Gibbs Process Research Articles

Bayesian Inference for Inverse Power Exponentiated Pareto Distribution Using Progressive Type-II Censoring with Application to Flood-Level Data Analysis

Progressive type-II (Prog-II) censoring schemes are gaining traction in estimating the parameters, and reliability characteristics of lifetime distributions. The focus of this paper is to enhance the accuracy and reliability of such estimations for the inverse power exponentiated Pareto (IPEP) distribution, a flexible extension of the exponentiated Pareto distribution suitable for modeling engineering and medical data. We aim to develop novel statistical inference methods applicable under Prog-II censoring, leading to a deeper understanding of failure time behavior, improved decision-making, and enhanced overall model reliability. Our investigation employs both classical and Bayesian approaches. The classical technique involves constructing maximum likelihood estimators of the model parameters and their bootstrap covariance intervals. Using the Gibbs process constructed by the Metropolis–Hasting sampler technique, the Markov chain Monte Carlo method provides Bayesian estimates of the unknown parameters. In addition, an actual data analysis is carried out to examine the estimation process’s performance under this ideal scheme.

A central limit theorem for a classical gas

For a class of translation-invariant pair potentials ϕ in (mathbb{R}^{d},zlambda ) satisfying a stability and regularity condition, we choose z so small that the associated collection mathcal{ G}(phi,zlambda ) of Gibbs processes contains at least the stationary process G, which is a Gibbs process in the sense of DLR and is given by the limiting Gibbs process with empty boundary conditions. Using an abstract version of the method of cluster expansions and Dobrushin’s approach to the central limit theorem, we present a central limit theorem for the particle numbers of G.

BEYONDPLANCK

We present a Gibbs sampling solution to the mapmaking problem for cosmic microwave background (CMB) measurements that builds on existing destriping methodology. Gibbs sampling breaks the computationally heavy destriping problem into two separate steps: noise filtering and map binning. Considered as two separate steps, both are computationally much cheaper than solving the combined problem. This provides a huge performance benefit as compared to traditional methods and it allows us, for the first time, to bring the destriping baseline length to a single sample. Here, we applied the Gibbs procedure to simulatedPlanck30 GHz data. We find that gaps in the time-ordered data are handled efficiently by filling them in with simulated noise as part of the Gibbs process. The Gibbs procedure yields a chain of map samples, from which we are able to compute the posterior mean as a best-estimate map. The variation in the chain provides information on the correlated residual noise, without the need to construct a full noise covariance matrix. However, if only a single maximum-likelihood frequency map estimate is required, we find that traditional conjugate gradient solvers converge much faster than a Gibbs sampler in terms of the total number of iterations. The conceptual advantages of the Gibbs sampling approach lies in statistically well-defined error propagation and systematic error correction. This methodology thus forms the conceptual basis for the mapmaking algorithm employed in the BEYONDPLANCKframework, which implements the first end-to-end Bayesian analysis pipeline for CMB observations.

Structural Properties of Gibbsian Point Processes in Abstract Spaces

In the language of random counting measures, many structural properties of the Poisson process can be studied in arbitrary measurable spaces. We provide a similarly general treatise of Gibbs processes. With the GNZ equations as a definition of these objects, Gibbs processes can be introduced in abstract spaces without any topological structure. In this general setting, partition functions, Janossy densities, and correlation functions are studied. While the definition covers finite and infinite Gibbs processes alike, the finite case allows, even in abstract spaces, for an equivalent and more explicit characterization via a familiar series expansion. Recent generalizations of factorial measures to arbitrary measurable spaces, where counting measures cannot be written as sums of Dirac measures, likewise allow to generalize the concept of Hamiltonians. The DLR equations, which completely characterize a Gibbs process, as well as basic results for the local convergence topology, are also formulated in full generality. We prove a new theorem on the extraction of locally convergent subsequences from a sequence of point processes and use this statement to provide existence results for Gibbs processes in general spaces with potentially infinite range of interaction. These results are used to guarantee the existence of Gibbs processes with cluster-dependent interactions and to prove a recent conjecture concerning the existence of Gibbsian particle processes.

