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The manuscript considers multivariate functional data analysis with a known graphical model among the functional variables representing their conditional relationships (e.g., brain region-level fMRI data with a prespecified connectivity graph among brain regions). Functional Gaussian graphical models (GGM) used for analyzing multivariate functional data customarily estimate an unknown graphical model, and cannot preserve knowledge of a given graph. We propose a method for 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 recently for constructing optimal multivariate covariance functions that retain a given graphical model. The theoretical connection helps to design a new algorithm that leverages Dempster’s covariance selection for obtaining 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 respecting the graph and retaining computational scalability. The benefits of the proposed algorithms are illustrated using empirical experiments and a neuroimaging application.
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Robust Inverse Regression for Multivariate Elliptical Functional Data
Functional data have received significant attention as they frequently appear in modern applications, such as functional magnetic resonance imaging (fMRI) and natural language processing. The infinite-dimensional nature of functional data makes it necessary to use dimension reduction techniques. Most existing techniques, however, rely on the covariance operator, which can be affected by heavy-tailed data and unusual observations. Therefore, in this paper, we consider a robust sliced inverse regression for multivariate elliptical functional data. For that reason, we introduce a new statistical linear operator, called the conditional spatial sign Kendall’s tau covariance operator, which can be seen as an extension of the multivariate Kendall’s tau to both the conditional and functional settings. The new operator is robust to heavy-tailed data and outliers, and hence can provide a robust estimate of the sufficient predictors. We also derive the convergence rates of the proposed estimators for both completely and partially observed data. Finally, we demonstrate the finite sample performance of our estimator using simulation examples and a real dataset based on fMRI.
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- Award ID(s):
- 2213140
- PAR ID:
- 10529032
- Publisher / Repository:
- Statistica Sinica
- Date Published:
- Journal Name:
- Statistica Sinica
- ISSN:
- 1017-0405
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.5705/ss.202023.0341
