Bayesian analysis of deformed tessellation models

We define a class of tessellation models based on perturbing or deforming standard tessellations such as the Voronoi tessellation. We show how distributions over this class of ‘deformed’ tessellations can be used to define prior distributions for models based on tessellations, and how inference for such models can be carried out using Markov chain Monte Carlo methods; stability properties of the algorithms are investigated. Our approach applies not only to fixed dimension problems, but also to variable dimension problems, in which the number of cells in the tessellation is unknown. We illustrate our methods with two real examples. The first relates to reconstructing animal territories, represented by the individual cells of a tessellation, from observation of an inhomogeneous Poisson point process. The second example involves the analysis of an image of a cross-section through a sample of metal, with the tessellation modelling the micro-crystalline structure of the metal.

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Fitting NOvA cross-section parameters with Markov Chain Monte Carlo

10.2172/1827879 ◽

1900 ◽

Author(s):

Michael Dolce

Keyword(s):

Monte Carlo ◽

Markov Chain ◽

Markov Chain Monte Carlo ◽

Cross Section

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Approximate decomposition of some modulated-Poisson Voronoi tessellations

Advances in Applied Probability ◽

10.1239/aap/1067436322 ◽

2003 ◽

Vol 35 (4) ◽

pp. 847-862 ◽

Cited By ~ 6

Author(s):

Bartłomiej Błaszczyszyn ◽

René Schott

Keyword(s):

Point Process ◽

Poisson Point Process ◽

Voronoi Tessellation ◽

Distribution Functions ◽

Voronoi Tessellations ◽

Intensity Measures ◽

Voronoi Cells ◽

True Value ◽

Homogeneous Models ◽

Inhomogeneous Poisson Point Process

We consider the Voronoi tessellation of Euclidean space that is generated by an inhomogeneous Poisson point process whose intensity takes different constant values on sets of some finite partition of the space. Considering the Voronoi cells as marks associated with points of the point process, we prove that the intensity measure (mean measure) of the marked Poisson point process admits an approximate decomposition formula. The true value is approximated by a mixture of respective intensity measures for homogeneous models, while the explicit upper bound for the remainder term can be computed numerically for a large class of practical examples. By the Campbell formula, analogous approximate decompositions are deduced for the Palm distributions of individual cells. This approach makes possible the analysis of a wide class of inhomogeneous-Poisson Voronoi tessellations, by means of formulae and estimates already established for homogeneous cases. Our analysis applies also to the Poisson process modulated by an independent stationary random partition, in which case the error of the approximation of the double-stochastic-Poisson Voronoi tessellation depends on some integrated linear contact distribution functions of the boundaries of the partition elements.

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SAR IMAGE SEGMENTATION WITH UNKNOWN NUMBER OF CLASSES COMBINED VORONOI TESSELLATION AND RJMCMC ALGORITHM

ISPRS Annals of Photogrammetry Remote Sensing and Spatial Information Sciences ◽

10.5194/isprsannals-iii-7-119-2016 ◽

2016 ◽

Vol III-7 ◽

pp. 119-124 ◽

Cited By ~ 1

Author(s):

Q. H. Zhao ◽

Y. Li ◽

Y. Wang

Keyword(s):

Monte Carlo ◽

Markov Chain ◽

Markov Chain Monte Carlo ◽

Posterior Distribution ◽

Voronoi Tessellation ◽

Random Variable ◽

Model Parameters ◽

Sar Image ◽

Sar Images ◽

Number Of Classes

This paper presents a novel segmentation method for automatically determining the number of classes in Synthetic Aperture Radar (SAR) images by combining Voronoi tessellation and Reversible Jump Markov Chain Monte Carlo (RJMCMC) strategy. Instead of giving the number of classes <i>a priori</i>, it is considered as a random variable and subject to a Poisson distribution. Based on Voronoi tessellation, the image is divided into homogeneous polygons. By Bayesian paradigm, a posterior distribution which characterizes the segmentation and model parameters conditional on a given SAR image can be obtained up to a normalizing constant; Then, a Revisable Jump Markov Chain Monte Carlo(RJMCMC) algorithm involving six move types is designed to simulate the posterior distribution, the move types including: splitting or merging real classes, updating parameter vector, updating label field, moving positions of generating points, birth or death of generating points and birth or death of an empty class. Experimental results with real and simulated SAR images demonstrate that the proposed method can determine the number of classes automatically and segment homogeneous regions well.

