The ANZIAM Journal

Research Article

A MODIFIED PÓLYA URN PROCESS AND AN INDEX FOR SPATIAL DISTRIBUTIONS WITH VOLUME EXCLUSION

BENJAMIN J. BINDERa1 c1, EMILY J. HACKETT-JONESa2, JONATHAN TUKEa1 and KERRY A. LANDMANa2

a1 School of Mathematical Sciences, The University of Adelaide, South Australia 5005, Australia (email: benjamin.binder@adelaide.edu.au, simon.tuke@adelaide.edu.au)

a2 Department of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Australia (email: kerryl@unimelb.edu.au, emilyhackettjones@gmail.com)

Abstract

Spatial data sets can be analysed by counting the number of objects in equally sized bins. The bin counts are related to the Pólya urn process, where coloured balls (for example, white or black) are removed from the urn at random. If there are insufficient white or black balls for the prescribed number of trials, the Pólya urn process becomes untenable. In this case, we modify the Pólya urn process so that it continues to describe the removal of volume within a spatial distribution of objects. We determine when the standard formula for the variance of the standard Pólya distribution gives a good approximation to the true variance. The variance quantifies an index for assessing whether a spatial point data set is at its most randomly distributed state, called the complete spatial randomness (CSR) state. If the bin size is an order of magnitude larger than the size of the objects, then the standard formula for the CSR limit is indicative of when the CSR state has been attained. For the special case when the object size divides the bin size, the standard formula is in fact exact.

(Received February 09 2012)

(Revised April 04 2012)

2010 Mathematics subject classification

  • primary 62H11; secondary 47N30

Keywords and phrases

  • pólya urn;
  • spatial statistics;
  • volume exclusion

Correspondence:

c1 For correspondence; e-mail: benjamin.binder@adelaide.edu.au