Integral and differential characterizations of the Gibbs process

Nguyen, X. X.; Zessin, Hans

doi:10.2307/1426106

Cited by 36 publications

(50 citation statements)

References 3 publications

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“…Proposition 1 (Georgii (1976), Nguyen & Zessin (1979)). For any bounded set ƒ R 2 , a probability measure P on ƒ is the Quermass point process on ƒ with parameter Â 2 R 3 , intensitý > 0 and distribution of the radii (i.e.…”

Section: Quermass Model On the Whole Plane: The Markov Propertymentioning

confidence: 95%

“…In this section the Markov property of the Quermass model is displayed via the GNZ equation (Georgii (1976); Nguyen & Zessin (1979)). An alternative presentation, from a statistical physics point of view, can be found in Dereudre (2009).…”

Section: Quermass Model On the Whole Plane: The Markov Propertymentioning

confidence: 99%

“…In this paper, we estimate all parameters of the Quermass process, including the intensity, via a Takacs-Fiksel (TF) procedure, Takacs (1986);Fiksel (1988); Coeurjolly et al (2012). The TF contrast function is based on an empirical counterpart of the Georgii-Nguyen-Zessin (GNZ) equation (Georgii (1976); Nguyen & Zessin (1979)) and depends on the choice of test functions. In the context of point processes, the performance of the procedure depends on the latter choice.…”

Section: Introductionmentioning

confidence: 99%

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Estimation of the Intensity Parameter of the Germ‐Grain Quermass‐Interaction Model when the Number of Germs is not Observed

Dereudre

Lavancier

Helisová

2014

Scandinavian J Statistics

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The Quermass-interaction model allows to generalize the classical germ-grain Boolean model in adding a morphological interaction between the grains. It enables to model random structures with specific morphologies, which are unlikely to be generated from a Boolean model. The Quermass-interaction model depends in particular on an intensity parameter, which is impossible to estimate from classical likelihood or pseudo-likelihood approaches because the number of points is not observable from a germ-grain set. In this paper, we present a procedure based on the Takacs-Fiksel method, which is able to estimate all parameters of the Quermass-interaction model, including the intensity. An intensive simulation study is conducted to assess the efficiency of the procedure and to provide practical recommendations. It also illustrates that the estimation of the intensity parameter is crucial in order to identify the model. The Quermass-interaction model is finally fitted by our method to P. Diggle's heather data set.

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Section: Quermass Model On the Whole Plane: The Markov Propertymentioning

confidence: 95%

Section: Quermass Model On the Whole Plane: The Markov Propertymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Estimation of the Intensity Parameter of the Germ‐Grain Quermass‐Interaction Model when the Number of Germs is not Observed

Dereudre

Lavancier

Helisová

2014

Scandinavian J Statistics

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“…for any non-negative measurable function f (Papangelou, 1974), see also Georgii (1976) and Nguyen and Zessin (1979).…”

Section: Proof We Begin By Showing Thatmentioning

confidence: 99%

A J–function for inhomogeneous point processes

Lieshout

2011

Statistica Neerlandica

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We propose new summary statistics for intensity-reweighted moment stationary point processes, that is, point processes with translation invariant n-point correlation functions for all n ∈ N, that generalise the well known J -, empty space, and spherical Palm contact distribution functions. We represent these statistics in terms of generating functionals and relate the inhomogeneous J -function to the inhomogeneous reduced second moment function. Extensions to space time and marked point processes are briefly discussed.

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“…These point processes correspond to the Gibbs measures. The equilibrium in one dimension between the number of events and the integrated hazard rate may be replaced in higher dimension by the Campbell equilibrium equation or Georgii-Nguyen-Zessin formula (see Georgii (1976), Nguyen and Zessin (1979a) and Section 3), which is the basis for defining the class of h-residuals where h represents a test function. In particular, considered various choices of h, leading to the so-called raw residuals, inverse residuals and Pearson residuals, and showed that they share similarities with the residuals that are obtained for generalized linear models.…”

Section: Introductionmentioning

confidence: 99%

Residuals and Goodness-of-Fit Tests for Stationary Marked Gibbs Point Processes

Coeurjolly

Lavancier

2012

Journal of the Royal Statistical Society Series B: Statistical Methodology

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The inspection of residuals is a fundamental step to investigate the quality of adjustment of a parametric model to data. For spatial point processes, the concept of residuals has been recently proposed by Baddeley et al. as an empirical counterpart of the Campbell equilibrium equation for marked Gibbs point processes. The present paper focuses on stationary marked Gibbs point processes and deals with asymptotic properties of residuals for such processes. In particular, the consistency and the asymptotic normality are obtained for a wide class of residuals including the classical ones (raw residuals, inverse residuals, Pearson residuals). Based on these asymptotic results, we define goodness-of-fit tests with Type-I error theoretically controlled. One of these tests constitutes an extension of the quadrat counting test widely used to test the null hypothesis of a homogeneous Poisson point process.

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Integral and differential characterizations of the Gibbs process

Cited by 36 publications

References 3 publications

Estimation of the Intensity Parameter of the Germ‐Grain Quermass‐Interaction Model when the Number of Germs is not Observed

Estimation of the Intensity Parameter of the Germ‐Grain Quermass‐Interaction Model when the Number of Germs is not Observed

A J–function for inhomogeneous point processes

Residuals and Goodness-of-Fit Tests for Stationary Marked Gibbs Point Processes

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