Additive Models for Symmetric Positive-Definite Matrices and Lie Groups
Additive Models for Symmetric Positive-Definite Matrices and Lie Groups
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DOI：
10.1093/biomet/asac055
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发表时间：
2022-09
期刊：
影响因子：
2.7
通讯作者：
Z. Lin;H. Müller;B. U. Park
中科院分区：
文献类型：
--
作者：
Z. Lin;H. Müller;B. U. Park
本文提出并研究了一个对称正定矩阵值响应和多个标量预测变量的加性回归模型。该模型利用了从对称正定矩阵的对数乔莱斯基和对数欧几里德框架继承的阿贝尔群结构，并自然地扩展到一般阿贝尔李群。所提出的可加性模型被示出为连接到切空间上的可加性模型。这种连接不仅需要一个有效的算法来估计的分量函数，但也允许一个一般的黎曼流形的建议添加剂模型推广。最优渐近收敛速度和估计的分量函数的正态性和数值研究表明，该模型具有良好的数值性能，不受维数灾难时，有多个预测。通过对脑弥散张量成像数据的分析，证明了该模型的实用价值。
We propose and investigate an additive regression model for symmetric positive-definite matrix valued responses and multiple scalar predictors. The model exploits the abelian group structure inherited from either of the log-Cholesky and log-Euclidean frameworks for symmetric positive-definite matrices and naturally extends to general abelian Lie groups. The proposed additive model is shown to connect to an additive model on a tangent space. This connection not only entails an efficient algorithm to estimate the component functions but also allows one to generalize the proposed additive model to general Riemannian manifolds. Optimal asymptotic convergence rates and normality of the estimated component functions are established and numerical studies show that the proposed model enjoys good numerical performance and is not subject to the curse of dimensionality when there are multiple predictors. The practical merits of the proposed model are demonstrated through an analysis of brain diffusion tensor imaging data.