Spatial Process Gradients and Their Use in Sensitivity Analysis for Environmental Processes

• Activity Number: 81
• Type: Contributed
• Date/Time: Sunday, August 4, 2013 : 4:00 PM to 5:50 PM
• Sponsor: Section on Statistics and the Environment
• Author(s): Maria Terres*+ and Alan E. Gelfand
• Companies: Duke University and Duke University
• Keywords: Spatial Gradient ; Gaussian Process ; Matern Covariance

Abstract

Gradient analysis for spatial processes was originally developed in Banerjee et al. (2003). Assuming a Gaussian Process model for the variable of interest, distribution theory was obtained for the associated directional derivatives at each location. These ideas are extended in the current work to accommodate a continuous covariate whose directional derivatives are also of interest. The Gaussian structure of all variables, including the directional derivatives, allows for kriging across the spatial region using multivariate normal theory. The resulting distributions are completely determined through the data and model parameters, allowing for all gradient analysis to occur post-model fitting. Such methods allow us to explore questions regarding the relationship between the directions of maximum gradient and rates of changes of the variables. These techniques are demonstrated for a dataset from Duke Forest consisting of elevation across the study site and a point pattern indicating tree locations within the site. A log-Gaussian Cox process is used to model the intensity of the point pattern. Gradient analysis is then applied to compare elevation and the fitted intensity surface.