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B0725
Title: Fr\'echet integration and adaptive metric selection for covariance objects in functional data Authors:  Alexander Petersen - Brigham Young University (United States) [presenting]
Hans-Georg Mueller - University of California Davis (United States)
Abstract: For multivariate functional data recorded for a sample of subjects on a common domain, one is often interested in the covariance between pairs of the component functions. We generalize the straightforward approach of integrating the pointwise covariance matrices over the functional time domain by defining the Fr\'echet integral, which, in analogy to the Fr\'echet mean, depends on the metric chosen for the space of covariance matrices. This generalization is motivated by the class of power transformations on covariance matrices and the associated metrics that are more suitable for this nonlinear space. Data-adaptive metric selection with respect to a user-specified target criterion, for example fastest decline of the eigenvalues, is proposed. Asymptotic results of the functional covariance and optimal metric estimators are presented, and their practical utility is illustrated in a comparison of connectivity between brain voxels for normal subjects and Alzheimer's patients based on fMRI data.