Stochastic Expectation Maximization Algorithm Research Articles

Pharmacokinetic-pharmacodynamic modelling and simulation of methotrexate dosing in patients with rheumatoid arthritis.

To develop a non-linear mixed-effects population pharmacokinetic and pharmacodynamic (PK-PD) model describing the change in the concentration of methotrexate polyglutamates in erythrocytes (ery-MTX-PGn with "n" number of glutamate, representing PK component) and how this relates to modified 28-joint Disease Activity Score incorporating erythrocyte sedimentation rate (DAS-28-3) for rheumatoid arthritis (RA), representing PD component. An existing PK model was fitted to data from a study consisting of 117 RA patients. The estimation of population PK-PD parameters was performed using stochastic approximation expectation maximisation algorithm in Monolix 2021R2. The model was used to perform Monte Carlo simulations of a loading dose regimen (50mg subcutaneous methotrexate as loading doses, then 20mg weekly oral methotrexate) compared to a standard dosing regimen (10mg weekly oral methotrexate for 2 weeks, then 20mg weekly oral methotrexate). Every 40 nmol/L increase in ery-MTX-PG3-5 total concentration correlated with 1-unit reduction in DAS-28-3. Significant covariate effects on the therapeutic response of methotrexate included the use of prednisolone in the first 4 weeks (positive use correlated with 25% reduction in DAS-28-3 when other variables were constant) and patient age (every 10-year increase in age correlated with 3.4% increase in DAS-28-3 when other variables were constant). 4 methotrexate loading doses led to a higher percentage of patients achieving a good/moderate response compared to the standard regimen (Week 4: 87.6% vs. 39.8%; Week 10: 64.7% vs. 57.0%). A loading dose regimen was more likely to achieve higher ery-MTX-PG concentration and better therapeutic response after 4 weeks of methotrexate treatment.

Estimating Latent-Variable Panel Data Models Using Parameter-Expanded SEM Methods

This paper presents new estimation algorithms for three types of dynamic panel data models with latent variables: factor models, discrete choice models, and persistent-transitory quantile processes. The new methods combine the parameter expansion (PX) ideas in Liu et al. (1998) with the stochastic expectation-maximization (SEM) algorithm in likelihood and moment-based contexts. The goal is to facilitate convergence in models with a large space of latent variables by improving algorithmic efficiency. This is achieved by specifying expanded models within the M step. Effectively, we are proposing new estimators for the pseudo-data within iterations that take into account the fact that the model of interest is misspecified for draws based on parameter values far from the truth. We establish the asymptotic equivalence of the likelihood-based PX-SEM to an alternative SEM algorithm with a smaller expected fraction of missing information compared to the standard SEM based on the original model, implying a faster global convergence rate. Finally, in simulations we show that the new algorithms significantly improve the convergence speed relative to standard SEM algorithms, sometimes dramatically so, by reducing the total computing time from hours to a few minutes.

Uncertainty Computation at Finite Distance in Nonlinear Mixed Effects Models-a New Method Based on Metropolis-Hastings Algorithm.

The standard errors (SE) of the maximum likelihood estimates (MLE) of the population parameter vector in nonlinear mixed effect models (NLMEM) are usually estimated using the inverse of the Fisher information matrix (FIM). However, at a finite distance, i.e. far from the asymptotic, the FIM can underestimate the SE of NLMEM parameters. Alternatively, the standard deviation of the posterior distribution, obtained in Stan via the Hamiltonian Monte Carlo algorithm, has been shown to be a proxy for the SE, since, under some regularity conditions on the prior, the limiting distributions of the MLE and of the maximum a posterior estimator in a Bayesian framework are equivalent. In this work, we develop a similar method using the Metropolis-Hastings (MH) algorithm in parallel to the stochastic approximation expectation maximisation (SAEM) algorithm, implemented in the saemix R package. We assess this method on different simulation scenarios and data from a real case study, comparing it to other SE computation methods. The simulation study shows that our method improves the results obtained with frequentist methods at finite distance. However, it performed poorly in a scenario with the high variability and correlations observed in the real case study, stressing the need for calibration.

