A Third Order Point Process Characteristic

Second order characteristics, in particular Ripley's K‐function, are widely used for the statistical analysis of point patterns. We examine a third order analogue, namely the mean number T(r) of r‐close triples of points per unit area. Equivalently this is the expected number of r‐close point pairs in an r‐neighbourhood of the typical point. Various estimators for this function are proposed and compared, and we give an explicit formula for the isotropic edge correction. The theoretical value of T seems to be unobtainable for most point process models apart from the homogeneous Poisson process. However, simulation studies show that the function T discriminates well between different types of point processes. In particular it detects a clear difference between the Poisson process and the Baddeley–Silverman cell process whereas the K‐functions for these processes coincide.

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