Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-10T14:03:27.477Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Multivariate analysis

Published online by Cambridge University Press:  12 June 2012

BENJAMIN J. BINDER ,
EMILY J. HACKETT-JONES ,
JONATHAN TUKE  and
KERRY A. LANDMAN
BENJAMIN J. BINDER*
Affiliation:
School of Mathematical Sciences, The University of Adelaide, South Australia 5005, Australia (email: benjamin.binder@adelaide.edu.au, simon.tuke@adelaide.edu.au)
EMILY J. HACKETT-JONES
Affiliation:
Department of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Australia (email: kerryl@unimelb.edu.au, emilyhackettjones@gmail.com)
JONATHAN TUKE
Affiliation:
School of Mathematical Sciences, The University of Adelaide, South Australia 5005, Australia (email: benjamin.binder@adelaide.edu.au, simon.tuke@adelaide.edu.au)
KERRY A. LANDMAN
Affiliation:
Department of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Australia (email: kerryl@unimelb.edu.au, emilyhackettjones@gmail.com)
*
✉For correspondence; e-mail: benjamin.binder@adelaide.edu.au
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

pólya urnspatial statisticsvolume exclusion

MSC classification

Secondary: 62H11: Directional data; spatial statistics
Type
Research Article
Information
The ANZIAM Journal , Volume 53 , Issue 2 , October 2011 , pp. 122 - 133
Copyright
Copyright © Australian Mathematical Society 2012

References

[1]Binder, B. J., “Ghost rods adopting the role of withdrawn baffles in batch mixer designs”, Phys. Lett. A 374 (2010) 3483–3486; doi:10.1016/j.physleta.2010.06.039.CrossRefGoogle Scholar
[2]Binder, B. J. and Landman, K. A., “Exclusion processes on a growing domain”, J. Theor. Biol. 259 (2009) 541–551; doi:10.1016/j.jtbi.2009.04.025.CrossRefGoogle ScholarPubMed
[3]Binder, B. J. and Landman, K. A., “Quantifying evenly distributed states in exclusion and non-exclusion processes”, Phys. Rev. E. 83 (2011) 041914; doi:10.1103/PhysRevE.83.041914.CrossRefGoogle Scholar
[4]Chung, C. A., Lin, T.-H., Chen, S.-D. and Huang, H.-I., “Hybrid cellular automaton modeling of nutrient modulated cell growth in tissue engineering constructs”, J. Theoret. Biol. 262 (2010) 267–278; doi:10.1016/j.jtbi.2009.09.031.CrossRefGoogle ScholarPubMed
[5]Davies, O. L., “On asymptotic formulae for the hypergeometric series: I. Hypergeometric series in which the fourth element, x, is unity”, Biometrika 25 (1933) 295–322; doi:10.1093/biomet/25.3-4.295.CrossRefGoogle Scholar
[6]Davies, O. L., “On asympotic formulae for the hypergeometric series: II. Hypergeometric series in which the fourth element, x, is not necessarily unity”, Biometrika 26 (1934) 59–107; doi:10.1093/biomet/26.1-2.59.Google Scholar
[7]Diggle, P. J., Statistical analysis of spatial point patterns (Academic Press, London, 1983).Google Scholar
[8]Eggenberger, F. and Pólya, G., “Über die Statistik verketteter Vorgänge”, Z. angew. Math. Mech. 3 (1923) 279–289; doi:10.1002/zamm.19230030407.CrossRefGoogle Scholar
[9]Enderling, H., Hlatky, L. and Hahnfeldt, P., “Migration rules: Tumours are conglomerates of self-metastases”, Brit. J. Cancer 100 (2009) 1917–1925; doi:10.1038/sj.bjc.6605071.CrossRefGoogle ScholarPubMed
[10]Johnson, N. L., Kemp, A. W. and Kotz, S., Univariate discrete distributions, 3rd edn (John Wiley & Sons, Hoboken, NJ, 2005).CrossRefGoogle Scholar
[11]Johnson, N. L. and Kotz, S., Urn models and their applications (John Wiley & Sons, New York, 1977).Google Scholar
[12]Jones, S. W., “The enhancement of mixing by chaotic advection”, Phys. Fluids A. 3 (1991) 1081–1086; doi:10.1063/1.858089.CrossRefGoogle Scholar
[13]Kemp, C. D. and Kemp, A. W., “Generalized hypergeometric distributions”, J. R. Stat. Soc. B 18 (1956) 202–211; http://www.jstor.org/stable/2983704.Google Scholar
[14]Khain, E., Sander, L. M. and Schneider-Mizell, C. M., “The role of cell–cell adhesion in wound healing”, J. Stat. Phys. 128 (2007) 209–218; doi:10.1007/s10955-006-9194-8.CrossRefGoogle Scholar
[15]Kholfi, S. and Mahmoud, H. M., “The class of tenable zero-balanced Pólya urns with an initially dominant subset of colors”, Stat. Prob. Lett. 82 (2012) 49–57; doi:10.1016/j.spl.2011.08.006.CrossRefGoogle Scholar
[16]Li, T. and Manas-Zloczower, I., “A study of distributive mixing in counterrotating twin screw extruders”, Int. Polym. Process. 10 (1995) 314–320; http://www.polymer-process.com/IPP950314.CrossRefGoogle Scholar
[17]Mahmoud, H. M., Pólya urn models (Chapman & Hall/CRC, Boca Raton, FL, 2008).CrossRefGoogle Scholar
[18]Noack, A., “A class of random variables with discrete distributions”, Ann. Math. Stat. 21 (1950) 127–132; doi:10.1214/aoms/1177729894.CrossRefGoogle Scholar
[19]Phelps, J. H. and Tucker, C. L., “Lagrangian particle calculations of distributive mixing: Limitations and applications”, Chem. Eng. Sci. 61 (2006) 6826–6836; doi:10.1016/j.ces.2006.07.008.CrossRefGoogle Scholar
[20]Ripley, B. D., Spatial statistics (John Wiley & Sons, New York, 1981).CrossRefGoogle Scholar
[21]Zhang, D., Brinas, I. M., Binder, B. J., Landman, K. A. and Newgreen, D. F., “Neural crest regionalisation for enteric nervous system formation: implications for Hirschsprung’s disease and stem cell therapy”, Dev. Biol. 339 (2010) 280–294; doi:10.1016/j.ydbio.2009.12.014.CrossRefGoogle ScholarPubMed