A hierarchical model for space–time surveillance data on meningococcal disease incidence

Summary. We describe a model-based approach to analyse space–time surveillance data on meningococcal disease. Such data typically comprise a number of time series of disease counts, each representing a specific geographical area. We propose a hierarchical formulation, where latent parameters capture temporal, seasonal and spatial trends in disease incidence. We then add—for each area—a hidden Markov model to describe potential additional (autoregressive) effects of the number of cases at the previous time point. Different specifications for the functional form of this autoregressive term are compared which involve the number of cases in the same or in neighbouring areas. The two states of the Markov chain can be interpreted as representing an 'endemic' and a 'hyperendemic' state. The methodology is applied to a data set of monthly counts of the incidence of meningococcal disease in the 94 départements of France from 1985 to 1997. Inference is carried out by using Markov chain Monte Carlo simulation techniques in a fully Bayesian framework. We emphasize that a central feature of our model is the possibility of calculating—for each region and each time point—the posterior probability of being in a hyperendemic state, adjusted for global spatial and temporal trends, which we believe is of particular public health interest.     

Nonparametric Bayesian analysis for multi-site hidden Markov model

The hidden Markov model (HMM) provides an attractive framework for modeling long-term persistence in a variety of applications including pattern recognition. Unlike typical mixture models, hidden Markov states can represent the heterogeneity in data and it can be extended to a multivariate case using a hierarchical Bayesian approach. This article provides a nonparametric Bayesian modeling approach to the multi-site HMM by considering stick-breaking priors for each row of an infinite state transition matrix. This extension has many advantages over a parametric HMM. For example, it can provide more flexible information for identifying the structure of the HMM than parametric HMM analysis, such as the number of states in HMM. We exploit a simulation example and a real dataset to evaluate the proposed approach.     

A Bayesian HMM with random effects and an unknown number of states for DNA copy number analysis

Hidden Markov models (HMMs) have been shown to be a flexible tool for modelling complex biological processes. However, choosing the number of hidden states remains an open question and the inclusion of random effects also deserves more research, as it is a recent addition to the fixed-effect HMM in many application fields. We present a Bayesian mixed HMM with an unknown number of hidden states and fixed covariates. The model is fitted using reversible-jump Markov chain Monte Carlo, avoiding the need to select the number of hidden states. We show through simulations that the estimations produced are more precise than those from a fixed-effect HMM and illustrate its practical application to the analysis of DNA copy number data, a field where HMMs are widely used.     

VARIATIONAL BAYESIAN ANALYSIS FOR HIDDEN MARKOV MODELS

The variational approach to Bayesian inference enables simultaneous estimation of model parameters and model complexity. An interesting feature of this approach is that it also leads to an automatic choice of model complexity. Empirical results from the analysis of hidden Markov models with Gaussian observation densities illustrate this. If the variational algorithm is initialized with a large number of hidden states, redundant states are eliminated as the method converges to a solution, thereby leading to a selection of the number of hidden states. In addition, through the use of a variational approximation, the deviance information criterion for Bayesian model selection can be extended to the hidden Markov model framework. Calculation of the deviance information criterion provides a further tool for model selection, which can be used in conjunction with the variational approach.     

Predictive Inference for Big,Spatial, Non‐Gaussian Data: MODIS Cloud Data and its Change‐of‐Support

Remote sensing of the earth with satellites yields datasets that can be massive in size, nonstationary in space, and non‐Gaussian in distribution. To overcome computational challenges, we use the reduced‐rank spatial random effects (SRE) model in a statistical analysis of cloud‐mask data from NASA's Moderate Resolution Imaging Spectroradiometer (MODIS) instrument on board NASA's Terra satellite. Parameterisations of cloud processes are the biggest source of uncertainty and sensitivity in different climate models’ future projections of Earth's climate. An accurate quantification of the spatial distribution of clouds, as well as a rigorously estimated pixel‐scale clear‐sky‐probability process, is needed to establish reliable estimates of cloud‐distributional changes and trends caused by climate change. Here we give a hierarchical spatial‐statistical modelling approach for a very large spatial dataset of 2.75 million pixels, corresponding to a granule of MODIS cloud‐mask data, and we use spatial change‐of‐Support relationships to estimate cloud fraction at coarser resolutions. Our model is non‐Gaussian; it postulates a hidden process for the clear‐sky probability that makes use of the SRE model, EM‐estimation, and optimal (empirical Bayes) spatial prediction of the clear‐sky‐probability process. Measures of prediction uncertainty are also given.     

