Spatial Clustering Tests Based on the Domination Number of a New Random Digraph Family

Abstract

We use the domination number of a parametrized random digraph family called proportional-edge proximity catch digraphs (PCDs) for testing multivariate spatial point patterns. This digraph family is based on relative positions of data points from various classes. We extend the results on the distribution of the domination number of proportional-edge PCDs, and use the domination number as a statistic for testing segregation and association against complete spatial randomness. We demonstrate that the domination number of the PCD has binomial distribution when size of one class is fixed while the size of the other (whose points constitute the vertices of the digraph) tends to infinity and has asymptotic normality when sizes of both classes tend to infinity. We evaluate the finite sample performance of the test by Monte Carlo simulations and prove the consistency of the test under the alternatives. We find the optimal parameters for testing each of the segregation and association alternatives. Furthermore, the methodology discussed in this article is valid for data in higher dimensions also.

Keywords:

• Association
• Complete spatial randomness
• Consistency
• Delaunay triangulation
• Proximity catch digraph
• Proximity map
• Segregation

Mathematics Subject Classification:

• Primary 62H30
• Secondary 60D05, 05C80, 05C20, 62G10, 62G20, 62H11

Acknowledgments

Supported by DARPA as administered by the Air Force Office of Scientific Research under contract DOD F49620-99-1-0213 and by ONR Grant N00014-95-1-0777 and by TUBITAK Kariyer Project Grant 107T647. We also thank anonymous referees, whose constructive comments and suggestions greatly improved the presentation and flow of this article.