Class Of Markov Random Fields Research Articles

Stable limit theorems for empirical processes under conditional neighborhood dependence

This paper introduces a new concept of stochastic dependence among many random variables which we call conditional neighborhood dependence (CND). Suppose that there are a set of random variables and a set of sigma algebras where both sets are indexed by the same set endowed with a neighborhood system. When the set of random variables satisfies CND, any two non-adjacent sets of random variables are conditionally independent given sigma algebras having indices in one of the two sets’ neighborhood. Random variables with CND include those with conditional dependency graphs and a class of Markov random fields with a global Markov property. The CND property is useful for modeling cross-sectional dependence governed by a complex, large network. This paper provides two main results. The first result is a stable central limit theorem for a sum of random variables with CND. The second result is a Donsker-type result of stable convergence of empirical processes indexed by a class of functions satisfying a certain bracketing entropy condition when the random variables satisfy CND.

Stable Limit Theorems for Empirical Processes under Conditional Neighborhood Dependence

This paper introduces a new concept of stochastic dependence among many random variables which we call conditional neighborhood dependence (CND). Suppose that there are a set of random variables and a set of sigma algebras where both sets are indexed by the same set endowed with a neighborhood system. When the set of random variables satisfies CND, any two non-adjacent sets of random variables are conditionally independent given sigma algebras having indices in one of the two sets’ neighborhood. Random variables with CND include those with conditional dependency graphs and a class of Markov random fields with a global Markov property. The CND property is useful for modeling cross-sectional dependence governed by a complex, large network. This paper provides two main results. The first result is a stable central limit theorem for a sum of random variables with CND. The second result is a Donsker-type result of stable convergence of empirical processes indexed by a class of functions satisfying a certain bracketing entropy condition when the random variables satisfy CND. When there are high-degree vertices, they potentially hamper normal approximation by causing widespread dependence among the random variables. Thus we generalize the results by approximating the sum of random variables by the sum conditioned on high-degree vertices so that stable limit results continue to hold even when the maximum degree of the neighborhood system diverges to infinity as the size of the system grows.

MAP entropy estimation: applications in robust image filtering

We introduce a new approach for image filtering in a Bayesian framework. In this case the probability density function (pdf) of the likelihood function is approximated using the concept of non-parametric or kernel estimation. The method is based on the generalized Gaussian Markov random fields (GGMRF), a class of Markov random fields which are used as prior information into the Bayesian rule, which principal objective is to eliminate those effects caused by the excessive smoothness on the reconstruction process of images which are rich in contours or edges. Accordingly to the hypothesis made for the present work, it is assumed a limited knowledge of the noise pdf, so the idea is to use a non-parametric estimator to estimate such a pdf and then apply the entropy to construct the cost function for the likelihood term. The previous idea leads to the construction of Maximum a posteriori (MAP) robust estimators, since the real systems are always exposed to continuous perturbations of unknown nature. Some promising results of three new MAP entropy estimators (MAPEE) for image filtering are presented, together with some concluding remarks.

Sparse matrix decompositions and graph characterizations

Zeros in positive definite correlation matrices arise frequently in probability and statistics, and are intimately related to the notion of stochastic independence. The question of when zeros (i.e., sparsity) in a positive definite matrix A are preserved in its Cholesky decomposition, and vice versa, was addressed by Paulsen et al. [V.I. Paulsen, S.C. Power, R.R. Smith, Schur products and matrix completions, J. Funct. Anal. 85 (1989) 151–178]. In particular, they prove that for the pattern of zeros in A to be retained in the Cholesky decomposition of A, the pattern of zeros in A has to necessarily correspond to a chordal (or decomposable) graph associated with a specific type of vertex ordering. This result therefore also yields a characterization of chordal graphs in terms of sparse positive definite matrices, and has proved to be extremely useful in probabilistic and statistical analysis of Markov random fields. Now, consider a positive definite matrix A and its Cholesky decomposition given by A=LDLT, where L is lower triangular with unit diagonal entries, and D a diagonal matrix with positive entries. In this paper, we prove that a necessary and sufficient condition for zeros (i.e., sparsity) in a positive definite matrix A to be preserved in its associated Cholesky matrix L, and in addition also preserved in the inverse of the Cholesky matrix L-1, is that the pattern of zeros corresponds to a co-chordal or homogeneous graph associated with a specific type of vertex ordering. We proceed to provide a second characterization of this class of graphs in terms of determinants of submatrices that correspond to cliques in the graph. These results add to the growing body of literature in the field of sparse matrix decompositions, and also prove to be critical ingredients in the probabilistic analysis of an important class of Markov random fields.

A Maximum Entropy Solution of the Covariance Extension Problem for Reciprocal Processes

Stationary reciprocal processes defined on a finite interval of the integer line can be seen as a special class of Markov random fields restricted to one dimension. Nonstationary reciprocal processes have been extensively studied in the past especially by Jamison et al. The specialization of the nonstationary theory to the stationary case, however, does not seem to have been pursued in sufficient depth in the literature. Stationary reciprocal processes (and reciprocal stochastic models) are potentially useful for describing signals which naturally live in a finite region of the time (or space) line. Estimation or identification of these models starting from observed data seems still to be an open problem which can lead to many interesting applications in signal and image processing. In this paper, we discuss a class of reciprocal processes which is the acausal analog of auto-regressive (AR) processes, familiar in control and signal processing. We show that maximum likelihood identification of these processes leads to a covariance extension problem for block-circulant covariance matrices. This generalizes the famous covariance band extension problem for stationary processes on the integer line. As in the usual stationary setting on the integer line, the covariance extension problem turns out to be a basic conceptual and practical step in solving the identification problem. We show that the maximum entropy principle leads to a complete solution of the problem.

