The covariance structure of multivariate functional data can be highly complex, especially if the multivariate dimension is large, making extensions of statistical methods for standard multivariate data to the functional data setting challenging. For example, Gaussian graphical models have recently been extended to the setting of multivariate functional data by applying multivariate methods to the coefficients of truncated basis expansions. However, a key difficulty compared to multivariate data is that the covariance operator is compact, and thus not invertible. The methodology in this paper addresses the general problem of covariance modeling for multivariate functional data, and functional Gaussian graphical models in particular. As a first step, a new notion of separability for the covariance operator of multivariate functional data is proposed, termed partial separability, leading to a novel Karhunen-Lo\`eve-type expansion for such data. Next, the partial separability structure is shown to be particularly useful in order to provide a well-defined functional Gaussian graphical model that can be identified with a sequence of finite-dimensional graphical models, each of identical fixed dimension. This motivates a simple and efficient estimation procedure through application of the joint graphical lasso. Empirical performance of the method for graphical model estimation is assessed through simulation and analysis of functional brain connectivity during a motor task. %Empirical performance of the method for graphical model estimation is assessed through simulation and analysis of functional brain connectivity during a motor task.
翻译:多变量功能数据共变结构可能非常复杂,特别是如果多变量维度巨大,则多变量功能数据共变结构可能非常复杂,使标准多变量数据的统计方法扩展至功能性数据设置具有挑战性。例如,高斯图形模型最近通过应用多变量基扩展系数的多变量方法,扩展至多变量功能数据的多变量功能性数据设置。然而,与多变量数据相比的一个关键困难是,共变量操作器是紧凑的,因此不可倒置。本文件中的方法解决了多变量功能数据共变模型和功能性高斯图形模型的一般问题,特别是功能性高斯图形模型。第一步,提出了多变量功能性数据共变操作者可分离的新概念,称为部分分离方法,导致这类数据出现新颖的Karhunen-Lo ⁇ eve-类型扩展。此外,部分分离性模型被证明特别有用,以便提供一个定义明确的功能性高斯图形模型,可以与多变量功能性功能性数据模型相匹配,特别是功能性图形模型排序,可被确定为可计量的多变量图形化图形模型,每个是通用的模型,通过通用的模型,通过模型,通过数字性分析方法,通过数字性估算,通过数字性能评估,通过通用性能评估。