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Non-Stationary Spatial Modeling
Dave Higdon, Jenise Swall, John Kern
TL;DR
Stationary geostatistical models can fail when spatial dependence varies across a region, motivating models that represent non-stationary covariance. This paper uses a hierarchical process-convolution model with spatially evolving kernels, applies it to dioxin concentrations, and estimates uncertainty in the degree of non-stationarity. The application indicates that the inferred process supports spatially varying dependence while allowing the stationary case as τψ increases.
Problem
Stationary spatial models assume constant dependence across a region, but some settings require heterogeneous spatial covariance and prior formulations do not account for uncertainty in its specification.
Method
A hierarchical process-convolution model uses Gaussian moving-average kernels that evolve smoothly with location, producing a valid non-stationary covariance structure.
Results
A 95% credible interval for τψ is (31, 36), while τψ = 50 produces a solution effectively equivalent to the stationary case.
Takeaways & Limitations
The approach accounts for uncertainty in the extent of non-stationarity without requiring repeated realizations and was practically estimated with MCMC in the application.
Takeaways & Limitations
The prior specification is not unique, the range-parameter priors are somewhat clumsy, and prior-sensitivity investigations are ongoing.
Abstract
from arXiv · showhide
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.
1 INTRODUCTION
Geostatistical analyses commonly model spatial dependence with stationary Gaussian processes whose covariance depends only on distance. Because spatial covariance can vary across a region, the paper proposes a hierarchical moving-average model for location-dependent covariance and uncertainty.
- Geostatistical analyses commonly model spatial data with Gaussian processes and represent dependence through a covariogram based on distance.This specification produces a stationary random field whose distribution is invariant under spatial shifts.
- Stationary and isotropic assumptions can be inappropriate when spatial covariance is heterogeneous across the sampling region.Prior non-stationary approaches generally rely on repeated measurements, whereas Haas (1990) fits such a model from a single realization.
- The paper proposes a hierarchical moving-average Gaussian-process model whose kernel evolves over location to represent heterogeneous spatial covariance.The hierarchy allows uncertainty about the spatial dependence specification to enter inference.
2 SPECIFYING SPATIAL COVARIANCE
The paper constructs non-stationary Gaussian processes by allowing spatially varying smoothing kernels within a process-convolution framework. Spatially evolving kernel parameters define changing covariance structure while preserving a valid correlation function.
- Process convolution extends moving-average Gaussian-process models by allowing smoothing kernels to vary with spatial location.The approach first considers stationary moving-average formulations, then permits covariance structure to change gradually over space.
- The resulting spatial correlation is not a correlogram because it depends on locations s and s′, not distance alone.
- The constructive moving-average formulation guarantees a positive-definite, valid correlation function for any suitably bounded family of kernels.Working with kernels avoids the difficulty of ensuring symmetry and positive definiteness directly for every pair of locations.
- For the application, each location-specific kernel is normal, centered at s, and assigned a spatially varying covariance matrix Σ(s).The kernels are required to evolve smoothly over space, and the correlation formula is computationally tractable once the covariance parameters are specified.
- A geometrically based parameterization represents Σ(s) through spatially varying ellipse foci, encoding local kernel scaling, stretching, and rotation.Independent Gaussian fields generate ψx(s) and ψy(s), whose focus pairs define ellipses and corresponding covariance matrices.
- Figure 2 illustrates one realization by plotting spatially evolving kernel ellipses and the conditional surface they generate.The one-standard-deviation ellipses are shrunk by a factor of 10 for visibility; contour and perspective views show the realization.
3 APPLICATION IN ENVIRONMENTAL MONITORING
The Piazza Road application uses a hierarchical non-stationary spatial model to estimate dioxin concentrations while accounting for location-dependent covariance and uncertainty in its extent. MCMC explores kernel parameters and posterior spatial dependence using 60 measurements from the pilot area.
- Application motivation: Dioxin concentrations are modeled with location-dependent spatial dependence because contamination is higher near a winding stream channel.The expected dependence is stronger along the channel, motivating a spatially varying model.
- Data and model: The analysis uses n = 60 log concentration measurements from a restricted subset of the pilot area.The observations are decomposed into an overall mean, spatial trend, and i.i.d. Gaussian error.
- Prior specification: The spatial trend uses the non-stationary Gaussian-process prior, while the error precision receives a gamma prior.The spatial range parameters are constrained to produce a proper posterior.
- Computation: Posterior inference is conducted with MCMC, primarily using Metropolis and Metropolis-Hastings steps after integrating out the spatial trend.Most full conditional distributions do not have simple forms.
- Results: The parameter τψ controls non-stationarity, with spatially distributed kernel estimates examined at τψ = 25 and 50.The posterior distribution gives a 95% credible interval for τψ of (31, 36).
4 DISCUSSION
The paper develops a spatial process model for non-stationary dependence and demonstrates MCMC-based estimation in an environmental application. It incorporates uncertainty about heterogeneity, while acknowledging that the approach remains at an early development stage and depends on debatable prior choices.
- Discussion: The developed spatial process model allows non-stationary spatial dependence and supports practical estimation through MCMC.The authors present an application demonstrating this estimation strategy.
- Discussion: The approach accounts for uncertainty in the extent of non-stationarity without requiring repeated realizations from the spatial process.The authors view uncertainty about estimated spatial heterogeneity as important when evaluating model extensions.
- Discussion: The modeling approach remains in its initial stages, and its effectiveness in serious applications remains to be tested.The paper describes the initial indications as promising.
- Prior sensitivity: Uniform priors over fixed intervals for τz and τψ are described as somewhat clumsy, although the chosen intervals cover remotely plausible values for this application.Alternative prior formulations are possible in the hierarchical specification.
- Prior sensitivity: The data are unlikely to contain detailed information about the structure of the ψs, and prior-sensitivity investigations are ongoing.A possible alternative would shrink the ψs toward a common value rather than toward the origin.