This paper presents a Bayesian reformulation of covariate-assisted principal (CAP) regression of Zhao et al. (2021), which aims to identify components in the covariance of response signal that are associated with covariates in a regression framework. We introduce a geometric formulation and reparameterization of individual covariance matrices in their tangent space. By mapping the covariance matrices to the tangent space, we leverage Euclidean geometry to perform posterior inference. This approach enables joint estimation of all parameters and uncertainty quantification within a unified framework, fusing dimension reduction for covariance matrices with regression model estimation. We validate the proposed method through simulation studies and apply it to analyze associations between covariates and brain functional connectivity, utilizing data from the Human Connectome Project.

翻译：本文提出了Zhao等人(2021)提出的协变量辅助主回归方法的贝叶斯重表述，旨在识别响应信号协方差中与回归框架下协变量相关的成分。我们引入了单个协方差矩阵在切空间中的几何表述和重新参数化。通过将协方差矩阵映射到切空间，我们利用欧几里得几何进行后验推断。该方法能够在一个统一框架内联合估计所有参数并进行不确定性量化，将协方差矩阵的降维与回归模型估计相结合。我们通过模拟研究验证了所提方法，并利用人类连接组项目的数据，将其应用于分析协变量与脑功能连接之间的关联。