Variable selection is fundamental to high-dimensional statistical modeling. Many variable selection techniques may be implemented by maximum penalized likelihood using various penalty functions. Optimizing the penalized likelihood function is often challenging because it may be nondifferentiable and/or nonconcave. This article proposes a new class of algorithms for finding a maximizer of the penalized likelihood for a broad class of penalty functions. These algorithms operate by perturbing the penalty function slightly to render it differentiable, then optimizing this differentiable function using a minorize-maximize (MM) algorithm. MM algorithms are useful extensions of the well-known class of EM algorithms, a fact that allows us to analyze the local and global convergence of the proposed algorithm using some of the techniques employed for EM algorithms. In particular, we prove that when our MM algorithms converge, they must converge to a desirable point; we also discuss conditions under which this convergence may be guaranteed. We exploit the Newton-Raphson-like aspect of these algorithms to propose a sandwich estimator for the standard errors of the estimators. Our method performs well in numerical tests.
Stability Selection was recently introduced by Meinshausen and Buhlmann (2010) as a very general technique designed to improve the performance of a variable selection algorithm. It is based on aggregating the results of applying a selection procedure to subsamples of the data. We introduce a variant, called Complementary Pairs Stability Selection (CPSS), and derive bounds both on the expected number of variables included by CPSS that have low selection probability under the original procedure, and on the expected number of high selection probability variables that are excluded. These results r…
We propose a method for variable selection in multiple regression with random predictors. This method is based on a criterion that permits to reduce the variable selection problem to a problem of estimating suitable permutation and dimensionality. Then, estimators for these parameters are proposed and the resulting method for selecting variables is shown to be consistent. A simulation study that permits to gain understanding of the performances of the proposed approach and to compare it with an existing method is given.
The paper revisits the Bayesian group lasso and uses spike and slab priors for group variable selection. In the process, the connection of our model with penalized regression is demonstrated, and the role of posterior median for thresholding is pointed out. We show that the posterior median estimator has the oracle property for group variable selection and estimation under orthogonal designs, while the group lasso has suboptimal asymptotic estimation rate when variable selection consistency is achieved. Next we consider bi-level selection problem and propose the Bayesian sparse group selection…
In the context of a linear model with a sparse coefficient vector, exponential weights methods have been shown to be achieve oracle inequalities for prediction. We show that such methods also succeed at variable selection and estimation under the necessary identifiability condition on the design matrix, instead of much stronger assumptions required by other methods such as the Lasso or the Dantzig Selector. The same analysis yields consistency results for Bayesian methods and BIC-type variable selection under similar conditions.
We consider varying coefficient Cox models with high-dimensional covariates. We apply the group Lasso method to these models and propose a variable selection procedure. Our procedure copes with variable selection and structure identification from a high dimensional varying coefficient model to a semivarying coefficient model simultaneously. We derive an oracle inequality and closely examine restrictive eigenvalue conditions, too. In this paper, we give the details for Cox models with time-varying coefficients. The theoretical results on variable selection can be easily extended to some other i…
We consider a finite mixture of Gaussian regression model for high- dimensional data, where the number of covariates may be much larger than the sample size. We propose to estimate the unknown conditional mixture density by a maximum likelihood estimator, restricted on relevant variables selected by an 1-penalized maximum likelihood estimator. We get an oracle inequality satisfied by this estimator with a Jensen-Kullback-Leibler type loss. Our oracle inequality is deduced from a general model selection theorem for maximum likelihood estimators with a random model collection. We can derive the …
In this article we propose a new variable selection method for analyzing data collected from longitudinal sample surveys. The procedure is based on the survey-weighted quadratic inference function, which was recently introduced as an alternative to the survey-weighted generalized estimating function. Under the joint model-design framework, we introduce the penalized survey-weighted quadratic inference estimator and obtain sufficient conditions for the existence, weak consistency, sparsity and asymptotic normality. To illustrate the finite sample performance of the model selection procedure, we…
We consider the problem of variable selection in high-dimensional sparse additive models. We focus on the case that the components belong to nonparametric classes of functions. The proposed method is motivated by geometric considerations in Hilbert spaces and consists of comparing the norms of the projections of the data onto various additive subspaces. Under minimal geometric assumptions, we prove concentration inequalities which lead to new conditions under which consistent variable selection is possible. As an application, we establish conditions under which a single component can be estima…
