Logistic回归的双层变量选择研究

变量选择是统计建模的重要环节,选择合适的变量可以建立结构简单、预测精准的稳健模型。本文在logistic回归下提出了新的双层变量选择惩罚方法——adaptive Sparse Group Lasso(adSGL),其独特之处在于基于变量的分组结构作筛选,实现了组内和组间双层选择。该方法的优点是对各单个系数和组系数采取不同程度的惩罚,避免了过度惩罚大系数,从而提高了模型的估计和预测精度。求解的难点是惩罚似然函数不是严格凸的,因此本文基于组坐标下降法求解模型,并建立了调整参数的选取准则。模拟分析表明,对比现有代表性方法Sparse Group Lasso、Group Lasso及Lasso,adSGL法不仅提高了双层选择精度,而且降低了模型误差。最后本文将adSGL法应用到信用卡信用评分研究,对比logistic回归,它具有更高的分类精度和稳健性。     

SGL-SVM方法研究及其在财务困境预测中的应用

针对分类问题,本文提出了稀疏组Lasso支持向量机方法（Sparse group lasso SVM, SGL-SVM）,即在SVM模型的损失函数中引入SGL惩罚函数,能同时进行组间变量和组内变量的筛选。由于SGL-SVM的目标函数求解比较复杂,本文又提出了一种快速的双层坐标下降算法。通过模拟实验,发现SGL-SVM方法在预测效果和变量选择上均要好于其他方法,对于变量具有自然分组结构且组内是稀疏的数据,本文方法在提高变量选择效果的同时又能提高模型的预测精度。最后,将本文提出的SGL-SVM方法应用到我国制造业上市公司财务困境预测中。     

General Sparse Boosting: Improving Feature Selection of L2 Boosting by Correlation-Based Penalty Family

In high-dimensional setting, componentwise L₂boosting has been used to construct sparse model that performs well, but it tends to select many ineffective variables. Several sparse boosting methods, such as, SparseL₂Boosting and Twin Boosting, have been proposed to improve the variable selection of L₂boosting algorithm. In this article, we propose a new general sparse boosting method (GSBoosting). The relations are established between GSBoosting and other well known regularized variable selection methods in the orthogonal linear model, such as adaptive Lasso, hard thresholds, etc. Simulation results show that GSBoosting has good performance in both prediction and variable selection.     

Robust adaptive Lasso for variable selection

The adaptive least absolute shrinkage and selection operator (Lasso) and least absolute deviation (LAD)-Lasso are two attractive shrinkage methods for simultaneous variable selection and regression parameter estimation. While the adaptive Lasso is efficient for small magnitude errors, LAD-Lasso is robust against heavy-tailed errors and severe outliers. In this article, we consider a data-driven convex combination of these two modern procedures to produce a robust adaptive Lasso, which not only enjoys the oracle properties, but synthesizes the advantages of the adaptive Lasso and LAD-Lasso. It fully adapts to different error structures including the infinite variance case and automatically chooses the optimal weight to achieve both robustness and high efficiency. Extensive simulation studies demonstrate a good finite sample performance of the robust adaptive Lasso. Two data sets are analyzed to illustrate the practical use of the procedure.     

Variable selection for semiparametric regression models with iterated penalization

Semiparametric regression models with multiple covariates are commonly encountered. When there are covariates not associated with response variable, variable selection may lead to sparser models, more lucid interpretations and more accurate estimation. In this study, we adopt a sieve approach for the estimation of nonparametric covariate effects in semiparametric regression models. We adopt a two-step iterated penalization approach for variable selection. In the first step, a mixture of the Lasso and group Lasso penalties are employed to conduct the first-round variable selection and obtain the initial estimate. In the second step, a mixture of the weighted Lasso and weighted group Lasso penalties, with weights constructed using the initial estimate, are employed for variable selection. We show that the proposed iterated approach has the variable selection consistency property, even when number of unknown parameters diverges with sample size. Numerical studies, including simulation and analysis of a diabetes dataset, show satisfactory performance of the proposed approach.     

