Markov Dependence Research Articles

The Specification of Dynamic Discrete-Time Two-State Panel Data Models

This paper examines dynamic binary response and multi-spell duration model approaches to analyzing longitudinal discrete-time binary outcomes. Prototypical dynamic binary response models specify low-order Markovian state dependence and restrict the effects of observed and unobserved heterogeneity on the probability of transitioning into and out of a state to have the same magnitude and opposite signs. In contrast, multi-spell duration models typically allow for state-specify c duration dependence, and allow the probability of entry into and exit from a state to vary flexibly. We show that both of these approaches are special cases within a general framework. We compare specific dynamic binary response and multi-spell duration models empirically using a case study of poverty transitions. In this example, both the specification of state dependence and the restrictions on the state-specific transition probabilities imposed by the simpler dynamic binary response models are severely rejected against the more flexible multi-spell duration models. Consistent with recent literature, we conclude that the standard dynamic binary response model is unacceptably restrictive in this context.

A random variable whose digits in the $\widetilde {L}$-representation have the Markovian dependence

A random variable whose digits in the $\widetilde {L}$-representation have the Markovian dependence

Hidden Gibbs random fields model selection using Block Likelihood Information Criterion

Performing model selection between Gibbs random fields is a very challenging task. Indeed, because of the Markovian dependence structure, the normalizing constant of the fields cannot be computed using standard analytical or numerical methods. Furthermore, such unobserved fields cannot be integrated out, and the likelihood evaluation is a doubly intractable problem. This forms a central issue to pick the model that best fits an observed data. We introduce a new approximate version of the Bayesian Information Criterion. We partition the lattice into contiguous rectangular blocks, and we approximate the probability measure of the hidden Gibbs field by the product of some Gibbs distributions over the blocks. On that basis, we estimate the likelihood and derive the Block Likelihood Information Criterion (BLIC) that answers model choice questions such as the selection of the dependence structure or the number of latent states. We study the performances of BLIC for those questions. In addition, we present a comparison with ABC algorithms to point out that the novel criterion offers a better trade‐off between time efficiency and reliable results. Copyright © 2016 John Wiley & Sons, Ltd.

Transition Logic Regression Method to Identify Interactions in Binary Longitudinal Data

Logic regression is an adaptive regression method which searches for Boolean (logic) combinations of binary variables that best explain the variability in the outcome, and thus, it reveals interaction effects which are associated with the response. In this study, we extended logic regression to longitudinal data with binary response and proposed “Transition Logic Regression Method” to find interactions related to response. In this method, interaction effects over time were found by Annealing Algorithm with AIC (Akaike Information Criterion) as the score function of the model. Also, first and second orders Markov dependence were allowed to capture the correlation among successive observations of the same individual in longitudinal binary response. Performance of the method was evaluated with simulation study in various conditions. Proposed method was used to find interactions of SNPs and other risk factors related to low HDL over time in data of 329 participants of longitudinal TLGS study.

Carnivores in corridors: estimating tiger occupancy in Kanha–Pench corridor, Madhya Pradesh, India

We assessed occupancy for tigers Panthera tigris across Kanha–Pench corridor (KPC) in Madhya Pradesh, India. Tiger presence information was recorded by sign detection data generated from occupancy sampling of 17 grids, each of 100 km2. We used spatially replicated survey to explicitly address imperfect detections. We modelled occupancy (ψ) as a function of proportion of forested habitat, relative occurrence of prey species, and anthropogenic disturbances. Models were developed using a first-order Markovian dependence model implemented in program PRESENCE. Our results show that density of prey and levels of human disturbance were key determinants affecting tiger presence. We estimated that of the 2200 km2 potential tiger habitat available in KPC, tigers occupy 84 %, or an area of 1848 km2 (SE = 332.64 km2). Model averaging resulted in a replicate level detection probability p (SE) = 0.33 (0.08) for signs and a tiger habitat occupancy estimate of ψ (SE) = 0.84 (0.18). Landscape-level occupancy sign surveys are useful to assess large carnivore spatial distributions and determine factors governing their presence or absence. Occupancy results support earlier findings that prey presence and absence of human disturbance were key determinants for survival of tigers. We highlight the importance of corridors, habitats, and linkages which are important lifelines for large carnivores, tigers in particular. We discuss current infrastructural development pressure in the area and provide recommendations on critical local linkages to focus conservation efforts to maintain and improve current habitat connectivity for tigers and other carnivores in the area.

