The Loss Rank Criterion for Variable Selection in Linear Regression Analysis

Abstract. Lasso and other regularization procedures are attractive methods for variable selection, subject to a proper choice of shrinkage parameter. Given a set of potential subsets produced by a regularization algorithm, a consistent model selection criterion is proposed to select the best one among this preselected set. The approach leads to a fast and efficient procedure for variable selection, especially in high‐dimensional settings. Model selection consistency of the suggested criterion is proven when the number of covariates d is fixed. Simulation studies suggest that the criterion still enjoys model selection consistency when d is much larger than the sample size. The simulations also show that our approach for variable selection works surprisingly well in comparison with existing competitors. The method is also applied to a real data set.     

Bayesian generalized fused lasso modeling via NEG distribution

The fused lasso penalizes a loss function by the L₁ norm for both the regression coefficients and their successive differences to encourage sparsity of both. In this paper, we propose a Bayesian generalized fused lasso modeling based on a normal-exponential-gamma (NEG) prior distribution. The NEG prior is assumed into the difference of successive regression coefficients. The proposed method enables us to construct a more versatile sparse model than the ordinary fused lasso using a flexible regularization term. Simulation studies and real data analyses show that the proposed method has superior performance to the ordinary fused lasso.     

Regression with outlier shrinkage

We propose a robust regression method called regression with outlier shrinkage (ROS) for the traditional n>p$n>p$ cases. It improves over the other robust regression methods such as least trimmed squares (LTS) in the sense that it can achieve maximum breakdown value and full asymptotic efficiency simultaneously. Moreover, its computational complexity is no more than that of LTS. We also propose a sparse estimator, called sparse regression with outlier shrinkage (SROS), for robust variable selection and estimation. It is proven that SROS can not only give consistent selection but also estimate the nonzero coefficients with full asymptotic efficiency under the normal model. In addition, we introduce a concept of nearly regression equivariant estimator for understanding the breakdown properties of sparse estimators, and prove that SROS achieves the maximum breakdown value of nearly regression equivariant estimators. Numerical examples are presented to illustrate our methods.     

Bayesian variable selection in Poisson change-point regression analysis

In this article, we develop a Bayesian variable selection method that concerns selection of covariates in the Poisson change-point regression model with both discrete and continuous candidate covariates. Ranging from a null model with no selected covariates to a full model including all covariates, the Bayesian variable selection method searches the entire model space, estimates posterior inclusion probabilities of covariates, and obtains model averaged estimates on coefficients to covariates, while simultaneously estimating a time-varying baseline rate due to change-points. For posterior computation, the Metropolis-Hastings within partially collapsed Gibbs sampler is developed to efficiently fit the Poisson change-point regression model with variable selection. We illustrate the proposed method using simulated and real datasets.     

BAYESIAN HYPER‐LASSOS WITH NON‐CONVEX PENALIZATION

The Lasso has sparked interest in the use of penalization of the log‐likelihood for variable selection, as well as for shrinkage. We are particularly interested in the more‐variables‐than‐observations case of characteristic importance for modern data. The Bayesian interpretation of the Lasso as the maximum a posteriori estimate of the regression coefficients, which have been given independent, double exponential prior distributions, is adopted. Generalizing this prior provides a family of hyper‐Lasso penalty functions, which includes the quasi‐Cauchy distribution of Johnstone and Silverman as a special case. The properties of this approach, including the oracle property, are explored, and an EM algorithm for inference in regression problems is described. The posterior is multi‐modal, and we suggest a strategy of using a set of perfectly fitting random starting values to explore modes in different regions of the parameter space. Simulations show that our procedure provides significant improvements on a range of established procedures, and we provide an example from chemometrics.     

