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Distance methods applied to a semi-deterministic clustering process

Published online by Cambridge University Press:  01 July 2016

Peter Diggle
Peter Diggle*
Affiliation:
University of Newcastle upon Tyne
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Extract

Aggregated spatial patterns may be generated by a clustering process (see, for example, Bartlett (1964)) in which ‘parent’ events are distributed completely at random, and produce, independently, random numbers of ‘offspring’ according to some distribution Pn; the position of each offspring relative to its parent is governed, independently, by a given bivariate distribution. Parents and offspring are assumed indistinguishable. For such a process, Bartlett (1974) shows that the distribution function F of the distance, X say, from a randomly selected point to the nearest event is given by where ρ denotes the mean number of parents per unit area, A is the circle with centre the origin and radius x, and E(ds) denotes the event ‘no offspring in A from parent in ds’.

Type
Spatial Pattern
Information
Advances in Applied Probability , Volume 7 , Issue 3 , September 1975 , pp. 450 - 451
Copyright
Copyright © Applied Probability Trust 1975 

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References

Bartlett, M. S. (1964) Spectral analysis of two-dimensional point processes. Biometrika 51, 299–311.CrossRefGoogle Scholar
Bartlett, M. S. (1974) The statistical analysis of spatial pattern. Adv. Appl. Prob. 6, 336–358.CrossRefGoogle Scholar
Diggle, P. J. (1975) Robust density estimation using distance methods. Biometrika 62, 39–48.CrossRefGoogle Scholar