Reliability modeling and evaluation for degradation data with heterogeneous initiation time

In engineering practice, some products start deteriorating only after a period of usage. Moreover, the degradation-free periods of different products show a phenomenon of clustering, that is, there are several heterogeneous subpopulations. Both the degradation-free period and its subpopulation label are often unobservable, which poses great challenges to the reliability modeling and evaluation. In this paper, a randomly delayed degradation model considering the heterogeneous initiation time is developed, where the mixture lognormal distribution and the Wiener process are integrated to characterize the overall degradation process. The model parameters are estimated by the stochastic expectation maximization algorithm which also produces the estimator of the degradation-free period for each product simultaneously. In addition, inferences on the choice of parameters’ initial values, interval estimation, determination of the number of subpopulations and classification of products are also presented. Finally, simulation studies are carried out to validate the proposed formulas and methods, and a practical example is provided for illustration.

Categorising the world into local climate zones: towards quantifying labelling uncertainty for machine learning models

Abstract Image classification is often prone to labelling uncertainty. To generate suitable training data, images are labelled according to evaluations of human experts. This can result in ambiguities, which will affect subsequent models. In this work, we aim to model the labelling uncertainty in the context of remote sensing and the classification of satellite images. We construct a multinomial mixture model given the evaluations of multiple experts. This is based on the assumption that there is no ambiguity of the image class, but apparently in the experts’ opinion about it. The model parameters can be estimated by a stochastic expectation maximisation algorithm. Analysing the estimates gives insights into sources of label uncertainty. Here, we focus on the general class ambiguity, the heterogeneity of experts, and the origin city of the images. The results are relevant for all machine learning applications where image classification is pursued and labelling is subject to humans.

On the estimation of interval censored destructive negative binomial cure model.

In this article, a competitive risk survival model is considered in which the initial number of risks, assumed to follow a negative binomial distribution, is subject to a destructive mechanism. Assuming the population of interest to have a cure component, the form of the data as interval-censored, and considering both the number of initial risks and risks remaining active after destruction to be missing data, we develop two distinct estimation algorithms for this model. Making use of the conditional distributions of the missing data, we develop an expectation maximization (EM) algorithm, in which the conditional expected complete log-likelihood function is decomposed into simpler functions which are then maximized independently. A variation of the EM algorithm, called the stochastic EM (SEM) algorithm, is also developed with the goal of avoiding the calculation of complicated expectations and improving performance at parameter recovery. A Monte Carlo simulation study is carried out to evaluate the performance of both estimation methods through calculated bias, root mean square error, and coverage probability of the asymptotic confidence interval. We demonstrate the proposed SEM algorithm as the preferred estimation method through simulation and further illustrate the advantage of the SEM algorithm, as well as the use of a destructive model, with data from a children's mortality study.

Inference Based on the Stochastic Expectation Maximization Algorithm in a Kumaraswamy Model with an Application to COVID-19 Cases in Chile

Extensive research has been conducted on models that utilize the Kumaraswamy distribution to describe continuous variables with bounded support. In this study, we examine the trapezoidal Kumaraswamy model. Our objective is to propose a parameter estimation method for this model using the stochastic expectation maximization algorithm, which effectively tackles the challenges commonly encountered in the traditional expectation maximization algorithm. We then apply our results to the modeling of daily COVID-19 cases in Chile.

Estimation for two Gompertz populations under a balanced joint progressive Type-II censoring scheme

Comparative lifetime experiments are remarkable when the study is to ascertain the relative merits of two competing products regarding the duration of their service life. This paper considers the comparative lifetime experiments of two Gompertz populations under a balanced joint progressive Type-II censoring scheme. The lifetime distributions of the units are assumed to follow the Gompertz distribution with a common shape but different scale parameters. The maximum likelihood estimates of the unknown parameters are derived. The existence of the maximum likelihood estimates is proved. Expectation-maximization and stochastic expectation-maximization algorithms are provided to calculate the estimates. The bootstrap-p, bootstrap-t, and approximate confidence intervals are established. To obtain the Bayesian estimates, it is assumed that the prior of scale parameters is a Beta-Gamma distribution and the prior of the common shape parameter is an independent Gamma distribution. Under squared error loss and LINEX loss functions, the Metropolis-Hastings algorithm is provided to compute the Bayes estimates and the credible intervals. Further, the statistical inferences with order restriction are studied when it is known a priori that the expectation of the lifespan of one population is shorter than that of the other population. A wide range of simulation experiments is conducted to evaluate the performance of the proposed methods. Finally, the lifetimes of white organic light-emitting diodes and the breaking strengths of jute fiber of gauge lengths are analyzed to illustrate the practical application of the proposed model and methods.