Model selection and parameter estimation of a multinomial logistic regression model

In the multinomial regression model, we consider the methodology for simultaneous model selection and parameter estimation by using the shrinkage and LASSO (least absolute shrinkage and selection operation) [R. Tibshirani, Regression shrinkage and selection via the LASSO, J. R. Statist. Soc. Ser. B 58 (1996), pp. 267–288] strategies. The shrinkage estimators (SEs) provide significant improvement over their classical counterparts in the case where some of the predictors may or may not be active for the response of interest. The asymptotic properties of the SEs are developed using the notion of asymptotic distributional risk. We then compare the relative performance of the LASSO estimator with two SEs in terms of simulated relative efficiency. A simulation study shows that the shrinkage and LASSO estimators dominate the full model estimator. Further, both SEs perform better than the LASSO estimators when there are many inactive predictors in the model. A real-life data set is used to illustrate the suggested shrinkage and LASSO estimators.     

Efficient Bayesian Multivariate Surface Regression

Methods for choosing a fixed set of knot locations in additive spline models are fairly well established in the statistical literature. The curse of dimensionality makes it nontrivial to extend these methods to nonadditive surface models, especially when there are more than a couple of covariates. We propose a multivariate Gaussian surface regression model that combines both additive splines and interactive splines, and a highly efficient Markov chain Monte Carlo algorithm that updates all the knot locations jointly. We use shrinkage prior to avoid overfitting with different estimated shrinkage factors for the additive and surface part of the model, and also different shrinkage parameters for the different response variables. Simulated data and an application to firm leverage data show that the approach is computationally efficient, and that allowing for freely estimated knot locations can offer a substantial improvement in out‐of‐sample predictive performance.     

ACCELERATED FAILURE TIME MODELS WITH NONLINEAR COVARIATES EFFECTS

As a flexible alternative to the Cox model, the accelerated failure time (AFT) model assumes that the event time of interest depends on the covariates through a regression function. The AFT model with non‐parametric covariate effects is investigated, when variable selection is desired along with estimation. Formulated in the framework of the smoothing spline analysis of variance model, the proposed method based on the Stute estimate ( Stute, 1993 [Consistent estimation under random censorship when covariables are present, J. Multivariate Anal. 45 , 89–103]) can achieve a sparse representation of the functional decomposition, by utilizing a reproducing kernel Hilbert norm penalty. Computational algorithms and theoretical properties of the proposed method are investigated. The finite sample size performance of the proposed approach is assessed via simulation studies. The primary biliary cirrhosis data is analyzed for demonstration.     

Application of shrinkage estimation in linear regression models with autoregressive errors

In this paper, we consider the shrinkage and penalty estimation procedures in the linear regression model with autoregressive errors of order p when it is conjectured that some of the regression parameters are inactive. We develop the statistical properties of the shrinkage estimation method including asymptotic distributional biases and risks. We show that the shrinkage estimators have a significantly higher relative efficiency than the classical estimator. Furthermore, we consider the two penalty estimators: least absolute shrinkage and selection operator (LASSO) and adaptive LASSO estimators, and numerically compare their relative performance with that of the shrinkage estimators. A Monte Carlo simulation experiment is conducted for different combinations of inactive predictors and the performance of each estimator is evaluated in terms of the simulated mean-squared error. This study shows that the shrinkage estimators are comparable to the penalty estimators when the number of inactive predictors in the model is relatively large. The shrinkage and penalty methods are applied to a real data set to illustrate the usefulness of the procedures in practice.     

Cluster‐Specific Variable Selection for Product Partition Models

We propose a random partition model that implements prediction with many candidate covariates and interactions. The model is based on a modified product partition model that includes a regression on covariates by favouring homogeneous clusters in terms of these covariates. Additionally, the model allows for a cluster‐specific choice of the covariates that are included in this evaluation of homogeneity. The variable selection is implemented by introducing a set of cluster‐specific latent indicators that include or exclude covariates. The proposed model is motivated by an application to predicting mortality in an intensive care unit in Lisboa, Portugal.     

Simultaneous estimation of Cronbach’s alpha coefficients

The simultaneous estimation of Cronbachs alpha coefficients from q populations under the compound symmetry assumption is considered. In a multi-sample scenario, it is suspected that all the Cronbachs alpha coefficients are identical. Consequently, the inclusion of non-sample information (NSI) on the homogeneity of Cronbachs alpha coefficients in the estimation process may improve precision. We propose improved estimators based on the linear shrinkage, preliminary test, and the Steins type shrinkage strategies, to incorporate available NSI into the estimation. Their asymptotic properties are derived and discussed using the concepts of bias and risk. Extensive Monte-Carlo simulations were conducted to investigate the performance of the estimators.     

