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Abstract:
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Functional data analysis provides tools for the analysis of data samples that can be viewed as being generated by repeated realizations of an underlying stochastic process. The application of this methodology to multivariate processes will be illustrated by quantifying resting state fMRI connectivity through functional covariance measures between all pairs of time signals, extending the notion of a covariance matrix for multivariate data to the functional case. We generalize the straightforward approach of integrating pointwise covariance matrices over the time domain by defining the Fr\'echet integral, which depends on the metric chosen for the space of covariance matrices. We discuss a class of power metrics, for which the Fr\'echet integral is computed by transforming the covariance matrices with the chosen power, applying the classical Riemann integral and finally applying the inverse transformation to return to the original scale. We also propose data-adaptive metric selection, and illustrate its utility via a comparison of connectivity between brain voxels for normal subjects and Alzheimer's patients based on fMRI data.
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