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Abstract:
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Motived by functional magnetic resonance imaging studies, we study the effect of clinical/demographic variables on the dynamic functional structures. To this end, we propose the supervised principal component regression for functional data with possibly high dimensional clinical variables. Compared with its classical counterpart, the principal component regression, the proposed methodology relies on a newly proposed integrated residual sum of squares for functional data and makes use of the association information directly. It can be formulated as a sequence of nonconvex generalized Rayleigh quotient optimization problems, which turn out to be NP-hard and thus computational intractable. Utilizing the invariance property of linear subspaces under rotations, we then reformulate the problem into a simultaneous sparse linear regression problem. Formally, we show that the reformulated problem can recover the same subspace as if the original sequence of nonconvex problems were solved. Statistically, the rate of convergence for the obtained estimators is established. Numerical studies and an application to the Human Connectome Project lend further support to the proposed methodology.
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