Summary statistics for measuring the relationship among three types of points in multivariate point patterns

In multivariate spatial point patterns' statistical analysis, conventional summary statistics can only detect the dependence between two types of points, and cannot be used to detect the dependence among three types of points. New summary statistics are proposed which can be used to detect the influence of the presence the kth type points on the relationship between the ith and the jth type points when the relationship between the ith and the jth type points is positive correlation (or negative correlation, or no spatial interaction), can also be used to infer information about the type of correlation and the range of interaction in multivariate point patterns. In order to reduce the edge-effects the border method to estimate the proposed summary statistics is applied. A simulation and a real example are used to illustrate the proposed methodologies.