Stochastic EM Algorithm Research Articles

Sequential Modeling of Topic Dynamics with Multiple Timescales

We propose an online topic model for sequentially analyzing the time evolution of topics in document collections. Topics naturally evolve with multiple timescales. For example, some words may be used consistently over one hundred years, while other words emerge and disappear over periods of a few days. Thus, in the proposed model, current topic-specific distributions over words are assumed to be generated based on the multiscale word distributions of the previous epoch. Considering both the long- and short-timescale dependency yields a more robust model. We derive efficient online inference procedures based on a stochastic EM algorithm, in which the model is sequentially updated using newly obtained data; this means that past data are not required to make the inference. We demonstrate the effectiveness of the proposed method in terms of predictive performance and computational efficiency by examining collections of real documents with timestamps.

Maximum likelihood estimation in discrete mixed hidden Markov models using the SAEM algorithm

Mixed hidden Markov models have been recently defined in the literature as an extension of hidden Markov models for dealing with population studies. The notion of mixed hidden Markov models is particularly relevant for modeling longitudinal data collected during clinical trials, especially when distinct disease stages can be considered. However, parameter estimation in such models is complex, especially due to their highly nonlinear structure and the presence of unobserved states. Moreover, existing inference algorithms are extremely time consuming when the model includes several random effects. New inference procedures are proposed for estimating population parameters, individual parameters and sequences of hidden states in mixed hidden Markov models. The main contribution consists of a specific version of the stochastic approximation EM algorithm coupled with the Baum–Welch algorithm for estimating population parameters. The properties of this algorithm are investigated via a Monte-Carlo simulation study, and an application of mixed hidden Markov models to the description of daily seizure counts in epileptic patients is presented.

Comparison of EM and SEM Algorithms in Poisson Regression Models: A Simulation Study

In this work, we propose to compare two algorithms to compute maximum likelihood estimates for the parameters of a mixture Poisson regression model: the EM algorithm and the Stochastic EM algorithm. The comparison of the two procedures was done through a simulation study of the performance of these approaches on simulated data sets and real data sets. Simulation results show that the choice of the approach depends essentially on the overlap of the regression lines. In the real data case, we show that the Stochastic EM algorithm resulted in model estimates that best fit the regression model.

An empirical Bayes procedure for the selection of Gaussian graphical models

A new methodology for model determination in decomposable graphical Gaussian models (Dawid and Lauritzen in Ann. Stat. 21(3), 1272---1317, 1993) is developed. The Bayesian paradigm is used and, for each given graph, a hyper-inverse Wishart prior distribution on the covariance matrix is considered. This prior distribution depends on hyper-parameters. It is well-known that the models's posterior distribution is sensitive to the specification of these hyper-parameters and no completely satisfactory method is registered. In order to avoid this problem, we suggest adopting an empirical Bayes strategy, that is a strategy for which the values of the hyper-parameters are determined using the data. Typically, the hyper-parameters are fixed to their maximum likelihood estimations. In order to calculate these maximum likelihood estimations, we suggest a Markov chain Monte Carlo version of the Stochastic Approximation EM algorithm. Moreover, we introduce a new sampling scheme in the space of graphs that improves the add and delete proposal of Armstrong et al. (Stat. Comput. 19(3), 303---316, 2009). We illustrate the efficiency of this new scheme on simulated and real datasets.

Nonlinear methods for inverse statistical problems

In the uncertainty treatment framework considered, the intrinsic variability of the inputs of a physical simulation model is modelled by a multivariate probability distribution. The objective is to identify this probability distribution–the dispersion of which is independent of the sample size since intrinsic variability is at stake–based on observation of some model outputs. Moreover, in order to limit the number of (usually burdensome) physical model runs inside the inversion algorithm to a reasonable level, a nonlinear approximation methodology making use of Kriging and a stochastic EM algorithm is presented. It is compared with iterated linear approximation on the basis of numerical experiments on simulated data sets coming from a simplified but realistic modelling of a dyke overflow. Situations where this nonlinear approach is to be preferred to linearisation are highlighted.

