Analyzing longitudinal circular data by projected normal models: a semi-parametric approach based on finite mixture models

The analysis of circular data has been recently the focus of a wide range of literature, with the general objective of providing reliable parameter estimates in the presence of heterogeneity and/or dependence among observations under a longitudinal setting. In this paper, we extend the variance component model approach to the analysis of longitudinal circular data, defining a mixed effects model for radial projections onto the circle and introducing dependence between projections through a set of correlated random coefficients. Estimation is carried out by numerical integration through an expectation-maximization algorithm without parametric assumptions upon the random coefficients distribution. The resulting model is a finite mixture of projected normal distributions. A simulation study has been carried out to investigate the behavior of the proposed model in a series of empirical situations. The proposed model is computationally parsimonious and, when applied to a real dataset on animal orientation, produces novel results.