  Angular measurements are often modeled as circular random variables, where
there are natural circular analogues of moments, including correlation. Because
a product of circles is a torus, a d-dimensional vector of circular random
variables lies on a d-dimensional torus. For such vectors we present here a
class of graphical models, which we call torus graphs, based on the full
exponential family with pairwise interactions. The topological distinction
between a torus and Euclidean space has several important consequences.
  Our development was motivated by the problem of identifying phase coupling
among 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 of
24 phase angles measured repeatedly across 840 experimental trials
(replications) during a memory task, where the electrodes were in 4 distinct
brain regions, all known to be active while memories are being stored or
retrieved. In realistic numerical simulations, we found that a standard
pairwise assessment, known as phase locking value, is unable to describe
multivariate phase interactions, but that torus graphs can accurately identify
conditional associations. Torus graphs generalize several more restrictive
approaches that have appeared in various scientific literatures, and produced
intuitive results in the data we analyzed. Torus graphs thus unify multivariate
analysis of circular data and present fertile territory for future research.