Empirical Performance of Cross-Validation With Oracle Methods in a Genomics Context

When employing model selection methods with oracle properties such as the smoothly clipped absolute deviation (SCAD) and the Adaptive Lasso, it is typical to estimate the smoothing parameter by m-fold cross-validation, for example, m = 10. In problems where the true regression function is sparse and the signals large, such cross-validation typically works well. However, in regression modeling of genomic studies involving Single Nucleotide Polymorphisms (SNP), the true regression functions, while thought to be sparse, do not have large signals. We demonstrate empirically that in such problems, the number of selected variables using SCAD and the Adaptive Lasso, with 10-fold cross-validation, is a random variable that has considerable and surprising variation. Similar remarks apply to non-oracle methods such as the Lasso. Our study strongly questions the suitability of performing only a single run of m-fold cross-validation with any oracle method, and not just the SCAD and Adaptive Lasso.     

Empirical Performance of Cross-Validation With Oracle Methods in a Genomics Context

When employing model selection methods with oracle properties such as the smoothly clipped absolute deviation (SCAD) and the Adaptive Lasso, it is typical to estimate the smoothing parameter by m-fold cross-validation, for example, m = 10. In problems where the true regression function is sparse and the signals large, such cross-validation typically works well. However, in regression modeling of genomic studies involving Single Nucleotide Polymorphisms (SNP), the true regression functions, while thought to be sparse, do not have large signals. We demonstrate empirically that in such problems, the number of selected variables using SCAD and the Adaptive Lasso, with 10-fold cross-validation, is a random variable that has considerable and surprising variation. Similar remarks apply to non-oracle methods such as the Lasso. Our study strongly questions the suitability of performing only a single run of m-fold cross-validation with any oracle method, and not just the SCAD and Adaptive Lasso.     

Overlapping group lasso for high-dimensional generalized linear models

Abstract

Structured sparsity has recently been a very popular technique to deal with the high-dimensional data. In this paper, we mainly focus on the theoretical problems for the overlapping group structure of generalized linear models (GLMs). Although the overlapping group lasso method for GLMs has been widely applied in some applications, the theoretical properties about it are still unknown. Under some general conditions, we presents the oracle inequalities for the estimation and prediction error of overlapping group Lasso method in the generalized linear model setting. Then, we apply these results to the so-called Logistic and Poisson regression models. It is shown that the results of the Lasso and group Lasso procedures for GLMs can be recovered by specifying the group structures in our proposed method. The effect of overlap and the performance of variable selection of our proposed method are both studied by numerical simulations. Finally, we apply our proposed method to two gene expression data sets: the p53 data and the lung cancer data.     

基于随机化适应性Lasso的高维变量选择

Lasso等惩罚变量选择方法选入模型的变量数受到样本量限制。文献中已有研究变量系数显著性的方法舍弃了未选入模型的变量含有的信息。本文在变量数大于样本量即p>n的高维情况下,使用随机化bootstrap方法获得变量权重,在计算适应性Lasso时构建选择事件的条件分布并剔除系数不显著的变量,以得到最终估计结果。本文的创新点在于提出的方法突破了适应性Lasso可选变量数的限制,当观测数据含有大量干扰变量时能够有效地识别出真实变量与干扰变量。与现有的惩罚变量选择方法相比,多种情境下的模拟研究展示了所提方法在上述两个问题中的优越性。实证研究中对NCI-60癌症细胞系数据进行了分析,结果较以往文献有明显改善。     

Detection of multiple undocumented change-points using adaptive Lasso

The problem of detecting multiple undocumented change-points in a historical temperature sequence with simple linear trend is formulated by a linear model. We apply adaptive least absolute shrinkage and selection operator (Lasso) to estimate the number and locations of change-points. Model selection criteria are used to choose the Lasso smoothing parameter. As adaptive Lasso may overestimate the number of change-points, we perform post-selection on change-points detected by adaptive Lasso using multivariate t simultaneous confidence intervals. Our method is demonstrated on the annual temperature data (year: 1902–2000) from Tuscaloosa, Alabama.     

Quantile regression and variable selection for the single-index model

In this paper, we propose a new full iteration estimation method for quantile regression (QR) of the single-index model (SIM). The asymptotic properties of the proposed estimator are derived. Furthermore, we propose a variable selection procedure for the QR of SIM by combining the estimation method with the adaptive LASSO penalized method to get sparse estimation of the index parameter. The oracle properties of the variable selection method are established. Simulations with various non-normal errors are conducted to demonstrate the finite sample performance of the estimation method and the variable selection procedure. Furthermore, we illustrate the proposed method by analyzing a real data set.     

