Techniques for nonlinear least squares and robust regression

Recently, the authors and others have made considerable progress in developing algorithms for solving certain large-residual nonlinear least-squares problems where Gauss-Newton (GN) methods can be expected to perform poorly. These methods take account of the term in the Hessian ignored by the GN methods and use quasi-Newton procedures to update this term explicitly. This paper reviews these new approaches and discusses how they can be modified to give good performance on nonlinear models with robust loss functions where lack of scale invariance causes several new problems to arise.     

Simultaneous confidence band for the difference of segmented linear models

Consider comparing between two treatments a response variable, whose expectation depends on the value of a continuous covariate in some nonlinear fashion. We fit separate segmented linear models to each treatment to approximate the nonlinear relationship. For this setting, we provide a simultaneous confidence band for the difference between treatments of the expected value functions. The treatments are said to differ significantly on intervals of the covariate where the simultaneous confidence band does not contain zero. We consider segmented linear models where the locations of the changepoints are both known and unknown. The band is obtained from asymptotic results.     

Model‐based clustering of longitudinal data

A new family of mixture models for the model‐based clustering of longitudinal data is introduced. The covariance structures of eight members of this new family of models are given and the associated maximum likelihood estimates for the parameters are derived via expectation–maximization (EM) algorithms. The Bayesian information criterion is used for model selection and a convergence criterion based on the Aitken acceleration is used to determine the convergence of these EM algorithms. This new family of models is applied to yeast sporulation time course data, where the models give good clustering performance. Further constraints are then imposed on the decomposition to allow a deeper investigation of the correlation structure of the yeast data. These constraints greatly extend this new family of models, with the addition of many parsimonious models. The Canadian Journal of Statistics 38:153–168; 2010 © 2010 Statistical Society of Canada     

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.     

Semiparametric transformation models with Bayesian P-splines

In this paper, we aim to develop a semiparametric transformation model. Nonparametric transformation functions are modeled with Bayesian P-splines. The transformed variables can be fitted to a general nonlinear mixed model, including linear or nonlinear regression models, mixed effect models, factor analysis models, and other latent variable models as special cases. Markov chain Monte Carlo algorithms are implemented to estimate transformation functions and unknown quantities in the model. The performance of the developed methodology is demonstrated with a simulation study. Its application to a real study on polydrug use is presented.     

Multiple Imputation of Missing or Faulty Values Under Linear Constraints

Many statistical agencies, survey organizations, and research centers collect data that suffer from item nonresponse and erroneous or inconsistent values. These data may be required to satisfy linear constraints, for example, bounds on individual variables and inequalities for ratios or sums of variables. Often these constraints are designed to identify faulty values, which then are blanked and imputed. The data also may exhibit complex distributional features, including nonlinear relationships and highly nonnormal distributions. We present a fully Bayesian, joint model for modeling or imputing data with missing/blanked values under linear constraints that (i) automatically incorporates the constraints in inferences and imputations, and (ii) uses a flexible Dirichlet process mixture of multivariate normal distributions to reflect complex distributional features. Our strategy for estimation is to augment the observed data with draws from a hypothetical population in which the constraints are not present, thereby taking advantage of computationally expedient methods for fitting mixture models. Missing/blanked items are sampled from their posterior distribution using the Hit-and-Run sampler, which guarantees that all imputations satisfy the constraints. We illustrate the approach using manufacturing data from Colombia, examining the potential to preserve joint distributions and a regression from the plant productivity literature. Supplementary materials for this article are available online.     

A Novel Estimation Approach for Mixture Transition Distribution Model in High-Order Markov Chains

A transformation is proposed to convert the nonlinear constraints of the parameters in the mixture transition distribution (MTD) model into box-constraints. The proposed transformation removes the difficulties associated with the maximum likelihood estimation (MLE) process in the MTD modeling so that the MLEs of the parameters can be easily obtained via a hybrid algorithm from the evolutionary algorithms and/or quasi-Newton algorithms for global optimization. Simulation studies are conducted to demonstrate MTD modeling by the proposed novel approach through a global search algorithm in R environment. Finally, the proposed approach is used for the MTD modelings of three real data sets.     