Stochastic Expectation Maximization Algorithm for Linear Mixed-Effects Model with Interactions in the Presence of Incomplete Data.

The purpose of this paper is to propose a new algorithm based on stochastic expectation maximization (SEM) to deal with the problem of unobserved values when multiple interactions in a linear mixed-effects model (LMEM) are present. We test the effectiveness of the proposed algorithm with the stochastic approximation expectation maximization (SAEM) and Monte Carlo Markov chain (MCMC) algorithms. This comparison is implemented to highlight the importance of including the maximum effects that can affect the model. The applications are made on both simulated psychological and real data. The findings demonstrate that our proposed SEM algorithm is highly preferable to the other competitor algorithms.

Comparison of simulation-based algorithms for parameter estimation and state reconstruction in nonlinear state-space models

<p style='text-indent:20px;'>This study aims at comparing simulation-based approaches for estimating both the state and unknown parameters in nonlinear state-space models. Numerical results on different toy models show that the combination of a Conditional Particle Filter (CPF) with Backward Simulation (BS) smoother and a Stochastic Expectation-Maximization (SEM) algorithm is a promising approach. The CPFBS smoother run with a small number of particles allows to explore efficiently the state-space and simulate relevant trajectories of the state conditionally to the observations. When combined with the SEM algorithm, this algorithm provides accurate estimates of the state and the parameters in nonlinear models, where the application of EM algorithms combined with a standard particle smoother or an ensemble Kalman smoother is limited.</p>

Analyzing the Rainfall Pattern in Honduras Through Non-Homogeneous Hidden Markov Models

One of the major climatic interests of the last decades has been to understand and describe the rainfall patterns of specific areas of the world as functions of other climate covariates. We do it for the historical climate monitoring data from Tegucigalpa, Honduras, using non-homogeneous hidden Markov models (NHMMs), which are dynamic models usually used to identify and predict heterogeneous regimes. For estimating the NHMM in an efficient and scalable way, we propose the stochastic Expectation-Maximization (EM) algorithm and a Bayesian method, and compare their performance in synthetic data. Although these methodologies have already been used for estimating several other statistical models, it is not the case of NHMMs which are still widely fitted by the traditional EM algorithm. We observe that, under tested conditions, the performance of the Bayesian and stochastic EM algorithms is similar and discuss their slight differences. Analyzing the Honduras rainfall data set, we identify three heterogeneous rainfall periods and select temperature and humidity as relevant covariates for explaining the dynamic relation among these periods.

Stress-strength reliability estimation for the inverted exponentiated Rayleigh distribution under unified progressive hybrid censoring with application

<abstract> <p>In this paper, we studied the estimation of a stress-strength reliability model ($R = P(X>Y)$) based on inverted exponentiated Rayleigh distribution under the unified progressive hybrid censoring scheme (unified PHCS). The maximum likelihood estimates of the unknown parameters were obtained using the stochastic expectation-maximization algorithm (stochastic EMA). The asymptotic confidence intervals were also created. Under squared error and Linex and generalized entropy loss functions, the Gibbs sampler together with Metropolis-Hastings algorithm was provided to compute the Bayes estimates and the credible intervals. Extensive simulations were performed to see the effectiveness of the proposed estimation methods. Also, parallel to the development of reliability studies, it is necessary to study its application in different sciences such as engineering. Therefore, droplet splashing data under two nozzle pressures were proposed to exemplify the theoretical outcomes.</p> </abstract>

Estimations and optimal censoring schemes for the unified progressive hybrid gamma-mixed Rayleigh distribution