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Bayesian Radial Basis Functions of Variable Dimension

Neural Computation ◽

10.1162/089976698300017421 ◽

1998 ◽

Vol 10 (5) ◽

pp. 1217-1233 ◽

Cited By ~ 58

Author(s):

C. C. Holmes ◽

B. K. Mallick

Keyword(s):

Monte Carlo ◽

Markov Chain ◽

Markov Chain Monte Carlo ◽

Radial Basis Functions ◽

Basis Function ◽

Basis Functions ◽

Reversible Jump ◽

Variable Dimension ◽

Modeling Process ◽

Radial Basis

A Bayesian framework for the analysis of radial basis functions (RBF) is proposed that accommodates uncertainty in the dimension of the model. A distribution is defined over the space of all RBF models of a given basis function, and posterior densities are computed using reversible jump Markov chain Monte Carlo samplers (Green, 1995). This alleviates the need to select the architecture during the modeling process. The resulting networks are shown to adjust their size to the complexity of the data.

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Extended stochastic gradient Markov chain Monte Carlo for large-scale Bayesian variable selection

Biometrika ◽

10.1093/biomet/asaa029 ◽

2020 ◽

Vol 107 (4) ◽

pp. 997-1004

Author(s):

Qifan Song ◽

Yan Sun ◽

Mao Ye ◽

Faming Liang

Keyword(s):

Monte Carlo ◽

Markov Chain ◽

Markov Chain Monte Carlo ◽

Latent Variables ◽

Large Scale ◽

Bayesian Variable Selection ◽

Stochastic Gradient ◽

Monte Carlo Algorithm ◽

Monte Carlo Algorithms ◽

Fixed Dimension

Summary Stochastic gradient Markov chain Monte Carlo algorithms have received much attention in Bayesian computing for big data problems, but they are only applicable to a small class of problems for which the parameter space has a fixed dimension and the log-posterior density is differentiable with respect to the parameters. This paper proposes an extended stochastic gradient Markov chain Monte Carlo algorithm which, by introducing appropriate latent variables, can be applied to more general large-scale Bayesian computing problems, such as those involving dimension jumping and missing data. Numerical studies show that the proposed algorithm is highly scalable and much more efficient than traditional Markov chain Monte Carlo algorithms.

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Approximate decomposition of some modulated-Poisson Voronoi tessellations

Advances in Applied Probability ◽

10.1017/s0001867800012611 ◽

2003 ◽

Vol 35 (04) ◽

pp. 847-862

Author(s):

Bartłomiej Błaszczyszyn ◽

René Schott

Keyword(s):

Point Process ◽

Poisson Point Process ◽

Voronoi Tessellation ◽

Distribution Functions ◽

Voronoi Tessellations ◽

Intensity Measures ◽

Voronoi Cells ◽

True Value ◽

Homogeneous Models ◽

Inhomogeneous Poisson Point Process

We consider the Voronoi tessellation of Euclidean space that is generated by an inhomogeneous Poisson point process whose intensity takes different constant values on sets of some finite partition of the space. Considering the Voronoi cells as marks associated with points of the point process, we prove that the intensity measure (mean measure) of the marked Poisson point process admits an approximate decomposition formula. The true value is approximated by a mixture of respective intensity measures for homogeneous models, while the explicit upper bound for the remainder term can be computed numerically for a large class of practical examples. By the Campbell formula, analogous approximate decompositions are deduced for the Palm distributions of individual cells. This approach makes possible the analysis of a wide class of inhomogeneous-Poisson Voronoi tessellations, by means of formulae and estimates already established for homogeneous cases. Our analysis applies also to the Poisson process modulated by an independent stationary random partition, in which case the error of the approximation of the double-stochastic-Poisson Voronoi tessellation depends on some integrated linear contact distribution functions of the boundaries of the partition elements.

