New nonparametric functions for measuring interactions in point patterns
by
Jianbao Chen
Department of Mathematics and Statistics, Curtin University of Technology
Coauthors: Adrian Baddeley (Department of Mathematics and Statistics, the University of Western Australia), Gopal Nair(Department of Mathematics and Statistics, Curtin University of Technology)

We propose functions F^* , G^* and J^*=(1-G^*)/(1-F^*) to quantify the strength and range of interpoint interactions in a point pattern, where F^* and G^* are distance distribution functions of r-close triples of arbitrary point and the typical pattern point, respectively. Function J^*(r) is identically equal to 1 for a Poisson point pattern. Values of J^*(r) greater or smaller than 1 suggest regularity or clustering respectively. The theoretical results of J^*(r) for some point patterns are derived. Edge-corrected estimators of F^*, G^* and J^* are constructed by using border method. We compute estimators of F^* , G^* and J^* for some standard real datasets and large sample simulated datasets and provide a Monte Carlo test for complete spatial randomness. Our simulation results show that G^* and J^* are sensitive to cluster and large sample regular point patterns but not to small sample regular point patterns. In addition, they can discriminate the large sample cell point patterns with homogeneous Poisson point patterns at short distance but function K can't. Therefore, they can be used as summary statistics to detect the types of large sample point patterns. Our results also show that F^* is not sensitive enough to different point patterns and can't be as a summary statistic.

Date received: April 5, 2002