence in the observed data. In some instances, this assumption that the spatial dependence structure is constant throughout the sampling region is clearly violated. We present a spatial model which allows the spatial dependence structure to vary as a function of location. Unlike previous formulations which do not account for uncertainty in the speciﬁcation of this non-stationarity (eg. Sampson and Guttorp (1992)), we develop a hierarchical model which can incorporate this uncertainty in the resulting inference. The non-stationary spatial dependence is explained through a constructive ”process-convolution” approach, which ensures that the resulting covariance structure is valid. We apply this method to an example in toxic waste remediation.

Subfields: Environmental Engineering, Management Science and Operations Research, Statistics and Probability

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Environmental Engineering

In Management Science and Operations Research

In Statistics and Probability

Citations in the corpus

Taking the Human Out of the Loop: A Review of Bayesian Optimization (2016, background): “There are many other examples of nonstationary covariance functions [3], [22], [54], [70], [118], [121], [126], [132] that have been proposed for GP regression along with closely related output warping techniques [141] that can also model certain kinds of nonstationary processes.”
Input Warping for Bayesian Optimization of Non-Stationary Functions (2014, background): “Various non-stationary covariance functions have been proposed [e.g., Higdon et al., 1998, Rasmussen and Williams, 2006].”
Variable Noise and Dimensionality Reduction for Sparse Gaussian processes (2006, methodology): “There have been a number of previous approaches to developing non-stationary GP models, e.g. Paciorek and Schervish [2004], Higdon et al. [1999].”
Near-optimal sensor placements in Gaussian processes (2005, neutral): “If the bandwidth of a Gaussian basis f is allowed to vary with its position, this yields the model of Gibbs (1997), and with fewer restrictions, (Higdon et al., 1998).”
Interpolation of nonstationary air pollution processes: a spatial spectral approach (2002, background): “Ver Hoef and Barry (1998) and Higdon et al. (1999) give a model for accounting for heterogeneity in the spatial covariance function of a spatial process, using a moving average speci cation of a Gaussian process.”
Gaussian predictive process models for large spatial data sets (2008, methodology): “The integral representation permits kernel convolution with spatially varying kernels as in Higdon et al. (1999). It can also replace Brownian motion with a stationary Gaussian process on 2. Finite approximations allow replacing the white noise realizations ui with stationary process realizations su”
Space and Space-Time Modeling using Process Convolutions (2002, background): “The fine component refines the coarse specification by a factor of two in both spatial coordinates.”
Modeling Nonstationary Processes Through Dimension Expansion (2010, background): “The process–convolution approach (Higdon (1998), Higdon et al. (1999)) relies on the notion that a wide range of stationary Gaussian processes may be expressed as a kernel convolved with a Gaussian white noise process, with the kernel being allowed to vary spatially to account for nonstationarity.”
Computationally Efficient Convolved Multiple Output Gaussian Processes (2011, background): “Applications include the construction of nonstationary covariances (Higdon, 1998; Higdon et al., 1998; Fuentes, 2002a,b; Pciorek and Schervish, 2004) and spatiotemporal covariances (Wikle et al., 1998; Wikle, 2002, 2003).”
Dynamic Trees for Learning and Design (2009, background): “However, nonstationary GP modeling schemes (e.g., Higdon, Swall, and Kern 1999) require even more computational effort than their stationary counterparts.”
Bayesian inference for non‐stationary spatial covariance structure via spatial deformations (2003, methodology): “Higdon et al. (1999) proposed an alternative model for heterogeneity in the spatial covariance function, which is based on a moving average specification of a Gaussian process.”
A Spatio-Temporal Model for Mean, Anomaly, and Trend Fields of North Atlantic Sea Surface Temperature (2009, methodology): “To ensure the kernels vary smoothly over space, we also allow the processes to evolve with location, according to isotropic DPCs. Unlike Higdon et al. (1999), however, we do not use any hyperparameter to model these kernels, to accelerate convergence of the others in the MCMC.”
Spatial modelling using a new class of nonstationary covariance functions (2006, methodology): “An alternative to the eigendecomposition parameterization is to represent the Gaussian kernels as ellipses of constant probability density, parameterized by the focus and size of the ellipse, and to have the focal coordinates and ellipse sizes vary smoothly over space (Higdon et al. 1999).”
Spatial statistical models that use flow and stream distance (2006, neutral): “There is increasing use of these moving average models to construct valid autocovariances when confronted with new problems, such as multivariate models (Ver Hoef and Barry 1998; Ver Hoef et al. 2004), and nonstationary models (Higdon 1998; Higdon et al. 1999; Fuentes 2002).”
Bayesian Nonparametric Spatial Modeling With Dirichlet Process Mixing (2005, methodology): “These include spatially varying kernel convolution ideas as in Higdon, Swall and Kern (1999) and Higdon (2002) as well as a local stationarity approach as in Fuentes and Smith (2001).”
Generalized spatial dirichlet process models (2007, methodology): “Flexible and computationally tractable modelling to eliminate the stationarity assumption includes the spatially varying kernel approach of Higdon et al. (1999) and the local stationarity approach described in an unpublished North Carolina State University technical report by M. Fuentes and R.L.…”