Fitting mixtures of linear regressions

In most applications, the parameters of a mixture of linear regression models are estimated by maximum likelihood using the expectation maximization (EM) algorithm. In this article, we propose the comparison of three algorithms to compute maximum likelihood estimates of the parameters of these models: the EM algorithm, the classification EM algorithm and the stochastic EM algorithm. The comparison of the three procedures was done through a simulation study of the performance (computational effort, statistical properties of estimators and goodness of fit) of these approaches on simulated data sets.

Simulation results show that the choice of the approach depends essentially on the configuration of the true regression lines and the initialization of the algorithms.     

Quantile Regression via the EM Algorithm

The three-parameter asymmetric Laplace distribution (ALD) has received increasing attention in the field of quantile regression due to an important feature between its location and asymmetric parameters. On the basis of the representation of the ALD as a normal-variance–mean mixture with an exponential mixing distribution, this article develops EM and generalized EM algorithms, respectively, for computing regression quantiles of linear and nonlinear regression models. It is interesting to show that the proposed EM algorithm and the MM (Majorization–Minimization) algorithm for quantile regressions are really the same in terms of computation, since the updating formula of them are the same. This provides a good example that connects the EM and MM algorithms. Simulation studies show that the EM algorithm can successfully recover the true parameters in quantile regressions.     

A Note on parameter and standard error estimation in adaptive robust regression

This paper introduces practical methods of parameter and standard error estimation for adaptive robust regression where errors are assumed to be from a normal/independent family of distributions. In particular, generalized EM algorithms (GEM) are considered for the two cases of t and slash families of distributions. For the t family, a one step method is proposed to estimate the degree of freedom parameter. Use of empirical information is suggested for standard error estimation. It is shown that this choice leads to standard errors that can be obtained as a by-product of the GEM algorithm. The proposed methods, as discussed, can be implemented in most available nonlinear regression programs. Details of implementation in SAS NLIN are given using two specific examples.     

A class of finite mixture of quantile regressions with its applications

Mixture of linear regression models provide a popular treatment for modeling nonlinear regression relationship. The traditional estimation of mixture of regression models is based on Gaussian error assumption. It is well known that such assumption is sensitive to outliers and extreme values. To overcome this issue, a new class of finite mixture of quantile regressions (FMQR) is proposed in this article. Compared with the existing Gaussian mixture regression models, the proposed FMQR model can provide a complete specification on the conditional distribution of response variable for each component. From the likelihood point of view, the FMQR model is equivalent to the finite mixture of regression models based on errors following asymmetric Laplace distribution (ALD), which can be regarded as an extension to the traditional mixture of regression models with normal error terms. An EM algorithm is proposed to obtain the parameter estimates of the FMQR model by combining a hierarchical representation of the ALD. Finally, the iterated weighted least square estimation for each mixture component of the FMQR model is derived. Simulation studies are conducted to illustrate the finite sample performance of the estimation procedure. Analysis of an aphid data set is used to illustrate our methodologies.     

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.     

Estimation and diagnostics for heteroscedastic nonlinear regression models based on scale mixtures of skew-normal distributions

An extension of some standard likelihood based procedures to heteroscedastic nonlinear regression models under scale mixtures of skew-normal (SMSN) distributions is developed. This novel class of models provides a useful generalization of the heteroscedastic symmetrical nonlinear regression models (Cysneiros et al., 2010), since the random term distributions cover both symmetric as well as asymmetric and heavy-tailed distributions such as skew-t, skew-slash, skew-contaminated normal, among others. A simple EM-type algorithm for iteratively computing maximum likelihood estimates of the parameters is presented and the observed information matrix is derived analytically. In order to examine the performance of the proposed methods, some simulation studies are presented to show the robust aspect of this flexible class against outlying and influential observations and that the maximum likelihood estimates based on the EM-type algorithm do provide good asymptotic properties. Furthermore, local influence measures and the one-step approximations of the estimates in the case-deletion model are obtained. Finally, an illustration of the methodology is given considering a data set previously analyzed under the homoscedastic skew-t nonlinear regression model.     

Nonlinear regression modeling via the lasso-type regularization

We consider the problem of constructing nonlinear regression models with Gaussian basis functions, using lasso regularization. Regularization with a lasso penalty is an advantageous in that it estimates some coefficients in linear regression models to be exactly zero. We propose imposing a weighted lasso penalty on a nonlinear regression model and thereby selecting the number of basis functions effectively. In order to select tuning parameters in the regularization method, we use a deviance information criterion proposed by Spiegelhalter et al. (2002), calculating the effective number of parameters by Gibbs sampling. Simulation results demonstrate that our methodology performs well in various situations.     

