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??Hidden Markov model is widely used in statistical modeling of time, space and state transition data. The definition of hidden Markov multivariate normal distribution is given. The principle of using cluster analysis to determine the hidden state of observed variables is introduced. The maximum likelihood estimator of the unknown parameters in the model is derived. The simulated observation data set is used to test the estimation effect and stability of the method. The characteristic is simple classical statistical inference such as cluster analysis and maximum likelihood estimation. The method solves the parameter estimation problem of complex statistical models.     

Bayesian Monte Carlo estimation for profile hidden Markov models

Hidden Markov models are used as tools for pattern recognition in a number of areas, ranging from speech processing to biological sequence analysis. Profile hidden Markov models represent a class of so-called “left–right” models that have an architecture that is specifically relevant to classification of proteins into structural families based on their amino acid sequences. Standard learning methods for such models employ a variety of heuristics applied to the expectation-maximization implementation of the maximum likelihood estimation procedure in order to find the global maximum of the likelihood function. Here, we compare maximum likelihood estimation to fully Bayesian estimation of parameters for profile hidden Markov models with a small number of parameters. We find that, relative to maximum likelihood methods, Bayesian methods assign higher scores to data sequences that are distantly related to the pattern consensus, show better performance in classifying these sequences correctly, and continue to perform robustly with regard to misspecification of the number of model parameters. Though our study is limited in scope, we expect our results to remain relevant for models with a large number of parameters and other types of left–right hidden Markov models.     

Estimating Hidden Semi-Markov Chains From Discrete Sequences

This article addresses the estimation of hidden semi-Markov chains from nonstationary discrete sequences. Hidden semi-Markov chains are particularly useful to model the succession of homogeneous zones or segments along sequences. A discrete hidden semi-Markov chain is composed of a nonobservable state process, which is a semi-Markov chain, and a discrete output process. Hidden semi-Markov chains generalize hidden Markov chains and enable the modeling of various durational structures. From an algorithmic point of view, a new forward-backward algorithm is proposed whose complexity is similar to that of the Viterbi algorithm in terms of sequence length (quadratic in the worst case in time and linear in space). This opens the way to the maximum likelihood estimation of hidden semi-Markov chains from long sequences. This statistical modeling approach is illustrated by the analysis of branching and flowering patterns in plants.     

Regime-switching stochastic volatility model: estimation and calibration to VIX options

We develop and implement a method for maximum likelihood estimation of a regime-switching stochastic volatility model. Our model uses a continuous time stochastic process for the stock dynamics with the instantaneous variance driven by a Cox–Ingersoll–Ross process and each parameter modulated by a hidden Markov chain. We propose an extension of the EM algorithm through the Baum–Welch implementation to estimate our model and filter the hidden state of the Markov chain while using the VIX index to invert the latent volatility state. Using Monte Carlo simulations, we test the convergence of our algorithm and compare it with an approximate likelihood procedure where the volatility state is replaced by the VIX index. We found that our method is more accurate than the approximate procedure. Then, we apply Fourier methods to derive a semi-analytical expression of S&P500 and VIX option prices, which we calibrate to market data. We show that the model is sufficiently rich to encapsulate important features of the joint dynamics of the stock and the volatility and to consistently fit option market prices.     

基于EM算法隐Markov模型参数估计

描述最大似然参数估计问题,介绍如何用EM算法求解最大似然参数估计.首先给出EM算法的抽象形式,然后介绍EM算法的一个应用:求隐Markov模型中的参数估计.用EM算法推导出隐Markov模型中参数的迭代公式.     

Leroux's method for general hidden Markov models

The method introduced by Leroux [Maximum likelihood estimation for hidden Markov models, Stochastic Process Appl. 40 (1992) 127–143] to study the exact likelihood of hidden Markov models is extended to the case where the state variable evolves in an open interval of the real line. Under rather minimal assumptions, we obtain the convergence of the normalized log-likelihood function to a limit that we identify at the true value of the parameter. The method is illustrated in full details on the Kalman filter model.     

A simulation based approach to the parameter estimation for the three-parameter gamma distribution

The gamma distribution is one of the commonly used statistical distribution in reliability. While maximum likelihood has traditionally been the main method for estimation of gamma parameters, Hirose has proposed a continuation method to parameter estimation for the three-parameter gamma distribution. In this paper, we propose to apply Markov chain Monte Carlo techniques to carry out a Bayesian estimation procedure using Hirose’s simulated data as well as two real data sets. The method is indeed flexible and inference for any quantity of interest is readily available.     

