Premium
POINT PROCESSES WITH A CLUSTERING TRANSITION
Author(s) -
And David Gates,
Westcott Mark
Publication year - 1988
Publication title -
australian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.434
H-Index - 41
eISSN - 1467-842X
pISSN - 0004-9581
DOI - 10.1111/j.1467-842x.1988.tb00468.x
Subject(s) - statistical physics , cluster analysis , critical point (mathematics) , criticality , phase transition , mathematics , pairwise comparison , statistic , constant (computer programming) , transition point , dimension (graph theory) , planar , point (geometry) , distribution (mathematics) , physics , mathematical analysis , computer science , combinatorics , statistics , geometry , thermodynamics , computer graphics (images) , nuclear physics , programming language
summary The paper examines the properties of a simple pairwise interaction point process or distribution, when the potential is attractive. This leads to various degrees of clustering. Using measures such as the normalizing constant, number of close pairs and a generalized scan statistic, it is found that there are effectively three regimes for the model, determined by the relative magnitudes of numbers of points N , the point domain V and strength of interaction v. As N v increases, abrupt transitions between these regimes occur with the middle “critical” one holding for only a relatively small range of N v ‐values. It is possible to simulate the process reliably in one dimension, unlike the planar version, and confirmatory results are reported. A discrete version of the model, more susceptible to analysis, is introduced; it leads to the same conclusions and exhibits a criticality resembling, but distinct from, that in processes with phase transitions. Possible application of the discrete model to urbanization is described.