An EM Algorithm for Nonlinear Random Effects Models

Effective prognosis of lithium-ion batteries involves the inclusion of the influences of uncertainties that can be incorporated through random effect parameters in a nonlinear mixed effect degradation model framework. This study is geared towards the estimation of the reliability of lithium-ion batteries, using parametric effects determination involving uncertainty, using a multiphase decay patterned sigmoidal model, experimental data and the Weibull distribution function. The random effect model, which uses Maximum Likelihood Estimation (MLE) and Stochastic Approximation Expectation Maximization (SAEM) algorithm to predict the parametric values, was found to estimate the remaining useful life (RUL) to an accuracy of more than 98%. The State-of-Health (SOH) of the batteries was estimated using the Weibull distribution function, which is found to be an appropriate formulation to use.

In a meta-analysis of diagnostic accuracy studies, the sensitivities and specificities of a diagnostic test may depend on the disease prevalence since the severity and definition of disease may differ from study to study due to the design and the population considered. In this paper, we extend the bivariate nonlinear random effects model on sensitivities and specificities to jointly model the disease prevalence, sensitivities and specificities using trivariate nonlinear random-effects models. Furthermore, as an alternative parameterization, we also propose jointly modeling the test prevalence and the predictive values, which reflect the clinical utility of a diagnostic test. These models allow investigators to study the complex relationship among the disease prevalence, sensitivities and specificities; or among test prevalence and the predictive values, which can reveal hidden information about test performance. We illustrate the proposed two approaches by reanalyzing the data from a meta-analysis of radiological evaluation of lymph node metastases in patients with cervical cancer and a simulation study. The latter illustrates the importance of carefully choosing an appropriate normality assumption for the disease prevalence, sensitivities and specificities, or the test prevalence and the predictive values. In practice, it is recommended to use model selection techniques to identify a best-fitting model for making statistical inference. In summary, the proposed trivariate random effects models are novel and can be very useful in practice for meta-analysis of diagnostic accuracy studies.

Nonlinear random effects mixture models: Maximum likelihood estimation via the EM algorithm

Nonlinear random effects models are considered from the Bayesian point of view. The method of analysis follows closely that of Lindley and Smith (1972, Journal of the Royal Statistical Society, Series B 34, 1-42). The numerical method is related to the EM algorithm.

Clustering is a fundamental problem in statistics and machine learning, whose solution is commonly computed by the Expectation-Maximization (EM) method, which finds a locally optimal solution for an objective function called log-likelihood. Since the surface of the log-likelihood function is non convex, a stochastic search with Markov Chain Monte Carlo (MCMC) methods can help escaping locally optimal solutions. In this article, we tackle two fundamental conflicting goals: Finding higher quality solutions and achieving faster convergence. With that motivation in mind, we introduce an efficient algorithm that combines elements of the EM and MCMC methods to find clustering solutions that are qualitatively better than those found by the standard EM method. Moreover, our hybrid algorithm allows tuning model parameters and understanding the uncertainty in their estimation. The main issue with MCMC methods is that they generally require a very large number of iterations to explore the posterior of each model parameter. Convergence is accelerated by several algorithmic improvements which include sufficient statistics, simplified model parameter priors, fixing covariance matrices and iterative sampling from small blocks of the data set. A brief experimental evaluation shows promising results.

Computation: Expectation-Maximization Algorithm

Pharmacokinetic-Pharmacodynamic Modeling and Simulation

A stochastic approximation expectation maximum (EM) algorithm is proposed to obtain Maximum likelihood (ML) estimation of nonlinear random effect models in which the manifest variables are distributed as a reproductive dispersion model (RDM) and may be missing with ignorable missingness mechanism in this paper. A method composed of simulation step as well as stochastic approximation step is used to obtain the conditional expectation, whereas the M-step is executed via the method of conditional maximization. The most attractive point of this approach is that it is novel and non-trivial, which can be used to obtain the ML estimates and the estimation of standard errors simultaneously. Moreover, A model selection criterion is developed. Empirical results are used to illustrate the usefulness of the methodologies proposed above.

