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Title: Spherical autoregressive models, with application to distributional and compositional time series Authors:  Changbo Zhu - University of California, Davis (United States) [presenting]
Hans-Georg Mueller - University of California Davis (United States)
Abstract: A new class of autoregressive models is introduced for spherical time series, where the dimension of the spheres on which the observations of the time series are situated may be finite-dimensional or infinite-dimensional as in the case of a general Hilbert sphere. Spherical time series arise in various settings. We focus on distributional and compositional time series. Applying a square root transformation to the densities of the observations of a distributional time series maps the distributional observations to the Hilbert sphere, equipped with the Fisher-Rao metric. The challenge in modeling such time series lies in the intrinsic non-linearity of spheres and Hilbert spheres. To address this difficulty, we consider rotation operators to map observations on the sphere. Specifically, we introduce a class of skew-symmetric operators such that the associated exponential operators are rotation operators that for each given pair of points on the sphere map one of the points to the other one. We exploit the fact that the space of skew-symmetric operators is Hilbertian to develop autoregressive modeling of geometric differences that correspond to rotations of spherical and distributional time series. We showcase the models with a time series of yearly observations of bivariate distributions of the minimum/maximum temperatures for a period of 120 days during each summer for the years 1990-2018 at Los Angeles (LAX) and John F. Kennedy (JFK) international airports.