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The Watanabe-Strogatz and Ott-Antonsen theories provided a seminal frameworkfor rigorous and comprehensive studies of collective phenomena in a broad classof paradigmatic models for ensembles of coupled oscillators. Recently, a"circular cumulant" approach was suggested for constructing the perturbationtheory for the Ott-Antonsen approach. In this paper, we derive the relationsbetween the distribution of Watanabe-Strogatz phases and the circular cumulantsof the original phases. These relations are important for the interpretation ofthe circular cumulant approach in the context of the Watanabe-Strogatz andOtt-Antonsen theories. Special attention is paid to the case of hierarchy ofcircular cumulants, which is generally relevant for constructing perturbationtheories for the Watanabe-Strogatz and Ott-Antonsen approaches.

Relationships between the Distribution of Watanabe-Strogatz Variables and Circular Cumulants for Ensembles of Phase Elements