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.     

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.     

Adaptive lasso for accelerated hazards models

The important feature of the accelerated hazards (AH) model is that it can capture the gradual effect of treatment. Because of the complexity in its estimation, few discussion has been made on the variable selection of the AH model. The Bayesian non-parametric prior, called the transformed Bernstein polynomial prior, is employed for simultaneously robust estimation and variable selection in sparse AH models. We first introduce a naive lasso-type accelerated hazards model, and later, in order to reduce estimation bias and improve variable selection accuracy, we further consider an adaptive lasso AH model as a direct extension of the naive lasso-type model. Through our simulation studies, we obtain that the adaptive lasso AH model performs better than the lasso-type model with respect to the variable selection and prediction accuracy. We also illustrate the performance of the proposed methods via a brain tumour study.     

New Robust Variable Selection Methods for Linear Regression Models

Motivated by an entropy inequality, we propose for the first time a penalized profile likelihood method for simultaneously selecting significant variables and estimating unknown coefficients in multiple linear regression models in this article. The new method is robust to outliers or errors with heavy tails and works well even for error with infinite variance. Our proposed approach outperforms the adaptive lasso in both theory and practice. It is observed from the simulation studies that (i) the new approach possesses higher probability of correctly selecting the exact model than the least absolute deviation lasso and the adaptively penalized composite quantile regression approach and (ii) exact model selection via our proposed approach is robust regardless of the error distribution. An application to a real dataset is also provided.     

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.     

On the robustness of the generalized fused lasso to prior specifications

Using networks as prior knowledge to guide model selection is a way to reach structured sparsity. In particular, the fused lasso that was originally designed to penalize differences of coefficients corresponding to successive features has been generalized to handle features whose effects are structured according to a given network. As any prior information, the network provided in the penalty may contain misleading edges that connect coefficients whose difference is not zero, and the extent to which the performance of the method depend on the suitability of the graph has never been clearly assessed. In this work we investigate the theoretical and empirical properties of the adaptive generalized fused lasso in the context of generalized linear models. In the fixed \(p\) setting, we show that, asymptotically, adding misleading edges in the graph does not prevent the adaptive generalized fused lasso from enjoying asymptotic oracle properties, while forgetting suitable edges can be more problematic. These theoretical results are complemented by an extensive simulation study that assesses the robustness of the adaptive generalized fused lasso against misspecification of the network as well as its applicability when theoretical coefficients are not exactly equal. Our contribution is also to evaluate the applicability of the generalized fused lasso for the joint modeling of multiple sparse regression functions. Illustrations are provided on two real data examples.     

Group and within-group variable selection for competing risks data

Variable selection in the presence of grouped variables is troublesome for competing risks data: while some recent methods deal with group selection only, simultaneous selection of both groups and within-group variables remains largely unexplored. In this context, we propose an adaptive group bridge method, enabling simultaneous selection both within and between groups, for competing risks data. The adaptive group bridge is applicable to independent and clustered data. It also allows the number of variables to diverge as the sample size increases. We show that our new method possesses excellent asymptotic properties, including variable selection consistency at group and within-group levels. We also show superior performance in simulated and real data sets over several competing approaches, including group bridge, adaptive group lasso, and AIC / BIC-based methods.     

Sparsity identification in ultra-high dimensional quantile regression models with longitudinal data

Abstract

In this paper, we propose a variable selection method for quantile regression model in ultra-high dimensional longitudinal data called as the weighted adaptive robust lasso (WAR-Lasso) which is double-robustness. We derive the consistency and the model selection oracle property of WAR-Lasso. Simulation studies show the double-robustness of WAR-Lasso in both cases of heavy-tailed distribution of the errors and the heavy contaminations of the covariates. WAR-Lasso outperform other methods such as SCAD and etc. A real data analysis is carried out. It shows that WAR-Lasso tends to select fewer variables and the estimated coefficients are in line with economic significance.     

Bayesian model-averaged regularization for Gaussian graphical models

The graphical lasso has now become a useful tool to estimate high-dimensional Gaussian graphical models, but its practical applications suffer from the problem of choosing regularization parameters in a data-dependent way. In this article, we propose a model-averaged method for estimating sparse inverse covariance matrices for Gaussian graphical models. We consider the graphical lasso regularization path as the model space for Bayesian model averaging and use Markov chain Monte Carlo techniques for the regularization path point selection. Numerical performance of our method is investigated using both simulated and real datasets, in comparison with some state-of-art model selection procedures.     

Variable selection approach for zero-inflated count data via adaptive lasso

This article proposes a variable selection approach for zero-inflated count data analysis based on the adaptive lasso technique. Two models including the zero-inflated Poisson and the zero-inflated negative binomial are investigated. An efficient algorithm is used to minimize the penalized log-likelihood function in an approximate manner. Both the generalized cross-validation and Bayesian information criterion procedures are employed to determine the optimal tuning parameter, and a consistent sandwich formula of standard errors for nonzero estimates is given based on local quadratic approximation. We evaluate the performance of the proposed adaptive lasso approach through extensive simulation studies, and apply it to analyze real-life data about doctor visits.     