Gibbs process distinguishes survival and reveals contact-inhibition genes in Glioblastoma multiforme.

Tumor growth is a spatiotemporal birth-and-death process with loss of heterotypic contact-inhibition of locomotion (CIL) of tumor cells promoting invasion and metastasis. Therefore, representing tumor cells as two-dimensional points, we can expect the tumor tissues in histology slides to reflect realizations of spatial birth-and-death process which can be mathematically modeled to reveal molecular mechanisms of CIL, provided the mathematics models the inhibitory interactions. Gibbs process as an inhibitory point process is a natural choice since it is an equilibrium process of the spatial birth-and-death process. That is if the tumor cells maintain homotypic contact inhibition, the spatial distributions of tumor cells will result in Gibbs hard core process over long time scales. In order to verify if this is the case, we applied the Gibbs process to 411 TCGA Glioblastoma multiforme patient images. Our imaging dataset included all cases for which diagnostic slide images were available. The model revealed two groups of patients, one of which - the "Gibbs group," showed the convergence of the Gibbs process with significant survival difference. Further smoothing the discretized (and noisy) inhibition metric, for both increasing and randomized survival time, we found a significant association of the patients in the Gibbs group with increasing survival time. The mean inhibition metric also revealed the point at which the homotypic CIL establishes in tumor cells. Besides, RNAseq analysis between patients with loss of heterotypic CIL and intact homotypic CIL in the Gibbs group unveiled cell movement gene signatures and differences in Actin cytoskeleton and RhoA signaling pathways as key molecular alterations. These genes and pathways have established roles in CIL. Taken together, our integrated analysis of patient images and RNAseq data provides for the first time a mathematical basis for CIL in tumors, explains survival as well as uncovers the underlying molecular landscape for this key tumor invasion and metastatic phenomenon.

Gibbs spatial process for characterization of filament interaction in ReRAM devices via photon emission microscopy

In this work, we investigate spatial statistical properties of filament patterns in resistive random-access memory (ReRAM) devices measured from a newly developed near-infrared photon emission microscopy (PEM) [Stellari et al., IEEE Electron Device Lett. 42, 828 (2021); Stellari et al., in Proceedings of the 47th International Symposium for Testing and Failure Analysis Conference (ISTFA) (ASM International, 2021), pp. 115–121]. Unlike previous reports on uncorrelated filaments [Stellari et al., IEEE Electron Device Lett. 42, 828 (2021); Wu et al., Appl. Phys. Lett. 99, 093502 (2011)], we report a strong clustering and non-Poisson pattern of filaments constructed from individual devices. A Poisson-mixture model incorporating the clustering (attractive) effect is introduced with an excellent agreement with the PEM data for global and nearest-neighbor spatial statistics. On the other hand, a two-filament pattern is also detected within the ReRAM devices. We found that both attractive and repulsive interactions among the filaments are required in a Gibbs process to explain the filament spatial distribution. We implemented a birth-death algorithm using a Markov-chain Monte Carlo approach and achieve good agreement with the PEM data using a generalized Morse potential.

Phytoconstituents of Methanolic Extract of Fenugreek Seeds as a Green Corrosion Inhibitor of Metals (Fe, Al and Cu)

DFT calculations were carried out on significant Fenugreek seed compounds (1-Methylpyridinium-3-carboxylate MPC, Apigenin-8-C-glucose ACG, and 2-Isobutyl-3-methoxypyrazine IBMP) as a green source of ecologically friendly Fe, Al, and Cu metal corrosion inhibitors. Complete geometry optimizations were performed by DFT-B3LYP/6-31G* to determine any relationship between the chemical structure and corrosion inhibition, mostly on metals' surfaces. Global computational parameters of the inhibitors and thermodynamic Gibbs process of adsorption of metals were calculated and used to evaluate each corrosion inhibitor's performance. Our findings showed that MPC has the maximum anti-corrosion efficiency across all molecules with a physical adsorption mechanism. It exhibited significant inhibition efficiency with Cu when compared with Fe and Al, based on the highest electrophilicity index (ω) values compared to other inhibitors and its impact on metals in the following manner: Cu>Fe>Al. In regards, the range of inhibitors increased by the following order: IBMP>ACG>MPC. Remarkable corrosion inhibition of MPC is demonstrated by its unique high electrophilicity, softness (σ), and lowest ∆Egap on the metal surface. These outcomes are close to the experimental

Multiscale processes to describe the Eastern Sicily Seismic Sequences

In this paper a version of hybrid of Gibbs process model is proposed as method to characterize the multiscale interaction structure of several seismic sequences occurred in Eastern Sicily in the last decade.

