Bayesian Modeling with Spatial Curvature Processes

Abstract

Spatial process models are widely used for modeling point-referenced variables arising from diverse scientific domains. Analyzing the resulting random surface provides deeper insights into the nature of latent dependence within the studied response. We develop Bayesian modeling and inference for rapid changes on the response surface to assess directional curvature along a given trajectory. Such trajectories or curves of rapid change, often referred to as wombling boundaries, occur in geographic space in the form of rivers in a flood plain, roads, mountains or plateaus or other topographic features leading to high gradients on the response surface. We demonstrate fully model based Bayesian inference on directional curvature processes to analyze differential behavior in responses along wombling boundaries. We illustrate our methodology with a number of simulated experiments followed by multiple applications featuring the Boston Housing data; Meuse river data; and temperature data from the Northeastern United States.

Keywords: Bayesian modeling; Directional Curvature; Gaussian Processes; Wombling.

Figure 1:

Spatial plots for synthetic patterns, from Pattern 1 (left) and Pattern 2 (right). Scales are shown in the legend alongside.

Figure 2:

(left) shows color coded directional gradients for segments, (center) shows color coded directional curvature for segments in the direction normal to the curve, (right) shows curves selected for performing curvature wombling. green indicates a positive significance, cyan indicates negative significance and white indicates no significance.

Figure 3:

Plots showing (left) probability density of median house prices (in USD 1000) (right) spatial plot of median owner occupied house prices in Boston.

Figure 4:

Plots (left to right) showing fitted process, divergence and Laplacian for the median house price surface.

Figure 5:

Curvature wombling on the Boston Housing Data.

Figure 6:

Plots showing heavy metal concentrations in the topsoil of a flood plain at 155 locations for (from left to right) Cadmium (Cd), Copper (Cu), Lead (Pb) and Zinc (Zn) (in mg/kg of soil).

Figure 7:

Plots showing results for curvature wombling on the Meuse river for Cadmium (Cd) concentration. Plots showing (left) the resulting fitted process (center) the contiguous segments that display significant gradients (right) the contiguous segments with significant curvature.