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General Departmental Seminar Series

Hongtu Zhu
Department of Biostatistics,
University of North Carolina

Friday, October 3rd, 2008
12:00 pm - 1:00 pm
5275 MSC

Intrinsic Semiparametric and Nonparametric Models for Positive-definite
Matrices with Applications to Diffusion Tensor Imaging

We develop intrinsic semiparametric and nonparametric models for the analysis of positive-definite matrices as responses in a Riemannian manifold and their association with a set of covariates, such as age and gender, in a Euclidean space. The primary motivation and application of the proposed methodology is in medical imaging. Because the set of positive-definite matrices do not form a vector space, directly applying classical multivariate regression may be inadequate in establishing the relationship between positive-definite matrices and covariates of interest, such as age and gender, in real applications.

Our intrinsic regression model, as a semiparametric model, uses a link function to map from the Euclidean space of covariates to the Riemannian manifold of positive-definite matrices. We develop an estimation procedure to calculate parameter estimates and establish their limiting distribution. We develop score statistics to test linear hypotheses on unknown parameters and develop a test procedure based on a resampling method to simultaneously assess the statistical significance of linear hypotheses across a large region of interest.

Simulation studies are used to demonstrate the methodology and examine the finite performance of the test procedure for controlling the family-wise error rate. We apply our methods to the detection of statistical significance of diagnostic effects on the integrity of white matter in a diffusion tensor study of human immunodeficiency virus.

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