Statistical Depth in Spatial Point Process

Abstract

Statistical depth is widely used as a powerful tool to measure the center-outward rank of multivariate and functional data. Recent studies have introduced the notion of depth to the temporal point process, which exhibits randomness in the cardinality as well as distribution in the observed events. The proposed methods can well capture the rank of a point process in a given time interval, where a critical step is to measure the rank by using inter-arrival events. In this paper, we propose to extend the depth concept to multivariate spatial point process. In this case, the observed process is in a multi-dimensional location and there are no conventional inter-arrival events in the temporal process. We adopt the newly developed depth in metric space by defining two different metrics, namely the penalized metric and the smoothing metric, to fully explore the depth in the spatial point process. The mathematical properties and the large sample theory, as well as depth-based hypothesis testings, are thoroughly discussed. We then use several simulations to illustrate the effectiveness of the proposed depth method. Finally, we apply the new method in a real-world dataset and obtain desirable ranking performance.