Penalized Composite Quasi-Likelihood for Ultrahigh-Dimensional Variable Selection
01 Jun 2011-Journal of The Royal Statistical Society Series B-statistical Methodology (NIH Public Access)-Vol. 73, Iss: 3, pp 325-349
TL;DR: A data‐driven weighted linear combination of convex loss functions, together with weighted L1‐penalty is proposed and established a strong oracle property of the method proposed that has both the model selection consistency and estimation efficiency for the true non‐zero coefficients.
Abstract: In high-dimensional model selection problems, penalized least-square approaches have been extensively used. This paper addresses the question of both robustness and efficiency of penalized model selection methods, and proposes a data-driven weighted linear combination of convex loss functions, together with weighted L1-penalty. It is completely data-adaptive and does not require prior knowledge of the error distribution. The weighted L1-penalty is used both to ensure the convexity of the penalty term and to ameliorate the bias caused by the L1-penalty. In the setting with dimensionality much larger than the sample size, we establish a strong oracle property of the proposed method that possesses both the model selection consistency and estimation efficiency for the true non-zero coefficients. As specific examples, we introduce a robust method of composite L1-L2, and optimal composite quantile method and evaluate their performance in both simulated and real data examples.
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TL;DR: A two-step model averaging procedure for improving prediction accuracy of the true conditional mean of a censored response variable is proposed, adapting a delete-one Mallows criterion, where the standard constraint that weights sum to one is removed.
Abstract: This article considers ultrahigh dimensional prediction problems with censored response vari- ables. We propose a two-step model averaging procedure for improving prediction accuracy of the true conditional mean of a censored response variable. The first step is to construct a class of candidate models, each with low-dimensional covariates. For this, a feature screening procedure is developed to separate the active and inactive predictors through a fused mean- variance index and group covariates with similar size of index together to form regression models with censored response variables. The new model-free screening method can easi- ly deal with many types of predictors and response variables, such as discrete, categorical and continuous variables, still works well when predictors have heavy-tailed distributions or strongly dependend on each other, and enjoys rank consistency properties under mild regularity conditions. The second step is to find the optimal model weights for averaging by adapting a delete-one Mallows criterion, where the standard constraint that weights sum to one is removed. The theoretical results show that the delete-one Mallows criterion achieves the lowest possible prediction loss asymptotically. Numerical studies demonstrate the su- perior performance of the proposed variable screening and model averaging procedures over existing methods.
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Cites background from "Penalized Composite Quasi-Likelihoo..."
...It is commonly assumed that only a small number of covariates actually contributes to survival models considered, which leads to the well-known sparse survival models for helping interpretation and improving prediction accuracy (Bradic, Fan, and Wang 2011)....
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TL;DR: A new class of regression estimation methods by combining many candidate models with possibly different dimensions to address the issue of tapering effect estimation is proposed and an information-weighted composite likelihood is proposed.
Abstract: In this paper, we propose a new class of regression estimation methods by combining many candidate models with possibly different dimensions to address the issue of tapering effect estimation. An i...
1 citations
Cites background from "Penalized Composite Quasi-Likelihoo..."
...However, both CQR and composite L1-L2 are composite of different models with same model dimensions....
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...CONTACT Weixin Yao weixin.yao@ucr.edu 2019 Taylor & Francis Group, LLC Composite model estimation received increasing attentions recently, e.g., composite quantile regression (CQR) (Zou and Yuan 2008; Jiang et al. 2016; Guo et al. 2017), and composite L1-L2 (Bradic et al. 2011)....
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TL;DR: In this article , a robust post-selection inference method based on the Huber loss for the regression coefficients, when the error distribution is heavy-tailed and asymmetric in a high-dimensional linear model with an intercept term, was proposed.
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Journal Article•
TL;DR: Correlation of various social factors with what is about the contribution of the pharmacist as patient educator in pharmacy is analyzed to estimate the evaluation of patient satisfaction of pharmaceutical services using correlation of social factors.
