On the circular correlation coefficients for bivariate von Mises distributions on a torus

Bayesian analysis of coupled cellular and nuclear trajectories for cell migration

“…In particular, $$\rho _{\text{JS}}, \rho _{\text{FL}} \in [-1, 1]$$; they are equal to 1 (−1) under perfect positive (negative) toroidal‐linear ( T‐linear ) relationship, and they are 0 under independence, although the reverse implication may not necessarily hold. For the von Mises cosine distribution, both $$\rho _{\text{JS}}$$ and $$\rho _{\text{FL}}$$ can be written as functions of κ 1 , κ 2 , and κ 3 using partial derivatives of the normalizing constant (Chakraborty and Wong, 2018).…”

“…Remark If $$\kappa _3 \ge 0$$ in (1), then the random angle pair $$(\theta _1, \theta _2)$$ has a nonnegative T‐linear association as measured through $$\rho _{\text{JS}}(\theta _1, \theta _2)$$ (Chakraborty and Wong, 2018). …”