Journal of Data Science

Bootstrapping is commonly used as a tool for non-parametric statistical inference to assess the quality of estimators in variable selection models. However, for a massive dataset, the computational requirement when using bootstrapping in variable selection models (BootVS) can be crucial. In this study, we propose a novel framework using a bag of little bootstraps variable selection (BLBVS) method with a ridge hybrid procedure to assess the quality of estimators in generalized linear models with a regularized term, such as lasso and group lasso penalties. The proposed method can be easily and naturally implemented with distributed computing, and thus has significant computational advantages for massive datasets. The simulation results show that our novel BLBVS method performs excellently in both accuracy and efficiency when compared with BootVS. Real data analyses including regression on a bike sharing dataset and classification of a lending club dataset are presented to illustrate the computational superiority of BLBVS in large-scale datasets.

Penalized regression provides an automated approach to preform simultaneous variable selection and parameter estimation and is a popular method to analyze high-dimensional data. Since the conception of the LASSO in the mid-to-late 1990s, extensive research has been done to improve penalized regression. The LASSO, and several of its variations, performs penalization symmetrically around zero. Thus, variables with the same magnitude are shrunk the same regardless of the direction of effect. To the best of our knowledge, sign-based shrinkage, preferential shrinkage based on the sign of the coefficients, has yet to be explored under the LASSO framework. We propose a generalization to the LASSO, asymmetric LASSO, that performs sign-based shrinkage. Our method is motivated by placing an asymmetric Laplace prior on the regression coefficients, rather than a symmetric Laplace prior. This corresponds to an asymmetric ${\ell _{1}}$ penalty under the penalized regression framework. In doing so, preferential shrinkage can be performed through an auxiliary tuning parameter that controls the degree of asymmetry. Our numerical studies indicate that the asymmetric LASSO performs better than the LASSO when effect sizes are sign skewed. Furthermore, in the presence of positively-skewed effects, the asymmetric LASSO is comparable to the non-negative LASSO without the need to place an a priori constraint on the effect estimates and outperforms the non-negative LASSO when negative effects are also present in the model. A real data example using the breast cancer gene expression data from The Cancer Genome Atlas is also provided, where the asymmetric LASSO identifies two potentially novel gene expressions that are associated with BRCA1 with a minor improvement in prediction performance over the LASSO and non-negative LASSO.

Multi-classification is commonly encountered in data science practice, and it has broad applications in many areas such as biology, medicine, and engineering. Variable selection in multiclass problems is much more challenging than in binary classification or regression problems. In addition to estimating multiple discriminant functions for separating different classes, we need to decide which variables are important for each individual discriminant function as well as for the whole set of functions. In this paper, we address the multi-classification variable selection problem by proposing a new form of penalty, supSCAD, which first groups all the coefficients of the same variable associated with all the discriminant functions altogether and then imposes the SCAD penalty on the supnorm of each group. We apply the new penalty to both soft and hard classification and develop two new procedures: the supSCAD multinomial logistic regression and the supSCAD multi-category support vector machine. Our theoretical results show that, with a proper choice of the tuning parameter, the supSCAD multinomial logistic regression can identify the underlying sparse model consistently and enjoys oracle properties even when the dimension of predictors goes to infinity. Based on the local linear and quadratic approximation to the non-concave SCAD and nonlinear multinomial log-likelihood function, we show that the new procedures can be implemented efficiently by solving a series of linear or quadratic programming problems. Performance of the new methods is illustrated by simulation studies and real data analysis of the Small Round Blue Cell Tumors and the Semeion Handwritten Digit data sets.