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Abstract:
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In recent years, samples of time-varying object data such as time-varying networks that are not in a vector space have been increasingly collected. These data can be viewed as elements of a general metric space that lacks local or global linear structure and therefore common approaches that have been used with great success for the analysis of functional data cannot be applied directly. In this talk, I will propose some recent advances along this direction. First, I will discuss ways to obtain dominant modes of variations in time varying object data using metric covariance, a novel association measure for paired object data lying in a metric space, that we use to define a metric auto-covariance function for a sample of random object-valued curves. The eigenfunctions of the metric auto-covariance operator can be used as building blocks for an object functional principal component analysis. Finally I will describe Fréchet integrals which serve as analogues of functional principal components for time-varying objects and are obtained by applying Fréchet means and projections of distance functions of the random object trajectories in the directions of the eigenfunctions.
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