The dual problem of testing the predictive significance of a particular covariate, and identification of the set of relevant covariates is common in applied research and methodological investigations. To study this problem in the context of functional linear regression models with predictor variables observed over a grid and a scalar response, we consider basis expansions of the functional covariates and apply the likelihood ratio test. Based on p-values from testing each predictor, we propose a new variable selection method, which is consistent in selecting the relevant predictors from set of…
The pattern of zero entries in the inverse covariance matrix of a multivariate normal distribution corresponds to conditional independence restrictions between variables. Covariance selection aims at estimating those structural zeros from data. We show that neighborhood selection with the Lasso is a computationally attractive alternative to standard covariance selection for sparse high-dimensional graphs. Neighborhood selection estimates the conditional independence restrictions separately for each node in the graph and is hence equivalent to variable selection for Gaussian linear models. We s…
Contamination can severely distort an estimator unless the estimation procedure is suitably robust. This is a well-known issue and has been addressed in Robust Statistics, however, the relation of contamination and distorted variable selection has been rarely considered in literature. As for variable selection, many methods for sparse model selection have been proposed, including the Stability Selection which is a meta-algorithm based on some variable selection algorithm in order to immunize against particular data configurations. We introduce the variable selection breakdown point that quanti…
Lasso is a celebrated method for variable selection in linear models, but it faces challenges when the variables are moderately or strongly correlated. This motivates alternative approaches such as using a non-convex penalty, adding a ridge regularization, or conducting a post-Lasso thresholding. In this paper, we compare Lasso with 5 other methods: Elastic net, SCAD, forward selection, thresholded Lasso, and forward backward selection. We measure their performances theoretically by the expected Hamming error, assuming that the regression coefficients are iid drawn from a two-point mixture and…
We propose a method for variable selection in discriminant analysis with mixed categorical and continuous variables. This method is based on a criterion that permits to reduce the variable selection problem to a problem of estimating suitable permutation and dimensionality. Then, estimators for these parameters are proposed and the resulting method for selecting variables is shown to be consistent. A simulation study that permits to study several poperties of the proposed approach and to compare it with an existing method is given.
This paper deals with variable selection in the regression and binary classification frameworks. It proposes an automatic and exhaustive procedure which relies on the use of the CART algorithm and on model selection via penalization. This work, of theoretical nature, aims at determining adequate penalties, i.e. penalties which allow to get oracle type inequalities justifying the performance of the proposed procedure. Since the exhaustive procedure can not be executed when the number of variables is too big, a more practical procedure is also proposed and still theoretically validated. A simula…
We propose a method for incorporating variable selection into local polynomial regression. This can improve the accuracy of the regression by extending the bandwidth in directions corresponding to those variables judged to be are unimportant. It also increases our understanding of the dataset by highlighting areas where these variables are redundant. The approach has the potential to effect complete variable removal as well as perform partial removal when a variable redundancy applies only to particular regions of the data. We define a nonparametric oracle property and show that this is more t…
INRIA（法国国家信息与自动化研究所）；Laboratoire d’Informatique de l’Ecole Normale Supérieure（巴黎高等师范学院计算机实验室）；CNRS（法国国家科学研究中心）；ENS（巴黎高等师范学院）
We consider the problem of high-dimensional non-linear variable selection for supervised learning. Our approach is based on performing linear selection among exponentially many appropriately defined positive definite kernels that characterize non-linear interactions between the original variables. To select efficiently from these many kernels, we use the natural hierarchical structure of the problem to extend the multiple kernel learning framework to kernels that can be embedded in a directed acyclic graph; we show that it is then possible to perform kernel selection through a graph-adapted sp…
Gaussian graphical models (GGMs) are well-established tools for probabilistic exploration of dependence structures using precision matrices. We develop a Bayesian method to incorporate covariate information in this GGMs setup in a nonlinear seemingly unrelated regression framework. We propose a joint predictor and graph selection model and develop an efficient collapsed Gibbs sampler algorithm to search the joint model space. Furthermore, we investigate its theoretical variable selection properties. We demonstrate our method on a variety of simulated data, concluding with a real data set from …
The interest in variable selection for clustering has increased recently due to the growing need in clustering high-dimensional data. Variable selection allows in particular to ease both the clustering and the interpretation of the results. Existing approaches have demonstrated the efficiency of variable selection for clustering but turn out to be either very time consuming or not sparse enough in high-dimensional spaces. This work proposes to perform a selection of the discriminative variables by introducing sparsity in the loading matrix of the Fisher-EM algorithm. This clustering method has…
