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Angular measurements are often modeled as circular random variables, wherethere are natural circular analogues of moments, including correlation. Becausea product of circles is a torus, a d-dimensional vector of circular randomvariables lies on a d-dimensional torus. For such vectors we present here aclass of graphical models, which we call torus graphs, based on the fullexponential family with pairwise interactions. The topological distinctionbetween a torus and Euclidean space has several important consequences.  Our development was motivated by the problem of identifying phase couplingamong oscillatory signals recorded from multiple electrodes in the brain:oscillatory phases across electrodes might tend to advance or recede together,indicating coordination across brain areas. The data analyzed here consisted of24 phase angles measured repeatedly across 840 experimental trials(replications) during a memory task, where the electrodes were in 4 distinctbrain regions, all known to be active while memories are being stored orretrieved. In realistic numerical simulations, we found that a standardpairwise assessment, known as phase locking value, is unable to describemultivariate phase interactions, but that torus graphs can accurately identifyconditional associations. Torus graphs generalize several more restrictiveapproaches that have appeared in various scientific literatures, and producedintuitive results in the data we analyzed. Torus graphs thus unify multivariateanalysis of circular data and present fertile territory for future research.

Torus Graphs for Multivariate Phase Coupling Analysis