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θ-stationary point processes and their second-order analysis

Published online by Cambridge University Press:  14 July 2016

Karen Byth
Karen Byth*
Affiliation:
The Australian National University
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Abstract

The concept of θ-stationarity for a simple second-order point process in R2 is introduced. This concept is closely related to that of isotropy. Some θ-stationary processes are defined. Techniques are given for simulating realisations of these processes. The second-order analysis of these processes which have an obvious point of reference or origin is considered. Methods are suggested for modelling spatial patterns which are realisations of such processes. These methods are illustrated using simulated data. The ideas are extended to multitype point processes.

Keywords

ISOTROPYMOMENT MEASURESθ-STATIONARY POINT PROCESSESSPATIAL POINT PROCESSES
Type
Research Papers
Information
Journal of Applied Probability , Volume 18 , Issue 4 , December 1981 , pp. 864 - 878
Copyright
Copyright © Applied Probability Trust 1981 

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References

Byth, K. (1981) An analysis of the patterns formed by sporophores growing about a tree.Google Scholar
Byth, K. (1982) On kernel methods of estimating marginal radial and angular probability density functions. To appear.CrossRefGoogle Scholar
Diggle, P. J. (1979) On parameter estimation and goodness-of-fit testing for spatial point patterns. Biometrics 35, 87–101.CrossRefGoogle Scholar
Ford, E. D., Pelham, P. A. and Mason, J. (1980) Spatial patterns of sporophore distribution around a young birch tree in three successive years. Trans. British Mycological Soc .CrossRefGoogle Scholar
Kelly, F. P. and Ripley, B. D. (1976) A note on Strauss's model for clustering. Biometrika 63, 357–360.CrossRefGoogle Scholar
Lotwick, H. W. and Silverman, B. W. (1980) On the analysis of processes of points of several types. Submitted for publication.Google Scholar
Matérn, B. (1971) Doubly stochastic Poisson processes in the plane. In Statistical Ecology , ed. Patil, G. P., Pielou, E. C. and Waters, W. E., Pennsylvania State University Press, University Park, PA.Google Scholar
Ripley, B. D. (1976a) The second order analysis of stationary point processes. J. Appl. Prob. 13, 255–266.Google Scholar
Ripley, B. D. (1976b) On stationarity and superposition of point processes. Ann. Prob. 4, 999–1005.CrossRefGoogle Scholar
Ripley, B. D. (1976C) The disintegration of invariant measures. Math. Proc. Camb. Phil. Soc. 79, 337–341.CrossRefGoogle Scholar
Ripley, B. D. (1977) Modelling spatial patterns. J. R. Statist. Soc. B 39, 172–212.Google Scholar
Ripley, B. D. (1979a) Tests of ‘randomness’ for spatial point patterns. J. R. Statist. Soc. B 41, 368–374.Google Scholar
Ripley, B. D. (1979b) Simulating spatial patterns: dependent samples from a multivariate density. Appl. Statist. 28, 109–112.CrossRefGoogle Scholar