Abstract: Objective: Pharmacists' professional roles to include provision of information and pharmaceutical care services have obtained in a focus on collaborative pharmacist-patient professional relationships. We analyzed correlation of various social factors with what is about the contribution of the pharmacist as patient educator in pharmacy. The need for pharmacist services is discussed, as communicate effectively with patients that are quality of the scientific evidence to support their efficacy and reduce toxicity. Pharmacist appropriate interaction with patients taking the wrong medication can largely solve. The use of statistical methods in many areas of applied healthcare research has grown considerably in recent years. Methods: In this paper, have been used a multivariate analysis with latent variables for the estimation of the evaluation of patient satisfaction of pharmaceutical services using correlation of social factors. Also, in this paper for the factor analysis (FA) approach, have been used the principal component analysis (PCA) jointly to compute the coefficients of variables and dimension reduction. Results: The results have shown the four components were significant and with our results were agreement. Conclusion: Exploratory factor analysis was used to examine the covariance structure of the data reported by approximately 500 members. Further research is needed to develop and test instruments based on theoretical frameworks, to test satisfaction go to pharmacies and to choose a model for data determined in satisfaction over time.
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Cites methods from "Penalized Composite Quasi-Likelihoo..."
...Also, a set of new methods to select variables in the correlated data are proposed [24]....
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Posted Content•
TL;DR: In this article, a regression relationship between initial estimators and values of model-independent parameters is proposed to improve the performance of composite estimators for both estimation efficiency and bias reduction.
Abstract: Composition methodologies in the current literature are mainly to promote estimation efficiency via direct composition, either, of initial estimators or of objective functions. In this paper, composite estimation is investigated for both estimation efficiency and bias reduction. To this end, a novel method is proposed by utilizing a regression relationship between initial estimators and values of model-independent parameter in an asymptotic sense. The resulting estimators could have smaller limiting variances than those of initial estimators, and for nonparametric regression estimation, could also have faster convergence rate than the classical optimal rate that the corresponding initial estimators can achieve. The simulations are carried out to examine its performance in finite sample situations.
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Cites background from "Penalized Composite Quasi-Likelihoo..."
...For a different problem (Fan and Wang 2011), the choice of the optimal weights was discussed, but, the theoretical justification in the scenario under study was not explored before....
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References
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TL;DR: A new method for estimation in linear models called the lasso, which minimizes the residual sum of squares subject to the sum of the absolute value of the coefficients being less than a constant, is proposed.
Abstract: SUMMARY We propose a new method for estimation in linear models. The 'lasso' minimizes the residual sum of squares subject to the sum of the absolute value of the coefficients being less than a constant. Because of the nature of this constraint it tends to produce some coefficients that are exactly 0 and hence gives interpretable models. Our simulation studies suggest that the lasso enjoys some of the favourable properties of both subset selection and ridge regression. It produces interpretable models like subset selection and exhibits the stability of ridge regression. There is also an interesting relationship with recent work in adaptive function estimation by Donoho and Johnstone. The lasso idea is quite general and can be applied in a variety of statistical models: extensions to generalized regression models and tree-based models are briefly described.
40,785 citations
TL;DR: In this article, penalized likelihood approaches are proposed to handle variable selection problems, and it is shown that the newly proposed estimators perform as well as the oracle procedure in variable selection; namely, they work as well if the correct submodel were known.
Abstract: Variable selection is fundamental to high-dimensional statistical modeling, including nonparametric regression. Many approaches in use are stepwise selection procedures, which can be computationally expensive and ignore stochastic errors in the variable selection process. In this article, penalized likelihood approaches are proposed to handle these kinds of problems. The proposed methods select variables and estimate coefficients simultaneously. Hence they enable us to construct confidence intervals for estimated parameters. The proposed approaches are distinguished from others in that the penalty functions are symmetric, nonconcave on (0, ∞), and have singularities at the origin to produce sparse solutions. Furthermore, the penalty functions should be bounded by a constant to reduce bias and satisfy certain conditions to yield continuous solutions. A new algorithm is proposed for optimizing penalized likelihood functions. The proposed ideas are widely applicable. They are readily applied to a variety of ...
8,314 citations
Stanford University1, Cleveland Clinic2, University of Toronto3, Centre national de la recherche scientifique4, Université Paris-Saclay5, University of Paris-Sud6, Avaya7, Rutgers University8, RAND Corporation9, IBM10, University of Pennsylvania11, University of Western Australia12, University of Minnesota13
TL;DR: A publicly available algorithm that requires only the same order of magnitude of computational effort as ordinary least squares applied to the full set of covariates is described.
