对称正定矩阵时间序列自回归模型的统计推断  

Statistical Inference of Autoregressive Model for Symmetric Positive-Definite Matrix Time Series

作　　者：胡盛铝 黄涛 HU Sheng-chang;HUANG Tao(School of Statistics and Management,Shanghai University of Finance and Economics,Shanghai 200433,China)

机构地区：[1]上海财经大学统计与管理学院,上海200433

出　　处：《数理统计与管理》2025年第2期239-252,共14页Journal of Applied Statistics and Management

基　　金：国家自然科学基金资助项目(11871323);国家自然科学基金资助项目(12371271);教育部社科基金(21YJA910001);上海财经大学创新团队项目;上海市数据科技与决策前沿科学研究基地。

摘　　要：对称正定矩阵作为非欧随机对象数据中的一种典型数据,具有对称性和正定性的特定约束,且处于黎曼流形当中,使得所有基于欧式度量空间而建立的传统统计模型、方法和理论都不再适用。为此,本文将结合黎曼流形的几何结构,引入对称正定矩阵的黎曼均值、方差、协方差的定义,并针对对称正定矩阵时间序列构建自回归模型,同时给出所提模型的渐近理论和预测方法,最后通过模拟研究和实际数据分析来展示所提模型和方法的有效性和实用性。Samples of dynamic or time-varying random object data such as symmetric positive definite(SPD)matrix time series are increasingly encountered in data science.Common time series methods are infeasible when observations are time courses of sPD matrices that are elements in Riemannian manifold.In this paper,we first introduce the definitions of Riemannian mean,variance and covariance of SPD matrix,and then propose an autoregressive model for SPD matrix time series.Asymptotic theory for the proposed model and forecasting method are also developed.We finally demonstrate the desirable properties of the proposed method via simulations and a real data analysis.

关 键 词：对称正定矩阵 随机对象数据 自回归模型 黎曼流形 切线空间 

分 类 号：O212[理学—概率论与数理统计]