Sufficient Dimension Reduction for Spatial Point Processes Directed by Gaussian Random Fields

Abstract

We develop a sufficient dimension reduction paradigm for inhomogeneous spatial point processes driven by a Gaussian random fields. Specifically, we introduce the notion of the kth-order Central Intensity Subspace. We show that a Central Subspace (Cook, 1998) can be defined as the combination of all Central Intensity Subspaces. For many commonly used spatial point process models, we find that the Central Subspace is equivalent to the first-order Central Intensity Subspace. To estimate the latter, we propose a flexible framework under which most existing benchmark inverse regression methods can be extended to the spatial point process setting. We develop novel graphical and formal testing methods to determine the structural dimension of the Central Subspace. These methods are extremely versatile in that they do not require any specific model assumption on the correlation structures of the covariates and the spatial point process. To illustrate the practical use of the proposed methods, we apply them to both simulated data and two real examples.