FACADE INTERPRETATION USING A MARKED POINT PROCESS

Our objective is the interpretation of facade images in a top-down manner, using a Markov marked point process formulated as a Gibbs process. Given single rectified facade images, we aim at the accurate detection of relevant facade objects as windows and entrances, using prior knowledge about their possible configurations within facade images. We represent facade objects by a simplified rectangular object model and present an energy model, which evaluates the agreement of a proposed configuration with the given image and the statistics about typical configurations, which we learned from training data. We show promising results on different datasets and provide a qualitative evaluation, which demonstrates the capability of complete and accurate detection of facade objects.

FACADE INTERPRETATION USING A MARKED POINT PROCESS

Abstract. Our objective is the interpretation of facade images in a top-down manner, using a Markov marked point process formulated as a Gibbs process. Given single rectified facade images, we aim at the accurate detection of relevant facade objects as windows and entrances, using prior knowledge about their possible configurations within facade images. We represent facade objects by a simplified rectangular object model and present an energy model, which evaluates the agreement of a proposed configuration with the given image and the statistics about typical configurations, which we learned from training data. We show promising results on different datasets and provide a qualitative evaluation, which demonstrates the capability of complete and accurate detection of facade objects.

Gibbs point process approximation: Total variation bounds using Stein’s method

We obtain upper bounds for the total variation distance between the distributions of two Gibbs point processes in a very general setting. Applications are provided to various well-known processes and settings from spatial statistics and statistical physics, including the comparison of two Lennard-Jones processes, hard core approximation of an area interaction process and the approximation of lattice processes by a continuous Gibbs process. Our proof of the main results is based on Stein's method. We construct an explicit coupling between two spatial birth-death processes to obtain Stein factors, and employ the Georgii-Nguyen-Zessin equation for the total bound.

Bayesian analysis of a Gibbs hard-core point pattern model with varying repulsion range

A Bayesian solution is suggested for the modelling of spatial point patterns with inhomogeneous hard-core radius using Gaussian processes in the regularization. The key observation is that a straightforward use of the finite Gibbs hard-core process likelihood together with a log-Gaussian random field prior does not work without penalisation towards high local packing density. Instead, a nearest neighbour Gibbs process likelihood is used. This approach to hard-core inhomogeneity is an alternative to the transformation inhomogeneous hard-core modelling. The computations are based on recent Markovian approximation results for Gaussian fields. As an application, data on the nest locations of Sand Martin (Riparia riparia) colony11Dataset is attached to the online version. on a vertical sand bank are analysed.

Spatial organisation of a bimodal forest stand

Many populations have a bimodal size distribution, even when composed of a single cohort. In developing forest stands, this pattern is usually attributed to asymmetric competition at canopy closure among trees which have access to the upper canopy and those which have failed to reach it. Nevertheless, alternative explanations for bimodality exist, and in sessile organisms spatial pattern analysis can be used to compare their predictions. A 0.25-ha plot was created in a maturing stand of Asiatic white birch (Betula platyphylla Sukacz.) in Central Kamchatka. All stems >1 cm diameter at breast height (DBH) were fully mapped. Mark correlation analysis revealed size compensation among stems up to 3.5 m apart, providing evidence that competition affected the distribution of stem sizes. The spatial pattern of trees was analysed using the pair correlation function g(r). Large trees (>20 cm DBH) had a dispersed distribution to which a Strauss soft-core Gibbs process model was fitted. This suggested that large trees interacted at scales up to 4.16 m. Small trees (1–20 cm DBH) were distributed randomly, but a cross-pair correlation analysis revealed a greater likelihood of occurrence beginning at 4.3 m from large trees, closely matching the modelled interaction distance. These results are consistent with the hypothesis that asymmetric competition is structuring this bimodal cohort of trees: large stems tend towards a dispersed pattern, exerting competitive effects at scales up to approximately 4 m, whereas smaller stems are more commonly found in the interstices within the pattern of large trees.

