# Multiple regression in directional / circular statistics?

I'm trying to develop a predictive model for an angular dependent variable (on $[0,2\pi])$ using several independent measurements – also angular variables, on $[0,2\pi]$ – as predictors. Each predictor is significantly but not extremely strongly correlated with the dependent variable. How can I combine the predictors to determine a predictive model for the dependent variable that is optimal in some sense? And how can I rigorously identify the strongest predictor(s)?

For variables on Euclidean space(s), I'd employ multiple regression (or similar) and principal components analysis. But the periodicity of all variables mucks with these approaches, e.g., 0.02 must be highly correlated with 6.26, but not with 3.14. How are "the usual" procedures generalized to directional / circular statistics? Any insights or cites to useful references would be useful. (I'm already aware of the texts by N. Fisher and Mardia & Jupp, but don't have handy access to these.)

In the book that I have it says that only recently some papers have begun to explore multivariate regression where one or more variables are circular. I have not checked them myself, but relevant sources seem to be:

Bhattacharya, S. and SenGupta, A. (2009). Bayesian analysis of semiparametric linear-circular models. Journal of Agricultural, Biological and Environmental Statistics, 14, 33-65.

Lund, U. (1999). Least circular distance regression for directional data. Journal of Applied Statistics, 26, 723-733

Lund, U. (2002). Tree-based regression or a circular response. Communications in Statistics - Theory and Methods, 31, 1549-1560.

Qin, X., Zhang, J.-S., and Yan, X.-D. (2011). A nonparametric circular-linear multivariate regression model with a rule of thumb bandwidth selector. Computers and Mathematics with Applications, 62, 3048-3055.

In case for a circular response you have only a single circular regressor (which I understand that is not the case for you, but maybe separate regressions would be of interest as well) there is a way to estimate the model. [1] recommend fitting general linear model

$$\cos(\Theta_j) = \gamma_0^c + \sum_{k=1}^m\left(\gamma_{ck}^c\cos(k\psi_j)+\gamma_{sk}^c\sin(k\psi_j)\right)+\varepsilon_{1j},$$ $$\sin(\Theta_j) = \gamma_0^s + \sum_{k=1}^m\left(\gamma_{ck}^s\cos(k\psi_j)+\gamma_{sk}^s\sin(k\psi_j)\right)+\varepsilon_{2j}.$$

The good thing is that this model can be estimated using the function lm.circular from the R library circular.

[1] Jammalamadaka, S. R. and SenGupta, A. (2001). Topics in Circular Statistics. World Scientific, Singapore.

You can take a look at these articles that deal with multiple regression when the dependent variable is circular, or spherical. The approach is based on the projected normal distribution.

Hernandez-Stumpfhauser, Daniel, F. Jay Breidt, and Mark J. van der Woerd. "The general projected normal distribution of arbitrary dimension: modeling and Bayesian inference." Bayesian Analysis 12.1 (2017): 113-133.

Wang, Fangpo, and Alan E. Gelfand. "Directional data analysis under the general projected normal distribution." Statistical methodology 10.1 (2013): 113-127

Nuñez-Antonio, Gabriel, Eduardo Gutiérrez-Peña, and Gabriel Escarela. "A Bayesian regression model for circular data based on the projected normal distribution." Statistical Modelling 11.3 (2011): 185-201.

Presnell, Brett, Scott P. Morrison, and Ramon C. Littell. "Projected multivariate linear models for directional data." Journal of the American Statistical Association 93.443 (1998): 1068-1077.

This last one was the first one to come out using this projected normal approach