A continuous-time HMM approach to modeling the magnitude-frequency distribution of earthquakes

The magnitude-frequency distribution (MFD) of earthquake is a fundamental statistic in seismology. The so-called b-value in the MFD is of particular interest in geophysics. A continuous time hidden Markov model (HMM) is proposed for characterizing the variability of b-values. The HMM-based approach to modeling the MFD has some appealing properties over the widely used sliding-window approach. Often, large variability appears in the estimation of b-value due to window size tuning, which may cause difficulties in interpretation of b-value heterogeneities. Continuous-time hidden Markov models (CT-HMMs) are widely applied in various fields. It bears some advantages over its discrete time counterpart in that it can characterize heterogeneities appearing in time series in a finer time scale, particularly for highly irregularly-spaced time series, such as earthquake occurrences. We demonstrate an expectation–maximization algorithm for the estimation of general exponential family CT-HMM. In parallel with discrete-time hidden Markov models, we develop a continuous time version of Viterbi algorithm to retrieve the overall optimal path of the latent Markov chain. The methods are applied to New Zealand deep earthquakes. Before the analysis, we first assess the completeness of catalogue events to assure the analysis is not biased by missing data. The estimation of b-value is stable over the selection of magnitude thresholds, which is ideal for the interpretation of b-value variability.     

BAYESIAN HIDDEN MARKOV MODELS FOR LONGITUDINAL COUNTS

This paper describes a Bayesian approach to modelling carcinogenity in animal studies where the data consist of counts of the number of tumours present over time. It compares two autoregressive hidden Markov models. One of them models the transitions between three latent states: an inactive transient state, a multiplying state for increasing counts and a reducing state for decreasing counts. The second model introduces a fourth tied state to describe non‐zero observations that are neither increasing nor decreasing. Both these models can model the length of stay upon entry of a state. A discrete constant hazards waiting time distribution is used to model the time to onset of tumour growth. Our models describe between‐animal‐variability by a single hierarchy of random effects and the within‐animal variation by first‐order serial dependence. They can be extended to higher‐order serial dependence and multi‐level hierarchies. Analysis of data from animal experiments comparing the influence of two genes leads to conclusions that differ from those of Dunson (2000). The observed data likelihood defines an information criterion to assess the predictive properties of the three‐ and four‐state models. The deviance information criterion is appropriately defined for discrete parameters.     

Robust fitting of hidden Markov regression models under a longitudinal setting

We propose a robust estimation procedure for the analysis of longitudinal data including a hidden process to account for unobserved heterogeneity between subjects in a dynamic fashion. We show how to perform estimation by an expectation–maximization-type algorithm in the hidden Markov regression literature. We show that the proposed robust approaches work comparably to the maximum-likelihood estimator when there are no outliers and the error is normal and outperform it when there are outliers or the error is heavy tailed. A real data application is used to illustrate our proposal. We also provide details on a simple criterion to choose the number of hidden states.     

Estimation with right‐censored observations under a semi‐Markov model

The semi‐Markov process often provides a better framework than the classical Markov process for the analysis of events with multiple states. The purpose of this paper is twofold. First, we show that in the presence of right censoring, when the right end‐point of the support of the censoring time is strictly less than the right end‐point of the support of the semi‐Markov kernel, the transition probability of the semi‐Markov process is nonidentifiable, and the estimators proposed in the literature are inconsistent in general. We derive the set of all attainable values for the transition probability based on the censored data, and we propose a nonparametric inference procedure for the transition probability using this set. Second, the conventional approach to constructing confidence bands is not applicable for the semi‐Markov kernel and the sojourn time distribution. We propose new perturbation resampling methods to construct these confidence bands. Different weights and transformations are explored in the construction. We use simulation to examine our proposals and illustrate them with hospitalization data from a recent cancer survivor study. The Canadian Journal of Statistics 41: 237–256; 2013 © 2013 Statistical Society of Canada     