Variational Characterisation of Gibbs Measures with Delaunay Triangle Interaction

This paper deals with stationary Gibbsian point processes on the plane with an interaction that depends on the tiles of the Delaunay triangulation of points via a bounded triangle potential. It is shown that the class of these Gibbs processes includes all minimisers of the associated free energy density and is therefore nonempty. Conversely, each such Gibbs process minimises the free energy density, provided the potential satisfies a weak long-range assumption.

The approach to statistical equilibrium in collisionless wave-particle interactions

The long-time behaviour of the wave-particle interaction is investigated through a Fermi-like model that describes the interaction as the results of a large number of collisions between an ensemble of particles and two infinitely massive barriers with variable amplitude. The results are compared with those of particle in cell (PIC) simulations. The system approaches statistical equilibrium through a phase space filamentation that is analogous to a laminar Gibbs process in a conservative system.

Neutral models for patchy landscapes

The landscapes currently studied in ecology are either “discontinuous” (category-based or patch-based), as in the case of mosaics of agricultural units, or of more “continuous” type (raster lattices), as used for representing elevation or other ecological gradients. The main landscape models either involve explicit processes or are neutral, recreating spatial patterns in the absence of studied processes (using statistical rules). This article presents neutral models suitable for the creation and handling of patchy landscapes. These models (Patchy Landscape Neutral Models) adapt the Gibbs process already used successfully in forestry and biology to describe the local interactions between landscape units. These interactions can be either ecological, if justified, for example, by natural mechanisms of dispersal (plant species dynamics), or crop successions in anthropized landscapes, or statistical (geometrical). We define a global “cost function” representative of the landscape to be simulated by summing the “pair function” that expresses the interactions between units for the whole landscape. This generic approach makes it possible to reconstruct different kinds of patchy landscape compositions (land cover) and opens the way to study changes in landscape configuration (unit arrangements) as well as an analytical description of landscapes.

Modelling the microstructure of concrete with spherical grains

The modelling of the geometrical structure of a special kind of concrete is discussed. It is a matter of self-flowing refractory castables with spherical grains. The system of hard spheres is compared to several well-established models of spatial statistics and statistical physics as well as to a model used by concrete engineers. It turns out that the structure can be well described by the latter model and a further newly developed model, which is similar to a marked Gibbs process. The result is based on the comparison of common summary statistics like the pair correlation function and the spherical contact distribution function.

Non-Homogeneous Gibbs Process Models for Forestry — A Case Study

Forestry statistics needs realistic models for non-homogeneously distributed trees in forests. For elder trees, non-homogeneous Poisson or Cox processes are not realistic models. Instead, non-homogeneous Gibbs processes are suggested, which are described by a fixed pair potential describing the short-range interaction of trees and a location dependent intensity function describing long-range variation of tree density. The application of this model is demonstrated for a hickory forest in North Carolina and an artificial forest stand.

Statistical Inference for a Gibbs Point Process of Mutually Non‐Intersecting Discs

AbstractFor a Gibbs point process of mutually non‐intersecting discs a parameter estimation method is suggested. It is applied to a pattern of positions of beadlet anemones, for which, until now, no appropriate Gibbs process model has been found.

The distribution of interparticle distance and its application in finite-element modelling of composite materials

The distribution of interparticle distance, based on a Voronoi tessellation, is found approximately for a hard-core Gibbs process. The moments of this distribution are then used as input for finite-element analysis of the region surrounding a single filler sphere within a composite material. Statistical analysis provides close bounds for overall elastic properties of the material. Results from finite-element analysis can therefore be applied to real composite materials; reasonable agreement is found with a particular set of experimental data.