Spherical Autoregressive Models, With Application to Distributional and Compositional Time Series
球面自回归模型及其在分布与成分时间序列中的应用AI 解读 ↓
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球面自回归模型及其在分布与成分时间序列中的应用
- 研究问题
- 当时间序列观测为概率分布或成分向量时，如何在球面的非线性几何结构中定义动态变化并建立自回归模型？
- 方法与思路
- 作者以平方根变换将观测映射到有限维或 Hilbert 球面，用反对称算子的指数表示旋转，再对旋转对应的几何差异建模。
- 材料陈述的贡献
- 摘要称，提出适用于有限维和无限维球面的一类自回归模型，为分布及成分序列提供几何建模框架，并以机场气温数据为动机。
- 阅读关注点
- 应核对旋转算子的选取与唯一性、几何差异的定义，以及自回归参数的估计、动态性质和气温分布的构造方式。
- 阅读边界
- 这是 arXiv 预印本，不能据此认定已经同行评审或期刊接收；摘要未说明估计条件、预测评价或实证结果，不能判断相对方法表现。
原文摘要
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.