Statistical inference is considered for variables of interest, called primary variables, when auxiliary variables are observed along with the primary variables. We consider the setting of incomplete data analysis, where some primary variables are not observed. Utilizing a parametric model of joint distribution of primary and auxiliary variables, it is possible to improve the estimation of parametric model for the primary variables when the auxiliary variables are closely related to the primary variables. However, the estimation accuracy reduces when the auxiliary variables are irrelevant to th…
We compare two major approaches to variable selection in clustering: model selection and regularization. Based on previous results, we select the method of Maugis et al. (2009b), which modified the method of Raftery and Dean (2006), as a current state of the art model selection method. We select the method of Witten and Tibshirani (2010) as a current state of the art regularization method. We compared the methods by simulation in terms of their accuracy in both classification and variable selection. In the first simulation experiment all the variables were conditionally independent given clust…
This document is additional material to our previous study comparing several strategies for variable subset selection. Our recommended approach was to fit the full model with all the candidate variables and best possible prior information, and perform the variable selection using the projection predictive framework. Here we give an example of performing such an analysis, using Stan for fitting the model, and R for the variable selection.
An econometric or statistical model may undergo a marginal gain if we admit a new variable to the model, and a marginal loss if we remove an existing variable from the model. Assuming equality of opportunity among all candidate variables, we derive a valuation framework by the expected marginal gain and marginal loss in all potential modeling scenarios. However, marginal gain and loss are not symmetric; thus, we introduce three unbiased solutions. When used in variable selection, our new approaches significantly outperform several popular methods used in practice. The results also explore some…
Variable selection in high-dimensional scenarios is of great interested in statistics. One application involves identifying differentially expressed genes in genomic analysis. Existing methods for addressing this problem have some limits or disadvantages. In this paper, we propose distance based variable importance measures to deal with these problems, which is inspired by the Multi-Response Permutation Procedure (MRPP). The proposed variable importance assessments can effectively measure the importance of an individual dimension by quantifying its influence on the differences between multivar…
Relevant methods of variable selection have been proposed in model-based clustering and classification. These methods are making use of backward or forward procedures to define the roles of the variables. Unfortunately, these stepwise procedures are terribly slow and make these variable selection algorithms inefficient to treat large data sets. In this paper, an alternative regularization approach of variable selection is proposed for model-based clustering and classification. In this approach, the variables are first ranked with a lasso-like procedure in order to avoid painfully slow stepwise…
Variable selection, or more generally, model reduction is an important aspect of the statistical workflow aiming to provide insights from data. In this paper, we discuss and demonstrate the benefits of using a reference model in variable selection. A reference model acts as a noise-filter on the target variable by modeling its data generating mechanism. As a result, using the reference model predictions in the model selection procedure reduces the variability and improves stability leading to improved model selection performance. Assuming that a Bayesian reference model describes the true dist…
Variable selection is an important statistical problem. This problem becomes more challenging when the candidate predictors are of mixed type (e.g. continuous and binary) and impact the response variable in nonlinear and/or non-additive ways. In this paper, we review existing variable selection approaches for the Bayesian additive regression trees (BART) model, a nonparametric regression model, which is flexible enough to capture the interactions between predictors and nonlinear relationships with the response. An emphasis of this review is on the capability of identifying relevant predictors.…
We propose a new variable selection algorithm, subsample-ordered least-angle regression (solar), and its coordinate descent generalization, solar-cd. Solar re-constructs lasso paths using the $L_0$ norm and averages the resulting solution paths across subsamples. Path averaging retains the ranking information of the informative variables while averaging out sensitivity to high dimensionality, improving variable selection stability, efficiency, and accuracy. We prove that: (i) with a high probability, path averaging perfectly separates informative variables from redundant variables on the avera…
Sparse linear prediction methods suffer from decreased prediction accuracy when the predictor variables have cluster structure (e.g. there are highly correlated groups of variables). To improve prediction accuracy, various methods have been proposed to identify variable clusters from the data and integrate cluster information into a sparse modeling process. But none of these methods achieve satisfactory performance for prediction, variable selection and variable clustering simultaneously. This paper presents Variable Cluster Principal Component Regression (VC-PCR), a prediction method that sup…
In this article, we advocate the ensemble approach for variable selection. We point out that the stochastic mechanism used to generate the variable-selection ensemble (VSE) must be picked with care. We construct a VSE using a stochastic stepwise algorithm, and compare its performance with numerous state-of-the-art algorithms.