<abstract> <p>Censoring is a common occurrence in reliability engineering tests. This article considers estimation of the model parameters and the reliability characteristics of the gamma-mixed Rayleigh distribution based on a novel unified progressive hybrid censoring scheme (UPrgHyCS), where experimenters are allowed more flexibility in designing the test and higher efficiency. The maximum likelihood estimates of the model parameters and reliability are provided using the stochastic expectation–maximization algorithm based on the UPrgHyCS. Further, the Bayesian inference associated with any parametric function of the model parameters is considered using the Markov chain Monte Carlo method with the Metropolis-Hastings (M-H) algorithm. Asymptotic confidence and credible intervals of the proposed quantities are also created. The maximum a posteriori estimates of the model parameters are acquired. Due to the importance of determining the optimal censoring scheme for reliability problems, different optimality criteria are proposed and derived to find it. This method can help to design experiments and get more information about unknown parameters for a given sample size. Finally, comprehensive simulation experiments are provided to investigate the performances of the considered estimates, and a real dataset is analyzed to elucidate the practical application and the optimality criterion work in real life scenarios. The Bayes estimates using the M-H technique show the best performance in terms of error values.</p> </abstract>

Parameter-wise co-clustering for high-dimensional data

In recent years, data dimensionality has increasingly become a concern, leading to many parameter and dimension reduction techniques being proposed in the literature. A parameter-wise co-clustering model, for (possibly high-dimensional) data modelled via continuous random variables, is presented. The proposed model, although allowing more flexibility, still maintains the very high degree of parsimony and interpretability achieved by traditional co-clustering. More precisely, the keystone consists of dramatically increasing the number of column-clusters while expressing each as a combination of a limited number of mean-dependent and variance-dependent column-clusters. A stochastic expectation-maximization algorithm along with a Gibbs sampler is used for parameter estimation and an integrated complete log-likelihood criterion is used for model selection. Simulated and real datasets are used for illustration and comparison with traditional co-clustering.

Estimation of site frequency spectra from low-coverage sequencing data using stochastic EM reduces overfitting, runtime, and memory usage.

The site frequency spectrum is an important summary statistic in population genetics used for inference on demographic history and selection. However, estimation of the site frequency spectrum from called genotypes introduces bias when working with low-coverage sequencing data. Methods exist for addressing this issue but sometimes suffer from 2 problems. First, they can have very high computational demands, to the point that it may not be possible to run estimation for genome-scale data. Second, existing methods are prone to overfitting, especially for multidimensional site frequency spectrum estimation. In this article, we present a stochastic expectation-maximization algorithm for inferring the site frequency spectrum from NGS data that address these challenges. We show that this algorithm greatly reduces runtime and enables estimation with constant, trivial RAM usage. Furthermore, the algorithm reduces overfitting and thereby improves downstream inference. An implementation is available at github.com/malthesr/winsfs.

Multi-center federated learning: clients clustering for better personalization

Personalized decision-making can be implemented in a Federated learning (FL) framework that can collaboratively train a decision model by extracting knowledge across intelligent clients, e.g. smartphones or enterprises. FL can mitigate the data privacy risk of collaborative training since it merely collects local gradients from users without access to their data. However, FL is fragile in the presence of statistical heterogeneity that is commonly encountered in personalized decision making, e.g., non-IID data over different clients. Existing FL approaches usually update a single global model to capture the shared knowledge of all users by aggregating their gradients, regardless of the discrepancy between their data distributions. By comparison, a mixture of multiple global models could capture the heterogeneity across various clients if assigning the client to different global models (i.e., centers) in FL. To this end, we propose a novel multi-center aggregation mechanism to cluster clients using their models’ parameters. It learns multiple global models from data as the cluster centers, and simultaneously derives the optimal matching between users and centers. We then formulate it as an optimization problem that can be efficiently solved by a stochastic expectation maximization (EM) algorithm. Experiments on multiple benchmark datasets of FL show that our method outperforms several popular baseline methods. The experimental source codes are publicly available on the Github repository (GitHub repository: https://github.com/mingxuts/multi-center-fed-learning).