Variable selection for semiparametric proportional hazards model under progressive Type-II censoring

Variable selection is an effective methodology for dealing with models with numerous covariates. We consider the methods of variable selection for semiparametric Cox proportional hazards model under the progressive Type-II censoring scheme. The Cox proportional hazards model is used to model the influence coefficients of the environmental covariates. By applying Breslow’s “least information” idea, we obtain a profile likelihood function to estimate the coefficients. Lasso-type penalized profile likelihood estimation as well as stepwise variable selection method are explored as means to find the important covariates. Numerical simulations are conducted and Veteran’s Administration Lung Cancer data are exploited to evaluate the performance of the proposed method.     

Efficient estimation for time series following generalized linear models

In this paper, we consider James–Stein shrinkage and pretest estimation methods for time series following generalized linear models when it is conjectured that some of the regression parameters may be restricted to a subspace. Efficient estimation strategies are developed when there are many covariates in the model and some of them are not statistically significant. Statistical properties of the pretest and shrinkage estimation methods including asymptotic distributional bias and risk are developed. We investigate the relative performances of shrinkage and pretest estimators with respect to the unrestricted maximum partial likelihood estimator (MPLE). We show that the shrinkage estimators have a lower relative mean squared error as compared to the unrestricted MPLE when the number of significant covariates exceeds two. Monte Carlo simulation experiments were conducted for different combinations of inactive covariates and the performance of each estimator was evaluated in terms of its mean squared error. The practical benefits of the proposed methods are illustrated using two real data sets.     

Hierarchical priors for Bayesian CART shrinkage

The Bayesian CART (classification and regression tree) approach proposed by Chipman, George and McCulloch (1998) entails putting a prior distribution on the set of all CART models and then using stochastic search to select a model. The main thrust of this paper is to propose a new class of hierarchical priors which enhance the potential of this Bayesian approach. These priors indicate a preference for smooth local mean structure, resulting in tree models which shrink predictions from adjacent terminal node towards each other. Past methods for tree shrinkage have searched for trees without shrinking, and applied shrinkage to the identified tree only after the search. By using hierarchical priors in the stochastic search, the proposed method searches for shrunk trees that fit well and improves the tree through shrinkage of predictions.     

Non-penalty shrinkage estimation of random effect models for longitudinal data with AR(1) errors

In this paper, we consider the non-penalty shrinkage estimation method of random effect models with autoregressive errors for longitudinal data when there are many covariates and some of them may not be active for the response variable. In observational studies, subjects are followed over equally or unequally spaced visits to determine the continuous response and whether the response is associated with the risk factors/covariates. Measurements from the same subject are usually more similar to each other and thus are correlated with each other but not with observations of other subjects. To analyse this data, we consider a linear model that contains both random effects across subjects and within-subject errors that follows autoregressive structure of order 1 (AR(1)). Considering the subject-specific random effect as a nuisance parameter, we use two competing models, one includes all the covariates and the other restricts the coefficients based on the auxiliary information. We consider the non-penalty shrinkage estimation strategy that shrinks the unrestricted estimator in the direction of the restricted estimator. We discuss the asymptotic properties of the shrinkage estimators using the notion of asymptotic biases and risks. A Monte Carlo simulation study is conducted to examine the relative performance of the shrinkage estimators with the unrestricted estimator when the shrinkage dimension exceeds two. We also numerically compare the performance of the shrinkage estimators to that of the LASSO estimator. A longitudinal CD4 cell count data set will be used to illustrate the usefulness of shrinkage and LASSO estimators.     

