29,828 research outputs found
Structured variable selection in support vector machines
When applying the support vector machine (SVM) to high-dimensional
classification problems, we often impose a sparse structure in the SVM to
eliminate the influences of the irrelevant predictors. The lasso and other
variable selection techniques have been successfully used in the SVM to perform
automatic variable selection. In some problems, there is a natural hierarchical
structure among the variables. Thus, in order to have an interpretable SVM
classifier, it is important to respect the heredity principle when enforcing
the sparsity in the SVM. Many variable selection methods, however, do not
respect the heredity principle. In this paper we enforce both sparsity and the
heredity principle in the SVM by using the so-called structured variable
selection (SVS) framework originally proposed in Yuan, Joseph and Zou (2007).
We minimize the empirical hinge loss under a set of linear inequality
constraints and a lasso-type penalty. The solution always obeys the desired
heredity principle and enjoys sparsity. The new SVM classifier can be
efficiently fitted, because the optimization problem is a linear program.
Another contribution of this work is to present a nonparametric extension of
the SVS framework, and we propose nonparametric heredity SVMs. Simulated and
real data are used to illustrate the merits of the proposed method.Comment: Published in at http://dx.doi.org/10.1214/07-EJS125 the Electronic
Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Regression on manifolds: Estimation of the exterior derivative
Collinearity and near-collinearity of predictors cause difficulties when
doing regression. In these cases, variable selection becomes untenable because
of mathematical issues concerning the existence and numerical stability of the
regression coefficients, and interpretation of the coefficients is ambiguous
because gradients are not defined. Using a differential geometric
interpretation, in which the regression coefficients are interpreted as
estimates of the exterior derivative of a function, we develop a new method to
do regression in the presence of collinearities. Our regularization scheme can
improve estimation error, and it can be easily modified to include lasso-type
regularization. These estimators also have simple extensions to the "large ,
small " context.Comment: Published in at http://dx.doi.org/10.1214/10-AOS823 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Penalized variable selection procedure for Cox models with semiparametric relative risk
We study the Cox models with semiparametric relative risk, which can be
partially linear with one nonparametric component, or multiple additive or
nonadditive nonparametric components. A penalized partial likelihood procedure
is proposed to simultaneously estimate the parameters and select variables for
both the parametric and the nonparametric parts. Two penalties are applied
sequentially. The first penalty, governing the smoothness of the multivariate
nonlinear covariate effect function, provides a smoothing spline ANOVA
framework that is exploited to derive an empirical model selection tool for the
nonparametric part. The second penalty, either the
smoothly-clipped-absolute-deviation (SCAD) penalty or the adaptive LASSO
penalty, achieves variable selection in the parametric part. We show that the
resulting estimator of the parametric part possesses the oracle property, and
that the estimator of the nonparametric part achieves the optimal rate of
convergence. The proposed procedures are shown to work well in simulation
experiments, and then applied to a real data example on sexually transmitted
diseases.Comment: Published in at http://dx.doi.org/10.1214/09-AOS780 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Rodeo: Sparse Nonparametric Regression in High Dimensions
We present a greedy method for simultaneously performing local bandwidth
selection and variable selection in nonparametric regression. The method starts
with a local linear estimator with large bandwidths, and incrementally
decreases the bandwidth of variables for which the gradient of the estimator
with respect to bandwidth is large. The method--called rodeo (regularization of
derivative expectation operator)--conducts a sequence of hypothesis tests to
threshold derivatives, and is easy to implement. Under certain assumptions on
the regression function and sampling density, it is shown that the rodeo
applied to local linear smoothing avoids the curse of dimensionality, achieving
near optimal minimax rates of convergence in the number of relevant variables,
as if these variables were isolated in advance
Optimal cross-validation in density estimation with the -loss
We analyze the performance of cross-validation (CV) in the density estimation
framework with two purposes: (i) risk estimation and (ii) model selection. The
main focus is given to the so-called leave--out CV procedure (Lpo), where
denotes the cardinality of the test set. Closed-form expressions are
settled for the Lpo estimator of the risk of projection estimators. These
expressions provide a great improvement upon -fold cross-validation in terms
of variability and computational complexity. From a theoretical point of view,
closed-form expressions also enable to study the Lpo performance in terms of
risk estimation. The optimality of leave-one-out (Loo), that is Lpo with ,
is proved among CV procedures used for risk estimation. Two model selection
frameworks are also considered: estimation, as opposed to identification. For
estimation with finite sample size , optimality is achieved for large
enough [with ] to balance the overfitting resulting from the
structure of the model collection. For identification, model selection
consistency is settled for Lpo as long as is conveniently related to the
rate of convergence of the best estimator in the collection: (i) as
with a parametric rate, and (ii) with some
nonparametric estimators. These theoretical results are validated by simulation
experiments.Comment: Published in at http://dx.doi.org/10.1214/14-AOS1240 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
A Selective Review of Group Selection in High-Dimensional Models
Grouping structures arise naturally in many statistical modeling problems.
Several methods have been proposed for variable selection that respect grouping
structure in variables. Examples include the group LASSO and several concave
group selection methods. In this article, we give a selective review of group
selection concerning methodological developments, theoretical properties and
computational algorithms. We pay particular attention to group selection
methods involving concave penalties. We address both group selection and
bi-level selection methods. We describe several applications of these methods
in nonparametric additive models, semiparametric regression, seemingly
unrelated regressions, genomic data analysis and genome wide association
studies. We also highlight some issues that require further study.Comment: Published in at http://dx.doi.org/10.1214/12-STS392 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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