Stochastic correlation coefficient ensembles for variable selection

In this paper, we propose a novel Max-Relevance and Min-Common-Redundancy criterion for variable selection in linear models. Considering that the ensemble approach for variable selection has been proven to be quite effective in linear regression models, we construct a variable selection ensemble (VSE) by combining the presented stochastic correlation coefficient algorithm with a stochastic stepwise algorithm. We conduct extensive experimental comparison of our algorithm and other methods using two simulation studies and four real-life data sets. The results confirm that the proposed VSE leads to promising improvement on variable selection and regression accuracy.     

Bayesian Model Averaging in Proportional Hazard Models: Assessing the Risk of a Stroke

In the context of the Cardiovascular Health Study, a comprehensive investigation into the risk factors for strokes, we apply Bayesian model averaging to the selection of variables in Cox proportional hazard models. We use an extension of the leaps-and-bounds algorithm for locating the models that are to be averaged over and make available S-PLUS software to implement the methods. Bayesian model averaging provides a posterior probability that each variable belongs in the model, a more directly interpretable measure of variable importance than a P -value. P -values from models preferred by stepwise methods tend to overstate the evidence for the predictive value of a variable and do not account for model uncertainty. We introduce the partial predictive score to evaluate predictive performance. For the Cardiovascular Health Study, Bayesian model averaging predictively outperforms standard model selection and does a better job of assessing who is at high risk for a stroke.     

Functional variable selection via Gram–Schmidt orthogonalization for multiple functional linear regression

ABSTRACT

Functional linear model is of great practical importance, as exemplified by applications in high-throughput studies such as meteorological and biomedical research. In this paper, we propose a new functional variable selection procedure, called functional variable selection via Gram–Schmidt (FGS) orthogonalization, for a functional linear model with a scalar response and multiple functional predictors. Instead of the regularization methods, FGS takes into account the similarity between the functional predictors in a data-driven way and utilizes the technique of Gram–Schmidt orthogonalization to remove the irrelevant predictors. FGS can successfully discriminate between the relevant and the irrelevant functional predictors to achieve a high true positive ratio without including many irrelevant predictors, and yield explainable models, which offers a new perspective for the variable selection method in the functional linear model. Simulation studies are carried out to evaluate the finite sample performance of the proposed method, and also a weather data set is analysed.     

Estimation and variable selection for generalised partially linear single-index models

In this paper, we study the problem of estimation and variable selection for generalised partially linear single-index models based on quasi-likelihood, extending existing studies on variable selection for partially linear single-index models to binary and count responses. To take into account the unit norm constraint of the index parameter, we use the ‘delete-one-component’ approach. The asymptotic normality of the estimates is demonstrated. Furthermore, the smoothly clipped absolute deviation penalty is added for variable selection of parameters both in the nonparametric part and the parametric part, and the oracle property of the variable selection procedure is shown. Finally, some simulation studies are carried out to illustrate the finite sample performance.     

A new model selection procedure for finite mixture regression models

Abstract

In this article, we propose a new penalized-likelihood method to conduct model selection for finite mixture of regression models. The penalties are imposed on mixing proportions and regression coefficients, and hence order selection of the mixture and the variable selection in each component can be simultaneously conducted. The consistency of order selection and the consistency of variable selection are investigated. A modified EM algorithm is proposed to maximize the penalized log-likelihood function. Numerical simulations are conducted to demonstrate the finite sample performance of the estimation procedure. The proposed methodology is further illustrated via real data analysis.     

RandGA: injecting randomness into parallel genetic algorithm for variable selection

Recently, the ensemble learning approaches have been proven to be quite effective for variable selection in linear regression models. In general, a good variable selection ensemble should consist of a diverse collection of strong members. Based on the parallel genetic algorithm (PGA) proposed in [41 M. Zhu and H.A. Chipman, Darwinian evolution in parallel universes: A parallel genetic algorithm for variable selection, Technometrics 48(4) (2006), pp. 491–502. doi: 10.1198/004017006000000093[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]], in this paper, we propose a novel method RandGA through injecting randomness into PGA with the aim to increase the diversity among ensemble members. Using a number of simulated data sets, we show that the newly proposed method RandGA compares favorably with other variable selection techniques. As a real example, the new method is applied to the diabetes data.     

Reversible jump Markov chain Monte Carlo algorithms for Bayesian variable selection in logistic mixed models

In this article, to reduce computational load in performing Bayesian variable selection, we used a variant of reversible jump Markov chain Monte Carlo methods, and the Holmes and Held (HH) algorithm, to sample model index variables in logistic mixed models involving a large number of explanatory variables. Furthermore, we proposed a simple proposal distribution for model index variables, and used a simulation study and real example to compare the performance of the HH algorithm with our proposed and existing proposal distributions. The results show that the HH algorithm with our proposed proposal distribution is a computationally efficient and reliable selection method.     

