统计学中微分几何的两例应用  

Two applications of differential geometry in statistics

作　　者：彭捷 Jie Peng

机构地区：[1]Department of Statistics,University of California Davis,Davis,CA 95616,USA

出　　处：《中国科学:数学》2025年第1期169-182,共14页Scientia Sinica:Mathematica

摘　　要：本文讨论两个统计问题,其中微分几何的概念和观念发挥了关键作用.第一个问题涉及函数型数据的主成分估计,其中一部分参数具有正交约束,因此寻找最大似然估计需要在Stiefel流形上进行优化.第二个问题涉及对从扩散磁共振成像数据中估计出来的扩散张量进行核平滑.由于这些张量是3×3的正定矩阵,所以平滑过程需要在3×3的正定矩阵空间上定义,并且依赖于这个空间上特定度量的选择.In this paper,we discuss two statistical problems where differential geometry concepts and notions play a crucial role.The first problem involves the estimation of principal components of functional data where a part of the parameters have orthogonal constraints and thus finding the maximum likelihood estimator requires optimization on a Stiefel manifold.In the second problem,the goal is to smooth diffusion tensors estimated from diffusion MRI measurements.Since these tensors are 33 positive definite matrices,smoothing needs to be defined on the space of 3×3 positive definite matrices and depends on the specific choice of metric on this space.

关 键 词：Stiefel流形 典型度量 NEWTON法 函数型数据主成分分析 自然约化齐性空间 扩散张量 核平滑 

分 类 号：O186.1[理学—数学]