Xiang Ye, Janet Van Niekerk, Håvard Rue
7 min
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.
Sam: Yes. They use a latent Gaussian process—a hidden layer of values following a bell-shaped pattern across data points, linking nearby observations, like weather at close times influencing each other. This feeds into the linear predictor for both circular and straight variables, handling fixed effects from measured inputs and random effects for smooth variations, all in a unified Bayesian setup.
Alex: Huh. And since it's joint, it captures back-and-forth relationships between angles and numbers without forcing one-way links.
Sam: Exactly. Separate models miss shared dependencies, but this borrows strength via the shared hidden layer. Simulation studies show stable fits where standard approaches fail due to dual peaks. Real applications, from bone fixator displacements to protein structures, benefit from this clarity.
Alex: So the adjustment fixes the math snag and scales to complex real-world mixes. Walk me through those simulations—how do they test if it recovers true patterns reliably?
Sam: In one setup, they simulate a circular outcome depending on two straight-number predictors, like wind direction from speed and temperature. Uncertainty ranges around estimates always include the true values. Predictions for new data match simulated truths closely. They also test a straight response regressed on a circular input via a smooth hidden pattern over time—the uncertainty captures the true links, and new predictions align accurately. In the most complex sim, two circular responses link to linear ones via random walks and autoregressive processes, plus covariates. Posterior estimates recover key effects, even at 1000 data points, with efficient runtimes.
Alex: So across these, the evidence points to stable recovery and good predictions without multimodality snags.
Alex: Those simulations build confidence. But how does it hold up with real-world data, like weather?
Sam: The paper applies it to hourly wind data from a New York airport over three months—over 2000 observations of speed, direction, and temperature. Speed uses a Gamma distribution for positive rates that can't go negative. Direction uses LAvM, with temperature feeding in and time-based patterns like smooth random walks for daily cycles. Posterior estimates show temperature shifts direction with a small negative effect, and direction dampens speed.
Alex: So time patterns and temperature link the outcomes jointly. How do predictions compare to older methods?
Sam: They compare to splitting direction into sine and cosine parts for speed alone. The new joint setup scores about 77 points lower on the information criterion for speed—sharper forecasts—and handles direction too. It tests 24-hour rolling forecasts, with uncertainty bands covering observations and catching daily swings.
Alex: Solid for weather. What about a different field, like mechanics?
Sam: Next is biomechanical data: 160 readings of three linear shifts and three angular ones from a bone fixator device. A shared hidden layer with a full covariance matrix captures links between all—like one angle tugging a displacement. Fitted correlations show strong ties, around 0.52 between two linear shifts, plus cross-links like one angle to linear at 0.37, suggesting ways to stabilize by controlling coupled motions. Runtime's just one second.
Alex: So in both cases, the joint structure uncovers practical dependencies efficiently.
Alex: Pulling it together, this framework delivers stable predictions across simulations and real data like wind and bone mechanics, without old ambiguity snags.
Sam: Yes. The link adjustment resolves multimodality for reliable Bayesian joint models, borrowing strength via shared latent processes. Simulations show consistent recovery of true parameters. Real applications reveal coupled dependencies, like temperature influencing wind direction while direction affects speed.
Alex: Right, and those bone correlations highlight how angles and shifts interconnect.
Sam: One limitation is it works best when circular data clusters rather than spreading evenly—uniform cases don't fit well, as the adjustment assumes concentration. It can hit issues at edges like plus or minus 180 degrees, so shifting data to center near zero helps. The paper is cautious, recommending it for typical directional datasets.
Alex: Makes sense—not one-size-fits-all, but targeted where data concentrates. Overall, a notable advance for mixed circular-linear challenges.
Sam: Exactly. It supports forecasting wind patterns, modeling protein folding angles, or analyzing gait steps—areas with clustered directions. An R package is in development.
Alex: A grounded step forward. Thanks, Sam—that's our look at this Bayesian framework for circular models.
Sam: My pleasure, Alex.