We develop a framework of canonical correlation analysis for distribution-valued functional data within the geometry of Wasserstein spaces. Specifically, we formulate an intrinsic concept of correlation between random distributions, propose estimation methods based on functional principal component analysis (FPCA) and Tikhonov regularization, respectively, for the correlation and its corresponding weight functions, and establish the minimax convergence rates of the estimators. The key idea is to extend the framework of tensor Hilbert spaces to distribution-valued functional data to overcome the challenging issue raised by nonlinearity of Wasserstein spaces. The finite-sample performance of the proposed estimators is illustrated via simulation studies, and the practical merit is demonstrated via a study on the association of distributions of brain activities between two brain regions.

翻译：我们为瓦森斯坦空间的几何内分布价值的功能数据开发了一种典型关联分析框架。具体地说,我们制定了随机分布之间相互关系的内在概念,分别根据功能性主要组成部分分析(FCCA)和Tikhonov的正规化,就相关性及其相应的重量功能提出了估算方法,并确定了估测员的小型趋同率。关键的想法是将Souror Hilbert空间框架扩展至分布价值的功能数据,以克服瓦西尔斯坦空间非线性引起的具有挑战性的问题。通过模拟研究说明了拟议测算员的有限性能,并通过研究两个脑区域之间脑活动分布的关联,展示了实际优点。