Particle filtering for partially observed Gaussian state space models

Summary. Solving Bayesian estimation problems where the posterior distribution evolves over time through the accumulation of data has many applications for dynamic models. A large number of algorithms based on particle filtering methods, also known as sequential Monte Carlo algorithms, have recently been proposed to solve these problems. We propose a special particle filtering method which uses random mixtures of normal distributions to represent the posterior distributions of partially observed Gaussian state space models. This algorithm is based on a marginalization idea for improving efficiency and can lead to substantial gains over standard algorithms. It differs from previous algorithms which were only applicable to conditionally linear Gaussian state space models. Computer simulations are carried out to evaluate the performance of the proposed algorithm for dynamic tobit and probit models.     

Sensitivity analysis in ordered and restricted parameter models

This paper deals with the problem of local sensitivity analysis in ordered parameter models. In addition to order restrictions, some constraints imposed on the parameters by the model and/or the data are considered. Measures for assessing how much a change in the data modifies the results and conclusions of a statistical analysis of these models are presented. The sensitivity measures are derived using recent results in mathematical programming. The estimation problem is formulated as a primal nonlinear programming problem, and the sensitivities of the parameter estimates as well as the objective function sensitivities with respect to data are obtained. They are very effective in revealing the influential observations in this type of models and in evaluating the changes due to changes in data values. The methods are illustrated by their application to a wide variety of examples of order-restricted models including ordered exponential family parameters, ordered multinomial parameters, ordered linear model parameters, ordered and data constrained parameters, and ordered functions of parameters.     

On sequential Monte Carlo sampling methods for Bayesian filtering

In this article, we present an overview of methods for sequential simulation from posterior distributions. These methods are of particular interest in Bayesian filtering for discrete time dynamic models that are typically nonlinear and non-Gaussian. A general importance sampling framework is developed that unifies many of the methods which have been proposed over the last few decades in several different scientific disciplines. Novel extensions to the existing methods are also proposed. We show in particular how to incorporate local linearisation methods similar to those which have previously been employed in the deterministic filtering literature; these lead to very effective importance distributions. Furthermore we describe a method which uses Rao-Blackwellisation in order to take advantage of the analytic structure present in some important classes of state-space models. In a final section we develop algorithms for prediction, smoothing and evaluation of the likelihood in dynamic models.     

Multilevel and nonlinear panel data models

Summary This paper presents a selective survey on panel data methods. The focus is on new developments. In particular, linear multilevel models, specific nonlinear, nonparametric and semiparametric models are at the center of the survey. In contrast to linear models there do not exist unified methods for nonlinear approaches. In this case conditional maximum likelihood methods dominate for fixed effects models. Under random effects assumptions it is sometimes possible to employ conventional maximum likelihood methods using Gaussian quadrature to reduce a T-dimensional integral. Alternatives are generalized methods of moments and simulated estimators. If the nonlinear function is not exactly known, nonparametric or semiparametric methods should be preferred. Helpful comments and suggestions from an unknown referee are gratefully acknowledged.     

Estimation of nonlinear CUB models via numerical optimization and EM algorithm

The analysis of human perceptions is often carried out by resorting to surveys and questionnaires, where respondents are asked to express ratings about the objects being evaluated. A class of mixture models, called CUB (Combination of Uniform and shifted Binomial), has been recently proposed in this context. This article focuses on a model of this class, the Nonlinear CUB, and investigates some computational issues concerning parameter estimation, which is performed by Maximum Likelihood. More specifically, we consider two main approaches to optimize the log-likelihood: the classical numerical methods of optimization and the EM algorithm. The classical numerical methods comprise the widely used algorithms Nelder–Mead, Newton–Raphson, Broyden–Fletcher–Goldfarb–Shanno (BFGS), Berndt–Hall–Hall–Hausman (BHHH), Simulated Annealing, Conjugate Gradients and usually have the advantage of a fast convergence. On the other hand, the EM algorithm deserves consideration for some optimality properties in the case of mixture models, but it is slower. This article has a twofold aim: first we show how to obtain explicit formulas for the implementation of the EM algorithm in nonlinear CUB models and we formally derive the asymptotic variance–covariance matrix of the Maximum Likelihood estimator; second, we discuss and compare the performance of the two above mentioned approaches to the log-likelihood maximization.     

Nonlinear Autoregression with Positive Innovations

Use of nonlinear models in analyzing time series data is becoming increasingly popular. This paper considers a broad class of nonlinear autoregressive models where the autoregressive part is additive and the terms are nonlinear functions of the past data. Also, the innovation distribution is supported on the non-negative reals and satisfies a tail regularity condition. The linear parameters of the autoregression are estimated using a linear programming recipe which yields much more accurate estimates than traditional methods such as conditional least squares. Limiting distribution of the linear programming estimators is obtained. Simulation studies validate the asymptotic results and reveal excellent small sample properties of the LPE estimator.     

