Probabilistic model for two dependent circular variables

By Harshinder Singh, V. Hnizdo, E. Demchuk. 2002

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Two‐stage U‐statistics for Hypothesis Testing (2006, background): “This statistic can also test independence of a particular fish’s spawning time and the low tide time in marine biology (Lund, 1999) or the paired dihedral angles of protein peptides (Singh et al., 2002).”
RaptorX-Angle: real-value prediction of protein backbone dihedral angles through a hybrid method of clustering and deep learning (2018, methodology): “Mixture bivariate von Mises distributions are successfully used to describe the local bias of torsion angle pair (φ , ψ) [21–23], we assume that angle pairs belong to the same cluster k obey a common bivariate von Mises distribution (cid:13) (cid:14) where μ k and ν k are the mean value of φ and ψ…”
Maximum likelihood estimation using composite likelihoods for closed exponential families (2009, background): “The bivariate von Mises distribution on the torus (Singh et al., 2002) is an interesting example of a nonclosed exponential family model which can be regarded as approximately closed.”