A Nonparametric Index for Determining the Numbers of Events in Clusters

The task of discriminating between heterogeneity and complete spatial randomness (CSR) for a given point process can be divided into three subtasks: the identification of the point pattern; the determination of the sizes of clusters; and the estimation of the numbers of events in dominant clusters. Many studies have been performed regarding the first and second subtasks. However, limited work has been done on the third aspect; hence, the determination of the number of events in each dominant cluster is still an unsolved problem. In this paper, we provide a solution by constructing a new index which is defined as the ratio between the variance of the (k+1)th nearest distance and that of the kth nearest distance. Our method can be divided into two phases: the detection of point pattern and the estimation of the numbers of events in dominant clusters. These phases can be estimated by the values at which the index abruptly decreases to be less than 1. A comparative study between the existing indices and our index shows the following: (i) our index can indicate the numbers of events in dominant clusters in a relatively objective way, which is different from the K-function revealing the sizes of clustered patterns; (ii) it is a nonparametric index and is easy to implement; and (iii) it demonstrates the highest detection power for differentiating between heterogeneity and CSR. The simulations and two seismic case studies also confirmed the correctness of our method.