3,306 research outputs found
Principal Fitted Components for Dimension Reduction in Regression
We provide a remedy for two concerns that have dogged the use of principal
components in regression: (i) principal components are computed from the
predictors alone and do not make apparent use of the response, and (ii)
principal components are not invariant or equivariant under full rank linear
transformation of the predictors. The development begins with principal fitted
components [Cook, R. D. (2007). Fisher lecture: Dimension reduction in
regression (with discussion). Statist. Sci. 22 1--26] and uses normal models
for the inverse regression of the predictors on the response to gain reductive
information for the forward regression of interest. This approach includes
methodology for testing hypotheses about the number of components and about
conditional independencies among the predictors.Comment: Published in at http://dx.doi.org/10.1214/08-STS275 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Critical Behavior of a Three-State Potts Model on a Voronoi Lattice
We use the single-histogram technique to study the critical behavior of the
three-state Potts model on a (random) Voronoi-Delaunay lattice with size
ranging from 250 to 8000 sites. We consider the effect of an exponential decay
of the interactions with the distance,, with , and
observe that this system seems to have critical exponents and
which are different from the respective exponents of the three-state Potts
model on a regular square lattice. However, the ratio remains
essentially the same. We find numerical evidences (although not conclusive, due
to the small range of system size) that the specific heat on this random system
behaves as a power-law for and as a logarithmic divergence for
and Comment: 3 pages, 5 figure
Testing predictor contributions in sufficient dimension reduction
We develop tests of the hypothesis of no effect for selected predictors in
regression, without assuming a model for the conditional distribution of the
response given the predictors. Predictor effects need not be limited to the
mean function and smoothing is not required. The general approach is based on
sufficient dimension reduction, the idea being to replace the predictor vector
with a lower-dimensional version without loss of information on the regression.
Methodology using sliced inverse regression is developed in detail
Contour projected dimension reduction
In regression analysis, we employ contour projection (CP) to develop a new
dimension reduction theory. Accordingly, we introduce the notions of the
central contour subspace and generalized contour subspace. We show that both of
their structural dimensions are no larger than that of the central subspace
Cook [Regression Graphics (1998b) Wiley]. Furthermore, we employ CP-sliced
inverse regression, CP-sliced average variance estimation and CP-directional
regression to estimate the generalized contour subspace, and we subsequently
obtain their theoretical properties. Monte Carlo studies demonstrate that the
three CP-based dimension reduction methods outperform their corresponding
non-CP approaches when the predictors have heavy-tailed elliptical
distributions. An empirical example is also presented to illustrate the
usefulness of the CP method.Comment: Published in at http://dx.doi.org/10.1214/08-AOS679 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Estimating sufficient reductions of the predictors in abundant high-dimensional regressions
We study the asymptotic behavior of a class of methods for sufficient
dimension reduction in high-dimension regressions, as the sample size and
number of predictors grow in various alignments. It is demonstrated that these
methods are consistent in a variety of settings, particularly in abundant
regressions where most predictors contribute some information on the response,
and oracle rates are possible. Simulation results are presented to support the
theoretical conclusion.Comment: Published in at http://dx.doi.org/10.1214/11-AOS962 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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