A copula-based partition Markov procedure

The number of parameters needed to specify a discrete multivariate Markov chain grows exponentially with the order and dimension of the chain, and when the size of the database is not large enough, it is not possibly a consistent estimation. In this paper, we introduce a strategy to estimate a multivariate process with an order greater than the order achieved using standard procedures. The new strategy consists in obtaining a partition of the state space which is constructed from a combination of the partitions corresponding to the marginal processes and the partitions corresponding to the multivariate Markov chain.     

Modeling dropouts by conditional distribution, a copula-based approach

In this paper the concept of copulas is implemented into the methodology for solving the imputation problem in correlated incomplete data. We use the Gaussian copula as alternative to the joint distribution for modeling the conditional distribution, conditioned by the observed values of measurements. The general formula for imputation and its application for compound symmetry correlation structure are given.     

A copula-based Markov chain model for the analysis of binary longitudinal data

A fully parametric first-order autoregressive (AR(1)) model is proposed to analyse binary longitudinal data. By using a discretized version of a copula, the modelling approach allows one to construct separate models for the marginal response and for the dependence between adjacent responses. In particular, the transition model that is focused on discretizes the Gaussian copula in such a way that the marginal is a Bernoulli distribution. A probit link is used to take into account concomitant information in the behaviour of the underlying marginal distribution. Fixed and time-varying covariates can be included in the model. The method is simple and is a natural extension of the AR(1) model for Gaussian series. Since the approach put forward is likelihood-based, it allows interpretations and inferences to be made that are not possible with semi-parametric approaches such as those based on generalized estimating equations. Data from a study designed to reduce the exposure of children to the sun are used to illustrate the methods.     

R routines for performing estimation and statistical process control under copula-based time series models

Modeling serial dependence in time series is an important step in statistical process control. We provide a set of automatic routines useful for simulating and analyzing time series under a copula-based serial dependence. First, we introduce routines that generate time series data under a given copula. Second, we provide fully automated routines for obtaining maximum likelihood estimates for given time series data and then drawing a Shewhart-type control chart. Finally, real data are analyzed for illustration. We make the routines available as “Copula.Markov” package in R.     

Residual life estimation based on bivariate non-stationary gamma degradation process

Due to the growing importance in maintenance scheduling, the issue of residual life (RL) estimation for some high reliable products based on degradation data has been studied quite extensively. However, most of the existing work only deals with one-dimensional degradation data, which may not be realistic in some cases. Here, an adaptive method of RL estimation is developed based on two-dimensional degradation data. It is assumed that a product has two performance characteristics (PCs) and that the degradation of each PC over time is governed by a non-stationary gamma degradation process. From a practical consideration, it is further assumed that these two PCs are dependent and that their dependency can be characterized by a copula function. As the likelihood function in such a situation is complicated and computationally quite intensive, a two-stage method is used to estimate the unknown parameters of the model. Once new degradation information of the product being monitored becomes available, random effects are first updated by using the Bayesian method. Following that, the RL at current time is estimated accordingly. As the degradation data information accumulates, the RL can be re-estimated in an adaptive manner. Finally, a numerical example about fatigue cracks is presented in order to illustrate the proposed model and the developed inferential method.     

Statistical monitoring of a web server for error rates: a bivariate time-series copula-based modeling approach

The monitoring of web servers through statistical frameworks is of utmost important in order to verify possible suspicious anomalies in network traffic or misuse actions that compromise integrity, confidentiality, and availability of information. In this paper, by considering the Plackett copula function, we propose a bivariate beta-autoregressive moving average time-series model for proportion data over time, which is the case for variables present in web server monitoring such as error rates. To illustrate the proposed methodology, we monitor a Brazilian web server's rate of connection synchronization and rejection errors in a web system, with error logging rate in the past 10?min. In essence, the entire methodology may be generalized to any number of time-series of error rates.     

Markov Chains and De-initializing Processes

We define a notion of de-initializing Markov chains. We prove that to analyse convergence of Markov chains to stationarity, it suffices to analyse convergence of a de-initializing chain. Applications are given to Markov chain Monte Carlo algorithms and to convergence diagnostics.     