Abstract: The purpose of model selection algorithms such as All Subsets, Forward Selection and Backward Elimination is to choose a linear model on the basis of the same set of data to which the model will be applied. Typically we have available a large collection of possible covariates from which we hope to select a parsimonious set for the efficient prediction of a response variable. Least Angle Regression (LARS), a new model selection algorithm, is a useful and less greedy version of traditional forward selection methods. Three main properties are derived: (1) A simple modification of the LARS algorithm implements the Lasso, an attractive version of ordinary least squares that constrains the sum of the absolute regression coefficients; the LARS modification calculates all possible Lasso estimates for a given problem, using an order of magnitude less computer time than previous methods. (2) A different LARS modification efficiently implements Forward Stagewise linear regression, another promising new model selection method; this connection explains the similar numerical results previously observed for the Lasso and Stagewise, and helps us understand the properties of both methods, which are seen as constrained versions of the simpler LARS algorithm. (3) A simple approximation for the degrees of freedom of a LARS estimate is available, from which we derive a Cp estimate of prediction error; this allows a principled choice among the range of possible LARS estimates. LARS and its variants are computationally efficient: the paper describes a publicly available algorithm that requires only the same order of magnitude of computational effort as ordinary least squares applied to the full set of covariates.
7,828 citations
"Penalized Composite Quasi-Likelihoo..." refers background or methods in this paper
...…(16) can be recast as a penalized weighted least square regression argmin β n∑ i=1 w1∣∣∣Yi −XTi β̂ (0) ∣∣∣ + w2 ( Yi −XTi β )2 + n p∑ j=1 γλ(|β(0)j |)|βj | which can be efficiently solved by pathwise coordinate optimization (Friedman et al., 2008) or least angle regression (Efron et al., 2004)....
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...) are all nonnegative. This class of problems can be solved with fast and efficient computational algorithms such as pathwise coordinate optimization (Friedman et al., 2008) and least angle regression (Efron et al., 2004). One particular example is the combination of L 1 and L 2 regressions, in which K= 2, ρ 1(t) = |t−b 0|andρ 2(t) = t2. Here b 0 denotes themedian of error distributionε. Iftheerror distribution is sym...
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...i=1 w 1 Yi −XT i βˆ (0) +w 2 Yi −XT i β 2 +n Xp j=1 γλ(|β (0) j |)|βj| which can be efficiently solved by pathwise coordinate optimization (Friedman et al., 2008) or least angle regression (Efron et al., 2004). If b 0 6= 0, the penalized least-squares problem ( 16) is somewhat different from (5) since we have an additional parameter b 0. Using the same arguments, and treating b 0 as an additional parameter ...
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...This class of problems can be solved with fast and efficient computational algorithms such as pathwise coordinate optimization (Friedman et al., 2008) and least angle regression (Efron et al., 2004)....
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TL;DR: A new version of the lasso is proposed, called the adaptive lasso, where adaptive weights are used for penalizing different coefficients in the ℓ1 penalty, and the nonnegative garotte is shown to be consistent for variable selection.
Abstract: The lasso is a popular technique for simultaneous estimation and variable selection. Lasso variable selection has been shown to be consistent under certain conditions. In this work we derive a necessary condition for the lasso variable selection to be consistent. Consequently, there exist certain scenarios where the lasso is inconsistent for variable selection. We then propose a new version of the lasso, called the adaptive lasso, where adaptive weights are used for penalizing different coefficients in the l1 penalty. We show that the adaptive lasso enjoys the oracle properties; namely, it performs as well as if the true underlying model were given in advance. Similar to the lasso, the adaptive lasso is shown to be near-minimax optimal. Furthermore, the adaptive lasso can be solved by the same efficient algorithm for solving the lasso. We also discuss the extension of the adaptive lasso in generalized linear models and show that the oracle properties still hold under mild regularity conditions. As a bypro...
6,765 citations
TL;DR: In this article, a new approach toward a theory of robust estimation is presented, which treats in detail the asymptotic theory of estimating a location parameter for contaminated normal distributions, and exhibits estimators that are asyptotically most robust (in a sense to be specified) among all translation invariant estimators.
Abstract: This paper contains a new approach toward a theory of robust estimation; it treats in detail the asymptotic theory of estimating a location parameter for contaminated normal distributions, and exhibits estimators—intermediaries between sample mean and sample median—that are asymptotically most robust (in a sense to be specified) among all translation invariant estimators. For the general background, see Tukey (1960) (p. 448 ff.)
5,628 citations