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A comparison between parametric and non-parametric approaches to the analysis of replicated spatial point patterns

Part of: Stochastic processes Geometric probability and stochastic geometry Inference from stochastic processes

Published online by Cambridge University Press:  01 July 2016

Peter J. Diggle ,
Jorge Mateu  and
Helen E. Clough
Peter J. Diggle*
Affiliation:
Lancaster University
Jorge Mateu*
Affiliation:
Universitat Jaume I
Helen E. Clough*
Affiliation:
University of Liverpool
*
∗ Postal address: Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF, UK. Email address: p.diggle@lancaster.ac.uk
∗∗ Postal address: Department of Mathematics, Universitat Jaume I, E-12071 Castellón, Spain.
∗∗ Postal address: Department of Mathematical Sciences, University of Liverpool, Liverpool L69 3BX, UK.
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Abstract

The paper compares non-parametric (design-based) and parametric (model-based) approaches to the analysis of data in the form of replicated spatial point patterns in two or more experimental groups. Basic questions for data of this kind concern estimating the properties of the underlying spatial point process within each experimental group, and comparing the properties between groups. A non-parametric approach, building on work by Diggle et. al. (1991), summarizes each pattern by an estimate of the reduced second moment measure or K-function (Ripley (1977)) and compares mean K-functions between experimental groups using a bootstrap testing procedure. A parametric approach fits particular classes of parametric model to the data, uses the model parameter estimates as summaries and tests for differences between groups by comparing fits with and without the assumption of common parameter values across groups. The paper discusses how either approach can be implemented in the specific context of a single-factor replicated experiment and uses simulations to show how the parametric approach can be more efficient when the underlying model assumptions hold, but potentially misleading otherwise.

Keywords

Expected significance levelsK-functionpseudo-likelihood functionreplicated spatial point patternsspatial analysis of variance

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60G55: Point processes 62M30: Spatial processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 32 , Issue 2 , June 2000 , pp. 331 - 343
Copyright
Copyright © Applied Probability Trust 2000 

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References

Baddeley, A. J. and Möller, J. (1989). Nearest-neighbour Markov point processes and random sets. Int. Statist. Rev. 57, 89–121.Google Scholar
Baddeley, A. J. and Silverman, B. W. (1984). A cautionary example on the use of second-order methods for analyzing point patterns. Biometrics 40, 1089–93.Google Scholar
Baddeley, A. J., Reid, S., Howard, V. and Boyde, A. (1985). Unbiased estimation of particle density in the tandem scanning reflected light microscope. J. Microsc. 138, 203–212.Google Scholar
Baddeley, A. J., Moyeed, R. A., Howard, C. V. and Boyde, A. (1993). Analysis of a three-dimensional point pattern with replication. J. Appl. Statist. 42, 641–668.Google Scholar
Besag, J. (1977). Some methods of statistical analysis for spatial data. Bull. Int. Statist. Inst. 47, 77–92.Google Scholar
Cressie, N. A. (1993). Statistics for Spatial Data, Revised edn. John Wiley, New York.Google Scholar
Daley, D. J. and Vere-Jones, D. (1988). Introduction to the Theory of Point Processes. Springer, New York.Google Scholar
Dempster, A. P. and Schatzoff, M. (1965). Expected significance level as a sensitivity index for test statistics. J. Amer. Statist. Assoc. 60, 420–436.Google Scholar
Diggle, P. J. (1983). Statistical Analysis of Spatial Point Patterns. Academic Press, London.Google Scholar
Diggle, P. J. (1986). Parametric and non-parametric estimation for pairwise interaction point processes. In Proc. 1st World Congr. Bernoulli Society.Google Scholar
Diggle, P. J., Lange, N. and Benes, F. (1991). Analysis of variance for replicated spatial point patterns in clinical neuroanatomy. J. Amer. Statist. Assoc. 86, 415, 618–625.Google Scholar
Diggle, P. J., Fiksel, T., Grabarnik, P., Ogata, Y., Stoyan, D. and Tanemura, M. (1994). On parameter estimation for pairwise interaction point processes. Int. Statist. Rev. 62, 99–117.CrossRefGoogle Scholar
Evans, S. M. and Gundersen, H. J. G. (1989). Estimation of spatial distributions using the nucleator. Acta Stereologica 8, 395–400.Google Scholar
Gates, D. J. and Westcott, M. (1986). Clustering estimates for spatial point distributions with unstable nopagebreak[4] potentials. [4] Ann. Inst. Statist. Math. 38, 123–135.CrossRefGoogle Scholar
Jensen, J. L. (1993). Asymptotic normality of estimates in spatial point processes. Scand. J. Statist. 20, 97–109.Google Scholar
Jensen, J. L. and Möller, J. (1991). Pseudo-likelihood for exponential family models of spatial point processes. Ann. Appl. Prob. 1, 445–461.CrossRefGoogle Scholar
Kelly, F. P. and Ripley, B. D. (1976). On Strauss' model for clustering. Biometrika 63, 357–60.Google Scholar
Möller, J., (1994). Markov chain Monte Carlo and spatial point processes. Rep. 293, University of Aarhus.Google Scholar
Möller, J., (1999). Markov chain Monte Carlo and spatial point processes. In Stochastic Geometry, Likelihood, and Computation, Boca Raton, eds Barndorff-Nielsen, O. E., Kendall, W. S. and van Lieshout, M. N. M. Chapman and Hall/CRC, London, pp. 141–172.Google Scholar
Ogata, Y. and Tanemura, M. (1981). Estimation of interaction potentials of spatial point patterns through the maximum likelihood procedure. Ann. Inst. Statist. Math. B, 33, 315–338.CrossRefGoogle Scholar
Ogata, Y. and Tanemura, M. (1984). Likelihood analysis of spatial point patterns. J. Roy. Statist. Soc. B, 46, 496–518.Google Scholar
Ogata, Y. and Tanemura, M. (1989). Likelihood estimation of soft-core interaction potentials for Gibbsian point patterns. Ann. Inst. Statist. Math. B, 41, 583–600.Google Scholar
Penttinen, A. (1984). Modelling Interaction in Spatial Point Patterns: Parameter Estimation by the Maximum Likelihood Method. Jÿvaskÿla Studies in Computer Science, Economics and Statistics, No. 7. Jÿvaskÿlan Yliopisto, Jÿvaskÿla, Finland.Google Scholar
Ripley, B. D. (1977). Modelling spatial patterns (with discussion). J. Roy. Statist. Soc. B, 39, 172–212.Google Scholar
Ripley, B. D. (1979). AS137 Simulating spatial patterns. Appl. Statist. 28, 109–112.Google Scholar
Ripley, B. D. (1981). Spatial Statistics. John Wiley, New York.Google Scholar
Ripley, B. D. (1988). Statistical Inference for Spatial Processes. Cambridge University Press.CrossRefGoogle Scholar
Ripley, B. D. and Kelly, F. P. (1977). Markov point processes. J. Lond. Math. Soc. 15, 188–192.Google Scholar
Strauss, D. J. (1975). A model for clustering. Biometrika 63, 467–475.Google Scholar
Wilson, H. E. (1998). Statistical analysis of replicated spatial point patterns. , University of Lancaster, UK.Google Scholar
Wilson, H. E., Diggle, P. J. and Howard, C. V. (1998). Methods for the analysis of replicated spatial point patterns in clinical neuroanatomy. Abstract, Proc. 9th International Workshop in Stereology, Stochastic Geometry and Image Analysis, Comillas, October 1997. Adv. Appl. Prob. 30, 292–293.Google Scholar