Model-based component-wise gradient boosting is a popular tool for data-driven variable selection. In order to improve its prediction and selection qualities even further, several modifications of the original algorithm have been developed, that mainly focus on different stopping criteria, leaving the actual variable selection mechanism untouched. We investigate different prediction-based mechanisms for the variable selection step in model-based component-wise gradient boosting. These approaches include Akaikes Information Criterion (AIC) as well as a selection rule relying on the component-wi…
Sparsity-inducing penalties are useful tools for variable selection and they are also effective for regression settings where the data are functions. We consider the problem of selecting not only variables but also decision boundaries in logistic regression models for functional data, using the sparse regularization. The functional logistic regression model is estimated by the framework of the penalized likelihood method with the sparse group lasso-type penalty, and then tuning parameters are selected using the model selection criterion. The effectiveness of the proposed method is investigated…
Parameter estimation and the variable selection are two pioneer issues in regression analysis. While traditional variable selection methods require prior estimation of the model parameters, the penalized methods simultaneously carry on parameter estimation and variable select. Therefore, penalized variable selection methods are of great interest and have been extensively studied in literature. However, most of the papers in literature are only limited to the regression models with uncorrelated error terms and normality assumption. In this study, we combine the parameter estimation and the vari…
We consider the problem of variable selection in linear models when $p$, the number of potential regressors, may exceed (and perhaps substantially) the sample size $n$ (which is possibly small).
In this paper, we propose two variable selection methods for adjusting the censoring information for survival times, such as the restricted mean survival time. To adjust for the influence of censoring, we consider an inverse survival probability weighting (ISPW) for subjects with events. We derive a least absolute shrinkage and selection operator (lasso)-type variable selection method, which considers an inverse weighting for of the squared losses, and an information criterion-type variable selection method, which applies an inverse weighting of the survival probability to the power of each de…
In high-dimensions, many variable selection methods, such as the lasso, are often limited by excessive variability and rank deficiency of the sample covariance matrix. Covariance sparsity is a natural phenomenon in high-dimensional applications, such as microarray analysis, image processing, etc., in which a large number of predictors are independent or weakly correlated. In this paper, we propose the covariance-thresholded lasso, a new class of regression methods that can utilize covariance sparsity to improve variable selection. We establish theoretical results, under the random design setti…
This paper computationally demonstrates a sharp improvement in predictive performance for $k$ nearest neighbors thanks to an efficient forward selection of the predictor variables. We show both simulated and real-world data that this novel repeatedly approaches outperformance regression models under stepwise selection
We propose discrete determinantal point processes (DPPs) for priors on the model parameter in Bayesian variable selection. By our variable selection method, collinear predictors are less likely to be selected simultaneously because of the repulsion property of discrete DPPs. Three types of DPP priors are proposed. We show the efficiency of the proposed priors through numerical experiments and applications to collinear datasets.
