Partial Separability and Functional Graphical Models for Multivariate Gaussian Processes

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 modelling 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–Loève-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.

A kernel method for higher temporal resolution MRI using the partial separability (PS) model

The partial separability (PS) model for spatiotemporal signals has been exploited effectively for sparse (k, t)-space sampling in dynamic magnetic resonance imaging (MRI). However, the training data for defining the temporal subspace is reordered by using a projection strategy in the conventional PS model-based method, which results in a suboptimal temporal resolution imaging. To address this issue, a kernel method was presented in this work to reorder the training data to realize a higher temporal resolution MRI. Numerical simulation results show that the MRI temporal resolution could be further improved and the dynamic change of motion object could be accurately captured by the proposed method. In vivo cardiac cine MRI results demonstrate that the proposed method can reconstruct better MR images with higher temporal resolution (up to 8.4 ms per snapshot). This study may find use in ultra-high resolution dynamic MRI.