Functional Gaussian graphical models (GGM) used for analyzing multivariate functional data customarily estimate an unknown graphical model representing the conditional relationships between the functional variables. However, in many applications of multivariate functional data, the graph is known and existing functional GGM methods cannot preserve a given graphical constraint. In this manuscript, we demonstrate how to conduct multivariate functional analysis that exactly conforms to a given inter-variable graph. We first show the equivalence between partially separable functional GGM and graphical Gaussian processes (GP), proposed originally for constructing optimal covariance functions for multivariate spatial data that retain the conditional independence relations in a given graphical model. The theoretical connection help design a new algorithm that leverages Dempster's covariance selection to calculate the maximum likelihood estimate of the covariance function for multivariate functional data under graphical constraints. We also show that the finite term truncation of functional GGM basis expansion used in practice is equivalent to a low-rank graphical GP, which is known to oversmooth marginal distributions. To remedy this, we extend our algorithm to better preserve marginal distributions while still respecting the graph and retaining computational scalability. The insights obtained from the new results presented in this manuscript will help practitioners better understand the relationship between these graphical models and in deciding on the appropriate method for their specific multivariate data analysis task. The benefits of the proposed algorithms are illustrated using empirical experiments and an application to functional modeling of neuroimaging data using the connectivity graph among regions of the brain.

翻译：用于分析多变量函数数据的函数高斯图模型（GGM）通常估计一个未知的图模型，以表示函数变量之间的条件关系。然而，在许多多变量函数数据的应用中，图是已知的，而现有的函数GGM方法无法保持给定的图约束。在本手稿中，我们展示了如何执行严格符合给定变量间图的多变量函数分析。我们首先证明了部分可分离的函数GGM与图高斯过程（GP）之间的等价性，后者最初用于构建多变量空间数据的最优协方差函数，并保留给定图模型中的条件独立关系。这一理论联系有助于设计一种新算法，该算法利用Dempster协方差选择，在图的约束下计算多变量函数数据协方差函数的最大似然估计。我们还表明，实践中使用的函数GGM基展开的有限项截断等价于低秩图GP，而后者已知会过度平滑边际分布。为解决这一问题，我们扩展了算法，在保持图约束和计算可扩展性的同时，更好地保留边际分布。本手稿提出的新结果所获得的见解将帮助从业者更好地理解这些图模型之间的关系，并为其特定的多变量数据分析任务选择适当的方法。通过实证实验以及利用大脑区域连接图进行神经影像数据函数建模的应用，展示了所提出算法的优势。