2,251 research outputs found
Fisher Lecture: Dimension Reduction in Regression
Beginning with a discussion of R. A. Fisher's early written remarks that
relate to dimension reduction, this article revisits principal components as a
reductive method in regression, develops several model-based extensions and
ends with descriptions of general approaches to model-based and model-free
dimension reduction in regression. It is argued that the role for principal
components and related methodology may be broader than previously seen and that
the common practice of conditioning on observed values of the predictors may
unnecessarily limit the choice of regression methodology.Comment: This paper commented in: [arXiv:0708.3776], [arXiv:0708.3777],
[arXiv:0708.3779]. Rejoinder in [arXiv:0708.3781]. Published at
http://dx.doi.org/10.1214/088342306000000682 in the Statistical Science
(http://www.imstat.org/sts/) by the Institute of Mathematical Statistics
(http://www.imstat.org
Penalized single-index quantile regression
This article is made available through the Brunel Open Access Publishing Fund. Copyright for this article is retained by the author(s), with first publication rights granted to the journal.
This is an open-access article distributed under the terms and conditions of the Creative Commons Attribution
license (http://creativecommons.org/licenses/by/3.0/).The single-index (SI) regression and single-index quantile (SIQ) estimation methods product linear combinations of all the original predictors. However, it is possible that there are many unimportant predictors within the original predictors. Thus, the precision of parameter estimation as well as the accuracy of prediction will be effected by the existence of those unimportant predictors when the previous methods are used. In this article, an extension of the SIQ method of Wu et al. (2010) has been proposed, which considers Lasso and Adaptive Lasso for estimation and variable selection. Computational algorithms have been developed in order to calculate the penalized SIQ estimates. A simulation study and a real data application have been used to assess the performance of the methods under consideration
High-Dimensional Regression with Gaussian Mixtures and Partially-Latent Response Variables
In this work we address the problem of approximating high-dimensional data
with a low-dimensional representation. We make the following contributions. We
propose an inverse regression method which exchanges the roles of input and
response, such that the low-dimensional variable becomes the regressor, and
which is tractable. We introduce a mixture of locally-linear probabilistic
mapping model that starts with estimating the parameters of inverse regression,
and follows with inferring closed-form solutions for the forward parameters of
the high-dimensional regression problem of interest. Moreover, we introduce a
partially-latent paradigm, such that the vector-valued response variable is
composed of both observed and latent entries, thus being able to deal with data
contaminated by experimental artifacts that cannot be explained with noise
models. The proposed probabilistic formulation could be viewed as a
latent-variable augmentation of regression. We devise expectation-maximization
(EM) procedures based on a data augmentation strategy which facilitates the
maximum-likelihood search over the model parameters. We propose two
augmentation schemes and we describe in detail the associated EM inference
procedures that may well be viewed as generalizations of a number of EM
regression, dimension reduction, and factor analysis algorithms. The proposed
framework is validated with both synthetic and real data. We provide
experimental evidence that our method outperforms several existing regression
techniques
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