Clustering of temporal event processes

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Abstract
A temporal point process is a sequence of points, each representing the occurrence time of an event. Each temporal point process is related to the behavior of an entity. As a result, clustering of temporal point processes can help differentiate between entities, thereby revealing patterns of behaviors. This study proposes a hierarchical cluster method for clustering temporal point processes based on the discrete Fréchet (DF) distance. The DF cluster method is divided into four steps: (1) constructing a DF similarity matrix between temporal point processes; (2) constructing a complete linkage hierarchical tree based on the DF similarity matrix; (3) clustering the point processes with a threshold determined by locating the local maxima on the curve of the pseudo-F statistic (an index which measures the separability between clusters and the compactness in clusters); and (4) identifying inner patterns for each cluster formed by a series of dense intervals, each of which contains at least one event of all processes of the cluster. The contributions of the article are: (1) the proposed DF cluster method can cluster temporal point processes into different groups and (2) more importantly, it can identify the inner pattern of each cluster. Two synthetic data sets were created to illustrate the DF distance between temporal point process clusters (the first data set) and validate the proposed DF cluster method (the second data set), respectively. An experiment and a comparison with a method based on dynamic time warping show that DF cluster successfully identifies the preconfigured patterns in the second synthetic data set. The cluster method was then applied to a population migration history data set for the Northern Plains of the United States, revealing some interesting population migration patterns.

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