A comparison of some criteria for states selection in the latent Markov model for longitudinal data

We compare different selection criteria to choose the number of latent states of a multivariate latent Markov model for longitudinal data. This model is based on an underlying Markov chain to represent the evolution of a latent characteristic of a group of individuals over time. Then, the response variables observed at different occasions are assumed to be conditionally independent given this chain. Maximum likelihood estimation of the model is carried out through an Expectation–Maximization algorithm based on forward–backward recursions which are well known in the hidden Markov literature for time series. The selection criteria we consider are based on penalized versions of the maximum log-likelihood or on the posterior probabilities of belonging to each latent state, that is, the conditional probability of the latent state given the observed data. Among the latter criteria, we propose an appropriate entropy measure tailored for the latent Markov models. We show the results of a Monte Carlo simulation study aimed at comparing the performance of the above states selection criteria on the basis of a wide set of model specifications.     

Parameter Estimation in Hidden Markov Models With Intractable Likelihoods Using Sequential Monte Carlo

We propose sequential Monte Carlo-based algorithms for maximum likelihood estimation of the static parameters in hidden Markov models with an intractable likelihood using ideas from approximate Bayesian computation. The static parameter estimation algorithms are gradient-based and cover both offline and online estimation. We demonstrate their performance by estimating the parameters of three intractable models, namely the α-stable distribution, g-and-k distribution, and the stochastic volatility model with α-stable returns, using both real and synthetic data.     

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In this paper, we investigate a competing risks model based on exponentiated Weibull distribution under Type-I progressively hybrid censoring scheme. To estimate the unknown parameters and reliability function, the maximum likelihood estimators and asymptotic confidence intervals are derived. Since Bayesian posterior density functions cannot be given in closed forms, we adopt Markov chain Monte Carlo method to calculate approximate Bayes estimators and highest posterior density credible intervals. To illustrate the estimation methods, a simulation study is carried out with numerical results. It is concluded that the maximum likelihood estimation and Bayesian estimation can be used for statistical inference in competing risks model under Type-I progressively hybrid censoring scheme.     

Maximum likelihood estimation for stochastic volatility in mean models with heavy‐tailed distributions

In this article, we introduce a likelihood‐based estimation method for the stochastic volatility in mean (SVM) model with scale mixtures of normal (SMN) distributions. Our estimation method is based on the fact that the powerful hidden Markov model (HMM) machinery can be applied in order to evaluate an arbitrarily accurate approximation of the likelihood of an SVM model with SMN distributions. Likelihood‐based estimation of the parameters of stochastic volatility models, in general, and SVM models with SMN distributions, in particular, is usually regarded as challenging as the likelihood is a high‐dimensional multiple integral. However, the HMM approximation, which is very easy to implement, makes numerical maximum of the likelihood feasible and leads to simple formulae for forecast distributions, for computing appropriately defined residuals, and for decoding, that is, estimating the volatility of the process. Copyright © 2017 John Wiley & Sons, Ltd.     

Conditional Logistic Regression With Longitudinal Follow-up and Individual-Level Random Coefficients: A Stable and Efficient Two-Step Estimation Method

The analysis of data generated by animal habitat selection studies, by family studies of genetic diseases, or by longitudinal follow-up of households often involves fitting a mixed conditional logistic regression model to longitudinal data composed of clusters of matched case-control strata. The estimation of model parameters by maximum likelihood is especially difficult when the number of cases per stratum is greater than one. In this case, the denominator of each cluster contribution to the conditional likelihood involves a complex integral in high dimension, which leads to convergence problems in the numerical maximization. In this article we show how these computational complexities can be bypassed using a global two-step analysis for nonlinear mixed effects models. The first step estimates the cluster-specific parameters and can be achieved with standard statistical methods and software based on maximum likelihood for independent data. The second step uses the EM-algorithm in conjunction with conditional restricted maximum likelihood to estimate the population parameters. We use simulations to demonstrate that the method works well when the analysis is based on a large number of strata per cluster, as in many ecological studies. We apply the proposed two-step approach to evaluate habitat selection by pairs of bison roaming freely in their natural environment. This article has supplementary material online.     

二阶离散隐马尔科夫模型的严格定义及等价性质

隐马氏模型作为一种具有双重随机过程的统计模型,具有可靠的概率统计理论基础和强有力的数学结构,已被广泛应用于语音识别、生物序列分析、金融数据分析等领域.由于传统的一阶隐马氏模型无法表示更远状态距离间的依赖关系,就可能会忽略很多有用的统计特征,故有人提出二阶隐马氏模型的概念,但此概念并不严格.本文给出二阶离散隐马尔科夫模型的严格定义,并研究了二阶离散隐马尔科夫模型的两个等价性质.     