Fast and approximate exhaustive variable selection for generalised linear models with APES

We present APproximated Exhaustive Search (APES), which enables fast and approximated exhaustive variable selection in Generalised Linear Models (GLMs). While exhaustive variable selection remains as the gold standard in many model selection contexts, traditional exhaustive variable selection suffers from computational feasibility issues. More precisely, there is often a high cost associated with computing maximum likelihood estimates (MLE) for all subsets of GLMs. Efficient algorithms for exhaustive searches exist for linear models, most notably the leaps‐and‐bound algorithm and, more recently, the mixed integer optimisation (MIO) algorithm. The APES method learns from observational weights in a generalised linear regression super‐model and reformulates the GLM problem as a linear regression problem. In this way, APES can approximate a true exhaustive search in the original GLM space. Where exhaustive variable selection is not computationally feasible, we propose a best‐subset search, which also closely approximates a true exhaustive search. APES is made available in both as a standalone R package as well as part of the already existing mplot package.     

Bayesian variable selection with related predictors

In data sets with many predictors, algorithms for identifying a good subset of predictors are often used. Most such algorithms do not allow for any relationships between predictors. For example, stepwise regression might select a model containing an interaction AB but neither main effect A or B. This paper develops mathematical representations of this and other relations between predictors, which may then be incorporated in a model selection procedure. A Bayesian approach that goes beyond the standard independence prior for variable selection is adopted, and preference for certain models is interpreted as prior information. Priors relevant to arbitrary interactions and polynomials, dummy variables for categorical factors, competing predictors, and restrictions on the size of the models are developed. Since the relations developed are for priors, they may be incorporated in any Bayesian variable selection algorithm for any type of linear model. The application of the methods here is illustrated via the stochastic search variable selection algorithm of George and McCulloch (1993), which is modified to utilize the new priors. The performance of the approach is illustrated with two constructed examples and a computer performance dataset.     

Quantile regression for robust estimation and variable selection in partially linear varying-coefficient models

In this paper, we develop a new estimation procedure based on quantile regression for semiparametric partially linear varying-coefficient models. The proposed estimation approach is empirically shown to be much more efficient than the popular least squares estimation method for non-normal error distributions, and almost not lose any efficiency for normal errors. Asymptotic normalities of the proposed estimators for both the parametric and nonparametric parts are established. To achieve sparsity when there exist irrelevant variables in the model, two variable selection procedures based on adaptive penalty are developed to select important parametric covariates as well as significant nonparametric functions. Moreover, both these two variable selection procedures are demonstrated to enjoy the oracle property under some regularity conditions. Some Monte Carlo simulations are conducted to assess the finite sample performance of the proposed estimators, and a real-data example is used to illustrate the application of the proposed methods.     

Analysis and correction of bias in Total Decrease in Node Impurity measures for tree-based algorithms

Variable selection is one of the main problems faced by data mining and machine learning techniques. These techniques are often, more or less explicitly, based on some measure of variable importance. This paper considers Total Decrease in Node Impurity (TDNI) measures, a popular class of variable importance measures defined in the field of decision trees and tree-based ensemble methods, like Random Forests and Gradient Boosting Machines. In spite of their wide use, some measures of this class are known to be biased and some correction strategies have been proposed. The aim of this paper is twofold. Firstly, to investigate the source and the characteristics of bias in TDNI measures using the notions of informative and uninformative splits. Secondly, a bias-correction algorithm, recently proposed for the Gini measure in the context of classification, is extended to the entire class of TDNI measures and its performance is investigated in the regression framework using simulated and real data.     

Variable selection for varying dispersion beta regression model

The beta regression models are commonly used by practitioners to model variables that assume values in the standard unit interval (0, 1). In this paper, we consider the issue of variable selection for beta regression models with varying dispersion (VBRM), in which both the mean and the dispersion depend upon predictor variables. Based on a penalized likelihood method, the consistency and the oracle property of the penalized estimators are established. Following the coordinate descent algorithm idea of generalized linear models, we develop new variable selection procedure for the VBRM, which can efficiently simultaneously estimate and select important variables in both mean model and dispersion model. Simulation studies and body fat data analysis are presented to illustrate the proposed methods.     