Poisson-Driven Stationary Markov Models

ABSTRACT

We propose a simple yet powerful method to construct strictly stationary Markovian models with given but arbitrary invariant distributions. The idea is based on a Poisson-type transform modulating the dependence structure in the model. An appealing feature of our approach is the possibility to control the underlying transition probabilities and, therefore, incorporate them within standard estimation methods. Given the resulting representation of the transition density, a Gibbs sampler algorithm based on the slice method is proposed and implemented. In the discrete-time case, special attention is placed to the class of generalized inverse Gaussian distributions. In the continuous case, we first provide a brief treatment of the class of gamma distributions, and then extend it to cover other invariant distributions, such as the generalized extreme value class. The proposed approach and estimation algorithm are illustrated with real financial datasets. Supplementary materials for this article are available online.     

Specifying a Gaussian Markov Random Field by a Sparse Cholesky Triangle

ABSTRACT

This note discusses the approach of specifying a Gaussian Markov random field (GMRF) by the Cholesky triangle of the precision matrix. A such representation can be made extremely sparse using numerical techniques for incomplete sparse Cholesky factorization, and provide very computational efficient representation for simulating from the GMRF. However, we provide theoretical and empirical justification showing that the sparse Cholesky triangle representation is fragile when conditioning a GMRF on a subset of the variables or observed data, meaning that the computational cost increases.     

Fast sampling of Gaussian Markov random fields

This paper demonstrates how Gaussian Markov random fields (conditional autoregressions) can be sampled quickly by using numerical techniques for sparse matrices. The algorithm is general and efficient, and expands easily to various forms for conditional simulation and evaluation of normalization constants. We demonstrate its use by constructing efficient block updates in Markov chain Monte Carlo algorithms for disease mapping.     

Bayesian nonparametric estimation of a copula

A copula can fully characterize the dependence of multiple variables. The purpose of this paper is to provide a Bayesian nonparametric approach to the estimation of a copula, and we do this by mixing over a class of parametric copulas. In particular, we show that any bivariate copula density can be arbitrarily accurately approximated by an infinite mixture of Gaussian copula density functions. The model can be estimated by Markov Chain Monte Carlo methods and the model is demonstrated on both simulated and real data sets.     

Hierarchical Gaussian process mixtures for regression

As a result of their good performance in practice and their desirable analytical properties, Gaussian process regression models are becoming increasingly of interest in statistics, engineering and other fields. However, two major problems arise when the model is applied to a large data-set with repeated measurements. One stems from the systematic heterogeneity among the different replications, and the other is the requirement to invert a covariance matrix which is involved in the implementation of the model. The dimension of this matrix equals the sample size of the training data-set. In this paper, a Gaussian process mixture model for regression is proposed for dealing with the above two problems, and a hybrid Markov chain Monte Carlo (MCMC) algorithm is used for its implementation. Application to a real data-set is reported.     

Markov Beta and Gamma Processes for Modelling Hazard Rates

This paper generalizes the discrete time independent increment beta process of Hjort (1990 ), for modelling discrete failure times, and also generalizes the independent gamma process for modelling piecewise constant hazard rates ( Walker and Mallick, 1997 ). The generalizations are from independent increment to Markov increment prior processes allowing the modelling of smoothness. We derive posterior distributions and undertake a full Bayesian analysis.     

Statistical Modeling of Temporal Dependence in Financial Data via a Copula Function

In financial analysis it is useful to study the dependence between two or more time series as well as the temporal dependence in a univariate time series. This article is concerned with the statistical modeling of the dependence structure in a univariate financial time series using the concept of copula. We treat the series of financial returns as a first order Markov process. The Archimedean two-parameter BB7 copula is adopted to describe the underlying dependence structure between two consecutive returns, while the log-Dagum distribution is employed to model the margins marked by skewness and kurtosis. A simulation study is carried out to evaluate the performance of the maximum likelihood estimates. Furthermore, we apply the model to the daily returns of four stocks and, finally, we illustrate how its fitting to data can be improved when the dependence between consecutive returns is described through a copula function.     

Joint modelling of longitudinal biomarker and gap time between recurrent events: copula-based dependence

In this paper, we will extend the joint model of longitudinal biomarker and recurrent event via copula function for accounting the dependence between the two processes. The general idea of joining separate processes by allowing model-specific random effect may come from different families distribution. It is a main advantage of the proposed method that a copula construction does not constrain the choice of marginal distributions of random effects. A maximum likelihood estimation with importance sampling technique as a simple and easy understanding method is employed to model inference. To evaluate and verify the validation of the proposed joint model, a bootstrapping method as a model-based resampling is developed. Our proposed joint model is also applied to pemphigus disease data for assessing the effect of biomarker trajectory on risk of recurrence.     