Asymptotic properties of a stochastic EM algorithm for mixtures with censored data

Weak consistency and asymptotic normality is shown for a stochastic EM algorithm for censored data from a mixture of distributions under lognormal assumptions. The asymptotic properties hold for all parameters of the distributions, including the mixing parameter. In order to make parameter estimation meaningful it is necessary to know that the censored mixture distribution is identifiable. General conditions under which this is the case are given. The stochastic EM algorithm addressed in this paper is used for estimation of wood fibre length distributions based on optically measured data from cylindric wood samples (increment cores).

Pharmacogenetics and population pharmacokinetics: impact of the design on three tests using the SAEM algorithm

Pharmacogenetics is now widely investigated and health institutions acknowledge its place in clinical pharmacokinetics. Our objective is to assess through a simulation study, the impact of design on the statistical performances of three different tests used for analysis of pharmacogenetic information with nonlinear mixed effects models: (i) an ANOVA to test the relationship between the empirical Bayes estimates of the model parameter of interest and the genetic covariate, (ii) a global Wald test to assess whether estimates for the gene effect are significant, and (iii) a likelihood ratio test (LRT) between the model with and without the genetic covariate. We use the stochastic EM algorithm (SAEM) implemented in MONOLIX 2.1 software. The simulation setting is inspired from a real pharmacokinetic study. We investigate four designs with N the number of subjects and n the number of samples per subject: (i) N = 40/n = 4, similar to the original study, (ii) N = 80/n = 2 sorted in 4 groups, a design optimized using the PFIM software, (iii) a combined design, N = 20/n = 4 plus N = 80 with only a trough concentration and (iv) N = 200/n = 4, to approach asymptotic conditions. We find that the ANOVA has a correct type I error estimate regardless of design, however the sparser design was optimized. The type I error of the Wald test and LRT are moderatly inflated in the designs far from the asymptotic (<10%). For each design, the corrected power is analogous for the three tests. Among the three designs with a total of 160 observations, the design N = 80/n = 2 optimized with PFIM provides both the lowest standard error on the effect coefficients and the best power for the Wald test and the LRT while a high shrinkage decreases the power of the ANOVA. In conclusion, a correction method should be used for model-based tests in pharmacogenetic studies with reduced sample size and/or sparse sampling and, for the same amount of samples, some designs have better power than others.

Multi-Source Multi-Sensor Image Fusion Based on Bootstrap Approach and SEM Algorithm

Bootstrap approach and Stochastic EM algorithm combination applied for the improvement of the multisource and multi-sensor image fusion process; was presented in this research. Improvement concerned not only image quality and reducing processing execution time as mentioned in our previous Bootstrap EM algorithm (BEM), but also regarding ini- tialization dependence as well as fixed classes' number. Such interesting fusion algorithm for multisource and multi- sensor image using one stochastic phase, i.e. SEM algorithm, preceded by Bootstrap procedure was successfully imple- mented and tested for several prototype images. Targeted images were firstly split by an unsupervised Bayesian segmenta- tion approach in order to determine a joint region map for the fused image. The Bootstrap approach was then applied to the targeted multisource image in conjunction with the SEM algorithm, forming hence one Bootstrap SEM algorithm called BSEM. The procedure of such algorithm involved both statistical parameters' estimation from one representative Bootstrap sample of each source or sensor images.

A SAEM algorithm for the estimation of template and deformation parameters in medical image sequences

This paper is about object deformations observed throughout a sequence of images. We present a statistical framework in which the observed images are defined as noisy realizations of a randomly deformed template image. In this framework, we focus on the problem of the estimation of parameters related to the template and deformations. Our main motivation is the construction of estimation framework and algorithm which can be applied to short sequences of complex and highly-dimensional images. The originality of our approach lies in the representations of the template and deformations, which are defined on a common triangulated domain, adapted to the geometry of the observed images. In this way, we have joint representations of the template and deformations which are compact and parsimonious. Using such representations, we are able to drastically reduce the number of parameters in the model. Besides, we adapt to our framework the Stochastic Approximation EM algorithm combined with a Markov Chain Monte Carlo procedure which was proposed in 2004 by Kuhn and Lavielle. Our implementation of this algorithm takes advantage of some properties which are specific to our framework. More precisely, we use the Markovian properties of deformations to build an efficient simulation strategy based on a Metropolis-Hasting-Within-Gibbs sampler. Finally, we present some experiments on sequences of medical images and synthetic data.

