612 research outputs found
Contour regression: A general approach to dimension reduction
We propose a novel approach to sufficient dimension reduction in regression,
based on estimating contour directions of small variation in the response.
These directions span the orthogonal complement of the minimal space relevant
for the regression and can be extracted according to two measures of variation
in the response, leading to simple and general contour regression (SCR and GCR)
methodology. In comparison with existing sufficient dimension reduction
techniques, this contour-based methodology guarantees exhaustive estimation of
the central subspace under ellipticity of the predictor distribution and mild
additional assumptions, while maintaining \sqrtn-consistency and computational
ease. Moreover, it proves robust to departures from ellipticity. We establish
population properties for both SCR and GCR, and asymptotic properties for SCR.
Simulations to compare performance with that of standard techniques such as
ordinary least squares, sliced inverse regression, principal Hessian directions
and sliced average variance estimation confirm the advantages anticipated by
the theoretical analyses. We demonstrate the use of contour-based methods on a
data set concerning soil evaporation.Comment: Published at http://dx.doi.org/10.1214/009053605000000192 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Sparse Fr\'echet Sufficient Dimension Reduction with Graphical Structure Among Predictors
Fr\'echet regression has received considerable attention to model
metric-space valued responses that are complex and non-Euclidean data, such as
probability distributions and vectors on the unit sphere. However, existing
Fr\'echet regression literature focuses on the classical setting where the
predictor dimension is fixed, and the sample size goes to infinity. This paper
proposes sparse Fr\'echet sufficient dimension reduction with graphical
structure among high-dimensional Euclidean predictors. In particular, we
propose a convex optimization problem that leverages the graphical information
among predictors and avoids inverting the high-dimensional covariance matrix.
We also provide the Alternating Direction Method of Multipliers (ADMM)
algorithm to solve the optimization problem. Theoretically, the proposed method
achieves subspace estimation and variable selection consistency under suitable
conditions. Extensive simulations and a real data analysis are carried out to
illustrate the finite-sample performance of the proposed method
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