Markov-switching generalized additive models

We consider Markov-switching regression models, i.e. models for time series regression analyses where the functional relationship between covariates and response is subject to regime switching controlled by an unobservable Markov chain. Building on the powerful hidden Markov model machinery and the methods for penalized B-splines routinely used in regression analyses, we develop a framework for nonparametrically estimating the functional form of the effect of the covariates in such a regression model, assuming an additive structure of the predictor. The resulting class of Markov-switching generalized additive models is immensely flexible, and contains as special cases the common parametric Markov-switching regression models and also generalized additive and generalized linear models. The feasibility of the suggested maximum penalized likelihood approach is demonstrated by simulation. We further illustrate the approach using two real data applications, modelling (i) how sales data depend on advertising spending and (ii) how energy price in Spain depends on the Euro/Dollar exchange rate.     

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.     

Estimating the order of a hidden markov model

While the estimation of the parameters of a hidden Markov model has been studied extensively, the consistent estimation of the number of hidden states is still an unsolved problem. The AIC and BIC methods are used most commonly, but their use in this context has not been justified theoretically. The author shows that for many common models, the penalized minimum‐distance method yields a consistent estimate of the number of hidden states in a stationary hidden Markov model. In addition to addressing the identifiability issues, she applies her method to a multiple sclerosis data set and assesses its performance via simulation.     

A simple,doubly robust,efficient estimator for survival functions using pseudo observations

Survival functions are often estimated by nonparametric estimators such as the Kaplan‐Meier estimator. For valid estimation, proper adjustment for confounding factors is needed when treatment assignment may depend on confounding factors. Inverse probability weighting is a commonly used approach, especially when there is a large number of potential confounders to adjust for. Direct adjustment may also be used if the relationship between the time‐to‐event and all confounders can be modeled. However, either approach requires a correctly specified model for the relationship between confounders and treatment allocation or between confounders and the time‐to‐event. We propose a pseudo‐observation–based doubly robust estimator, which is valid when either the treatment allocation model or the time‐to‐event model is correctly specified and is generally more efficient than the inverse probability weighting approach. The approach can be easily implemented using standard software. A simulation study was conducted to evaluate this approach under a number of scenarios, and the results are presented and discussed. The results confirm robustness and efficiency of the proposed approach. A real data example is also provided for illustration.     

Mixtures of autoregressive-autoregressive conditionally heteroscedastic models: semi-parametric approach

We propose data generating structures which can be represented as a mixture of autoregressive-autoregressive conditionally heteroscedastic models. The switching between the states is governed by a hidden Markov chain. We investigate semi-parametric estimators for estimating the functions based on the quasi-maximum likelihood approach and provide sufficient conditions for geometric ergodicity of the process. We also present an expectation–maximization algorithm for calculating the estimates numerically.     

A transitional Markov switching autoregressive model

ABSTRACT

This paper is concerned with properties of a transitional Markov switching autoregressive (TMSAR) model, together with its maximum-likelihood estimation and inference. We extend existing MSAR models by allowing dependence of AR parameters on hidden states at time points prior to the current time t. A stationary solution is given and expressions for the theoretical autocovariance function are derived. Two time series are analyzed and the new model outperforms two existing MSAR models in terms of maximized log-likelihood, residual correlations, and one-step-ahead forecasting performance. The new model also gives more regime changes in agreement with real events.     

