An algebraic analysis of the bimodality of the generalized von Mises distribution

Bimodal probability distributions for planar directions are relevant to many scientific fields. Besides mixtures of two unimodal circular distributions, there are relatively few other circular distributions that allow for bimodality. One of these is the generalized von Mises distribution, which can be symmetric or asymmetric, bimodal or unimodal. Moreover, the GvM model possesses many important theoretical properties that are not available with competing models. This article provides an algebraic analysis of the bimodality of the GvM distribution. The problem of bimodality is reduced to study the nature of the roots of a quartic. Four real roots over the interval [−1,1] of the quartic correspond to bimodality. Situations with complex roots, multiple real roots and real roots outside [−1,1] are analyzed. The set of all parameters of the GvM distribution that yields bimodality is identified.