1,721 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
Randomized Dimension Reduction on Massive Data
Scalability of statistical estimators is of increasing importance in modern
applications and dimension reduction is often used to extract relevant
information from data. A variety of popular dimension reduction approaches can
be framed as symmetric generalized eigendecomposition problems. In this paper
we outline how taking into account the low rank structure assumption implicit
in these dimension reduction approaches provides both computational and
statistical advantages. We adapt recent randomized low-rank approximation
algorithms to provide efficient solutions to three dimension reduction methods:
Principal Component Analysis (PCA), Sliced Inverse Regression (SIR), and
Localized Sliced Inverse Regression (LSIR). A key observation in this paper is
that randomization serves a dual role, improving both computational and
statistical performance. This point is highlighted in our experiments on real
and simulated data.Comment: 31 pages, 6 figures, Key Words:dimension reduction, generalized
eigendecompositon, low-rank, supervised, inverse regression, random
projections, randomized algorithms, Krylov subspace method
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