Modelling transport mode decisions using hierarchical logistic regression models with spatial and cluster effects

This work is motivated by a mobility study conducted in the city of Munich, Germany. The variable of interest is a binary response, which indicates whether public transport has been utilized or not. One of the central questions is to identify areas of low/high utilization of public transport after adjusting for explanatory factors such as trip, individual and household attributes. For the spatial effects a modification of a class of Markov random fields (MRF) models with proper joint distributions introduced by Pettitt et al. (2002) is developed. It contains the intrinsic MRF in the limit and allows for efficient Markov Chain Monte Carlo (MCMC) algorithms. Further cluster effects using group and individual approaches are taken into consideration. The first one models heterogeneity between clusters, while the second one models heterogeneity within clusters. A naive approach to include individual cluster effects results in an unidentifiable model. It is shown how a re-parametrization gives identifiable parameters. This provides a new approach for modeling heterogeneity within clusters. Finally, the proposed model classes are applied to the mobility study.

Approximate Labeling via Graph Cuts Based on Linear Programming

A new framework is presented for both understanding and developing graph-cut-based combinatorial algorithms suitable for the approximate optimization of a very wide class of Markov Random Fields (MRFs) that are frequently encountered in computer vision. The proposed framework utilizes tools from the duality theory of linear programming in order to provide an alternative and more general view of state-of-the-art techniques like the \alpha-expansion algorithm, which is included merely as a special case. Moreover, contrary to \alpha-expansion, the derived algorithms generate solutions with guaranteed optimality properties for a much wider class of problems, for example, even for MRFs with nonmetric potentials. In addition, they are capable of providing per-instance suboptimality bounds in all occasions, including discrete MRFs with an arbitrary potential function. These bounds prove to be very tight in practice (that is, very close to 1), which means that the resulting solutions are almost optimal. Our algorithms' effectiveness is demonstrated by presenting experimental results on a variety of low-level vision tasks, such as stereo matching, image restoration, image completion, and optical flow estimation, as well as on synthetic problems.

Nonparametric Bayes estimators for hazard functions based on right censored data

In this article, we analyse right censored survival data by modelling their common hazard function nonparametrically. The hazard rate is assumed to be a stochastic process, with sample paths taking the form of step functions. This process jumps at times that form a time-homogeneous Poisson process, and a class of Markov random fields is used to model the values of these sample paths. Features of the posterior distribution, such as the mean hazard rate and survival probabilities, are evaluated using the Metropolis--Hastings--Green algorithm. We illustrate our methodology by simulation examples.

A Dense Class of Markov Random Fields and Associated Parameter Estimation

In this paper, we address the problem of estimating parameters of Markov random fields (MRFs). We consider polynomial energy functions. We show both theoretically and by using simulations that this model represents a wide class of MRFs. We then develop a general scheme for estimating the MRF parameters associated with the gradient of such an energy function. The degree of the polynomial energy function as well as a super-set of the neighborhood of pixels are assumed to be known. This method, based on Euler's lemma, requires a partition of the space of neighborhood configurations. Simulations allow us to give a texture interpretation of the parameters associated with polynomial MRFs. Some results of the estimation method are presented for short, middle, and long range interactions.

Markov modeling for Bayesian restoration of two-dimensional layered structures

Bayesian estimation of two-dimensional stratified structures is described. The major point addressed is the derivation of a statistical prior model that adequately describes such layered media. This problem is of interest in applications in which the data are generally processed in one dimension only. In order to take local interactions into account, a Markovian description is used. The model is derived so as to fulfill a set of constraints that summarize physical and geometrical characteristics of the problem as well as practical requirements. The resulting class of Markov random fields presents a unilateral structure on a nonrectangular lattice and a hierarchical organization which involves a line process. In addition, it is shown to be an extension of one-dimensional models already in use. The properties of the model are investigated, and its practicality is demonstrated by an application to seismic deconvolution. Simulation results show significant improvements with respect to the usual one-dimensional methods. >

Markov fields with polygonal realizations

We define a new class of Markov random fields with two-dimensional parameter z∈T⊂ℝ2, which take a finite number of values separated by random polygonal boundaries. In some special cases, these random fields are consistent in T and allow and alternative description in terms of the equilibrium evolution of a system of one-dimensional particles.

Bayes Smoothing Algorithms for Segmentation of Binary Images Modeled by Markov Random Fields

A new image segmentation algorithm is presented, based on recursive Bayes smoothing of images modeled by Markov random fields and corrupted by independent additive noise. The Bayes smoothing algorithm yields the a posteriori distribution of the scene value at each pixel, given the total noisy image, in a recursive way. The a posteriori distribution together with a criterion of optimality then determine a Bayes estimate of the scene. The algorithm presented is an extension of a 1-D Bayes smoothing algorithm to 2-D and it gives the optimum Bayes estimate for the scene value at each pixel. Computational concerns in 2-D, however, necessitate certain simplifying assumptions on the model and approximations on the implementation of the algorithm. In particular, the scene (noiseless image) is modeled as a Markov mesh random field, a special class of Markov random fields, and the Bayes smoothing algorithm is applied on overlapping strips (horizontal/vertical) of the image consisting of several rows (columns). It is assumed that the signal (scene values) vector sequence along the strip is a vector Markov chain. Since signal correlation in one of the dimensions is not fully used along the edges of the strip, estimates are generated only along the middle sections of the strips. The overlapping strips are chosen such that the union of the middle sections of the strips gives the whole image. The Bayes smoothing algorithm presented here is valid for scene random fields consisting of multilevel (discrete) or continuous random variables.

The Description of a Random Field by Means of Conditional Probabilities and Conditions of Its Regularity

The Description of a Random Field by Means of Conditional Probabilities and Conditions of Its Regularity