论文标题
球形自回旋模型，并应用于分布和组成时间序列
Spherical Autoregressive Models, With Application to Distributional and Compositional Time Series
论文作者
论文摘要
我们为球形时间序列介绍了新的自回归模型，其中时间序列的观察值的球的尺寸可能是有限的或无限二维的，例如希尔伯特·希尔伯特·史密斯（Hilbert Sphere）。球形时间序列在各种环境中出现。我们在这里专注于分布和组成时间序列。将平方根转换应用于分布时间序列观测值的密度，将分布观测值映射到配备Fisher-Rao公制的希尔伯特球体。同样，将平方根变换应用于组成时间序列的观测值的组成部分，将组成观测值映射到有限维球体，该球体配备了球体上的测量指标。建模这种时间序列的挑战在于球体和希尔伯特球体的内在非线性，在这些传统算术操作（例如加法或标量乘法）不再可用。为了解决这个困难，我们考虑旋转操作员在球体上绘制观测值。具体而言，我们介绍了一类偏斜的对称运算符，以使相关的指数运算符是旋转操作员，对于球体上的每个给定的点，旋转算子映射了一个点，一个点向另一个点。我们利用了一个事实，即偏斜算子的空间是希尔伯特式的，以开发与球形和分布时间序列旋转相对应的几何差异的自回归建模。激励我们的方法的数据包括每年一次的时间序列观察到1990 - 2018年在洛杉矶（LAX）和约翰·肯尼迪（John F. Kennedy）和肯尼迪（JFK）国际机场的每年夏季120天的最低/最高温度分布。
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.