A Mixed Stochastic Approximation EM (MSAEM) Algorithm for the Estimation of the Four-Parameter Normal Ogive Model.

In recent years, the four-parameter model (4PM) has received increasing attention in item response theory. The purpose of this article is to provide more efficient and more reliable computational tools for fitting the 4PM. In particular, this article focuses on the four-parameter normal ogive model (4PNO) model and develops efficient stochastic approximation expectation maximization (SAEM) algorithms to compute the marginalized maximum a posteriori estimator. First, a data augmentation scheme is used for the 4PNO model, which makes the complete data model be an exponential family, and then, a basic SAEM algorithm is developed for the 4PNO model. Second, to overcome the drawback of the SAEM algorithm, we develop an improved SAEM algorithm for the 4PNO model, which is called the mixed SAEM (MSAEM). Results from simulation studies demonstrate that: (1) the MSAEM provides more accurate or comparable estimates as compared with the other estimation methods, while computationally more efficient; (2) the MSAEM is more robust to the choices of initial values and the priors for item parameters, which is a valuable property for practice use. Finally, a real data set is analyzed to show the good performance of the proposed methods.

Inference for a discretized stochastic logistic differential equation and its application to biological growth

In this paper, we present a method to adjust a stochastic logistic differential equation (SLDE) to a set of highly sparse real data. We assume that the SLDE have two unknown parameters to be estimated. We calculate the Maximum Likelihood Estimator (MLE) to estimate the intrinsic growth rate. We prove that the MLE is strongly consistent and asymptotically normal. For estimating the diffusion parameter, the quadratic variation of the data is used. We validate our method with several types of simulated data. For more realistic cases in which we observe discretizations of the solution, we use diffusion bridges and the stochastic expectation-maximization algorithm to estimate the parameters. Furthermore, when we observe only one point for each path for a given number of trajectories we were still able to estimate the parameters of the SLDE. As far as we know, this is the first attempt to fit stochastic differential equations (SDEs) to these types of data. Finally, we apply our method to real data coming from fishery. The proposed adjustment method can be applied to other examples of SDEs and is highly applicable in several areas of science, especially in situations of sparse data.

A simplified stochastic EM algorithm for cure rate model with negative binomial competing risks: An application to breast cancer data.

In this article, a long-term survival model under competing risks is considered. The unobserved number of competing risks is assumed to follow a negative binomial distribution that can capture both over- and under-dispersion. Considering the latent competing risks as missing data, a variation of the well-known expectation maximization (EM) algorithm, called the stochastic EM algorithm (SEM), is developed. It is shown that the SEM algorithm avoids calculation of complicated expectations, which is a major advantage of the SEM algorithm over the EM algorithm. The proposed procedure also allows the objective function to be split into two simpler functions, one corresponding to the parameters associated with the cure rate and the other corresponding to the parameters associated with the progression times. The advantage of this approach is that each simple function, with lower parameter dimension, can be maximized independently. An extensive Monte Carlo simulation study is carried out to compare the performances of the SEM and EM algorithms. Finally, a breast cancer survival data is analyzed and it is shown that the SEM algorithm performs better than the EM algorithm.

Statistical inference of generalized progressive hybrid censored step-stress accelerated dependent competing risks model for Marshall-Olkin bivariate Weibull distribution

In this article, a step-stress accelerated life test based on the generalized Khamis-Higgins model in the presence of Marshall-Olkin bivariate Weibull distributed dependent competing risks is considered under generalized progressive hybrid censoring. It is assumed that the stress is changed when a pre-specified number of failures take place, and four different failure causes are discussed. The classical and Bayesian statistical methods are used to estimate the unknown parameters. In addition to the maximum likelihood estimation, the stochastic expectation-maximization algorithm is also applied. The Monte Carlo Markov Chain samples with importance sampling are used to obtain the Bayesian estimates under two loss functions, and the Bayesian highest posterior density credible intervals are constructed. Then the parameters and reliability function under normal use stress level are calculated. The Monte Carlo method is applied to compare the performance of the estimation methods. Finally, a real-life data set is analyzed for illustrative purposes.