Statistical inference on the Hilbert sphere with application to random densities

Dai, Xiongtao

doi:10.1214/21-ejs1942

Cited by 7 publications

(1 citation statement)

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“…Using fpsr, distributional data correspond to the elements of a segment of the Hilbert sphere S ∞ equipped with the Fisher-Rao metric (Dai 2022) and distributional time series then are accordingly represented as S ∞ -valued time series. For two-dimensional distributions, of daily maximum and minimum temperatures recorded for 24 hours over the summer months at airports in the U.S.…”

Section: Introductionmentioning

confidence: 99%

Spherical Autoregressive Models, With Application to Distributional and Compositional Time Series

Zhu¹,

Müller²

2022

Preprint

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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. Differences expressed in terms of rotations can be taken between the Fréchet mean and the observations or between consecutive observations of the time series. We derive theoretical properties of the ensuing autoregressive models and showcase these approaches with several motivating data. These 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. A second data application concerns a compositional time series with annual observations of compositions of energy sources for power generation in the U.S..

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Section: Introductionmentioning

confidence: 99%