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 distribution of future data well, the theoretically preferred usage of the reference model is to project its predictive distribution to a reduced model leading to projection predictive variable selection approach. Alternatively, reference models may also be used in an ad-hoc manner in combination with common variable selection methods. In several numerical experiments, we investigate the performance of the projective prediction approach as well as alternative variable selection methods with and without reference models. Our results indicate that the use of reference models generally translates into better and more stable variable selection. Additionally, we demonstrate that the projection predictive approach shows superior performance as compared to alternative variable selection methods independently of whether or not they use reference models.
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…
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.
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.
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…
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…
This paper considers the problem of variable selection in regression models in the case of functional variables that may be mixed with other type of variables (scalar, multivariate, directional, etc.). Our proposal begins with a simple null model and sequentially selects a new variable to be incorporated into the model based on the use of distance correlation proposed by \cite{Szekely2007}. For the sake of simplicity, this paper only uses additive models. However, the proposed algorithm may assess the type of contribution (linear, non linear, ...) of each variable. The algorithm has shown quit…
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…
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).
Gaussian stochastic process (GaSP) has been widely used in two fundamental problems in uncertainty quantification, namely the emulation and calibration of mathematical models. Some objective priors, such as the reference prior, are studied in the context of emulating (approximating) computationally expensive mathematical models. In this work, we introduce a new class of priors, called the jointly robust prior, for both the emulation and calibration. This prior is designed to maintain various advantages from the reference prior. In emulation, the jointly robust prior has an appropriate tail dec…
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…
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.
Thompson sampling is a heuristic algorithm for the multi-armed bandit problem which has a long tradition in machine learning. The algorithm has a Bayesian spirit in the sense that it selects arms based on posterior samples of reward probabilities of each arm. By forging a connection between combinatorial binary bandits and spike-and-slab variable selection, we propose a stochastic optimization approach to subset selection called Thompson Variable Selection (TVS). TVS is a framework for interpretable machine learning which does not rely on the underlying model to be linear. TVS brings together …
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 …
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.
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…
Factors are categorical variables, and the values which these variables assume are called levels. In this paper, we consider the variable selection problem where the set of potential predictors contains both factors and numerical variables. Formally, this problem is a particular case of the standard variable selection problem where factors are coded using dummy variables. As such, the Bayesian solution would be straightforward and, possibly because of this, the problem, despite its importance, has not received much attention in the literature. Nevertheless, we show that this perception is illu…
When fitting statistical models, some predictors are often found to be correlated with each other, and functioning together. Many group variable selection methods are developed to select the groups of predictors that are closely related to the continuous or categorical response. These existing methods usually assume the group structures are well known. For example, variables with similar practical meaning, or dummy variables created by categorical data. However, in practice, it is impractical to know the exact group structure, especially when the variable dimensional is large. As a result, the…
Variable selection is of increasing importance to address the difficulties of high dimensionality in many scientific areas. In this paper, we demonstrate a property for distance covariance, which is incorporated in a novel feature screening procedure together with the use of distance correlation. The approach makes no distributional assumptions for the variables and does not require the specification of a regression model, and hence is especially attractive in variable selection given an enormous number of candidate attributes without much information about the true model with the response. Th…
We propose trace pursuit for model-free variable selection under the sufficient dimension reduction paradigm. Two distinct algorithms are proposed: stepwise trace pursuit and forward trace pursuit. Stepwise trace pursuit achieves selection consistency with fixed p, and is readily applicable in the challenging setting with p>n. Forward trace pursuit can serve as an initial screening step to speed up the computation in the case of ultrahigh dimensionality. The screening consistency property of forward trace pursuit based on sliced inverse regression is established. Finite sample performances of …