Gene Regulation Network Inference With Joint Sparse Gaussian Graphical Models

Revealing biological networks is one key objective in systems biology. With microarrays, researchers now routinely measure expression profiles at the genome level under various conditions, and such data may be used to statistically infer gene regulation networks. Gaussian graphical models (GGMs) have proven useful for this purpose by modeling the Markovian dependence among genes. However, a single GGM may not be adequate to describe the potentially differing networks across various conditions, and hence it is more natural to infer multiple GGMs from such data. In this article we propose a class of nonconvex penalty functions aiming at the estimation of multiple GGMs with a flexible joint sparsity constraint. We illustrate the property of our proposed nonconvex penalty functions by simulation study. We then apply the method to a gene expression dataset from the GenCord Project, and show that our method can identify prominent pathways across different conditions. Supplementary materials for this article are available online.

On the Growth Rate of a Linear Stochastic Recursion with Markovian Dependence

We consider the linear stochastic recursion \(x_{i+1} = a_{i}x_{i}+b_{i}\) where the multipliers \(a_i\) are random and have Markovian dependence given by the exponential of a standard Brownian motion and \(b_{i}\) are i.i.d. positive random noise independent of \(a_{i}\). Using large deviations theory we study the growth rates (Lyapunov exponents) of the positive integer moments \(\lambda _q = \lim _{n\rightarrow \infty } \frac{1}{n} \log \mathbb {E}[(x_n)^q]\) with \(q\in \mathbb {Z}_+\). We show that the Lyapunov exponents \(\lambda _q\) exist, under appropriate scaling of the model parameters, and have non-analytic behavior manifested as a phase transition. We study the properties of the phase transition and the critical exponents using both analytic and numerical methods.

Multi-scale statistical signal processing of cutting force in cutting tool condition monitoring

In a machining system, accurate tool wear condition monitoring is paramount for guaranteeing the quality of the workpiece and tool life. The cutting force signal has been proved to be the most sensitive signal to depict the tool wear variation during the machining process. This paper introduces a data-driven modeling framework for tool wear monitoring in a machining process, which is based on statistical processing of cutting force wavelet transform by a hidden Markov tree. As a kind of data-driven prognostic approach, this method exploits the tool wear states feature from a deeply data mining perspective while the Markov dependence of wavelet transformation at different frequencies or scales is captured. With lathe turning as the research object, a detailed study on the statistical analysis of cutting force in different tool conditions is presented. A two phases monitoring process that assesses the tool wear conditions from generated model by using the statistical features of cutting force wavelet transform is built. Compared to the traditional classifiers, which usually have a difficult condition in distinguishing tool wear states when given a limited amount of samples, the proposed approach make more efficient use of the training data with high sensitivity to the tool wear conditions. Experimental studies of Inconel 718 cutting show that this approach is robust and makes the cutting force information exploited effectively in data mining. Based on the experimental results, the proposed method for tool condition monitoring outperforms the traditional used Hidden Markov model and Gaussian mixture model approach.

Binomial Reliability Demonstration Tests with Dependent Data

ABSTRACTBinomial reliability demonstration tests are a way to demonstrate the product or process capability where binary data are available. It is easy to compute the sample size for a binomial reliability demonstration test (BDRT) when the assumption of independent samples holds. However, the assumption of independence may not always be valid; thus, it is desirable to account for this dependence in a sample size calculation. We show how a Markov dependence model can be used to calculate the sample size. We provide tables of sample sizes and give an example for which a BRDT can be performed with dependent data.

Modified Synthetic Control Chart for One-Step Markov-Dependent Processes

Quality measurement by the attributes method often does not result in independent random variables for the count of non conformities or non conforming units. A model based on Markovian dependence can be considered as a practical option in such situations. For a Markovian production process, Adnaik et al. (in press) have proposed the “Single Attribute Control Charts” (SACCs) using two criteria. The first one is by controlling the error probabilities being (α*, β*) and the second, by using the ATS criterion. In this article, we propose a “Modified Synthetic Single Attribute Control Chart” for one-step Markovian Dependent process using ATS criterion (MSyn-SACC-MD).Zero state as well as the steady-state performance of the chart is studied. It is numerically illustrated that the proposed chart is more efficient than SACC-MD-A as proposed by Adnaik et al. (in press).

A Data Driven Runs Test to Identify First Order Positive Markovian Dependence

We propose a data driven test to identify first order positive Markovian dependence in a Bernoulli sequence, based on a combination of two runs tests: a well known runs test for the same purpose conditional on the numbers of ones in the sequence, and a modified runs test independent of the number of ones. We give analytic expressions for the exact distribution of the modified runs test statistic and for its power; also we built an algorithm to calculate it explicitly. To compare the power of the tests, we calculated these for some values of the proportion of ones and the success probability. We show that there are some intervals for the success probability in which the new runs test surpasses the power of the conditional test, and that the data driven test improves the power of the two runs tests, when they are considered separately.