A new method for the estimation of variance matrix with prescribed zeros in nonlinear mixed effects models

We propose a new method for the Maximum Likelihood Estimator (MLE) of nonlinear mixed effects models when the variance matrix of Gaussian random effects has a prescribed pattern of zeros (PPZ). The method consists of coupling the recently developed Iterative Conditional Fitting (ICF) algorithm with the Expectation Maximization (EM) algorithm. It provides positive definite estimates for any sample size, and does not rely on any structural assumption concerning the PPZ. It can be easily adapted to many versions of EM.     

Comparisons of computational methods for clustered binary data

Clustered binary data are common in medical research and can be fitted to the logistic regression model with random effects which belongs to a wider class of models called the generalized linear mixed model. The likelihood-based estimation of model parameters often has to handle intractable integration which leads to several estimation methods to overcome such difficulty. The penalized quasi-likelihood (PQL) method is the one that is very popular and computationally efficient in most cases. The expectation–maximization (EM) algorithm allows to estimate maximum-likelihood estimates, but requires to compute possibly intractable integration in the E-step. The variants of the EM algorithm to evaluate the E-step are introduced. The Monte Carlo EM (MCEM) method computes the E-step by approximating the expectation using Monte Carlo samples, while the Modified EM (MEM) method computes the E-step by approximating the expectation using the Laplace's method. All these methods involve several steps of approximation so that corresponding estimates of model parameters contain inevitable errors (large or small) induced by approximation. Understanding and quantifying discrepancy theoretically is difficult due to the complexity of approximations in each method, even though the focus is on clustered binary data. As an alternative competing computational method, we consider a non-parametric maximum-likelihood (NPML) method as well. We review and compare the PQL, MCEM, MEM and NPML methods for clustered binary data via simulation study, which will be useful for researchers when choosing an estimation method for their analysis.     

The one-step-late PXEM algorithm

The EM algorithm is a popular method for computing maximum likelihood estimates or posterior modes in models that can be formulated in terms of missing data or latent structure. Although easy implementation and stable convergence help to explain the popularity of the algorithm, its convergence is sometimes notoriously slow. In recent years, however, various adaptations have significantly improved the speed of EM while maintaining its stability and simplicity. One especially successful method for maximum likelihood is known as the parameter expanded EM or PXEM algorithm. Unfortunately, PXEM does not generally have a closed form M-step when computing posterior modes, even when the corresponding EM algorithm is in closed form. In this paper we confront this problem by adapting the one-step-late EM algorithm to PXEM to establish a fast closed form algorithm that improves on the one-step-late EM algorithm by insuring monotone convergence. We use this algorithm to fit a probit regression model and a variety of dynamic linear models, showing computational savings of as much as 99.9%, with the biggest savings occurring when the EM algorithm is the slowest to converge.     

Likelihood-based approaches for multivariate linear models under inequality constraints for incomplete data

In this paper, we consider a multivariate linear model with complete/incomplete data, where the regression coefficients are subject to a set of linear inequality restrictions. We first develop an expectation/conditional maximization (ECM) algorithm for calculating restricted maximum likelihood estimates of parameters of interest. We then establish the corresponding convergence properties for the proposed ECM algorithm. Applications to growth curve models and linear mixed models are presented. Confidence interval construction via the double-bootstrap method is provided. Some simulation studies are performed and a real example is used to illustrate the proposed methods.     

The Performance of a Robust Multistage Estimator in Nonlinear Regression with Heteroscedastic Errors

In this article, a robust multistage parameter estimator is proposed for nonlinear regression with heteroscedastic variance, where the residual variances are considered as a general parametric function of predictors. The motivation is based on considering the chi-square distribution for the calculated sample variance of the data. It is shown that outliers that are influential in nonlinear regression parameter estimates are not necessarily influential in calculating the sample variance. This matter persuades us, not only to robustify the estimate of the parameters of the models for both the regression function and the variance, but also to replace the sample variance of the data by a robust scale estimate.     

A SEMIPARAMETRIC REGRESSION MODEL WITH MISSING COVARIATES IN CONTINUOUS-TIME CAPTURE-RECAPTURE STUDIES

Covariate data were missing when a semiparametric regression model was used to study bird abundance in the Mai Po Sanctuary, Hong Kong. This paper proposes an EM‐type algorithm to estimate the regression parameters for that study. Analytical calculation of the expectation in the EM method is difficult, or even impossible, especially when missing covariates are continuous. A Monte Carlo method is used in the EM algorithm to ease the calculation complexity. Asymptotic variances of the parameter estimates are also derived. Properties of the proposed estimators are assessed through numerical simulations and a real example.     