Doubly truncated data arise in many areas such as astronomy, econometrics, and medical studies. For the regression analysis with doubly truncated response variables, the existence of double truncation may bring bias for estimation as well as affect variable selection. We propose a simultaneous estimation and variable selection procedure for the doubly truncated regression, allowing a diverging number of regression parameters. To remove the bias introduced by the double truncation, a Mann-Whitney-type loss function is used. The adaptive LASSO penalty is then added into the loss function to achi…
The Zellner's g-prior and its recent hierarchical extensions are the most popular default prior choices in the Bayesian variable selection context. These prior set-ups can be expressed power-priors with fixed set of imaginary data. In this paper, we borrow ideas from the power-expected-posterior (PEP) priors in order to introduce, under the g-prior approach, an extra hierarchical level that accounts for the imaginary data uncertainty. For normal regression variable selection problems, the resulting power-conditional-expected-posterior (PCEP) prior is a conjugate normal-inverse gamma prior whic…
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.…
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…
As data sets continue to grow in size and complexity, effective and efficient techniques are needed to target important features in the variable space. Many of the variable selection techniques that are commonly used alongside clustering algorithms are based upon determining the best variable subspace according to model fitting in a stepwise manner. These techniques are often computationally intensive and can require extended periods of time to run; in fact, some are prohibitively computationally expensive for high-dimensional data. In this paper, a novel variable selection technique is introd…
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…
Among the most popular variable selection procedures in high-dimensional regression, Lasso provides a solution path to rank the variables and determines a cut-off position on the path to select variables and estimate coefficients. In this paper, we consider variable selection from a new perspective motivated by the frequently occurred phenomenon that relevant variables are not completely distinguishable from noise variables on the solution path. We propose to characterize the positions of the first noise variable and the last relevant variable on the path. We then develop a new variable select…
In recent years, Ising prior with the network information for the "in" or "out" binary random variable in Bayesian variable selections has received more and more attentions. In this paper, we discover that even without the informative prior a Bayesian variable selection problem itself can be considered as a complete graph and described by a Ising model with random interactions. There are many advantages of treating variable selection as a graphical model, such as it is easy to employ the single site updating as well as the cluster updating algorithm, suitable for problems with small sample siz…
In many engineering and scientific applications, prediction variables are grouped, for example, in biological applications where assayed genes or proteins can be grouped by biological roles or biological pathways. Common statistical analysis methods such as ANOVA, factor analysis, and functional modeling with basis sets also exhibit natural variable groupings. Existing successful group variable selection methods such as Antoniadis and Fan (2001), Yuan and Lin (2006) and Zhao, Rocha and Yu (2009) have the limitation of selecting variables in an "all-in-all-out" fashion, i.e., when one variable …
For many practical problems, the regression models follow the strong heredity property (also known as the marginality), which means they include parent main effects when a second-order effect is present. Existing methods rely mostly on special penalty functions or algorithms to enforce the strong heredity in variable selection. We propose a novel hierarchical standardization procedure to maintain strong heredity in variable selection. Our method is effortless to implement and is applicable to any variable selection method for any type of regression. The performance of the hierarchical standard…
We propose an iterative variable selection scheme for high-dimensional data with binary outcomes. The scheme adopts a structured screen-and-select framework and uses non-local prior-based Bayesian model selection within the same. The structured screening is based on the association of the independent variables with the outcome which is measured in terms of the maximum marginal likelihood estimator. Performance comparison with several well-known methods in terms of true positive rate and false discovery rate shows that our proposed method stands to be a competitive alternative for sparse high-d…
The method of Bayesian variable selection via penalized credible regions separates model fitting and variable selection. The idea is to search for the sparsest solution within the joint posterior credible regions. Although the approach was successful, it depended on the use of conjugate normal priors. More recently, improvements in the use of global-local shrinkage priors have been made for high-dimensional Bayesian variable selection. In this paper, we incorporate global-local priors into the credible region selection framework. The Dirichlet-Laplace (DL) prior is adapted to linear regression…