Evaluation of a method for fitting a semi‐Markov process model in the presence of left‐censored spells using the Cardiovascular Health Study

We used a longitudinal data set covering 13 years from the Cardiovascular Health Study to evaluate the properties of a recently developed approach to deal with left censoring that fits a semi-Markov process (SMP) model by using an analog to the stochastic EM algorithm--the SMP-EM approach. It appears that the SMP-EM approach gives estimates of duration-dependent probabilities of health changes similar to those obtained by using SMP models that have the advantage of actual duration data. SMP-EM estimates of duration-dependent transition probabilities also appear more accurate and less variable than multi-state life table estimates.

A note on a validity test using the stochastic EM algorithm in order to assess the number of components in a finite mixture model

A validity test, based on the stochastic expectation-maximization (SEM) algorithm for assessing the performance of the EM algorithm or any other adequate method for detecting the number of components in mixture analyses was proposed in the literature. We prove in this note that the theoretical distribution of the test does not hold, and we derive the actual distribution. Finally, we give a few simulation results and make some comments on the methodology.

Parametric inference for mixed models defined by stochastic differential equations

Non-linear mixed models defined by stochastic differential equations (SDEs) are consid- ered: the parameters of the diffusion process are random variables and vary among the individuals. A maximum likelihood estimation method based on the Stochastic Approximation EM algorithm, is proposed. This estimation method uses the Euler-Maruyama approximation of the diffusion, achieved using latent auxiliary data introduced to complete the diffusion process between each pair of measure- ment instants. A tuned hybrid Gibbs algorithm based on conditional Brownian bridges simulations of the unobserved process paths is included in this algorithm. The convergence is proved and the error induced on the likelihood by the Euler-Maruyama approximation is bounded as a function of the step size of the approximation. Results of a pharmacokinetic simulation study illustrate the accuracy of this estimation method. The analysis of the Theophyllin real dataset illustrates the relevance of the SDE approach relative to the deterministic approach.

Estimation of parameters in incomplete data models defined by dynamical systems

Parametric incomplete data models defined by ordinary differential equations (ODEs) are widely used in biostatistics to describe biological processes accurately. Their parameters are estimated on approximate models, whose regression functions are evaluated by a numerical integration method. Accurate and efficient estimations of these parameters are critical issues. This paper proposes parameter estimation methods involving either a stochastic approximation EM algorithm (SAEM) in the maximum likelihood estimation, or a Gibbs sampler in the Bayesian approach. Both algorithms involve the simulation of non-observed data with conditional distributions using Hastings–Metropolis (H–M) algorithms. A modified H–M algorithm, including an original local linearization scheme to solve the ODEs, is proposed to reduce the computational time significantly. The convergence on the approximate model of all these algorithms is proved. The errors induced by the numerical solving method on the conditional distribution, the likelihood and the posterior distribution are bounded. The Bayesian and maximum likelihood estimation methods are illustrated on a simulated pharmacokinetic nonlinear mixed-effects model defined by an ODE. Simulation results illustrate the ability of these algorithms to provide accurate estimates.

Sensitivity analysis of longitudinal data with intermittent missing values

Several models for longitudinal data with nonrandom missingness are available. The selection model of Diggle and Kenward is one of these models. It has been mentioned by many authors that this model depends on untested modelling assumptions, such as the response distribution, from the observed data. So, a sensitivity analysis of the study’s conclusions for such assumptions is needed. The stochastic EM algorithm is proposed and developed to handle continuous longitudinal data with nonrandom intermittent missing values when the responses have non-normal distribution. This is a step in investigating the sensitivity of the parameter estimates to the change of the response distribution. The proposed technique is applied to real data from the International Breast Cancer Study Group.