Markov models of mite movements

Continuous time Markov models were used to analyse data from two bioassays to investigate the influence of β-fraction, a by-product of hop processing, on the two-spotted spider mite. The models were fitted to aggregate counts of the numbers of live and dead mites on treated and untreated halves of discs cut from leaves of hop and French bean plants. Some of the rate parameters were time dependent. Although not all parameters could be estimated precisely, the analysis enabled the quantitative effects of treatment over time to be estimated with reasonable precision. The estimated treatment effects were largely insensitive to the assumed values of other parameters. The first bioassay showed a progressive initial response to increasing concentration of β-fraction, although data at the intermediate concentration appeared anomalous. The second bioassay showed similar responses on hop and French bean leaves, with a stronger repellent effect on the lower leaf surface than on the upper surface.     

Modeling material stress using integrated Gaussian Markov random fields

The equations of a physical constitutive model for material stress within tantalum grains were solved numerically using a tetrahedrally meshed volume. The resulting output included a scalar vonMises stress for each of the more than 94,000 tetrahedra within the finite element discretization. In this paper, we define an intricate statistical model for the spatial field of vonMises stress which uses the given grain geometry in a fundamental way. Our model relates the three-dimensional field to integrals of latent stochastic processes defined on the vertices of the one- and two-dimensional grain boundaries. An intuitive neighborhood structure of the said boundary nodes suggested the use of a latent Gaussian Markov random field (GMRF). However, despite the potential for computational gains afforded by GMRFs, the integral nature of our model and the sheer number of data points pose substantial challenges for a full Bayesian analysis. To overcome these problems and encourage efficient exploration of the posterior distribution, a number of techniques are now combined: parallel computing, sparse matrix methods, and a modification of a block update strategy within the sampling routine. In addition, we use an auxiliary variables approach to accommodate the presence of outliers in the data.     

The occurrence of fingerprint characteristics as a two-dimensional poisson process

The individuality of n fingerprint is based on the configuration of occurences of the ten Galton characteristics ( ridge endings, forks, etc. ). A model ( Osterburg, Parthasarthy, Raghavan, Sclove, 1977 ) for the occurence of these characteristics, in terms of a grid of cells, is further developed. The occurence of the characteristics is modelled as a two-dimensional multivariate Poisson process. This approach allows one to treat multiple occurrences in a more satisfying way than in Osterburg, Parthasarathy, Raghavan and Sclove ( 1977 ) or Sclove ( 1978 )     

Estimation of the intensity function of an inhomogeneous Poisson process with a change‐point

Recent work on point processes includes studying posterior convergence rates of estimating a continuous intensity function. In this article, convergence rates for estimating the intensity function and change‐point are derived for the more general case of a piecewise continuous intensity function. We study the problem of estimating the intensity function of an inhomogeneous Poisson process with a change‐point using non‐parametric Bayesian methods. An Markov Chain Monte Carlo (MCMC) algorithm is proposed to obtain estimates of the intensity function and the change‐point which is illustrated using simulation studies and applications. The Canadian Journal of Statistics 47: 604–618; 2019 © 2019 Statistical Society of Canada     

Network-based genomic discovery: application and comparison of Markov random field models

Summary.  As biological knowledge accumulates rapidly, gene networks encoding genomewide gene–gene interactions have been constructed. As an improvement over the standard mixture model that tests all the genes identically and independently distributed a priori , Wei and co-workers have proposed modelling a gene network as a discrete or Gaussian Markov random field (MRF) in a mixture model to analyse genomic data. However, how these methods compare in practical applications is not well understood and this is the aim here. We also propose two novel constraints in prior specifications for the Gaussian MRF model and a fully Bayesian approach to the discrete MRF model. We assess the accuracy of estimating the false discovery rate by posterior probabilities in the context of MRF models. Applications to a chromatin immuno-precipitation–chip data set and simulated data show that the modified Gaussian MRF models have superior performance compared with other models, and both MRF-based mixture models, with reasonable robustness to misspecified gene networks, outperform the standard mixture model.