Gradient angle-based analysis for spatiotemporal point processes

Abstract

Spatiotemporal point processes (STPPs) are important in modeling randomly appeared events developed in space and time. Statistical methods of STPPs have been widely used in applications. In all of these methods, evaluations and inferences of intensity functions are the primary issues. The present article proposes a new method, which attempts to evaluate angles of gradient vectors of intensity functions rather than the intensity functions themselves. According to the nature of many natural and human phenomena, the evaluation of angle patterns of the gradient vectors is more important than the evaluation of their magnitude patterns because changes of angle patterns often indicate global changes of these phenomena. This issue is investigated by simulation studies, where significant variations of gradient angle patterns are identified only when modes of intensity functions are changed. To study these phenomena, the article proposes an analysis method for gradient angles of the first-order intensity function of STPPs. The proposed method is used to analyze aftershock earthquake activities caused by great mainshock earthquakes occurred in Japan 2011 and Indian Ocean 2004, respectively, where a significant global change in the second case is identified.