Advertisement

Mathematical Geology

, Volume 28, Issue 5, pp 601–624 | Cite as

A general family of counting distributions suitable for modeling cluster phenomena

  • Jef Caers
Article

Abstract

The two-dimensional spatial distribution of precious stones, such as diamonds in alluvial and coastal deposits, shows a high degree of clustering. Usually, stones tend to gather in relatively small clusters or traps, made by potholes, gullies, or small depressions in the rough bedcock. Therefore, when taking samples of such deposits, discrete distributions of the number of stones counted in each sample yield an extreme skewness. Most samples have no stones, whereas samples containing a few hundred stones are not unusual. This paper constructs a model and a method for fitting a new and general family of counting distributions based on the Neyman-Scott cluster model and the mixed Poisson process, which can be used to model a differing degree of clustering. General recursion equations for the discrete probabilities of these distributions are derived. Application of this model to simulated data shows that information such as cluster size, number of point events per cluster, and number of clusters per measurement unit can be extracted easily from this model. Fitting the model to data of two real diamond deposits of a totally different nature—small rich clusters of Namibia versus larger but less rich clusters of Guinea—demonstrates its flexibility.

Key words

discrete distribution Neyman-Scott cluster process mixed Poisson compound Poisson discrete random function diamond deposit 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Abramowitz, M., and Stegun, I. A., 1970, Handbook of mathematical functions: Dover Publ., Inc., New York, 1046 p.Google Scholar
  2. Agterberg, F. P., 1975, Spatial clustering and lognormal size distribution of volcanogenic massive sulphide deposits in the Bathurst area: Geol. Survey Canada. Paper 15-1C, p. 169–173.Google Scholar
  3. Agterberg, F. P., 1984, Discrete probability distributions for mineral deposits in cells: Utrecht VNU Science Press, Proceedings 27th Intern. Geol. Congress, v. 20, p. 205–225.Google Scholar
  4. Beirlant, J., Teugels, J. L., and Vynckier, P., 1993, Extremes in non-life insurance,in Galambos, J., ed., Extreme value theory and applications: Kluwer Academic Publ., Dordrecht, p. 489–510.Google Scholar
  5. Feller, W., 1943, On a general class of contagious distributions: Ann. Math. Stat., v. 14, no. 4, p. 389–400.Google Scholar
  6. Kleingeld, W., 1987, Geostatistics of the discrete particle: unpubl. doctorate dissertation, Ecole Nat. Sup. des Mines de Paris, Centre de Géostatistique, Fontainebleau, 275 p.Google Scholar
  7. Lajaunie, Ch., and Lantuejoul, Ch., 1989, Setting up the general methodology for discrete isofactorial models,in Armstrong, M., ed., Geostatistics: Kluwer Academic Publ., Dordrecht, v. 1, p. 323–334.Google Scholar
  8. Matern, B., 1971, Doubly stochastic Poisson process in the plain,in Patil, G. P., Pielou, E. C., and Waters, W. E., eds., Statistical ecology: Penn. State Univ., University Park, Pennsylvania, p. 195–213.Google Scholar
  9. Matheron, G., 1981a, Quatre familles discrètes: Research Rept. N-703, Centre de Géostatistique, Fontainebleau, 48 p.Google Scholar
  10. Matheron, G., 1981b, Deux autres familles, non moins discrètes, mais plus nombreuses: Research Rept. N-717, Centre de Géostatistique, Fontainebleau, 63 p.Google Scholar
  11. Neyman, J., 1939, On a new class of contagious distributions, applicable in entomology and bacteriology: Ann. Math. Stat., v. 10, no. 1, p. 35–57.Google Scholar
  12. Neyman, J., and Scott, E. L., 1958, Statistical approach to problems of cosmology: Jour. Roy. Stat. Soc., Ser. B, v. 20, no. 1, p. 1–29.Google Scholar
  13. Ogata, Y., 1988, Statistical models for earthquake occurrences and residual analysis for point processes: Jour. Am. Stat. Assoc., v. 83, no. 1, p. 9–27.Google Scholar
  14. Oosterveld, M. M., Campbell, D., and Hazell, K. R., 1987, Geology related to statistical evaluation parameters for a diamondiferous beach deposit: APCOM, 87, Proc. 20th Intern. Symp. Application of Computers and Mathematics in the Mineral Industries, v. 3: Geostatisties. South African Inst. Min. & Metall., Johannesburg, p. 129–136.Google Scholar
  15. Platt, W. J., Evans, G. W., and Rathbun, S. L., 1988, The population dynamics of a long lived conifer: Am. Naturalist, v. 131, no. 4, p. 419–525.Google Scholar
  16. Press, W. H., Flannery, B. P., Teukolsky, S. A., and Vetterling, W. T., 1990, Numerical recipes (FORTRAN), the art of scientific computing: Cambridge Univ. Press, Cambridge, 702 p.Google Scholar
  17. Rombouts, L., 1987, Evaluation of low grade/high value diamond deposits: Mining Magazine, v. 157, no. 3, p. 217–220.Google Scholar
  18. Rombouts, L., 1988, Geology and evaluation of the Guinean diamond deposits: Ann. Soc. Géol. Belg., v. 110, no. 3, p. 241–259.Google Scholar
  19. Sichel, H. S., 1973, Statistical valuation of diamondiferous deposits: Jour. South African Inst. Min & Met., v. 73, p. 235–243.Google Scholar
  20. Smith, R. L., 1989, Extreme value analysis of environmental time series: an application to trend detection in ground-level ozone: Statistical Science, v. 4, no. 4, p. 367–393.Google Scholar
  21. Vere-Jones, D., 1970, Stochastic models for earthquake occurrence: Jour. Roy. Stat. Soc., Ser. B, v. 32, no. 1, p. 1–45.Google Scholar

Copyright information

© International Association for Mathematical Geology 1996

Authors and Affiliations

  • Jef Caers
    • 1
  1. 1.Department of Civil EngineeringKU LeuvenHeverleeBelgium

Personalised recommendations