Disagreement Loop and Path Creation/Annihilation Algorithms for Binary Planar Markov Fields with Applications to Image Segmentation

Abstract. We introduce a class of Gibbs–Markov random fields built on regular tessellations that can be understood as discrete counterparts of Arak–Surgailis polygonal fields. We focus first on consistent polygonal fields, for which we show consistency, Markovianity and solvability by means of dynamic representations. Next, we develop disagreement loop as well as path creation and annihilation dynamics for their general Gibbsian modifications, which cover most lattice‐based Gibbs–Markov random fields subject to certain mild conditions. Applications to foreground–background image segmentation problems are discussed.     

Discrete Multicolour Random Mosaics with an Application to Network Extraction

We introduce a class of random fields that can be understood as discrete versions of multicolour polygonal fields built on regular linear tessellations. We focus first on a subclass of consistent polygonal fields, for which we show Markovianity and solvability by means of a dynamic representation. This representation is used to design new sampling techniques for Gibbsian modifications of such fields, a class which covers lattice‐based random fields. A flux‐based modification is applied to the extraction of the field tracks network from a Synthetic Aperture Radar image of a rural area.     

Bayesian inversion of geoelectrical resistivity data

Summary. Enormous quantities of geoelectrical data are produced daily and often used for large scale reservoir modelling. To interpret these data requires reliable and efficient inversion methods which adequately incorporate prior information and use realistically complex modelling structures. We use models based on random coloured polygonal graphs as a powerful and flexible modelling framework for the layered composition of the Earth and we contrast our approach with earlier methods based on smooth Gaussian fields. We demonstrate how the reconstruction algorithm may be efficiently implemented through the use of multigrid Metropolis–coupled Markov chain Monte Carlo methods and illustrate the method on a set of field data.     

Polygonal designs: some existence and non-existence results

Polygonal designs are introduced as a generalization of balanced incomplete block designs and as a specialization of partially balanced incomplete block designs. As in the case of balanced incomplete block designs, there is no hope of deciding the values of the parameters for which polygonal designs exist. We develop enough theory to reveal the structure, and thus, to resolve the existence problem for small polygonal designs, and derive necessary conditions for general cases. © 1998 Elsevier Science B.V. All rights reserved.     

Some theoretical results regarding the polygonal distribution

Polygonal distributions are a class of distributions that can be defined via the mixture of triangular distributions over the unit interval. We demonstrate that the densities of polygonal distributions are dense in the class of continuous and concave densities with bounded second derivatives. Furthermore, we prove that polygonal density functions provide O(g^{? 2}) approximations (where g is the number of triangular distribution components), in the supremum distance, to any density function from the hypothesized class. Parametric consistency and Hellinger consistency results for the maximum likelihood (ML) estimator are obtained. A result regarding model selection via penalized ML estimation is proved.     

Asymptotic Location of Largest Values of a Stationary Random Field

The study of the relationship between extreme values of dependent random fields and their locations has important practical applications, for instance, when dealing with censored data.

In this article we study the asymptotic behavior of the joint locations of the largest order statistics generated by a stationary random field with extremal index as well as the joint limiting distribution of the location of a high level exceedance nearest of the origin and the location of the maximum.     

Cyclic Polygonal Designs with Block Size 3 and Joint Distance α = 2

Polygonal designs are useful in survey sampling in terms of balanced sampling plans excluding contiguous units (BSECs) and balanced sampling plans excluding adjacent units (BSAs). In this article, the method of cyclic shifts has been used for the construction of cyclic polygonal designs (in terms of BSAs) with block size k = 3 and λ = 1, 2, 3, 4, 6, 12 for joint distance α = 2 and 51 new designs for treatments v ≤ 100 are given.     

On the detection of transitive clusters in undirected networks

A network cluster is defined as a set of nodes with ‘strong’ within group ties and ‘weak’ between group ties. Most clustering methods focus on finding groups of ‘densely connected’ nodes, where the dyad (or tie between two nodes) serves as the building block for forming clusters. However, since the unweighted dyad cannot distinguish strong relationships from weak ones, it then seems reasonable to consider an alternative building block, i.e. one involving more than two nodes. In the simplest case, one can consider the triad (or three nodes), where the fully connected triad represents the basic unit of transitivity in an undirected network. In this effort we propose a clustering framework for finding highly transitive subgraphs in an undirected/unweighted network, where the fully connected triad (or triangle configuration) is used as the building block for forming clusters. We apply our methodology to four real networks with encouraging results. Monte Carlo simulation results suggest that, on average, the proposed method yields good clustering performance on synthetic benchmark graphs, relative to other popular methods.     

Numerical method for means of linear Hawkes processes

Abstract

Linear Hawkes processes are widely used in many fields and means are the basic and critical information of them. However, there is little research on linear Hawkes processes’ means. In this paper, we present a numerical method based on the Laplace transform and inverse Laplace transform for means of linear Hawkes processes. The advantage of this method is that whatever the kernel function is, we can always obtain the numerical solutions of means for a linear Hawkes process. In addition, this numerical method provides the basic information of linear Hawkes processes by means. As an application, the numerical method is applied in a WeChat network model.     

