Spherical autoregressive models, with application to distributional and compositional time series
Spherical autoregressive models, with application to distributional and compositional time series
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球形自回归模型，适用于分布和组合时间序列
DOI：
10.1016/j.jeconom.2022.12.008
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发表时间：
2023
影响因子：
6.3
通讯作者：
Müller, Hans-Georg
中科院分区：
文献类型：
--
作者：
Zhu, Changbo;Müller, Hans-Georg
本文介绍了一类新的球面时间序列自回归模型。时间序列的观测值所在的球面的维数可以是有限维或无限维的，在后一种情况下，我们考虑希尔伯特球面。球形时间序列出现在各种设置中。我们在这里集中在分布和组成的时间序列。将平方根变换应用于分布时间序列的观测值的密度，将分布观测值映射到配备有Fisher-Rao度量的希尔伯特球。同样地，将平方根变换应用于组合时间序列的观测的分量，将组合观测映射到有限维球体，配备球体上的测地线度量。建模这种时间序列的挑战在于球体和希尔伯特球体的固有非线性，其中传统的算术运算，如加法或标量乘法不再可用。为了解决这一困难，我们考虑旋转算子映射观察球。具体来说，我们引入一类反对称算子，使得相关的指数算子是旋转算子，对于球面上的每个给定的点对，将该点对的第一点映射到该点对的第二点。我们利用这一事实，即空间的反对称运营商是希尔伯特开发自回归建模的几何差异，对应于旋转的球形和分布的时间序列。用旋转表示的差异可以在Fréchet平均值和观测值之间或时间序列的连续观测值之间进行。我们推导出随后的自回归模型的理论属性，并展示了这些方法与几个激励数据。其中包括1990-2018年洛杉矶（LAX）和John F.肯尼迪（JFK）国际机场第二个数据应用涉及一个组成的时间序列，每年观察美国发电的能源组成。
We introduce a new class of autoregressive models for spherical time series. The dimension of the spheres on which the observations of the time series are situated may be finite-dimensional or infinite-dimensional, where in the latter case we consider the 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 operators such that the associated exponential operators are rotation operators that for each given pair of points on the sphere map the first point of the pair to the second point of the pair. 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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影响因子：
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DOI：
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期刊：
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