A computational framework for variable selection in multivariate regression

Stepwise variable selection procedures are computationally inexpensive methods for constructing useful regression models for a single dependent variable. At each step a variable is entered into or deleted from the current model, based on the criterion of minimizing the error sum of squares (SSE). When there is more than one dependent variable, the situation is more complex. In this article we propose variable selection criteria for multivariate regression which generalize the univariate SSE criterion. Specifically, we suggest minimizing some function of the estimated error covariance matrix: the trace, the determinant, or the largest eigenvalue. The computations associated with these criteria may be burdensome. We develop a computational framework based on the use of the SWEEP operator which greatly reduces these calculations for stepwise variable selection in multivariate regression.     

Sparse optimization for nonconvex group penalized estimation

We consider a linear regression model where there are group structures in covariates. The group LASSO has been proposed for group variable selections. Many nonconvex penalties such as smoothly clipped absolute deviation and minimax concave penalty were extended to group variable selection problems. The group coordinate descent (GCD) algorithm is used popularly for fitting these models. However, the GCD algorithms are hard to be applied to nonconvex group penalties due to computational complexity unless the design matrix is orthogonal. In this paper, we propose an efficient optimization algorithm for nonconvex group penalties by combining the concave convex procedure and the group LASSO algorithm. We also extend the proposed algorithm for generalized linear models. We evaluate numerical efficiency of the proposed algorithm compared to existing GCD algorithms through simulated data and real data sets.     

Outlier detection and robust variable selection via the penalized weighted LAD-LASSO method

This paper studies the outlier detection and robust variable selection problem in the linear regression model. The penalized weighted least absolute deviation (PWLAD) regression estimation method and the adaptive least absolute shrinkage and selection operator (LASSO) are combined to simultaneously achieve outlier detection, and robust variable selection. An iterative algorithm is proposed to solve the proposed optimization problem. Monte Carlo studies are evaluated the finite-sample performance of the proposed methods. The results indicate that the finite sample performance of the proposed methods performs better than that of the existing methods when there are leverage points or outliers in the response variable or explanatory variables. Finally, we apply the proposed methodology to analyze two real datasets.     

Instrumental variable based variable selection for generalized linear models with endogenous covariates

We study the variable selection problem for a class of generalized linear models with endogenous covariates. Based on the instrumental variable adjustment technology and the smooth-threshold estimating equation (SEE) method, we propose an instrumental variable based variable selection procedure. The proposed variable selection method can attenuate the effect of endogeneity in covariates, and is easy for application in practice. Some theoretical results are also derived such as the consistency of the proposed variable selection procedure and the convergence rate of the resulting estimator. Further, some simulation studies and a real data analysis are conducted to evaluate the performance of the proposed method, and simulation results show that the proposed method is workable.     

On variable selection in generalized linear and related regression models

This paper is concerned with selection of explanatory variables in generalized linear models (GLM). The class of GLM's is quite large and contains e.g. the ordinary linear regression, the binary logistic regression, the probit model and Poisson regression with linear or log-linear parameter structure. We show that, through an approximation of the log likelihood and a certain data transformation, the variable selection problem in a GLM can be converted into variable selection in an ordinary (unweighted) linear regression model. As a consequence no specific computer software for variable selection in GLM's is needed. Instead, some suitable variable selection program for linear regression can be used. We also present a simulation study which shows that the log likelihood approximation is very good in many practical situations. Finally, we mention briefly possible extensions to regression models outside the class of GLM's.     

WLAD-LASSO method for robust estimation and variable selection in partially linear models

This paper focuses on robust estimation and variable selection for partially linear models. We combine the weighted least absolute deviation (WLAD) regression with the adaptive least absolute shrinkage and selection operator (LASSO) to achieve simultaneous robust estimation and variable selection for partially linear models. Compared with the LAD-LASSO method, the WLAD-LASSO method will resist to the heavy-tailed errors and outliers in the parametric components. In addition, we estimate the unknown smooth function by a robust local linear regression. Under some regular conditions, the theoretical properties of the proposed estimators are established. We further examine finite-sample performance of the proposed procedure by simulation studies and a real data example.     

Bayesian variable selection with strong heredity constraints

In this paper, we propose a Bayesian variable selection method for linear regression models with high-order interactions. Our method automatically enforces the heredity constraint, that is, a higher order interaction term can exist in the model only if both of its parent terms are in the model. Based on the stochastic search variable selection George and McCulloch (1993), we propose a novel hierarchical prior that fully considers the heredity constraint and controls the degree of sparsity simultaneously. We develop a Markov chain Monte Carlo (MCMC) algorithm to explore the model space efficiently while accounting for the heredity constraint by modifying the shotgun stochastic search algorithm Hans et al. (2007). The performance of the new model is demonstrated through comparisons with other methods. Numerical studies on both real data analysis and simulations show that our new method tends to find relevant variable more effectively when higher order interaction terms are considered.