A stochastic EM algorithm for a semiparametric mixture model

Recently, there has been a considerable interest in finite mixture models with semi-/non-parametric component distributions. Identifiability of such model parameters is generally not obvious, and when it occurs, inference methods are rather specific to the mixture model under consideration. Hence, a generalization of the EM algorithm to semiparametric mixture models is proposed. The approach is methodological and can be applied to a wide class of semiparametric mixture models. The behavior of the proposed EM type estimators is studied numerically not only through several Monte-Carlo experiments but also through comparison with alternative methods existing in the literature. In addition to these numerical experiments, applications to real data are provided, showing that the estimation method behaves well, that it is fast and easy to be implemented.

Analysis of Functional Status Transitions by Using a Semi-Markov Process Model in the Presence of Left-Censored Spells

Summary To analyse functional status transitions in the older population better, we fit a semi-Markov process model to data from the 1992–2002 Medicare Current Beneficiary Survey. We used an analogue of the stochastic EM algorithm to address the problem of left censoring of spells in longitudinal data. The iterative algorithm converged robustly under various initial values for the unobserved elapsed durations of spells in progress at base-line. Results on life expectancy and recovery from functional limitations based on the semi-Markov process model differ from those based on the traditional multistate life-table method. The proposed treatment of left-censored spells has the potential to expand the modelling capability that is available to researchers in fields where left censoring is a concern.

Adaptive basis selection for functional data analysis via stochastic penalization

We propose an adaptive method of analyzing a collection of curves which can be, individually, modeled as a linear combination of spline basis functions. Through the introduction of latent Bernoulli variables, the number of basis functions, the variance of the error measurements and the coefficients of the expansion are determined. We provide a modification of the stochastic EM algorithm for which numerical results show that the estimates are very close to the true curve in the sense of L2 norm.

Analysis of longitudinal data with intermittent missing values using the stochastic EM algorithm

Longitudinal data are not uncommon in many disciplines where repeated measurements on a response variable are collected for all subjects. Some intended measurements may not be available for some subjects resulting in a missing data pattern. Dropout pattern occurs when some subjects leave the study prematurely. The missing data pattern is defined as intermittent if a missing value followed by an observed value. When the probability of missingness depends on the missing value, and may be on the observed values, the missing data mechanism is termed as nonrandom. Ignoring the missing values in this case leads to biased inferences. The stochastic EM (SEM) algorithm is proposed and developed to find parameters estimates in the presence of intermittent missing values. Also, in this setting, the Monte Carlo method is developed to find the standard errors of parameters estimates. Finally, the proposed techniques are applied to a real data from the International Breast Cancer Study Group.

Cox proportional hazards survival regression in haplotype-based association analysis using the Stochastic-EM algorithm.

It is now widely recognized that haplotype information inferred from genotypes can be of great interest to better characterize the role of a candidate gene in the etiology of a complex trait in the context of association studies. Several works have recently advocated the simultaneous estimation of haplotype frequencies and haplotype effects in order to get a better efficiency in parameter estimation. Most of the available models can deal with a binary or a quantitative phenotype, but none has yet discussed the application of haplotype-based association analysis to a survival outcome. We describe how the recently proposed Stochastic-EM (SEM) algorithm can be applied to estimate haplotype effects in censored data analysis using a standard Cox proportional hazards formulation. This model has been implemented in the THESIAS software freely available at http://www.genecanvas.org.

Modelling Growth and Decline in Lung Function in Duchenne’s Muscular Dystrophy with an Augmented Linear Mixed Effects Model

Summary Longitudinal modelling of lung function in Duchenne’s muscular dystrophy is complicated by a mixture of both growth and decline in lung function within each subject, an unknown point of separation between these phases and significant heterogeneity between individual trajectories. Linear mixed effects models can be used, assuming a single changepoint for all cases; however, this assumption may be incorrect. The paper describes an extension of linear mixed effects modelling in which random changepoints are integrated into the model as parameters and estimated by using a stochastic EM algorithm. We find that use of this ‘mixture modelling’ approach improves the fit significantly.