A mixture of linear-linear regression models for a linear-circular regression

Abstract

We introduce a new approach to a linear-circular regression problem that relates multiple linear predictors to a circular response. We follow a modelling approach of a wrapped normal distribution that describes angular variables and angular distributions and advances them for a linear-circular regression analysis. Some previous works model a circular variable as projection of a bivariate Gaussian random vector on the unit square, and the statistical inference of the resulting model involves complicated sampling steps. The proposed model treats circular responses as the result of the modulo operation on unobserved linear responses. The resulting model is a mixture of multiple linear-linear regression models. We present two EM algorithms for maximum likelihood estimation of the mixture model, one for a parametric model and another for a nonparametric model. The estimation algorithms provide a great trade-off between computation and estimation accuracy, which was numerically shown using five numerical examples. The proposed approach was applied to a problem of estimating wind directions that typically exhibit complex patterns with large variation and circularity.