A SEMIPARAMETRIC BAYESIAN APPROACH TO NETWORK MODELLING USING DIRICHLET PROCESS PRIOR DISTRIBUTIONS

This paper considers the use of Dirichlet process prior distributions in the statistical analysis of network data. Dirichlet process prior distributions have the advantages of avoiding the parametric specifications for distributions, which are rarely known, and of facilitating a clustering effect, which is often applicable to network nodes. The approach is highlighted for two network models and is conveniently implemented using WinBUGS software.     

Some adjusted orthogonal,variance balanced,multidimensional block designs

A multidimensional block design (MBD) is an experimental design with d > 1 blocking criteria geometrically represented as a d-dimensional lattice with treatment varieties assigned to some or all nodes of the lattice. Intrablock analysis of variance tables for some special classes of two- and three-dimensional block designs with some empty nodes are given. Design plans and efficiencies for 31 two-dimensional designs, each universally optimal in defined classes of designs, and 7 three-dimensional designs, each nearly optimal in defined classes of designs, are listed in the appendices. A need for such designs is apparent when the blocking criteria are implemented successively and empty nodes do not represent wasted experimental units.     

Some partially variance balanced structurally incomplete row-column designs

Earlier results by the authors are used to provide the intrablock analysis for row-column designs that have observations at nodes of the row-column lattice, the design being structurally incomplete when some nodes are empty. Construction, properties, and intrablock analyses of some special b× b row-column designs with b empty nodes taken along the principal diagonal of the lattice are developed. The designs discussed have m > 1 associate classes and are said to be partially variance balanced. The special designs fall in two classes and are shown to be nearly optimal in a specified class of designs. A small catalog of designs constructed is provided and they should be useful when empty nodes do not represent wasted experimental units, perhaps because the row and column assignments of treatments are sequenced.     

What is the shape of a bundle? An analysis of Rosen's fibrous composites experiments using the chain‐of‐bundles model

In the 1960s, W. B. Rosen conducted some remarkable experiments on unidirectional fibrous composites that gave seminal insights into their failure under increasing tensile load. These insights led him to a grid system where the nodes in the grid were ineffective   length fibers and to model the composite as something he called a chain‐of‐bundles model (i.e., a series system of parallel subsystems of horizontal nodes that he referred to as bundles), where the chain fails when one of the bundles fails. A load‐sharing rule was used to quantify how the load is borne among the nodes. Here, Rosen's experiments are analyzed to determine the shape of a bundle. The analysis suggests that the bundles are not horizontal collection of nodes but rather small rectangular grid systems of nodes where the load‐sharing between nodes is local in its form. In addition, a Gibbs measure representation for the joint distribution of binary random variables is given. This is used to show how the system reliability for a reliability structure can be obtained from the partition function for the Gibbs measure and to illustrate how to assess the risk of failure of a bundle in the chain‐of‐bundle model.     

The multitype branching random walk,I

The limiting behaviour of the multitype branching random walk is studied. A limit theorem is proven for the supercritical process. Steady-state distributions are shown to exist for the subcritical process with immigration, and for the critical transient process beginning with Poisson random fields. An analogue of the exponential limit law is proven for the critical process whose migration process is Brownian motion in two dimensions.     

Asymptotic Throughput in Discrete‐Time Cyclic Networks with Queue‐Length‐Dependent Service Rates

Abstract

For a discrete‐time closed cyclic network of single server queues whose service rates are non‐decreasing in the queue length, we compute the queue‐length distribution at each node in terms of throughputs of related networks. For the asymptotic analysis, we consider sequences of networks where the number of nodes grows to infinity, service rates are taken only from a fixed finite set of non‐decreasing sequences, the ratio of customers to nodes has a limit, and the proportion of nodes for each possible service‐rate sequence has a limit. Under these assumptions, the asymptotic throughput exists and is calculated explicitly. Furthermore, the asymptotic queue‐length distribution at any node can be obtained in terms of the asymptotic throughput. The asymptotic throughput, regarded as a function of the limiting customer‐to‐node ratio, is strictly increasing for ratios up to a threshold value (possibly infinite) and is constant thereafter. For ratios less than the threshold, the asymptotic queue‐length distribution at each node has finite moments of all orders. However, at or above the threshold, bottlenecks (nodes with asymptotically‐infinite mean queue length) do occur, and we completely characterize such nodes.     

Improved estimation for the autocovariances of a Gaussian stationary process

For a Gaussian stationary process with mean μ and autocovariance function γ(·), we consider to improve the usual sample autocovariances with respect to the mean squares error (MSE) loss. For the cases μ=0 and μ≠0, we propose sort of empirical Bayes type estimators Γ? and Γ?, respectively. Then their MSE improvements upon the usual sample autocovariances are evaluated in terms of the spectral density of the process. Concrete examples for them are provided. We observe that if the process is near to a unit root process the improvement becomes quite large. Thus, consideration for estimators of this type seems important in many fields, e.g., econometrics.     

A random cluster algorithm for some continuous Markov random fields

Based on a random cluster representation, the Swendsen–Wang algorithm for the Ising and Potts distributions is extended to a class of continuous Markov random fields. The algorithm can be described briefly as follows. A given configuration is decomposed into clusters. Probabilities for flipping the values of the random variables in each cluster are calculated. According to these probabilities, values of all the random variables in each cluster will be either updated or kept unchanged and this is done independently across the clusters. A new configuration is then obtained. We will show through a simulation study that, like the Swendsen–Wang algorithm in the case of Ising and Potts distributions, the cluster algorithm here also outperforms the Gibbs sampler in beating the critical slowing down for some strongly correlated Markov random fields.