An efficient computational approach for prior sensitivity analysis and cross‐validation

Prior sensitivity analysis and cross‐validation are important tools in Bayesian statistics. However, due to the computational expense of implementing existing methods, these techniques are rarely used. In this paper, the authors show how it is possible to use sequential Monte Carlo methods to create an efficient and automated algorithm to perform these tasks. They apply the algorithm to the computation of regularization path plots and to assess the sensitivity of the tuning parameter in g‐prior model selection. They then demonstrate the algorithm in a cross‐validation context and use it to select the shrinkage parameter in Bayesian regression. The Canadian Journal of Statistics 38:47–64; 2010 © 2010 Statistical Society of Canada     

Asymptotic Normality of M-Estimators for Varying Coefficient Models with Longitudinal Data

This article considers a nonparametric varying coefficient regression model with longitudinal observations. The relationship between the dependent variable and the covariates is assumed to be linear at a specific time point, but the coefficients are allowed to change over time. A general formulation is used to treat mean regression, median regression, quantile regression, and robust mean regression in one setting. The local M-estimators of the unknown coefficient functions are obtained by local linear method. The asymptotic distributions of M-estimators of unknown coefficient functions at both interior and boundary points are established. Various applications of the main results, including estimating conditional quantile coefficient functions and robustifying the mean regression coefficient functions are derived. Finite sample properties of our procedures are studied through Monte Carlo simulations.     

Bayesian regularisation in structured additive regression: a unifying perspective on shrinkage, smoothing and predictor selection

This paper surveys various shrinkage, smoothing and selection priors from a unifying perspective and shows how to combine them for Bayesian regularisation in the general class of structured additive regression models. As a common feature, all regularisation priors are conditionally Gaussian, given further parameters regularising model complexity. Hyperpriors for these parameters encourage shrinkage, smoothness or selection. It is shown that these regularisation (log-) priors can be interpreted as Bayesian analogues of several well-known frequentist penalty terms. Inference can be carried out with unified and computationally efficient MCMC schemes, estimating regularised regression coefficients and basis function coefficients simultaneously with complexity parameters and measuring uncertainty via corresponding marginal posteriors. For variable and function selection we discuss several variants of spike and slab priors which can also be cast into the framework of conditionally Gaussian priors. The performance of the Bayesian regularisation approaches is demonstrated in a hazard regression model and a high-dimensional geoadditive regression model.     

Model selection in finite mixture of regression models: a Bayesian approach with innovative weighted g priors and reversible jump Markov chain Monte Carlo implementation

Finite mixture of regression (FMR) models are aimed at characterizing subpopulation heterogeneity stemming from different sets of covariates that impact different groups in a population. We address the contemporary problem of simultaneously conducting covariate selection and determining the number of mixture components from a Bayesian perspective that can incorporate prior information. We propose a Gibbs sampling algorithm with reversible jump Markov chain Monte Carlo implementation to accomplish concurrent covariate selection and mixture component determination in FMR models. Our Bayesian approach contains innovative features compared to previously developed reversible jump algorithms. In addition, we introduce component-adaptive weighted g priors for regression coefficients, and illustrate their improved performance in covariate selection. Numerical studies show that the Gibbs sampler with reversible jump implementation performs well, and that the proposed weighted priors can be superior to non-adaptive unweighted priors.     

Multivariate Bayes Wavelet shrinkage and applications

In recent years, wavelet shrinkage has become a very appealing method for data de-noising and density function estimation. In particular, Bayesian modelling via hierarchical priors has introduced novel approaches for Wavelet analysis that had become very popular, and are very competitive with standard hard or soft thresholding rules. In this sense, this paper proposes a hierarchical prior that is elicited on the model parameters describing the wavelet coefficients after applying a Discrete Wavelet Transformation (DWT). In difference to other approaches, the prior proposes a multivariate Normal distribution with a covariance matrix that allows for correlations among Wavelet coefficients corresponding to the same level of detail. In addition, an extra scale parameter is incorporated that permits an additional shrinkage level over the coefficients. The posterior distribution for this shrinkage procedure is not available in closed form but it is easily sampled through Markov chain Monte Carlo (MCMC) methods. Applications on a set of test signals and two noisy signals are presented.