A Bayesian regression framework for circular models with INLA

Regression models for circular variables are less developed, since the concept of building a linear predictor from linear combinations of covariates and various random effects, breaks the circular nature of the variable. In this paper, we introduce a new approach to rectify this issue, leading to well-defined regression models for circular responses when the data are concentrated. Our approach extends naturally to joint regression models where we can have several circular and non-circular responses, and allow us to handle a mix of linear covariates, circular covariates and various random effects. Our formulation aligns naturally with the integrated nested Laplace approximation (INLA), which provides fast and accurate Bayesian inference. We illustrate our approach through several simulated and real examples.

💡 Research Summary

This paper addresses a long‑standing difficulty in regression modelling with circular (directional) response variables: the multimodality of the likelihood that arises when a linear predictor is mapped onto the circle and the distance between observed angles and the predicted mean direction is measured directly on the circular manifold. Traditional approaches either use a link function such as the inverse‑tangent (g(z)=2 arctan z) or a scaled probit, then plug g(η) into the mean direction µ of a von Mises distribution. Because µ varies with each observation, the geodesic difference x − µ can fall outside the principal interval