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Summary 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, compared with multivariate data, a key difficulty is that the covariance operator is compact and thus not invertible. 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 providing 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 proposed method for graphical model estimation is assessed through simulation and analysis of functional brain connectivity during a motor task.
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Nonparanormal graph quilting with applications to calcium imaging
Abstract Probabilistic graphical models have become an important unsupervised learning tool for detecting network structures for a variety of problems, including the estimation of functional neuronal connectivity from two‐photon calcium imaging data. However, in the context of calcium imaging, technological limitations only allow for partially overlapping layers of neurons in a brain region of interest to be jointly recorded. In this case, graph estimation for the full data requires inference for edge selection when many pairs of neurons have no simultaneous observations. This leads to the graph quilting problem, which seeks to estimate a graph in the presence of block‐missingness in the empirical covariance matrix. Solutions for the graph quilting problem have previously been studied for Gaussian graphical models; however, neural activity data from calcium imaging are often non‐Gaussian, thereby requiring a more flexible modelling approach. Thus, in our work, we study two approaches for nonparanormal graph quilting based on the Gaussian copula graphical model, namely, a maximum likelihood procedure and a low rank‐based framework. We provide theoretical guarantees on edge recovery for the former approach under similar conditions to those previously developed for the Gaussian setting, and we investigate the empirical performance of both methods using simulations as well as real data calcium imaging data. Our approaches yield more scientifically meaningful functional connectivity estimates compared to existing Gaussian graph quilting methods for this calcium imaging data set.
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- PAR ID:
- 10468596
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- Stat
- Volume:
- 12
- Issue:
- 1
- ISSN:
- 2049-1573
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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