We develop a statistical framework for conducting inference on collections of time-varying covariance operators (covariance flows) over a general, possibly infinite dimensional, Hilbert space. We model the intrinsically non-linear structure of covariances by means of the Bures-Wasserstein metric geometry. We make use of the Riemmanian-like structure induced by this metric to define a notion of mean and covariance of a random flow, and develop an associated Karhunen-Lo\`eve expansion. We then treat the problem of estimation and construction of functional principal components from a finite collection of covariance flows, observed fully or irregularly. Our theoretical results are motivated by modern problems in functional data analysis, where one observes operator-valued random processes -- for instance when analysing dynamic functional connectivity and fMRI data, or when analysing multiple functional time series in the frequency domain. Nevertheless, our framework is also novel in the finite-dimensions (matrix case), and we demonstrate what simplifications can be afforded then. We illustrate our methodology by means of simulations and data analyses.

翻译：我们建立了一个统计框架，用于在一般（可能无限维）希尔伯特空间上对时变协方差算子集合（协方差流）进行统计推断。我们利用Bures-Wasserstein度量几何来建模协方差固有的非线性结构。通过该度量诱导的类黎曼结构，我们定义了随机流的均值与协方差概念，并建立了相应的Karhunen-Loève展开。随后，我们处理从有限个协方差流（完整观测或不规则观测）中估计和构建函数型主成分的问题。我们的理论成果受到现代函数型数据分析中实际问题的驱动——例如分析动态功能连接性与fMRI数据时，或分析频域中的多元函数型时间序列时，常会观测到算子值随机过程。尽管如此，该框架在有限维情形（矩阵情形）中同样具有新颖性，我们展示了此时可实现的简化。我们通过模拟与数据分析展示了所提方法。