A modified Pólya urn process and an index for spatial distributions with volume exclusion

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.