Adaptive Lasso Variable Selection for the Accelerated Failure Models

This article considers the adaptive lasso procedure for the accelerated failure time model with multiple covariates based on weighted least squares method, which uses Kaplan-Meier weights to account for censoring. The adaptive lasso method can complete the variable selection and model estimation simultaneously. Under some mild conditions, the estimator is shown to have sparse and oracle properties. We use Bayesian Information Criterion (BIC) for tuning parameter selection, and a bootstrap variance approach for standard error. Simulation studies and two real data examples are carried out to investigate the performance of the proposed method.     

Bayesian variable selection and estimation in maximum entropy quantile regression

Quantile regression has gained increasing popularity as it provides richer information than the regular mean regression, and variable selection plays an important role in the quantile regression model building process, as it improves the prediction accuracy by choosing an appropriate subset of regression predictors. Unlike the traditional quantile regression, we consider the quantile as an unknown parameter and estimate it jointly with other regression coefficients. In particular, we adopt the Bayesian adaptive Lasso for the maximum entropy quantile regression. A flat prior is chosen for the quantile parameter due to the lack of information on it. The proposed method not only addresses the problem about which quantile would be the most probable one among all the candidates, but also reflects the inner relationship of the data through the estimated quantile. We develop an efficient Gibbs sampler algorithm and show that the performance of our proposed method is superior than the Bayesian adaptive Lasso and Bayesian Lasso through simulation studies and a real data analysis.     

On the grouped selection and model complexity of the adaptive elastic net

Lasso proved to be an extremely successful technique for simultaneous estimation and variable selection. However lasso has two major drawbacks. First, it does not enforce any grouping effect and secondly in some situation lasso solutions are inconsistent for variable selection. To overcome this inconsistency adaptive lasso is proposed where adaptive weights are used for penalizing different coefficients. Recently a doubly regularized technique namely elastic net is proposed which encourages grouping effect i.e. either selection or omission of the correlated variables together. However elastic net is also inconsistent. In this paper we study adaptive elastic net which does not have this drawback. In this article we specially focus on the grouped selection property of adaptive elastic net along with its model selection complexity. We also shed some light on the bias-variance tradeoff of different regularization methods including adaptive elastic net. An efficient algorithm was proposed in the line of LARS-EN, which is then illustrated with simulated as well as real life data examples.     

Nonparametric Additive Instrumental Variable Estimator: A Group Shrinkage Estimation Perspective

In this article, we study a nonparametric approach regarding a general nonlinear reduced form equation to achieve a better approximation of the optimal instrument. Accordingly, we propose the nonparametric additive instrumental variable estimator (NAIVE) with the adaptive group Lasso. We theoretically demonstrate that the proposed estimator is root-n consistent and asymptotically normal. The adaptive group Lasso helps us select the valid instruments while the dimensionality of potential instrumental variables is allowed to be greater than the sample size. In practice, the degree and knots of B-spline series are selected by minimizing the BIC or EBIC criteria for each nonparametric additive component in the reduced form equation. In Monte Carlo simulations, we show that the NAIVE has the same performance as the linear instrumental variable (IV) estimator for the truly linear reduced form equation. On the other hand, the NAIVE performs much better in terms of bias and mean squared errors compared to other alternative estimators under the high-dimensional nonlinear reduced form equation. We further illustrate our method in an empirical study of international trade and growth. Our findings provide a stronger evidence that international trade has a significant positive effect on economic growth.     

Lasso-type estimation for covariate-adjusted linear model

Lasso is popularly used for variable selection in recent years. In this paper, lasso-type penalty functions including lasso and adaptive lasso are employed in simultaneously variable selection and parameter estimation for covariate-adjusted linear model, where the predictors and response cannot be observed directly and distorted by some observable covariate through some unknown multiplicative smooth functions. Estimation procedures are proposed and some asymptotic properties are obtained under some mild conditions. It deserves noting that under appropriate conditions, the adaptive lasso estimator correctly select covariates with nonzero coefficients with probability converging to one and that the estimators of nonzero coefficients have the same asymptotic distribution that they would have if the zero coefficients were known in advance, i.e. the adaptive lasso estimator has the oracle property in the sense of Fan and Li [6]. Simulation studies are carried out to examine its performance in finite sample situations and the Boston Housing data is analyzed for illustration.     