General mixed Poisson regression models with varying dispersion

A general class of mixed Poisson regression models is introduced. This class is based on a mixing between the Poisson distribution and a distribution belonging to the exponential family. With this, we unified some overdispersed models which have been studied separately, such as negative binomial and Poisson inverse gaussian models. We consider a regression structure for both the mean and dispersion parameters of the mixed Poisson models, thus extending, and in some cases correcting, some previous models considered in the literature. An expectation–maximization (EM) algorithm is proposed for estimation of the parameters and some diagnostic measures, based on the EM algorithm, are considered. We also obtain an explicit expression for the observed information matrix. An empirical illustration is presented in order to show the performance of our class of mixed Poisson models. This paper contains a Supplementary Material.     

Mixtures of Linear Regression with Measurement Errors

Existing research on mixtures of regression models are limited to directly observed predictors. The estimation of mixtures of regression for measurement error data imposes challenges for statisticians. For linear regression models with measurement error data, the naive ordinary least squares method, which directly substitutes the observed surrogates for the unobserved error-prone variables, yields an inconsistent estimate for the regression coefficients. The same inconsistency also happens to the naive mixtures of regression estimate, which is based on the traditional maximum likelihood estimator and simply ignores the measurement error. To solve this inconsistency, we propose to use the deconvolution method to estimate the mixture likelihood of the observed surrogates. Then our proposed estimate is found by maximizing the estimated mixture likelihood. In addition, a generalized EM algorithm is also developed to find the estimate. The simulation results demonstrate that the proposed estimation procedures work well and perform much better than the naive estimates.     

Generalized EM estimation for semi-parametric mixture distributions with discretized non-parametric component

We consider independent sampling from a two-component mixture distribution, where one component (called the parametric component) is from a known distributional family and the other component (called the non-parametric component) is unknown. This is a semi-parametric mixture distribution. We discretize the non-parametric component and estimate the parameters of this mixture model, namely the mixing proportion, the unknown parameters of the parametric component and the discretized non-parametric component. We define the maximum penalized likelihood (MPL) estimates of the mixture model parameters and then develop a generalized EM (GEM) iterative scheme to compute the MPL estimates. A simulation study and an example from biology are presented.     

Unsupervised learning of regression mixture models with unknown number of components

ABSTRACT

We propose a new unsupervised learning algorithm to fit regression mixture models with unknown number of components. The developed approach consists in a penalized maximum likelihood estimation carried out by a robust expectation–maximization (EM)-like algorithm. We derive it for polynomial, spline, and B-spline regression mixtures. The proposed learning approach is unsupervised: (i) it simultaneously infers the model parameters and the optimal number of the regression mixture components from the data as the learning proceeds, rather than in a two-fold scheme as in standard model-based clustering using afterward model selection criteria, and (ii) it does not require accurate initialization unlike the standard EM for regression mixtures. The developed approach is applied to curve clustering problems. Numerical experiments on simulated and real data show that the proposed algorithm performs well and provides accurate clustering results, and confirm its benefit for practical applications.     

The competing risks analysis for parallel and series systems using Type-II progressive censoring

Abstract

Recently, the study of the lifetime of systems in reliability and survival analysis in the presence of several causes of failure (competing risks) has attracted attention in the literature. In this paper, series and parallel systems with exponential lifetime for each item of the system are considered. Several causes of failure independently affect lifetime distributions and observations of failure times of the systems are considered under progressive Type-II censored scheme. For series systems, the maximum likelihood estimates of parameters are computed and confidence intervals for parameters of the model are obtained using Fisher information matrix. For parallel systems, the generalized EM algorithm which uses the Newton-Raphson algorithm inside the EM algorithm is used to compute the maximum likelihood estimates of parameters. Also, the standard errors of the maximum likelihood estimates are computed by using the supplemented EM algorithm. The simulation study confirms the good performance of the introduced approach.     

Regression analysis of autocorrelated poisson-distribution) data

A regression model assuming Poisson-dia distributed data. with autocorrelated errors falls into the class of regression models that; have the error structure which is both heteroscedastic and autocorrelated. In general, this class of regression models are not estimable. However, due to the properties of the Poisson distribution that the variance is equal to the mean, this regression model on Poisson-distributed data with autocorrelated. errors is estimable. In this note the special structure of the covarlance matrix of the model with the first order auto-correlated error Is derived utilizing this property, A method based on the least squares method of Frome, Kutner, and Beauchamp (1973), supplemented by steps for handling autocorrelation in studies of time series analysis, nonlinear regression, and econometrics is presented for obtaining generalized least squares estimates for the parameters of the model.     

LAD-Lasso variable selection for doubly censored median regression models

ABSTRACT

A variable selection procedure based on least absolute deviation (LAD) estimation and adaptive lasso (LAD-Lasso for short) is proposed for median regression models with doubly censored data. The proposed procedure can select significant variables and estimate the parameters simultaneously, and the resulting estimators enjoy the oracle property. Simulation results show that the proposed method works well.