A third order point process characteristic for multi-type point processes

The description and analysis of spatial point patterns have mainly been based on first- and second-order characteristics. However, and especially when analyzing complex and multivariate point patterns, the use of higher-order characteristics would be more informative. In this paper, we introduce a third-order characteristic for multi-type point processes, which is based on the number of r-close triples of points, where the three points are of three different types (species). This characteristic is useful, when the second-order characteristics indicate that the three point patterns are pairwise uncorrelated but there is some relationship between triples of points. Furthermore, we conjecture that the new statistic can be used to test independence between the three point processes