Covariate Selection for the Semiparametric Additive Risk Model

Abstract.  This paper considers covariate selection for the additive hazards model. This model is particularly simple to study theoretically and its practical implementation has several major advantages to the similar methodology for the proportional hazards model. One complication compared with the proportional model is, however, that there is no simple likelihood to work with. We here study a least squares criterion with desirable properties and show how this criterion can be interpreted as a prediction error. Given this criterion, we define ridge and Lasso estimators as well as an adaptive Lasso and study their large sample properties for the situation where the number of covariates p is smaller than the number of observations. We also show that the adaptive Lasso has the oracle property. In many practical situations, it is more relevant to tackle the situation with large p compared with the number of observations. We do this by studying the properties of the so-called Dantzig selector in the setting of the additive risk model. Specifically, we establish a bound on how close the solution is to a true sparse signal in the case where the number of covariates is large. In a simulation study, we also compare the Dantzig and adaptive Lasso for a moderate to small number of covariates. The methods are applied to a breast cancer data set with gene expression recordings and to the primary biliary cirrhosis clinical data.     

The Adaptive Gril Estimator with a Diverging Number of Parameters

We consider the problem of variables selection and estimation in linear regression model in situations where the number of parameters diverges with the sample size. We propose the adaptive Generalized Ridge-Lasso (\mboxAdaGril) which is an extension of the the adaptive Elastic Net. AdaGril incorporates information redundancy among correlated variables for model selection and estimation. It combines the strengths of the quadratic regularization and the adaptively weighted Lasso shrinkage. In this article, we highlight the grouped selection property for AdaCnet method (one type of AdaGril) in the equal correlation case. Under weak conditions, we establish the oracle property of AdaGril which ensures the optimal large performance when the dimension is high. Consequently, it achieves both goals of handling the problem of collinearity in high dimension and enjoys the oracle property. Moreover, we show that AdaGril estimator achieves a Sparsity Inequality, i.e., a bound in terms of the number of non-zero components of the “true” regression coefficient. This bound is obtained under a similar weak Restricted Eigenvalue (RE) condition used for Lasso. Simulations studies show that some particular cases of AdaGril outperform its competitors.     

VARIABLE SELECTION IN NONPARAMETRIC ADDITIVE MODELS

We consider a nonparametric additive model of a conditional mean function in which the number of variables and additive components may be larger than the sample size but the number of nonzero additive components is "small" relative to the sample size. The statistical problem is to determine which additive components are nonzero. The additive components are approximated by truncated series expansions with B-spline bases. With this approximation, the problem of component selection becomes that of selecting the groups of coefficients in the expansion. We apply the adaptive group Lasso to select nonzero components, using the group Lasso to obtain an initial estimator and reduce the dimension of the problem. We give conditions under which the group Lasso selects a model whose number of components is comparable with the underlying model, and the adaptive group Lasso selects the nonzero components correctly with probability approaching one as the sample size increases and achieves the optimal rate of convergence. The results of Monte Carlo experiments show that the adaptive group Lasso procedure works well with samples of moderate size. A data example is used to illustrate the application of the proposed method.     

Modeling association between multivariate correlated outcomes and high-dimensional sparse covariates: the adaptive SVS method

The problem of modeling the relationship between a set of covariates and a multivariate response with correlated components often arises in many areas of research such as genetics, psychometrics, signal processing. In the linear regression framework, such task can be addressed using a number of existing methods. In the high-dimensional sparse setting, most of these methods rely on the idea of penalization in order to efficiently estimate the regression matrix. Examples of such methods include the lasso, the group lasso, the adaptive group lasso or the simultaneous variable selection (SVS) method. Crucially, a suitably chosen penalty also allows for an efficient exploitation of the correlation structure within the multivariate response. In this paper we introduce a novel variant of such method called the adaptive SVS, which is closely linked with the adaptive group lasso. Via a simulation study we investigate its performance in the high-dimensional sparse regression setting. We provide a comparison with a number of other popular methods under different scenarios and show that the adaptive SVS is a powerful tool for efficient recovery of signal in such setting. The methods are applied to genetic data.