Hitachi (China) Research & Development Corporation（日立（中国）研究开发有限公司）
Variable selection is of significant importance for classification and regression tasks in machine learning and statistical applications where both predictability and explainability are needed. In this paper, a Copula Entropy (CE) based method for variable selection which use CE based ranks to select variables is proposed. The method is both model-free and tuning-free. Comparison experiments between the proposed method and traditional variable selection methods, such as Distance Correlation, Hilbert-Schmidt Independence Criterion, Stepwise Selection, regularized generalized linear models and A…
This study develops a data-driven group variable selection method for data envelopment analysis (DEA), a non-parametric linear programming approach to the estimation of production frontiers. The proposed method extends the group Lasso (least absolute shrinkage and selection operator) designed for variable selection on (often predefined) groups of variables in linear regression models to DEA models. In particular, a special constrained version of the group Lasso with the loss function suited for variable selection in DEA models is derived and solved by a new tailored algorithm based on the alte…
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 m…
Variable selection is an important problem in statistics and machine learning. Copula Entropy (CE) is a mathematical concept for measuring statistical independence and has been applied to variable selection recently. In this paper we propose to apply the CE-based method for variable selection to survival analysis. The idea is to measure the correlation between variables and time-to-event with CE and then select variables according to their CE value. Experiments on simulated data and two real cancer data were conducted to compare the proposed method with two related methods: random survival for…
We present a new variable selection method based on model-based gradient boosting and randomly permuted variables. Model-based boosting is a tool to fit a statistical model while performing variable selection at the same time. A drawback of the fitting lies in the need of multiple model fits on slightly altered data (e.g. cross-validation or bootstrap) to find the optimal number of boosting iterations and prevent overfitting. In our proposed approach, we augment the data set with randomly permuted versions of the true variables, so called shadow variables, and stop the step-wise fitting as soo…
The knockoff filter, recently developed by Barber and Candes, is an effective procedure to perform variable selection with a controlled false discovery rate (FDR). We propose a private version of the knockoff filter by incorporating Gaussian and Laplace mechanisms, and show that variable selection with controlled FDR can be achieved. Simulations demonstrate that our setting has reasonable statistical power.
Penalized regression models are popularly used in high-dimensional data analysis to conduct variable selection and model fitting simultaneously. Whereas success has been widely reported in literature, their performances largely depend on the tuning parameters that balance the trade-off between model fitting and model sparsity. Existing tuning criteria mainly follow the route of minimizing the estimated prediction error or maximizing the posterior model probability, such as cross-validation, AIC and BIC. This article introduces a general tuning parameter selection criterion based on a novel con…
Variable selection plays a fundamental role in high-dimensional data analysis. Various methods have been developed for variable selection in recent years. Well-known examples are forward stepwise regression (FSR) and least angle regression (LARS), among others. These methods typically add variables into the model one by one. For such selection procedures, it is crucial to find a stopping criterion that controls model complexity. One of the most commonly used techniques to this end is cross-validation (CV) which, in spite of its popularity, has two major drawbacks: expensive computational cost …
Bayesian variable selection is a powerful tool for data analysis, as it offers a principled method for variable selection that accounts for prior information and uncertainty. However, wider adoption of Bayesian variable selection has been hampered by computational challenges, especially in difficult regimes with a large number of covariates or non-conjugate likelihoods. Generalized linear models for count data, which are prevalent in biology, ecology, economics, and beyond, represent an important special case. Here we introduce an efficient MCMC scheme for variable selection in binomial and ne…
This paper considers linear model selection when the response is vector-valued and the predictors are randomly observed. We propose a new approach that decouples statistical inference from the selection step in a "post-inference model summarization" strategy. We study the impact of predictor uncertainty on the model selection procedure. The method is demonstrated through an application to asset pricing.
This paper arises from collaborative research the aim of which was to model clinical assessments of upper limb function after stroke using 3D kinematic data. We present a new nonlinear mixed-effects scalar-on-function regression model with a Gaussian process prior focusing on variable selection from large number of candidates including both scalar and function variables. A novel variable selection algorithm has been developed, namely functional least angle regression (fLARS). As they are essential for this algorithm, we studied the representation of functional variables with different methods …
Supervised classifying of biological samples based on genetic information, (e.g. gene expression profiles) is an important problem in biostatistics. In order to find both accurate and interpretable classification rules variable selection is indispensable. This article explores how an assessment of the individual importance of variables (effect size estimation) can be used to perform variable selection. I review recent effect size estimation approaches in the context of linear discriminant analysis (LDA) and propose a new conceptually simple effect size estimation method which is at the same ti…
The regularization approach for variable selection was well developed for a completely observed data set in the past two decades. In the presence of missing values, this approach needs to be tailored to different missing data mechanisms. In this paper, we focus on a flexible and generally applicable missing data mechanism, which contains both ignorable and nonignorable missing data mechanism assumptions. We show how the regularization approach for variable selection can be adapted to the situation under this missing data mechanism. The computational and theoretical properties for variable sele…