The goal of this article is to select important variables that can distinguish one class of data from another. A marginal variable selection method ranks the marginal effects for classification of individual variables, and is a useful and efficient approach for variable selection. Our focus here is to consider the bivariate effect, in addition to the marginal effect. In particular, we are interested in those pairs of variables that can lead to accurate classification predictions when they are viewed jointly. To accomplish this, we propose a permutation test called Significance test of Joint Ef…
We propose Stepwise cOnditional likelihood variable selection for Discriminant Analysis (SODA) to detect both main and quadratic interaction effects in logistic regression and quadratic discriminant analysis (QDA) models. In the forward stage, SODA adds in important predictors evaluated based on their overall contributions, whereas in the backward stage SODA removes unimportant terms so as to optimize the extended Bayesian Information Criterion (EBIC). Compared with existing methods on QDA variable selections, SODA can deal with high-dimensional data with the number of predictors much larger t…
Let X; Z be r and s-dimensional covariates, respectively, used to model the response variable Y as Y = m(X;Z) + σ(X;Z)ε. We develop an ANOVA-type test for the null hypothesis that Z has no influence on the regression function, based on residuals obtained from local polynomial fitting of the null model. Using p-values from this test, a group variable selection method based on multiple testing ideas is proposed. Simulations studies suggest that the proposed test procedure outperforms the generalized likelihood ratio test when the alternative is non-additive or there is heteroscedasticity. Additi…
Estimation of structure, such as in variable selection, graphical modelling or cluster analysis is notoriously difficult, especially for high-dimensional data. We introduce stability selection. It is based on subsampling in combination with (high-dimensional) selection algorithms. As such, the method is extremely general and has a very wide range of applicability. Stability selection provides finite sample control for some error rates of false discoveries and hence a transparent principle to choose a proper amount of regularisation for structure estimation. Variable selection and structure est…
In many scientific and engineering applications, covariates are naturally grouped. When the group structures are available among covariates, people are usually interested in identifying both important groups and important variables within the selected groups. Among existing successful group variable selection methods, some methods fail to conduct the within group selection. Some methods are able to conduct both group and within group selection, but the corresponding objective functions are non-convex. Such a non-convexity may require extra numerical effort. In this paper, we propose a novel Lo…
In many contexts, there is interest in selecting the most important variables from a very large collection, commonly referred to as support recovery or variable, feature or subset selection. There is an enormous literature proposing a rich variety of algorithms. In scientific applications, it is of crucial importance to quantify uncertainty in variable selection, providing measures of statistical significance for each variable. The overwhelming majority of algorithms fail to produce such measures. This has led to a focus in the scientific literature on independent screening methods, which exam…
High-dimensional sparse modeling with censored survival data is of great practical importance, and several methods have been proposed for variable selection based on different models. However, the impact of biased sample caused by left-truncation and covariates measurement error to variable selection is not fully explored. In this paper, we mainly focus on the additive hazards model and analyze the high-dimensional survival data subject to left-truncation and measurement error in covariates. We develop the three-stage procedure to correct the error effect, select variables, and estimate the pa…
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…
In high-dimensional data analysis, bi-level sparsity is often assumed when covariates function group-wisely and sparsity can appear either at the group level or within certain groups. In such cases, an ideal model should be able to encourage the bi-level variable selection consistently. Bi-level variable selection has become even more challenging when data have heavy-tailed distribution or outliers exist in random errors and covariates. In this paper, we study a framework of high-dimensional M-estimation for bi-level variable selection. This framework encourages bi-level sparsity through a com…
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…
A mathematical model for variable selection in functional regression models with scalar response is proposed. By "variable selection" we mean a procedure to replace the whole trajectories of the functional explanatory variables with their values at a finite number of carefully selected instants (or "impact points"). The basic idea of our approach is to use the Reproducing Kernel Hilbert Space (RKHS) associated with the underlying process, instead of the more usual L2[0,1] space, in the definition of the linear model. This turns out to be especially suitable for variable selection purposes, sin…
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…
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…
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…
