Boundary Detection Using a Bayesian Hierarchical Model for Multiscale Spatial Data

Spatial boundary analysis has attained considerable attention in several disciplines including engineering, shape analysis, spatial statistics, and computer science. The inferential question of interest is often to identify rapid surface change of an unobserved latent process. Curvilinear wombling and crisp wombling (or fuzzy) are two major approaches that have emerged in Bayesian spatial statistics literature. These methods are limited to a single spatial scale even though data with multiple spatial scales are often accessible. Thus, we propose a multiscale representation of the directional derivative Karhunen–Loéve expansion to perform directionally based boundary detection. Taking a multiscale spatial perspective allows us, for the first time, to consider the concept of curvilinear boundary fallacy (CBF) error, which is a boundary detection analog to the ecological fallacy that is often studied in spatial change of support literature. Furthermore, we propose a directionally based multiscale curvilinear boundary error criterion to quantify CBF. We refer to this metric as the criterion for boundary aggregation error (BAGE), and use it to perform boundary detection. Several theoretical results are derived to motivate BAGE. In particular, we show that no BAGE exists when the directional derivatives of eigenfunctions of a KL expansion are constant across spatial scales. We illustrate the use of our model through a simulated example and an analysis of Mediterranean wind measurements data.
Supplementary materials for this article are available online.