Single image defogging by multiscale depth fusion.

Restoration of fog images is important for the deweathering issue in computer vision. The problem is ill-posed and can be regularized within a Bayesian context using a probabilistic fusion model. This paper presents a multiscale depth fusion (MDF) method for defog from a single image. A linear model representing the stochastic residual of nonlinear filtering is first proposed. Multiscale filtering results are probabilistically blended into a fused depth map based on the model. The fusion is formulated as an energy minimization problem that incorporates spatial Markov dependence. An inhomogeneous Laplacian-Markov random field for the multiscale fusion regularized with smoothing and edge-preserving constraints is developed. A nonconvex potential, adaptive truncated Laplacian, is devised to account for spatially variant characteristics such as edge and depth discontinuity. Defog is solved by an alternate optimization algorithm searching for solutions of depth map by minimizing the nonconvex potential in the random field. The MDF method is experimentally verified by real-world fog images including cluttered-depth scene that is challenging for defogging at finer details. The fog-free images are restored with improving contrast and vivid colors but without over-saturation. Quantitative assessment of image quality is applied to compare various defog methods. Experimental results demonstrate that the accurate estimation of depth map by the proposed edge-preserved multiscale fusion should recover high-quality images with sharp details.

Long-run growth rate in a random multiplicative model

We consider the long-run growth rate of the average value of a random multiplicative process xi + 1 = aixi where the multipliers \documentclass[12pt]{minimal}\begin{document}$a_i=1+\rho \exp (\sigma W_i - \frac{1}{2}\sigma ^2 t_i)$\end{document}ai=1+ρexp(σWi−12σ2ti) have Markovian dependence given by the exponential of a standard Brownian motion Wi. The average value ⟨xn⟩ is given by the grand partition function of a one-dimensional lattice gas with two-body linear attractive interactions placed in a uniform field. We study the Lyapunov exponent \documentclass[12pt]{minimal}\begin{document}$\lambda =\lim _{n\rightarrow \infty } \frac{1}{n}\log \langle x_n\rangle$\end{document}λ=limn→∞1nlog⟨xn⟩, at fixed \documentclass[12pt]{minimal}\begin{document}$\beta = \frac{1}{2} \sigma^2 t_n n$\end{document}β=12σ2tnn, and show that it is given by the equation of state of the lattice gas in thermodynamical equilibrium. The Lyapunov exponent has discontinuous partial derivatives along a curve in the (ρ, β) plane ending at a critical point (ρC, βC) which is related to a phase transition in the equivalent lattice gas. Using the equivalence of the lattice gas with a bosonic system, we obtain the exact solution for the equation of state in the thermodynamical limit n → ∞.

The Distribution of Word Matches Between Markovian Sequences with Periodic Boundary Conditions

Word match counts have traditionally been proposed as an alignment-free measure of similarity for biological sequences. The D(2) statistic, which simply counts the number of exact word matches between two sequences, is a useful test bed for developing rigorous mathematical results, which can then be extended to more biologically useful measures. The distributional properties of the D(2) statistic under the null hypothesis of identically and independently distributed letters have been studied extensively, but no comprehensive study of the D(2) distribution for biologically more realistic higher-order Markovian sequences exists. Here we derive exact formulas for the mean and variance of the D(2) statistic for Markovian sequences of any order, and demonstrate through Monte Carlo simulations that the entire distribution is accurately characterized by a Pólya-Aeppli distribution for sequence lengths of biological interest. The approach is novel in that Markovian dependency is defined for sequences with periodic boundary conditions, and this enables exact analytic formulas for the mean and variance to be derived. We also carry out a preliminary comparison between the approximate D(2) distribution computed with the theoretical mean and variance under a Markovian hypothesis and an empirical D(2) distribution from the human genome.

A Bayesian statistical model for deriving the predictive distribution of hydroclimatic variables

Recent publications have provided evidence that hydrological processes exhibit a scaling behaviour, also known as the Hurst phenomenon. An appropriate way to model this behaviour is to use the Hurst-Kolmogorov stochastic process. The Hurst-Kolmogorov process entails high autocorrelations even for large lags, as well as high variability even at climatic scales. A problem that, thus, arises is how to incorporate the observed past hydroclimatic data in deriving the predictive distribution of hydroclimatic processes at climatic time scales. Here with the use of Bayesian techniques we create a framework to solve the aforementioned problem. We assume that there is no prior information for the parameters of the process and use a non-informative prior distribution. We apply this method with real-world data to derive the posterior distribution of the parameters and the posterior predictive distribution of various 30-year moving average climatic variables. The marginal distributions we examine are the normal and the truncated normal (for nonnegative variables). We also compare the results with two alternative models, one that assumes independence in time and one with Markovian dependence, and the results are dramatically different. The conclusion is that this framework is appropriate for the prediction of future hydroclimatic variables conditional on the observations.

