Abstract
Partial directed coherence (PDC) and directed coherence (DC) which describe complementary aspects of the directed information flow between pairs of univariate components that belong to a vector of simultaneously observed time series have recently been generalized as bPDC/bDC, respectively, to portray the relationship between subsets of component vectors (Takahashi, 2009; Faes and Nollo, 2013). This generalization is specially important for neuroscience applications as one often wishes to address the link between the set of time series from an observed ROI (region of interest) with respect to series from some other physiologically relevant ROI. bPDC/bDC are limited, however, in that several time series within a given subset may be irrelevant or may even interact opposingly with respect to one another leading to interpretation difficulties. To address this, we propose an alternative measure, termed cPDC/cDC, employing canonical decomposition to reveal the main frequency domain modes of interaction between the vector subsets. We also show bPDC/bDC and cPDC/cDC are related and possess mutual information rate interpretations. Numerical examples and a real data set illustrate the concepts. The present contribution provides what is seemingly the first canonical decomposition of information flow in the frequency domain.
Highlights
Human behavior is primarily thought as a property that emerges from the interaction of several brain areas, body parts, and the environment
We showed that bPDC/bDC introduced in Takahashi (2009) and Faes and Nollo (2013) are block coherences between properly chosen vector time series
When the time series are Gaussian, this implies that bPDC/bDC represent mutual information rates between well defined underlying vector time series
Summary
Human behavior is primarily thought as a property that emerges from the interaction of several brain areas, body parts, and the environment. The potential relevance of this type of question alone justifies looking for their deeper meaning in terms of information theoretical quantities Despite their practical importance, bPDC/bDC suffer from the limitation that several time series within a given subset may be irrelevant or interact in opposition to one another thereby posing interpretation difficulties. The aim of this article is twofold: (a) to provide a proper information theoretic interpretation for bPDC/bDC and (b) to introduce a canonical decomposition of information flows, termed, respectively, canonical PDC/DC (cPDC/cDC). These new decompositions allow us to closely mimic classical canonical correlation analysis so that different dynamically relevant interaction modes between brain areas can be exposed.
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