123,916 research outputs found
Deciding the dimension of effective dimension reduction space for functional and high-dimensional data
In this paper, we consider regression models with a Hilbert-space-valued
predictor and a scalar response, where the response depends on the predictor
only through a finite number of projections. The linear subspace spanned by
these projections is called the effective dimension reduction (EDR) space. To
determine the dimensionality of the EDR space, we focus on the leading
principal component scores of the predictor, and propose two sequential
testing procedures under the assumption that the predictor has an
elliptically contoured distribution. We further extend these procedures and
introduce a test that simultaneously takes into account a large number of
principal component scores. The proposed procedures are supported by theory,
validated by simulation studies, and illustrated by a real-data example. Our
methods and theory are applicable to functional data and high-dimensional
multivariate data.Comment: Published in at http://dx.doi.org/10.1214/10-AOS816 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Tensor Regression with Applications in Neuroimaging Data Analysis
Classical regression methods treat covariates as a vector and estimate a
corresponding vector of regression coefficients. Modern applications in medical
imaging generate covariates of more complex form such as multidimensional
arrays (tensors). Traditional statistical and computational methods are proving
insufficient for analysis of these high-throughput data due to their ultrahigh
dimensionality as well as complex structure. In this article, we propose a new
family of tensor regression models that efficiently exploit the special
structure of tensor covariates. Under this framework, ultrahigh dimensionality
is reduced to a manageable level, resulting in efficient estimation and
prediction. A fast and highly scalable estimation algorithm is proposed for
maximum likelihood estimation and its associated asymptotic properties are
studied. Effectiveness of the new methods is demonstrated on both synthetic and
real MRI imaging data.Comment: 27 pages, 4 figure
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