  Functional brain imaging through electroencephalography (EEG) relies upon the
analysis and interpretation of high-dimensional, spatially organized time
series. We propose to represent time-localized frequency domain
characterizations of EEG data as region-referenced functional data. This
representation is coupled with a hierarchical modeling approach to multivariate
functional observations. Within this familiar setting, we discuss how several
prior models relate to structural assumptions about multivariate covariance
operators. An overarching modeling framework, based on infinite factorial
decompositions, is finally proposed to balance flexibility and efficiency in
estimation. The motivating application stems from a study of implicit auditory
learning, in which typically developing (TD) children, and children with autism
spectrum disorder (ASD) were exposed to a continuous speech stream. Using the
proposed model, we examine differential band power dynamics as brain function
is interrogated throughout the duration of a computer-controlled experiment.
Our work offers a novel look at previous findings in psychiatry, and provides
further insights into the understanding of ASD. Our approach to inference is
fully Bayesian and implemented in a highly optimized Rcpp package.
