  We introduce a new class of autoregressive models 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 here 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. Likewise, applying a square root
transformation to the components of the observations of a compositional time
series maps the compositional observations to a finite-dimensional sphere,
equipped with the geodesic metric on spheres. The challenge in modeling such
time series lies in the intrinsic non-linearity of spheres and Hilbert spheres,
where conventional arithmetic operations such as addition or scalar
multiplication are no longer available. To address this difficulty, we consider
rotation operators to map observations on the sphere. Specifically, we
introduce a class of skew-symmetric operator 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. Motivating data for our methods include 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.