Background : Pompe disease is an autosomal-recessive glycogen storage disease due to deficiency of acid α-glucosidase (GAA). Clinical findings in infantile onset disease are due to glycogen accumulation leading to hypertrophic cardiomyopathy, skeletal myopathy, respiratory failure and death. Enzyme replacement therapy (ERT) with recombinant human GAA (rhGAA; Myozyme®) has been shown to prolong survival in infants. We assessed the hypothesis that myocardial response to rhGAA predicts clinical outcome in Pompe. Methods: Echos during one year of ERT were reviewed from patients enrolled in open label trials of rhGAA (n=38). A standardized echo protocol was used at specific time points (wks 0, 4, 8, 12, 26, 38, 52). A central reader made measurements off-line, blinded to study information. Variables including LV mass, volumes, and ejection fraction (EF) were determined. Clinical outcomes at study-end were analyzed in 2 groups: Group A, alive and free of assisted ventilation (AV) or Group B, died or requiring AV. Z-scores were calculated for all echo variables and median z-scores between outcome groups at wk 0 and wk 52 were compared using median scores tests. Non-linear random effects models were used to examine changes over time. Results: Mean age at first infusion = 10.9±9.8 months (range 1.2 – 43.1). At study onset, 7/38 received AV. At study end, mean time to last infusion = 115±44 weeks with 16/38 in Group A and 22/38 in Group B. For Group A vs. Group B, median z-scores at week 0 for LV mass = 6.11 vs. 7.41, mass to volume ratio (MVR) = 4.5 vs. 8.8, and EF = 0.8 for both groups; p≥ 0.12 for each measure. Median z-scores for the same outcome groups, respectively, at week 52 for LV mass = 0.57 vs. 3.41 (p=0.06), for MVR = 0.48 vs. 3.54 (p=0.001) and EF = −0.6 in both groups. In the subgroup not requiring AV at study onset (n=31), MVR is higher at all times in Group B (p≤ 0.03). Conclusion: rhGAA therapy can prolong life in young patients with Pompe, improving outcome compared to an untreated natural history cohort. Even with ERT, there is significant morbidity and mortality. Diminished myocardial response with persistence of significant LV hypertrophy predicts progression to poor clinical outcomes. These and other predictors are required to fully understand response to ERT in Pompe disease.

The restricted EM algorithm under inequality restrictions on the parameters

This paper proposes a method for obtaining nonlinear models of distribution grid based on available measurements from the power grid. We formulate a maximum likelihood estimation (MLE) problem that estimates unknown line parameters—specifically, the impedance between nodes—using measured voltage magnitudes and net active and reactive power injections at each node. The nonlinear model for the distribution grid uses a nonlinear approximation of the DistFlow model, which includes line losses and is parameterized by the unknown line impedances. We solve the resulting MLE problem using an expectation maximization (EM) algorithm, tailored for the nonlinear model, and provide a numerically robust implementation. The proposed method is demonstrated on the IEEE 37-node test network, and we compare it with the state-of-the-art methods. The proposed method achieves a 70% reduction in voltage error and an error for state variables that is more than 10000 times smaller. A final comparison uses data from a real network, and the proposed method achieves parameter estimates with errors 100 times smaller than competing methods.

Fusion of diverse data sources for flood extent mapping and risk assessment in Sindh: A comparative study of inundation mapping approaches

In the previous chapters, we discussed autoregressive linear mixed effects models. In this section, we discuss the relationships between the autoregressive linear mixed effects models and nonlinear mixed effects models, growth curves, and differential equations. The autoregressive model shows a profile approaching an asymptote, where the change is proportional to the distance remaining to the asymptote. Autoregressive models in discrete time correspond to monomolecular curves in continuous time. Autoregressive linear mixed effects models correspond to monomolecular curves with random effects in the baseline and asymptote, and special error terms. The autoregressive coefficient is a nonlinear parameter, but all random effects parameters in the model are linear. Therefore, autoregressive linear mixed effects models are nonlinear mixed effects models without nonlinear random effects and have a closed form of likelihood. When there are time-dependent covariates, autoregressive linear mixed effects models are represented by a differential equation and random effects. The monomolecular curve is one of the popular growth curves. We introduce other growth curves, such as the logistic curves and von Bertalanffy curves, and generalizations of growth curves. Re-parameterization is often performed in nonlinear models, and various representations of re-parameterization in monomolecular and other curves are provided herein.

In this paper, we offer generalized exponentiated gamma distribution. We consider the maximum likelihood and Bayesian estimators of unknown parameters and propose Markov chain Monte Carlo (MCMC) method to generate samples from the posterior distribution. The mean square error of estimations is computed and comparisons are made using Monte Carlo simulation. Key words: Maximum likelihood estimator, Bayesian estimator, squared error loss function, generalized exponentiated gamma distribution.

This paper proposes a high level language constituted of only a few primitives and macros for describing recursive maximum likelihood (ML) estimation algorithms. This language is applicable to estimation problems involving linear Gaussian models, or processes taking values in a finite set (only the first case is considered here). The use of high level primitive allows the development of highly modular ML estimation algorithms based on only few numerical blocks. The primitives, which correspond to the combination of different measurements, the extraction of sufficient statistics, and the conversion of the status of a variable from unknown to observed ones, or vice versa, are first defined for linear Gaussian relations specifying mixed deterministic/stochastic information about the system variables. These primitives are used to define other macros, and are illustrated by considering the filtering and smoothing problems for linear descriptor systems, as well as failure detection and isolation. >