Paromita Dubey, Assistant Professor, Data Sciences and Operations, USC Marshall School of Business
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, such as functional principal component analysis, 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. I will describe metric covariance, a new association measure for paired object data lying in a metric space (Ω, d) that we use to define a metric auto-covariance function for a sample of random Ω-valued curves, where Ω will not have a vector space or manifold structure. The proposed metric auto-covariance function is non-negative definite when the squared metric d^2 is of negative type. The eigenfunctions of the linear operator with the metric auto-covariance function as the kernel can be used as building blocks for an object functional principal component analysis for Ω-valued functional data, including time-varying probability distributions, covariance matrices and time-dynamic networks. Then I will describe how to obtain analogues of functional principal components for time-varying objects by applying weighted Fréchet means which serve as projections of the random object trajectories in the directions of the eigenfunctions, leading to Ω-valued Fréchet integrals. This talk is based on joint work with Hans-Georg Müller.