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SAR IMAGE SEGMENTATION WITH UNKNOWN NUMBER OF CLASSES COMBINED VORONOI TESSELLATION AND RJMCMC ALGORITHM

ISPRS Annals of Photogrammetry Remote Sensing and Spatial Information Sciences ◽

10.5194/isprs-annals-iii-7-119-2016 ◽

2016 ◽

Vol III-7 ◽

pp. 119-124

Author(s):

Q. H. Zhao ◽

Y. Li ◽

Y. Wang

Keyword(s):

Monte Carlo ◽

Markov Chain ◽

Markov Chain Monte Carlo ◽

Posterior Distribution ◽

Voronoi Tessellation ◽

Random Variable ◽

Model Parameters ◽

Sar Image ◽

Sar Images ◽

Number Of Classes

This paper presents a novel segmentation method for automatically determining the number of classes in Synthetic Aperture Radar (SAR) images by combining Voronoi tessellation and Reversible Jump Markov Chain Monte Carlo (RJMCMC) strategy. Instead of giving the number of classes <i>a priori</i>, it is considered as a random variable and subject to a Poisson distribution. Based on Voronoi tessellation, the image is divided into homogeneous polygons. By Bayesian paradigm, a posterior distribution which characterizes the segmentation and model parameters conditional on a given SAR image can be obtained up to a normalizing constant; Then, a Revisable Jump Markov Chain Monte Carlo(RJMCMC) algorithm involving six move types is designed to simulate the posterior distribution, the move types including: splitting or merging real classes, updating parameter vector, updating label field, moving positions of generating points, birth or death of generating points and birth or death of an empty class. Experimental results with real and simulated SAR images demonstrate that the proposed method can determine the number of classes automatically and segment homogeneous regions well.

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Low-voltage EDS of magnesium ferrite Dendrites in a FEG-SEM

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100164854 ◽

1996 ◽

Vol 54 ◽

pp. 478-479

Author(s):

Matthew T. Johnson ◽

Ian M. Anderson ◽

Jim Bentley ◽

C. Barry Carter

Keyword(s):

Monte Carlo ◽

Quantitative Analysis ◽

Monte Carlo Simulations ◽

Cross Section ◽

Spatial Resolution ◽

Low Voltage ◽

Magnesium Ferrite ◽

X Ray ◽

Interaction Volume ◽

Ferrite Spinel

Energy-dispersive X-ray spectrometry (EDS) performed at low (≤ 5 kV) accelerating voltages in the SEM has the potential for providing quantitative microanalytical information with a spatial resolution of ∼100 nm. In the present work, EDS analyses were performed on magnesium ferrite spinel [(MgxFe1−x)Fe2O4] dendrites embedded in a MgO matrix, as shown in Fig. 1. spatial resolution of X-ray microanalysis at conventional accelerating voltages is insufficient for the quantitative analysis of these dendrites, which have widths of the order of a few hundred nanometers, without deconvolution of contributions from the MgO matrix. However, Monte Carlo simulations indicate that the interaction volume for MgFe2O4 is ∼150 nm at 3 kV accelerating voltage and therefore sufficient to analyze the dendrites without matrix contributions.Single-crystal {001}-oriented MgO was reacted with hematite (Fe2O3) powder for 6 h at 1450°C in air and furnace cooled. The specimen was then cleaved to expose a clean cross-section suitable for microanalysis.

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Bayesian Applications in Auditory Research

Journal of Speech Language and Hearing Research ◽

10.1044/2018_jslhr-h-astm-18-0228 ◽

2019 ◽

Vol 62 (3) ◽

pp. 577-586 ◽

Cited By ~ 9

Author(s):

Garnett P. McMillan ◽

John B. Cannon

Keyword(s):

Monte Carlo ◽

Markov Chain ◽

Bayesian Methods ◽

Prior Distribution ◽

Interim Analysis ◽

Iterative Process ◽

Bayes Theorem ◽

Sound Therapy ◽

The Many ◽

Auditory Research

Purpose This article presents a basic exploration of Bayesian inference to inform researchers unfamiliar to this type of analysis of the many advantages this readily available approach provides. Method First, we demonstrate the development of Bayes' theorem, the cornerstone of Bayesian statistics, into an iterative process of updating priors. Working with a few assumptions, including normalcy and conjugacy of prior distribution, we express how one would calculate the posterior distribution using the prior distribution and the likelihood of the parameter. Next, we move to an example in auditory research by considering the effect of sound therapy for reducing the perceived loudness of tinnitus. In this case, as well as most real-world settings, we turn to Markov chain simulations because the assumptions allowing for easy calculations no longer hold. Using Markov chain Monte Carlo methods, we can illustrate several analysis solutions given by a straightforward Bayesian approach. Conclusion Bayesian methods are widely applicable and can help scientists overcome analysis problems, including how to include existing information, run interim analysis, achieve consensus through measurement, and, most importantly, interpret results correctly. Supplemental Material https://doi.org/10.23641/asha.7822592

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