A Bayesian regression framework for circular models with INLA

TL;DR

The paper addresses the difficulty of regressing circular outcomes by introducing a link-adjusted framework that measures distance in linear space before projecting to the circle, yielding a unimodal likelihood and stable inference. It develops a Bayesian regression system built on latent Gaussian processes and INLA, enabling circular responses, circular covariates, and joint modeling with fixed and random effects. Key contributions include the link-adjusted von Mises (LAC) family, the LAvM special case, and a unified joint circular regression approach that captures temporal/spatial dependence via latent fields, demonstrated through simulations and real data (wind and biomechanics). This approach offers fast, scalable, and coherent analysis for directional data, with practical impact for spatio-temporal directional modeling and multivariate directional-inference tasks.

Abstract

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.

Figures (12)

• Figure 1: Comparison of circular and linear subtraction between the variable $y$ and the GLM linear predictor. From left to right: (1) cosine values via circular subtraction; (2) cosine values via linear subtraction; (3) von Mises density via circular subtraction; and (4) von Mises density via linear subtraction, when $\kappa=1$.
• Figure 2: Densities of the vM (left) and LAvM (right) distributions with $\kappa=1$, For the vM distribution, $\mu = -\frac{\pi}{2},0,\frac{\pi}{2}$; for the LAvM distribution, $g\left(\eta\right) = -\frac{\pi}{2},0,\frac{\pi}{2}$.
• Figure 3: Log densities of the vM (left) and LAvM (right) distributions with $\kappa=1$, For the vM distribution, $\mu = -\frac{\pi}{2},0,\frac{\pi}{2}$; for the LAvM distribution, $g\left(\eta\right) = -\frac{\pi}{2},0,\frac{\pi}{2}$.
• Figure 4: Posterior summaries (left) and posterior predictive distribution (right) for the simulation in Section \ref{['sec4:circular_linear']}. In the posterior summaries, the red horizontal lines indicate the true values, the vertical bars represent the 95% credible intervals, and the blue dots show the posterior means ($\beta_{0}$, $\beta_{1}$, $\beta_{2}$) and mode ($\log\left(\kappa\right)$). In the posterior predictive density plot, the blue histogram and curve correspond to the posterior predictive distribution, while the orange ones denote the true density.
• Figure 5: Posterior summaries (left) and posterior predictive distributions (right) for the simulation in Section \ref{['sec4:linear_circular']}. The same plotting convention as in \ref{['fig:sim1']} is used. The blue dots in the left panel show the posterior means for $a_{0}$, $b_{0}$ and modes for $a_{1}$, $b_{1}$, $\log^{-1}\left(\kappa\right)$, $\log^{-1}\left(\tau\right)$.

Theorems & Definitions (2)

• Definition 1: Circular distance
• Definition 2: Link-adjusted circular distribution