Covariate construction of nonconvex windows for spatial point pattern data

Abstract

In the field of spatial statistics, window selection for point pattern data is a complex process. In some cases, the point pattern window is given a priori when a local phenomena is studied. In other cases, a researcher may choose this region using some objective means that reflects their view that the window may be representative of a larger region, or based on a probability sampling method. The common approaches used are the smallest rectangular bounding window and convex windows due to the obvious use of the Euclidean distance. The chosen window must however cover the true domain of the sampled point pattern data. Choosing a window too large results in estimation and inference in areas which are empty of observed data, but for which it has not been confirmed that observations could have occurred there. These holes in the domain could be regions where for some geographic (or other) reason the phenomena of interest does not occur. In this mini-dissertation a review of methods for spatial convex and nonconvex window estimation is provided, and an algorithm is proposed for selecting the point pattern domain without the restriction of convexity, allowing for a better fit to the true domain, and based on spatial covariate information. The effect of the window choice on spatial intensity estimates is illustrated by giving particular attention to the technique of smoothed kernel intensity estimation. The proposed algorithm is applied in the setting of rural villages in Tanzania's Mara province. As a spatial covariate, remotely sensed data based on the elevation of a point pattern is used in the form of a Digital Elevation Model (DEM) GTOPO30, specific to village house locations in this setting. Mathematical morphological operators are also used to extract physiographic features from the DEM and are included here as a preprocessing step in the spatial window domain modelling.