Series of univariate distributions indexed by equally spaced time points are ubiquitous in applications and their analysis constitutes one of the challenges of the emerging field of distributional data analysis. To quantify such distributional time series, we propose a class of intrinsic autoregressive models that operate in the space of optimal transport maps. The autoregressive transport models that we introduce here are based on regressing optimal transport maps on each other, where predictors can be transport maps from an overall barycenter to a current distribution or transport maps between past consecutive distributions of the distributional time series. Autoregressive transport models and their associated distributional regression models specify the link between predictor and response transport maps by moving along geodesics in Wasserstein space. These models emerge as natural extensions of the classical autoregressive models in Euclidean space. Unique stationary solutions of autoregressive transport models are shown to exist under a geometric moment contraction condition of Wu and Shao (2004), using properties of iterated random functions. We also discuss an extension to a varying coefficient model for first order autoregressive transport models. In addition to simulations, the proposed models are illustrated with distributional time series of house prices across U.S. counties and annual summer temperature distributions.

翻译：在等间距时间点上索引的单变量分布序列在应用中无处不在，其分析构成了分布数据分析这一新兴领域的挑战之一。为定量刻画此类分布时间序列，我们提出了一类在最优传输映射空间中运行的内在自回归模型。本文引入的自回归传输模型基于最优传输映射之间的回归：预测变量可以是自整体重心到当前分布的传输映射，也可以是分布时间序列中前后连续分布之间的传输映射。自回归传输模型及其相关的分布回归模型通过沿Wasserstein空间中的测地线移动来指定预测变量与响应变量传输映射之间的关联。这些模型可视为欧氏空间中经典自回归模型的自然推广。基于Wu与Shao（2004）的几何矩收缩条件，利用迭代随机函数的性质，我们证明了自回归传输模型存在唯一的平稳解。我们还讨论了针对一阶自回归传输模型的变系数模型扩展。除数值模拟外，所提模型还通过美国各县房价分布时间序列及年夏季气温分布实例进行了验证。