Hierarchical Bayesian Approach to a Multi-Site Hidden Markov Model

Multivariate data with a sequential or temporal structure occur in various fields of study. The hidden Markov model (HMM) provides an attractive framework for modeling long-term persistence in areas of pattern recognition through the extension of independent and identically distributed mixture models. Unlike in typical mixture models, the heterogeneity of data is represented by hidden Markov states. This article extends the HMM to a multi-site or multivariate case by taking a hierarchical Bayesian approach. This extension has many advantages over a single-site HMM. For example, it can provide more information for identifying the structure of the HMM than a single-site analysis. We evaluate the proposed approach by exploiting a spatial correlation that depends on the distance between sites.     

Nonparametric Bayesian modeling for monotonicity in catch ratio

This article proposes nonparametric Bayesian approaches to monotone function estimation. This approach uses a hierarchical Bayes framework and a characterization of stick-breaking process that allows unconstrained estimation of the monotone function. In order to avoid the limitation of parametric modeling, a general class of prior distributions, called stick-breaking priors, is considered. It accommodates much more flexible forms and can easily deal with skewness, multimodality, etc., of the dependent variable response. The proposed approach is incorporated to model the catch ratio based on automatic weather station (AWS) data.     

A simple hidden markov model for bayesian modeling with time dependent data

Consider a set of real valued observations collected over time. We pro¬pose a simple hidden Markow model for these realizations in which the the predicted distribution of the next future observation given the past is easily computed. The hidden or unobservable set of parameters is assumed to have a Markov structure of a special type. The model is quite flexible and can be used to incorporate different types of prior information in straightforward and sensible ways.     

A dynamic analysis of stock markets using a hidden Markov model

This paper proposes a framework to detect financial crises, pinpoint the end of a crisis in stock markets and support investment decision-making processes. This proposal is based on a hidden Markov model (HMM) and allows for a specific focus on conditional mean returns. By analysing weekly changes in the US stock market indexes over a period of 20 years, this study obtains an accurate detection of stable and turmoil periods and a probabilistic measure of switching between different stock market conditions. The results contribute to the discussion of the capabilities of Markov-switching models of analysing stock market behaviour. In particular, we find evidence that HMM outperforms threshold GARCH model with Student-t innovations both in-sample and out-of-sample, giving financial operators some appealing investment strategies.     

A non-homogeneous hidden Markov model for predicting the distribution of sea surface elevation

The prediction problem of sea state based on the field measurements of wave and meteorological factors is a topic of interest from the standpoints of navigation safety and fisheries. Various statistical methods have been considered for the prediction of the distribution of sea surface elevation. However, prediction of sea state in the transitional situation when waves are developing by blowing wind has been a difficult problem until now, because the statistical expression of the dynamic mechanism during this situation is very complicated. In this article, we consider this problem through the development of a statistical model. More precisely, we develop a model for the prediction of the time-varying distribution of sea surface elevation, taking into account a non-homogeneous hidden Markov model in which the time-varying structures are influenced by wind speed and wind direction. Our prediction experiments suggest the possibility that the proposed model contributes to an improvement of the prediction accuracy by using a homogenous hidden Markov model. Furthermore, we found that the prediction accuracy is influenced by the circular distribution of the circular hidden Markov model for the directional time series wind direction data.     

Semiparametric hidden Markov model with non-parametric regression

The hidden Markov model regression (HMMR) has been popularly used in many fields such as gene expression and activity recognition. However, the traditional HMMR requires the strong linearity assumption for the emission model. In this article, we propose a hidden Markov model with non-parametric regression (HMM-NR), where the mean and variance of emission model are unknown smooth functions. The new semiparametric model might greatly reduce the modeling bias and thus enhance the applicability of the traditional hidden Markov model regression. We propose an estimation procedure for the transition probability matrix and the non-parametric mean and variance functions by combining the ideas of the EM algorithm and the kernel regression. Simulation studies and a real data set application are used to demonstrate the effectiveness of the new estimation procedure.     

Bayesian inference in hidden Markov models through the reversible jump Markov chain Monte Carlo method

Hidden Markov models form an extension of mixture models which provides a flexible class of models exhibiting dependence and a possibly large degree of variability. We show how reversible jump Markov chain Monte Carlo techniques can be used to estimate the parameters as well as the number of components of a hidden Markov model in a Bayesian framework. We employ a mixture of zero-mean normal distributions as our main example and apply this model to three sets of data from finance, meteorology and geomagnetism.     