Uniform Accuracy of the Maximum Likelihood Estimates for Probabilistic Models of Biological Sequences

Probabilistic models for biological sequences (DNA and proteins) have many useful applications in bioinformatics. Normally, the values of parameters of these models have to be estimated from empirical data. However, even for the most common estimates, the maximum likelihood (ML) estimates, properties have not been completely explored. Here we assess the uniform accuracy of the ML estimates for models of several types: the independence model, the Markov chain and the hidden Markov model (HMM). Particularly, we derive rates of decay of the maximum estimation error by employing the measure concentration as well as the Gaussian approximation, and compare these rates.     

Parameter estimation in a condition-based maintenance model

A parameter estimation problem for a condition-based maintenance model is considered. We model a failing system that can be in a healthy or unhealthy operational state, or in a failure state. System deterioration is assumed to follow a hidden, three-state continuous time Markov process. Vector autoregressive data are obtained through condition monitoring at discrete time points, which gives partial information about the unobservable system state. Two kinds of data histories are considered: histories that end with observable system failure and histories that end when the system is suspended from operation but has not failed. Maximum likelihood estimates of the model parameters are obtained using the EM algorithm and a closed form expression for the pseudo-likelihood function is derived. Numerical results are provided which illustrate the estimation procedure.     

Basic problems and solution methods for two-dimensional continuous 3 × 3 order hidden Markov model

A novel model referred to as two-dimensional continuous 3 × 3 order hidden Markov model is put forward to avoid the disadvantages of the classical hypothesis of two-dimensional continuous hidden Markov model. This paper presents three equivalent definitions of the model, in which the state transition probability relies on not only immediate horizontal and vertical states but also immediate diagonal state, and in which the probability density of the observation relies on not only current state but also immediate horizontal and vertical states. The paper focuses on the three basic problems of the model, namely probability density calculation, parameters estimation and path backtracking. Some algorithms solving the questions are theoretically derived, by exploiting the idea that the sequences of states on rows or columns of the model can be viewed as states of a one-dimensional continuous 1 × 2 order hidden Markov model. Simulation results further demonstrate the performance of the algorithms. Because there are more statistical characteristics in the structure of the proposed new model, it can more accurately describe some practical problems, as compared to two-dimensional continuous hidden Markov model.     

Asymptotic risks of Viterbi segmentation

We consider the maximum likelihood (Viterbi) alignment of a hidden Markov model (HMM). In an HMM, the underlying Markov chain is usually hidden and the Viterbi alignment is often used as the estimate of it. This approach will be referred to as the Viterbi segmentation. The goodness of the Viterbi segmentation can be measured by several risks. In this paper, we prove the existence of asymptotic risks. Being independent of data, the asymptotic risks can be considered as the characteristics of the model that illustrate the long-run behavior of the Viterbi segmentation.     

Selecting hidden Markov model state number with cross-validated likelihood

The problem of estimating the number of hidden states in a hidden Markov model is considered. Emphasis is placed on cross-validated likelihood criteria. Using cross-validation to assess the number of hidden states allows to circumvent the well-documented technical difficulties of the order identification problem in mixture models. Moreover, in a predictive perspective, it does not require that the sampling distribution belongs to one of the models in competition. However, computing cross-validated likelihood for hidden Markov models for which only one training sample is available, involves difficulties since the data are not independent. Two approaches are proposed to compute cross-validated likelihood for a hidden Markov model. The first one consists of using a deterministic half-sampling procedure, and the second one consists of an adaptation of the EM algorithm for hidden Markov models, to take into account randomly missing values induced by cross-validation. Numerical experiments on both simulated and real data sets compare different versions of cross-validated likelihood criterion and penalised likelihood criteria, including BIC and a penalised marginal likelihood criterion. Those numerical experiments highlight a promising behaviour of the deterministic half-sampling criterion.     

Basic problems solving for two-dimensional discrete 3 × 4 order hidden markov model

A novel model is proposed to overcome the shortages of the classical hypothesis of the two-dimensional discrete hidden Markov model. In the proposed model, the state transition probability depends on not only immediate horizontal and vertical states but also on immediate diagonal state, and the observation symbol probability depends on not only current state but also on immediate horizontal, vertical and diagonal states. This paper defines the structure of the model, and studies the three basic problems of the model, including probability calculation, path backtracking and parameters estimation. By exploiting the idea that the sequences of states on rows or columns of the model can be seen as states of a one-dimensional discrete 1 × 2 order hidden Markov model, several algorithms solving the three questions are theoretically derived. Simulation results further demonstrate the performance of the algorithms. Compared with the two-dimensional discrete hidden Markov model, there are more statistical characteristics in the structure of the proposed model, therefore the proposed model theoretically can more accurately describe some practical problems.