Pseudolikelihood inference for Gibbsian T-tessellations ... and point processes

Recently a new class of planar tessellations, named T-tessellations, was introduced. Splits, merges and a third local modification named flip where shown to be sufficient for exploring the space of T-tessellations. Based on these local transformations and by analogy with point process theory, tools Campbell measures and a general simulation algorithm of Metropolis-Hastings-Green type were translated for random T-tessellations.The current report is concerned with parametric inference for Gibbs models of T-tessellations. The estimation criterion referred to as the pseudolikelihood is derived from Campbell measures of random T-tessellations and the Kullback-Leibler divergence. A detailed algorithm for approximating the pseudolikelihood maximum is provided. A simulation study seems to show that bias and variability of the pseudolikelihood maximum decrease when the tessellated domain grows in size.In the last part of the report, it is shown that an analogous approach based on the Campbell measure and the KL divergence when applied to point processes leads to the well-known pseudo-likelihood introduced by Besag. More surprisingly, the binomial regression method recently proposed by Baddeley and his co-authors for computing the pseudolikelihood maximum can be derived using the same approach starting from a slight modification of the Campbell measure

Stereological estimation of mean volume: precision of three simple sampling designs

Abstract New approximation formulae for the mean square error of stereological volume predictors are derived. The body under investigation is assumed to be an isotropic random compact set. Three simple systematic sampling probes are considered: lattice of points, serial sections, clustered sections. The derived formulae depend only on the mean body surface area and sampling parameters. The key argument is a refined result on the convergence of the spectral density of compact sets

A completely random T-tessellation model and Gibbsian extensions

A revised version of this paper has been published in Spatial Statistics, 2013, volume 6, pages 118- 138.In their 1993 paper, Arak, Clifford and Surgailis discussed a new model of random planar graph. As a particular case, that model yields tessellations with only T-vertices (T-tessellations). Using a similar approach involving Poisson lines, a new model of random T-tessellations is proposed. Campbell measures, Papangelou kernels and Georgii-Nguyen-Zessin formulae are translated from point process theory to random T-tessellations. It is shown that the new model shows properties similar to the Poisson point process and can therefore be considered as a completely random T-tessellation. Gibbs variants are introduced leading to models of random T-tessellations where selected features are controlled. Gibbs random T-tessellations are expected to better represent observed tessellations. As numerical experiments are a key tool for investigating Gibbs models, we derive a simulation algorithm of the Metropolis-Hastings-Green family

Precision of systematic sampling and transitive methods

International audienceThe use of the transitive methods for assessing the precision of systematic sampling is discussed. A key point of the transitive methods is the choice of a local model for the covariogram near the origin. The relationship between the regularity of the measurements and the regularity of their covariogram is given. This result is useful for choosing the appropriate covariogram model. A method forestimating the measurement regularity from discrete data is proposed for cases where it cannot be assessed a priori. Stereological applications where sampling is based on geometric probes such as serial sections, point or line grids are also discussed

COMPUTATION OF MINKOWSKI MEASURES ON 2D AND 3D BINARY IMAGES

Minkowski functionals encompass standard geometric parameters such as volume, area, length and the Euler-Poincaré characteristic. Software tools for computing approximations of Minkowski functionals on binary 2D or 3D images are now available based on mathematical methods due to Serra, Lang and Ohser. Minkowski functionals can not be used to describe spatial heterogeneity of structures. This description can be performed by using Minkowski measures, which are local versions of Minkowski functionals. In this paper, we discuss how to extend the computation of Minkowski functionals to the computation of Minkowski measures. Approximations of Minkowski measures are computed using fltering and look-up table transformations. The final result is represented as a grey-level image. Approximation errors are investigated based on numerical examples. Convergence and non convergence of the measure approximations are discussed. The measure of surface area is used to describe spatial heterogeneity of a synthetic structure, and of an image of tomato pericarp

Gibbsian T-tessellation model for agricultural landscape characterization

A new class of planar tessellations, named T-tessellations, was introduced in ([10]). A model was proposed to be considered as a completely random T-tessellation model (CRTT) and its Gibbsian variants were discussed. A general simulation algorithm of Metropolis-Hastings-Green type was derived for model simulation, involving three local transformations of T-tessellations. The current paper focuses on statistical inference for Gibbs models of T-tessellations. Statistical methods originated from point pattern analysis are implemented on the example of three agricultural landscapes approximated by T-tessellations. The choice of model statistics is guided by their capacity to highlight the differences between the landscape patterns. Model parameters are estimated by Monte Carlo Maximum Likelihood method, yielding a baseline for landscapes comparison. In the last part of the paper a global envelope test based on the empty-space function is proposed for assessing the goodness-of-fit of the model

Three Lectures on Systematic Geometric Sampling

International audienc

Etude de la répartition spatiale de phénomènes de fatigue des sols d'aspergeraies

Annexes 37 p. Directeur de stage Denis, J.B. et Vaillant, J. *INRA Laboratoire de Biométrie Versailles (FRA) Diffusion du document : INRA Laboratoire de Biométrie Versailles (FRA) Diplôme : DE

A coarea formula for multiple geometric integrals

International audienc

Second- and higher-order stereology

Diplôme : Ph. D