Bayesian clustering for continuous-time hidden Markov models

We develop clustering procedures for longitudinal trajectories based on a continuous-time hidden Markov model (CTHMM) and a generalized linear observation model. Specifically, in this article we carry out finite and infinite mixture model-based clustering for a CTHMM and achieve inference using Markov chain Monte Carlo (MCMC). For a finite mixture model with a prior on the number of components, we implement reversible-jump MCMC to facilitate the trans-dimensional move between models with different numbers of clusters. For a Dirichlet process mixture model, we utilize restricted Gibbs sampling split–merge proposals to improve the performance of the MCMC algorithm. We apply our proposed algorithms to simulated data as well as a real-data example, and the results demonstrate the desired performance of the new sampler.     

隐非齐次马尔可夫多元线性回归模型及其贝叶斯估计

隐马尔可夫模型对于异质纵向数据的处理有良好的效果,因此被广泛应用于工程技术、生物医学、经济管理等领域。文章引入了一种特殊的非齐次隐马尔可夫状态转移方式,并将其与经典的多元线性回归相结合,提出了隐非齐次马尔可夫多元线性回归模型,介绍了对该模型进行贝叶斯推断的方法原理和技术细节。最后,通过两个模拟实验说明了推断方法的结果是可靠的。     

MAP segmentation in Bayesian hidden Markov models: a case study

We consider the problem of estimating the maximum posterior probability (MAP) state sequence for a finite state and finite emission alphabet hidden Markov model (HMM) in the Bayesian setup, where both emission and transition matrices have Dirichlet priors. We study a training set consisting of thousands of protein alignment pairs. The training data is used to set the prior hyperparameters for Bayesian MAP segmentation. Since the Viterbi algorithm is not applicable any more, there is no simple procedure to find the MAP path, and several iterative algorithms are considered and compared. The main goal of the paper is to test the Bayesian setup against the frequentist one, where the parameters of HMM are estimated using the training data.     

Testing for two states in a hidden Markov model

The authors consider hidden Markov models (HMMs) whose latent process has m ≥ 2 states and whose state‐dependent distributions arise from a general one‐parameter family. They propose a test of the hypothesis m = 2. Their procedure is an extension to HMMs of the modified likelihood ratio statistic proposed by Chen, Chen & Kalbfleisch (2004) for testing two states in a finite mixture. The authors determine the asymptotic distribution of their test under the hypothesis m = 2 and investigate its finite‐sample properties in a simulation study. Their test is based on inference for the marginal mixture distribution of the HMM. In order to illustrate the additional difficulties due to the dependence structure of the HMM, they show how to test general regular hypotheses on the marginal mixture of HMMs via a quasi‐modified likelihood ratio. They also discuss two applications.     

Bayesian change points analysis for earthquakes body wave magnitude

Recently, the world has experienced an increased number of major earthquakes. The Zagros belt is among the most seismically active mountain ranges in the world. Due to Kuwait's location in the southwest of the Zagros belt, it is affected by relative tectonic movements in the neighboring region. It is vital to assess the Zagros seismic risks in Kuwait using recent data and coordinate with the competent authorities to reduce those risks. Using the body wave magnitude (Mb) data collected in Kuwait, we want to assess the recent changes in the magnitude of earthquakes and its variations in Kuwait's vicinity. We built a change point model to detect the significant changes in its parameters. This paper applies a hierarchical Bayesian technique and derives the marginal posterior density function for the Mb. Our interest lies in identifying a shift in the mean of a single or multiple change points as well as the changes in the variation. Building upon the model and its parameters for the 2002–2003 data, we detected three change points. The first, second and third change points occurred in September 2002, April 2003 and August 2003, respectively.     

Dynamic latent trait models with mixed hidden Markov structure for mixed longitudinal outcomes

We propose a general Bayesian joint modeling approach to model mixed longitudinal outcomes from the exponential family for taking into account any differential misclassification that may exist among categorical outcomes. Under this framework, outcomes observed without measurement error are related to latent trait variables through generalized linear mixed effect models. The misclassified outcomes are related to the latent class variables, which represent unobserved real states, using mixed hidden Markov models (MHMMs). In addition to enabling the estimation of parameters in prevalence, transition and misclassification probabilities, MHMMs capture cluster level heterogeneity. A transition modeling structure allows the latent trait and latent class variables to depend on observed predictors at the same time period and also on latent trait and latent class variables at previous time periods for each individual. Simulation studies are conducted to make comparisons with traditional models in order to illustrate the gains from the proposed approach. The new approach is applied to data from the Southern California Children Health Study to jointly model questionnaire-based asthma state and multiple lung function measurements in order to gain better insight about the underlying biological mechanism that governs the inter-relationship between asthma state and lung function development.     