A semiparametric approach to hidden Markov models under longitudinal observations

We propose a hidden Markov model for longitudinal count data where sources of unobserved heterogeneity arise, making data overdispersed. The observed process, conditionally on the hidden states, is assumed to follow an inhomogeneous Poisson kernel, where the unobserved heterogeneity is modeled in a generalized linear model (GLM) framework by adding individual-specific random effects in the link function. Due to the complexity of the likelihood within the GLM framework, model parameters may be estimated by numerical maximization of the log-likelihood function or by simulation methods; we propose a more flexible approach based on the Expectation Maximization (EM) algorithm. Parameter estimation is carried out using a non-parametric maximum likelihood (NPML) approach in a finite mixture context. Simulation results and two empirical examples are provided.     

Stability of Approximations of Average Run Length of Risk-Adjusted CUSUM Schemes Using the Markov Approach: Comparing Two Methods of Calculating Transition Probabilities

Risk-adjusted CUSUM schemes are designed to monitor the number of adverse outcomes following a medical procedure. An approximation of the average run length (ARL), which is the usual performance measure for a risk-adjusted CUSUM, may be found using its Markov property. We compare two methods of computing transition probability matrices where the risk model classifies patient populations into discrete, finite levels of risk. For the first method, a process of scaling and rounding off concentrates probability in the center of the Markov states, which are non overlapping sub-intervals of the CUSUM decision interval, and, for the second, a smoothing process spreads probability uniformly across the Markov states. Examples of risk-adjusted CUSUM schemes are used to show, if rounding is used to calculate transition probabilities, the values of ARLs estimated using the Markov property vary erratically as the number of Markov states vary and, on occasion, fail to converge for mesh sizes up to 3,000. On the other hand, if smoothing is used, the approximate ARL values remain stable as the number of Markov states vary. The smoothing technique gave good estimates of the ARL where there were less than 1,000 Markov states.     

A Bayesian model for multiple change point to extremes,with application to environmental and financial data

Abrupt changes often occur for environmental and financial time series. Most often, these changes are due to human intervention. Change point analysis is a statistical tool used to analyze sudden changes in observations along the time series. In this paper, we propose a Bayesian model for extreme values for environmental and economic datasets that present a typical change point behavior. The model proposed in this paper addresses the situation in which more than one change point can occur in a time series. By analyzing maxima, the distribution of each regime is a generalized extreme value distribution. In this model, the change points are unknown and considered parameters to be estimated. Simulations of extremes with two change points showed that the proposed algorithm can recover the true values of the parameters, in addition to detecting the true change points in different configurations. Also, the number of change points was a problem to be considered, and the Bayesian estimation can correctly identify the correct number of change points for each application. Environmental and financial data were analyzed and results showed the importance of considering the change point in the data and revealed that this change of regime brought about an increase in the return levels, increasing the number of floods in cities around the rivers. Stock market levels showed the necessity of a model with three different regimes.     

Modelling seasonally varying data: A case study for Sudden Infant Death Syndrome (SIDS)

Many time series are measured monthly, either as averages or totals, and such data often exhibit seasonal variability – the values of the series are consistently larger for some months of the year than for others. A typical series of this type is the number of deaths each month attributed to SIDS (Sudden Infant Death Syndrome). Seasonality can be modelled in a number of ways. This paper describes and discusses various methods for modelling seasonality in SIDS data, though much of the discussion is relevant to other seasonally varying data. There are two main approaches, either fitting a circular probability distribution to the data, or using regression-based techniques to model the mean seasonal behaviour. Both are discussed in this paper.     

Modelling extremes of time-dependent data by Markov-switching structures

We investigate the extremal clustering behaviour of stationary time series that possess two regimes, where the switch is governed by a hidden two-state Markov chain. We also suppose that the process is conditionally Markovian in each latent regime. We prove under general assumptions that above high thresholds these models behave approximately as a random walk in one (called dominant) regime and as a stationary autoregression in the other (dominated) regime. Based on this observation, we propose an estimation and simulation scheme to analyse the extremal dependence structure of such models, taking into account only observations above high thresholds. The properties of the estimation method are also investigated. Finally, as an application, we fit a model to high-level exceedances of water discharge data, simulate extremal events from the fitted model, and show that the (model-based) flood peak, flood duration and flood volume distributions match their observed counterparts.