Note on the Estimation of Embedded Hermitian Gaussian Graphical Models for MEEG Source Activity and Connectivity Analysis in the Frequency Domain. Part I: Single Frequency Comp
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This technical note presents an inference framework for hierarchically conditioned (embedded) Hermitian Gaussian Graphical Models (GGM). Our methodology, although extendible to several GGMs applications, is mainly centered on the specific neuroscientific context encompassing the estimation of brain sources activity and connectivity from Magnetoencephalography or Electroencephalography (MEEG) signals. This will be conceived in a series of papers with two parts. Part I: Analysis of a single frequency component from a single subject. Part II: Analysis of the Cross-Spectra for multiple subjects in a population. The strategy is based on Maximum Posterior Analysis, via the Expectation Maximization (EM) algorithm, of the Hermitian GGMs Parameters and Hyperparameters. The Hyperparameters set, which comprises MEEG signal noise covariance and source precision (connectivity) matrices of the MEEG generative model, is informed with Priors of special interest in the field and whose implementation in GGM estimation still poses a technical challenge. The theory presented here encapsulates two essential contributions. First: The formulation of a target function for the connectivity estimator that resembles an effective GGM of source activity, which is merged to the maximization step of the iterative EM strategy. Second: A new procedure for the estimation of the source activity effective GGM. These theoretical contributions provide for an explicit and highly optimal connectivity estimation procedure.
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