A Mixture of Linear-Linear Regression Models for Linear-Circular Regression

TL;DR

This paper presents a novel mixture of linear-linear regression models for linear-circular regression, utilizing EM algorithms for efficient maximum likelihood estimation, demonstrated on wind direction data.

Contribution

It introduces a new mixture model approach for linear-circular regression based on wrapped normal distribution, with two EM algorithms for estimation.

Findings

Effective estimation of wind directions with complex patterns.

Two EM algorithms offer a good balance between accuracy and computation.

Numerical examples validate the proposed approach.

Abstract

We introduce a new approach to a linear-circular regression problem that relates multiple linear predictors to a circular response. We follow a modeling approach of a wrapped normal distribution that describes angular variables and angular distributions and advances it 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 non-parametric 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.