Multi-scale decomposition of point process data

To automatically identify arbitrarily-shaped clusters in point data, a theory of point process decomposition based on kth Nearest Neighbour distance is proposed. We assume that a given set of point data is a mixture of homogeneous processes which can be separated according to their densities. Theoretically, the local density of a point is measured by its kth nearest distance. The theory is divided into three parts. First, an objective function of the kth nearest distance is constructed, where a point data set is modelled as a mixture of probability density functions (pdf) of different homogeneous processes. Second, we use two different methods to separate the mixture into different distinct pdfs, representing different homogeneous processes. One is the reversible jump Markov Chain Monte Carlo strategy, which simultaneously separates the data into distinct components. The other is the stepwise Expectation-Maximization algorithm, which divides the data progressively into distinct components. The clustering result is a binary tree in which each leaf represents a homogeneous process. Third, distinct clusters are generated from each homogeneous point process according to the density connectivity of the points. We use the Windowed Nearest Neighbour Expectation-Maximization (WNNEM) method to extend the theory and identify the spatiotemporal clusters. Our approach to point processes is similar to wavelet transformation in which any function can be seen as the summation of base wavelet functions. In our theory, any point process data set can be viewed as a mixture of a finite number of homogeneous point processes. The wavelet transform can decompose a function into components of different frequencies while our theory can separate point process data into homogeneous processes of different densities. Two experiments on synthetic data are provided to illustrate the theory. A case study on reservoir-induced earthquakes is also given to evaluate the theory. The results show the theory clearly reveals spatial point patterns of earthquakes in a reservoir area. The spatiotemporal relationship between the main earthquake and the clustered earthquake (namely, foreshocks and aftershocks) was also revealed.