Approximations to hard-core models and their application to statistical analysis

Westcott, Mark

doi:10.2307/3213567

Cited by 5 publications

(4 citation statements)

References 12 publications

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“…In Ogata and Tanemura (1981), an approximation for Z itself was applied and the feasibility of the procedure shown in Section 2 was demonstrated by some computer experiments. Gates and Westcott (1980) and Westcott (1982) use integrals (3.3) to fmd the bounds for the distribution of minimum interpoint distance of hard-core models. However, we should note that the rate of convergence of (3.2) depends on the potential <1>8 very much even if the density p is small.…”

Section: The Mayer Cluster Integralsmentioning

confidence: 99%

The Likelihood Analysis for Spatial Point Patterns

Ogata

Tanemura

1987

Jpn J Biomet

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The likelihood procedure for estimating the pairwise interaction potential function is developed for statistical analysis of homogeneous spatial point patterns. Approximation methods of the normalizing factor of Gibbs canonical distribution are discussed both to estimate a scale parameter and to measure the softness (or hardness) of repulsive interactions. The approximations are useful up to a considerably high density. The validity of our procedure is demonstrated by some computer experi-. ments. Some real data are analysed.

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Section: The Mayer Cluster Integralsmentioning

confidence: 99%

The Likelihood Analysis for Spatial Point Patterns

Ogata

Tanemura

1987

Jpn J Biomet

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“…For k = 2 , the algebra required to generate recursive formulas for the bounds is relatively simple (see Lieb 1963, Gates and Westcott 1980, and Westcott 1982. For k > 2 , however, more care is required to ensure that the inequalities are preserved during the reduction of order of the I's.…”

Section: Calculation Of Recursive Bounds For 6?mentioning

confidence: 99%

On the Distributions of Scan Statistics

Gates

Westcott

1984

Journal of the American Statistical Association

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This article considers the distribution of the scan statistic over some interval of the real line, both for a fixed number of independent uniform points and for a Poisson process. It provides a new series method of calculating this distribution in the Poisson case and a simple self-contained recursive approximation in the fixed case. These results work best at low densities, which is precisely the case when the closed formulas are computationally impractical. The results are compared with values tabulated by Neff and Naus (1980) and with a recent approximation due to Naus (1982). There is also an explanation of why Naus's approximation is good in the Poisson case and a derivation of recursive bounds in the fixed case.

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“…The term 'Gibbsian interaction', is derived from statistical mechanics, where these models have been used for nearly a century to describe the behavior of gases (Ripley, 1990;Cressie, 1991). Examples of spatial stochastic models that take into consideration the interaction among events include sequential packing models of non-overlapping discs by Matern (1980), Bartlett (1974), and Diggle et al (1976); Poisson cluster models by Matern (1980) and Diggle (1979); and Strausstype and hard-core models by Strauss (1975), Kelly and Ripley (1976), Gates and Westcott (1980) and Westcott (1982). While most of this work has been theoretical, the increase in computing power makes it possible to estimate the parameters of these models using theoretical approximations to the likelihood function or by using computer simulations (Fiksel, 1988;Ogata and Tanemura, 1981;1984;1985;Ord, 1990;Ripley, 1990).…”

Section: Introductionmentioning

confidence: 99%

Modeling small-scale spatial interaction of shortgrass prairie species

Reich

Bonham

Metzger

1997

Ecological Modelling

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Native grasses interact spatially with themselves and their environment and can therefore be thought of as a system of dependent random variables. One method of modeling the spatial dependence of a multi-species population is a Gibbsian pairwise potential model. Since natural selection operates at the level of individual plants, the information obtained from such a model should provide a greater understanding of the intraspecific interactions in plant populations, while providing a theoretical basis for determining a plants' 'competitive zone' of influence. In this paper we fit a pairwise potential model to describe the spatial dependency of dominant grasses and forbs measured on a 1.5 x 1.5 m study plot located on a shortgrass prairie site near Fort Collins, Colorado. Dominant grasses included blue grama (Bouteloua gracilis), western wheatgrass (Agropyron smithii), Indian ricegrass (Oryzopsis hymenoides), and needle-and-thread grass (Stipa comata). Procedures for introducing spatial heterogeneity in the model is also discussed. © 1997 Elsevier Science B.V.

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Approximations to hard-core models and their application to statistical analysis

Cited by 5 publications

References 12 publications

The Likelihood Analysis for Spatial Point Patterns

The Likelihood Analysis for Spatial Point Patterns

On the Distributions of Scan Statistics

Modeling small-scale spatial interaction of shortgrass prairie species

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