Nonlinear regression models for heterogeneous data with massive outliers

The income or expenditure-related data sets are often nonlinear, heteroscedastic, skewed even after the transformation, and contain numerous outliers. We propose a class of robust nonlinear models that treat outlying observations effectively without removing them. For this purpose, case-specific parameters and a related penalty are employed to detect and modify the outliers systematically. We show how the existing nonlinear models such as smoothing splines and generalized additive models can be robustified by the case-specific parameters. Next, we extend the proposed methods to the heterogeneous models by incorporating unequal weights. The details of estimating the weights are provided. Two real data sets and simulated data sets show the potential of the proposed methods when the nature of the data is nonlinear with outlying observations.     

Estimation of systems of equations subject to curvature constraints

Two computationally manageable estimation procedures for maintaining curvature constraints are compared. Using an efficient nonlinear optimizing algorithm, the well-known Cholesky factorization is compared with the eigenvalue decomposition method for estimating a system of input demand and output supply equations. Statistical tests are provided. The Cholesky factorization is superior when efficient optimizers are used. The eigenvalue decomposition may become a better choice when the user is limited to less efficient algorithms.     

Bayesian weighted inference from surveys

Data from large surveys are often supplemented with sampling weights that are designed to reflect unequal probabilities of response and selection inherent in complex survey sampling methods. We propose two methods for Bayesian estimation of parametric models in a setting where the survey data and the weights are available, but where information on how the weights were constructed is unavailable. The first approach is to simply replace the likelihood with the pseudo likelihood in the formulation of Bayes theorem. This is proven to lead to a consistent estimator but also leads to credible intervals that suffer from systematic undercoverage. Our second approach involves using the weights to generate a representative sample which is integrated into a Markov chain Monte Carlo (MCMC) or other simulation algorithms designed to estimate the parameters of the model. In the extensive simulation studies, the latter methodology is shown to achieve performance comparable to the standard frequentist solution of pseudo maximum likelihood, with the added advantage of being applicable to models that require inference via MCMC. The methodology is demonstrated further by fitting a mixture of gamma densities to a sample of Australian household income.     

Diagonal latent block model for binary data

This paper addresses the problem of co-clustering binary data in the latent block model framework with diagonal constraints for resulting data partitions. We consider the Bernoulli generative mixture model and present three new methods differing in the assumptions made about the degree of homogeneity of diagonal blocks. The proposed models are parsimonious and allow to take into account the structure of a data matrix when reorganizing it into homogeneous diagonal blocks. We derive algorithms for each of the presented models based on the classification expectation-maximization algorithm which maximizes the complete data likelihood. We show that our contribution can outperform other state-of-the-art (co)-clustering methods on synthetic sparse and non-sparse data. We also prove the efficiency of our approach in the context of document clustering, by using real-world benchmark data sets.     

Extending mixtures of multivariate <Emphasis Type="Italic">t</Emphasis>-factor analyzers

Model-based clustering typically involves the development of a family of mixture models and the imposition of these models upon data. The best member of the family is then chosen using some criterion and the associated parameter estimates lead to predicted group memberships, or clusterings. This paper describes the extension of the mixtures of multivariate t-factor analyzers model to include constraints on the degrees of freedom, the factor loadings, and the error variance matrices. The result is a family of six mixture models, including parsimonious models. Parameter estimates for this family of models are derived using an alternating expectation-conditional maximization algorithm and convergence is determined based on Aitken’s acceleration. Model selection is carried out using the Bayesian information criterion (BIC) and the integrated completed likelihood (ICL). This novel family of mixture models is then applied to simulated and real data where clustering performance meets or exceeds that of established model-based clustering methods. The simulation studies include a comparison of the BIC and the ICL as model selection techniques for this novel family of models. Application to simulated data with larger dimensionality is also explored.     

基于辅助粒子滤波的DSGE模型识别方法及应用

动态随机一般均衡模型中涵盖无法直接观测的变量,同时跨方程约束涉及复杂的非线性关系使方程的解析估计难以实现。在贝叶斯框架下识别动态随机一般均衡模型,基于状态空间方法建立度量方程和状态转移方程,采用辅助粒子滤波预测条件后验分布,建立贝叶斯误差带描述宏观经济变量脉冲响应函数的动态特征。实际数据分析验证了贝叶斯识别方法的有效性。