A hybrid Markov chain for the Bayesian analysis of the multinomial probit model

Bayesian inference for the multinomial probit model, using the Gibbs sampler with data augmentation, has been recently considered by some authors. The present paper introduces a modification of the sampling technique, by defining a hybrid Markov chain in which, after each Gibbs sampling cycle, a Metropolis step is carried out along a direction of constant likelihood. Examples with simulated data sets motivate and illustrate the new technique. A proof of the ergodicity of the hybrid Markov chain is also given.     

Equilibrium dynamics of ice streams: a Bayesian statistical analysis

Studies of the behaviors of glaciers, ice sheets, and ice streams rely heavily on both observations and physical models. Data acquired via remote sensing provide critical information on geometry and movement of ice over large sections of Antarctica and Greenland. However, uncertainties are present in both the observations and the models. Hence, there is a need for combining these information sources in a fashion that incorporates uncertainty and quantifies its impact on conclusions. We present a hierarchical Bayesian approach to modeling ice-stream velocities incorporating physical models and observations regarding velocity, ice thickness, and surface elevation from the North East Ice Stream in Greenland. The Bayesian model leads to interesting issues in model assessment and computation.     

Bayesian estimation for first-order autoregressive model with explanatory variables

In this article, we develop a Bayesian analysis in autoregressive model with explanatory variables. When σ² is known, we consider a normal prior and give the Bayesian estimator for the regression coefficients of the model. For the case σ² is unknown, another Bayesian estimator is given for all unknown parameters under a conjugate prior. Bayesian model selection problem is also being considered under the double-exponential priors. By the convergence of ρ-mixing sequence, the consistency and asymptotic normality of the Bayesian estimators of the regression coefficients are proved. Simulation results indicate that our Bayesian estimators are not strongly dependent on the priors, and are robust.     

Bayesian analysis of elapsed times in continuous‐time Markov chains

The authors consider Bayesian analysis for continuous‐time Markov chain models based on a conditional reference prior. For such models, inference of the elapsed time between chain observations depends heavily on the rate of decay of the prior as the elapsed time increases. Moreover, improper priors on the elapsed time may lead to improper posterior distributions. In addition, an infinitesimal rate matrix also characterizes this class of models. Experts often have good prior knowledge about the parameters of this matrix. The authors show that the use of a proper prior for the rate matrix parameters together with the conditional reference prior for the elapsed time yields a proper posterior distribution. The authors also demonstrate that, when compared to analyses based on priors previously proposed in the literature, a Bayesian analysis on the elapsed time based on the conditional reference prior possesses better frequentist properties. The type of prior thus represents a better default prior choice for estimation software.     

A Bayesian compartmental model for the evaluation of 1,3-butadiene metabolism

Summary. We propose a Bayesian model for physiologically based pharmacokinetics of 1,3-butadiene (BD). BD is classified as a suspected human carcinogen and exposure to it is common, especially through cigarette smoke as well as in urban settings. The main aim of the methodology and analysis that are presented here is to quantify variability in the rates of BD metabolism by human subjects. A three-compartmental model is described, together with informative prior distributions for the population parameters, all of which represent real physiological variables. The model is described in detail along with the meanings and interpretations of the associated parameters. A four-compartment model is also given for comparison. Markov chain Monte Carlo methods are described for fitting the model proposed. The model is fitted to toxicokinetic data obtained from 133 healthy subjects (males and females) from the four major racial groups in the USA, with ages ranging from 19 to 62 years. Subjects were exposed to 2 parts per million of BD for 20 min through a face mask by using a computer-controlled exposure and respiratory monitoring system. Stratification by ethnic group results in major changes in the physiological parameters. Sex and age were also tested but not found to have a significant effect.     

Nonparametric Bayesian inference for ergodic diffusions

The problem of nonparametric drift estimation for ergodic diffusions is studied from a Bayesian perspective. In particular, Gaussian process priors are exhibited that yield optimal contraction rates if the drift function belongs to a smoothness class.     

A non-homogeneous hidden Markov model for precipitation occurrence

A non-homogeneous hidden Markov model is proposed for relating precipitation occurrences at multiple rain-gauge stations to broad scale atmospheric circulation patterns (the so-called 'downscaling problem'). We model a 15-year sequence of winter data from 30 rain stations in south-western Australia. The first 10 years of data are used for model development and the remaining 5 years are used for model evaluation. The fitted model accurately reproduces the observed rainfall statistics in the reserved data despite a shift in atmospheric circulation (and, consequently, rainfall) between the two periods. The fitted model also provides some useful insights into the processes driving rainfall in this region.