On the circular correlation coefficients for bivariate von Mises distributions on a torus
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This paper studies circular correlations for the bivariate von Mises sine and cosine distributions. These are two simple and appealing models for bivariate angular data with five parameters each that have interpretations comparable to those in the ordinary bivariate normal model. However, the variability and association of the angle pairs cannot be easily deduced from the model parameters unlike the bivariate normal. Thus to compute such summary measures, tools from circular statistics are needed. We derive analytic expressions for the Jammalamadaka-Sarma and Fisher-Lee circular correlation coefficients for the von Mises sine and cosine models, and also provide their circular variances. These expressions can be easily evaluated in any standard statistical software, and we provide implementations in our R package *BAMBI*. We provide further results that lead to deeper insights into the Toroidial-linear association of the random coordinates in these models, and an interesting connection between these two correlations is also shown. We illustrate our results with numerical and visual examples computed using our R package.
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