CIRCULAR LOCATION
von Mises
Changing μ rotates the density; changing κ adjusts its concentration.
CIRCULAR VIEWOuter ring: 0.817 / rad
THE SAME DENSITY, UNWRAPPEDDensity / rad
Bayesian research / Circular response
Regression with a circular response
When the response is an angle, a regression model needs to respect the circle. For example, the outcome may be wind direction, while covariates and latent effects describe how that direction changes.
THE RESEARCH
Link-adjusted von Mises distribution
The link-adjusted von Mises (LAvM) distribution transforms a centred von Mises variable through a link adjustment. Its linear-scale parameter η changes both the location and shape of the circular density, while κ controls concentration in the underlying von Mises distribution. This provides a distributional basis for regression with a circular response; the accompanying tutorial develops its implementation in Stan.
RELATED PAPERA Bayesian regression framework for circular models with INLA
Ye, Van Niekerk & Rue (2026)
EXPLORE THE DISTRIBUTIONS
Move the sliders to see how the parameters change each density. In the circular view, zero density lies on the model circle and positive density extends outward. The curve below unwraps the same circle from −π to π.
LINEAR-SCALE LINK ADJUSTMENT
Link-adjusted von Mises
Changing η shifts and reshapes the density; η is not the mean direction.
CIRCULAR VIEWOuter ring: 1.44 / rad
THE SAME DENSITY, UNWRAPPEDDensity / rad
A useful comparison. Set η to 0: LAvM becomes a von Mises distribution centred at 0, when κ matches. For von Mises, κ = 0 is uniform. For LAvM, κ = 0 need not be uniform when η ≠ 0.
All angles are in radians, increasing counterclockwise from 0 on the right. These are continuous densities, not the binned counts of a rose diagram.