Fisher information matrix: A tool for dimension reduction, projection pursuit, independent component analysis, and more

AbstractHui & Lindsay (2010) proposed a new dimension reduction method for multivariate data. It was based on the so‐called white noise matrices derived from the Fisher information matrix. Their theory and empirical studies demonstrated that this method can detect interesting features from high‐dimensional data even with a moderate sample size. The theoretical emphasis in that paper was the detection of non‐normal projections. In this paper we show how to decompose the information matrix into non‐negative definite information terms in a manner akin to a matrix analysis of variance. Appropriate information matrices will be identified for diagnostics for such important modern modelling techniques as independent component models, Markov dependence models, and spherical symmetry. The Canadian Journal of Statistics 40: 712–730; 2012 © 2012 Statistical Society of Canada

On Delay Tomography: Fast Algorithms and Spatially Dependent Models

As an active branch of network tomography, delay tomography has received considerable attentions in recent years. However, most methods in the literature assume that the delays of different links are independent of each other, and pursuit sub-optimal estimate instead of the maximum likelihood estimate (MLE) due to computational challenges. In this paper, we propose a novel method to implement the EM algorithm widely used in delay tomography analysis for multicast networks. The proposed method makes use of a “delay pattern database” to avoid all redundant computations in the E-step, and is much faster than the traditional implementation. With the help of this new implementation, finding MLE for large networks, which was considered impractical previously, becomes an easy task. Taking advantage of this computational breakthrough, we further consider models for potential spatial dependence of links, and propose a novel adaptive spatially dependent model (ASDM) for delay tomography. In ASDM, Markov dependence among nearby links is allowed, and spatially dependent links (SDLs) can be automatically recognized via model selection. The superiority of the new methods is confirmed by simulation studies.

Markov chain model to study the occurrence of pre-monsoon thunderstorms over Bhubaneswar, India

The present work deals with pre-monsoon thunderstorms over Bhubaneswar belonging to the state of Orissa, India. A Markovian approach has been adopted to discern the probabilistic behavior of the time series of the occurrence and non-occurrence of this hazardous weather event by introducing a dichotomy within the time series. After a painstaking analysis through chi-square tests, we have identified serial independence in a few years and first-order two-state Markovian dependence in a few years (2000, 2001, 2004 and 2006). Finally, for the years of first-order two-state Markovian dependence, it has been observed that the probability of occurrence or non-occurrence of thunderstorm gets higher if the state of the previous day is similar to that of the current day. Furthermore, the probability of getting non-thunderstorm day followed by non-thunderstorm day is higher than the probability of getting thunderstorm day followed by thunderstorm day. It has been also observed that the unconditional climatological probability of the occurrence of severe pre-monsoon thunderstorm implied by the Markov chain is closely in agreement with the observed relative frequencies. However, it could be revealed that Markov chain cannot, in general, be suggested as a predictive tool for pre-monsoon thunderstorms under study without investigating the serial dependence inherent in the time series.

A transition model for quality‐of‐life data with non‐ignorable non‐monotone missing data

In this paper, we consider a full likelihood method to analyze continuous longitudinal responses with non-ignorable non-monotone missing data. We consider a transition probability model for the missingness mechanism. A first-order Markov dependence structure is assumed for both the missingness mechanism and observed data. This process fits the natural data structure in the longitudinal framework. Our main interest is in estimating the parameters of the marginal model and evaluating the missing-at-random assumption in the Effects of Public Information Study, a cancer-related study recently conducted at the University of Pennsylvania. We also present a simulation study to assess the performance of the model.

Exact distributions of constrained (k, ℓ) strings of failures between subsequent successes

Consider a sequence of binary (success–failure) random variables (RVs) ordered on a line. The number of strings with a constrained number of consecutive failures between two subsequent successes is studied under an overlapping enumeration scheme. The respective waiting time is examined as well. The study is first developed on sequences of independent and identically distributed RVs. It is extended then on sequences of dependent, exchangeability and Markovian dependency is considered, and independent, not necessarily identically distributed, RVs. Exact probabilities and moments are obtained by means of combinatorial analysis and via recursive schemes. An explicit expression of the mean value of the number of strings for both independent and dependent sequences is derived. An application in system reliability is provided.