MARS as an alternative approach of Gaussian graphical model for biochemical networks

The Gaussian graphical model (GGM) is one of the well-known modelling approaches to describe biological networks under the steady-state condition via the precision matrix of data. In literature there are different methods to infer model parameters based on GGM. The neighbourhood selection with the lasso regression and the graphical lasso method are the most common techniques among these alternative estimation methods. But they can be computationally demanding when the system's dimension increases. Here, we suggest a non-parametric statistical approach, called the multivariate adaptive regression splines (MARS) as an alternative of GGM. To compare the performance of both models, we evaluate the findings of normal and non-normal data via the specificity, precision, F-measures and their computational costs. From the outputs, we see that MARS performs well, resulting in, a plausible alternative approach with respect to GGM in the construction of complex biological systems.     

Regularization and variable selection via the elastic net

Summary.  We propose the elastic net, a new regularization and variable selection method. Real world data and a simulation study show that the elastic net often outperforms the lasso, while enjoying a similar sparsity of representation. In addition, the elastic net encourages a grouping effect, where strongly correlated predictors tend to be in or out of the model together. The elastic net is particularly useful when the number of predictors ( p ) is much bigger than the number of observations ( n ). By contrast, the lasso is not a very satisfactory variable selection method in the p ≫ n case. An algorithm called LARS-EN is proposed for computing elastic net regularization paths efficiently, much like algorithm LARS does for the lasso.     

Variable Selection for Semiparametric Varying Coefficient Partially Linear Errors-in-Variables (EV) Model with Missing Response

This paper focuses on the variable selection for semiparametric varying coefficient partially linear model when the covariates are measured with additive errors and the response is missing. An adaptive lasso estimator and the smoothly clipped absolute deviation estimator as a comparison for the parameters are proposed. With the proper selection of regularization parameter, the sampling properties including the consistency of the two procedures and the oracle properties are established. Furthermore, the algorithms and corresponding standard error formulas are discussed. A simulation study is carried out to assess the finite sample performance of the proposed methods.     

Structure identification and variable selection in geographically weighted regression models

Geographically weighted regression (GWR) is an important tool for exploring spatial non-stationarity of a regression relationship, in which whether a regression coefficient really varies over space is especially important in drawing valid conclusions on the spatial variation characteristics of the regression relationship. This paper proposes a so-called GWGlasso method for structure identification and variable selection in GWR models. This method penalizes the loss function of the local-linear estimation of the GWR model by the coefficients and their partial derivatives in the way of the adaptive group lasso and can simultaneously identify spatially varying coefficients, nonzero constant coefficients and zero coefficients. Simulation experiments are further conducted to assess the performance of the proposed method and the Dublin voter turnout data set is analysed to demonstrate its application.     

LAD-Lasso variable selection for doubly censored median regression models

ABSTRACT

A variable selection procedure based on least absolute deviation (LAD) estimation and adaptive lasso (LAD-Lasso for short) is proposed for median regression models with doubly censored data. The proposed procedure can select significant variables and estimate the parameters simultaneously, and the resulting estimators enjoy the oracle property. Simulation results show that the proposed method works well.     

Generalised Rank Regression Estimator with Standard Error Adjusted Lasso

One of the standard variable selection procedures in multiple linear regression is to use a penalisation technique in least‐squares (LS) analysis. In this setting, many different types of penalties have been introduced to achieve variable selection. It is well known that LS analysis is sensitive to outliers, and consequently outliers can present serious problems for the classical variable selection procedures. Since rank‐based procedures have desirable robustness properties compared to LS procedures, we propose a rank‐based adaptive lasso‐type penalised regression estimator and a corresponding variable selection procedure for linear regression models. The proposed estimator and variable selection procedure are robust against outliers in both response and predictor space. Furthermore, since rank regression can yield unstable estimators in the presence of multicollinearity, in order to provide inference that is robust against multicollinearity, we adjust the penalty term in the adaptive lasso function by incorporating the standard errors of the rank estimator. The theoretical properties of the proposed procedures are established and their performances are investigated by means of simulations. Finally, the estimator and variable selection procedure are applied to the Plasma Beta‐Carotene Level data set.     

Grouped penalization estimation of the osteoporosis data in the traditional Chinese medicine

Both continuous and categorical covariates are common in traditional Chinese medicine (TCM) research, especially in the clinical syndrome identification and in the risk prediction research. For groups of dummy variables which are generated by the same categorical covariate, it is important to penalize them group-wise rather than individually. In this paper, we discuss the group lasso method for a risk prediction analysis in TCM osteoporosis research. It is the first time to apply such a group-wise variable selection method in this field. It may lead to new insights of using the grouped penalization method to select appropriate covariates in the TCM research. The introduced methodology can select categorical and continuous variables, and estimate their parameters simultaneously. In our application of the osteoporosis data, four covariates (including both categorical and continuous covariates) are selected out of 52 covariates. The accuracy of the prediction model is excellent. Compared with the prediction model with different covariates, the group lasso risk prediction model can significantly decrease the error rate and help TCM doctors to identify patients with a high risk of osteoporosis in clinical practice. Simulation results show that the application of the group lasso method is reasonable for the categorical covariates selection model in this TCM osteoporosis research.     

Marginalized lasso in sparse regression

We propose marginalized lasso, a new nonconvex penalization for variable selection in regression problem. The marginalized lasso penalty is motivated from integrating out the penalty parameter in the original lasso penalty with a gamma prior distribution. This study provides a thresholding rule and a lasso-based iterative algorithm for parameter estimation in the marginalized lasso. We also provide a coordinate descent algorithm to efficiently optimize the marginalized lasso penalized regression. Numerical comparison studies are provided to demonstrate its competitiveness over the existing sparsity-inducing penalizations and suggest some guideline for tuning parameter selection.