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.
Highlights
The goal of the research is to develop a gradient-based approach to spatiotemporal point processes (STPPs) which can be used to describe the global trend of point occurrences
We have proposed a gradient angle-based analysis method for spatiotemporal point processes (STPPs)
The method is developed based on a gradient-based assumption, called the angle invariant assumption, for the first-order intensity function of STPPs
Summary
The goal of the research is to develop a gradient-based approach to spatiotemporal point processes (STPPs) which can be used to describe the global trend of point occurrences. A similar issue has been pointed out in infectious disease studies since the spread of an infectious disease is roughly symmetric about the original occurrence center [24] In these examples, the direction of the tendency is more important than the magnitude since the change of directions represents a global change while the change of magnitudes only represents a local change. For nonstationary spatiotemporal point patterns, the change of angle patterns of gradient vectors of intensity functions is more interesting than the only change of magnitude patterns since the previous one represents a global change while the latter one represents a local change. The proofs of our asymptotic properties are displayed in the Appendix
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