This paper is focused on the statistical analysis of data consisting of a collection of multiple series of probability measures that are indexed by distinct time instants and supported over a bounded interval of the real line. By modeling these time-dependent probability measures as random objects in the Wasserstein space, we propose a new auto-regressive model for the statistical analysis of multivariate distributional time series. Using the theory of iterated random function systems, results on the second order stationarity of the solution of such a model are provided. We also propose a consistent estimator for the auto-regressive coefficients of this model. Due to the simplex constraints that we impose on the model coefficients, the proposed estimator that is learned under these constraints, naturally has a sparse structure. The sparsity allows the application of the proposed model in learning a graph of temporal dependency from multivariate distributional time series. We explore the numerical performances of our estimation procedure using simulated data. To shed some light on the benefits of our approach for real data analysis, we also apply this methodology to two data sets, respectively made of observations from age distribution in different countries and those from the bike sharing network in Paris.

翻译：本文聚焦于对由多个概率测度序列组成的数据进行统计分析，这些序列由不同时间点索引，并支撑于实轴的有界区间上。通过将这些随时间变化的概率测度建模为Wasserstein空间中的随机对象，我们提出了一种用于多变量分布时间序列统计分析的新型自回归模型。利用迭代随机函数系统理论，我们给出了该模型解的二阶平稳性结果。同时，我们提出了该模型自回归系数的一致性估计量。由于对模型系数施加了单纯形约束，在此约束下学习的估计量天然具有稀疏结构。这种稀疏性使得所提模型能够应用于从多变量分布时间序列中学习时间依赖图。我们通过模拟数据探究了估计程序的数值性能。为阐明本方法在实际数据分析中的优势，我们还将该技术应用于两个数据集：分别包含不同国家年龄分布的观测数据以及巴黎共享单车网络的观测数据。