Circular-Linear Regression: Modeling Directional Outcomes with Bayesian Implementation

Modeling directional or angular outcomes presents unique statistical challenges due to the periodic nature of circular data. Traditional linear regression techniques are inadequate for such data structures, necessitating the use of circular-linear regression models. This study investigates the relationship between a linear predictor and a circular response variable using a model with an exponential link function. Both the classical approach (the Maximum likelihood estimation method and circular least squares method) and the Bayesian framework are employed to estimate the model parameters, allowing the incorporation of prior knowledge and a probabilistic interpretation of the estimates. Posterior distributions are obtained through Markov Chain Monte Carlo (MCMC) methods, and high-density intervals (HDIs) are used to assess convergence and credibility. The results demonstrate that the model effectively captures the non-linear and periodic relationship between wind direction and speed, and the Bayesian approach provides a robust framework for inference under uncertainty. This work highlights the importance and flexibility of Bayesian circular-linear regression in applications involving directional data, with potential implications across disciplines, such as meteorology, biology, neuroscience, and environmental science.