Directional
Statistics

XIANG YE / BAYESIAN RESEARCH

Directional
Statistics.

When the data have
a sense of direction.

Angles on a circle. Directions on a sphere. Statistical methods that respect the geometry of the data.

Explore the introduction Read the tutorials
01 / THE CIRCLE

359° and 1° are only 2° apart.
The end of the scale meets the beginning.

ANGULAR OBSERVATIONSS¹ / 001
Rose diagram of illustrative directionsSixteen equal angular bins. Sector area represents count. North is zero degrees; angles increase clockwise. An orange arrow shows the mean direction.0–22.5°: 6 observations22.5–45°: 13 observations45–67.5°: 29 observations67.5–90°: 29 observations90–112.5°: 13 observations112.5–135°: 6 observations135–157.5°: 4 observations157.5–180°: 3 observations180–202.5°: 2 observations202.5–225°: 1 observations225–247.5°: 2 observations247.5–270°: 2 observations270–292.5°: 1 observations292.5–315°: 2 observations315–337.5°: 3 observations337.5–360°: 4 observationsNESW0° / 360°90°180°270°
Frequency Mean direction Observations
Observations120
Mean direction · μ67.5°
Resultant length · R̄0.65
Illustrative data · 16 equal bins · sector area ∝ count
0° at north; angles increase clockwise.
View the counts and scale

Radius ∝ √count. Outer grid ring: 29 observations. Inner rings: 25%, 50%, and 75% of that count. The count scale adapts to the largest bin. At R̄ = 1 the observation marks overlap.

Counts in 22.5° angular bins
BearingCount
0.0–22.5°6
22.5–45.0°13
45.0–67.5°29
67.5–90.0°29
90.0–112.5°13
112.5–135.0°6
135.0–157.5°4
157.5–180.0°3
180.0–202.5°2
202.5–225.0°1
225.0–247.5°2
247.5–270.0°2
270.0–292.5°1
292.5–315.0°2
315.0–337.5°3
337.5–360.0°4
DIRECTIONAL DATA IN

Wind & weather

↗

Animal movement

↗

Geological orientation

↗

Time & phase

01 / UNDERSTAND THE GEOMETRY

A circle has
no natural beginning.

Compass bearings and times of day wrap around. Averaging their numerical labels can be misleading: the arithmetic mean of 359° and 1° is 180°, while their circular mean is 0°.

A rose diagram makes the pattern visible by grouping observations into angular sectors. In the diagram above, larger sector areas mean more observations.

How to read a rose diagram

ON A STRAIGHT SCALE

The numerical average is 180°.

A linear scale puts 1° and 359° at opposite ends An observation at 1 degree lies near the left end, and an observation at 359 degrees near the right end. Their arithmetic mean is 180 degrees in the middle. This ignores that the endpoints meet on a circle. 1°359° 180° · arithmetic mean 0°360°

(359° + 1°) / 2 = 180°

The calculation is correct on a line, but its result points away from both observations.

ON THE CIRCLE

The mean direction is 0°.

359° and 1° are neighbours on a circle Two observations lie one degree either side of north. Their shortest separation is two degrees. Their circular mean is north, zero degrees. The arithmetic mean, 180 degrees, points south, opposite the observations. Labels use leader lines; angular positions are drawn to scale. 359°1°N · 0°EW 2° apart 180° points the other way Angles drawn to scale
Same two observations, different geometry. Across north, 359° and 1° are separated by just 2°. The circular mean lies between them.

02 / BAYESIAN RESEARCH

From directions to models.

Research overview
01
π(κ)

Priors for circular models

Penalized complexity priors for the concentration of circular distributions.

Explore the research
02
Y ∈ S¹

Regression with a circular response

Link-adjusted von Mises models for angular outcomes and structured predictors.

Explore the research
03
(Y, θ)

Regression with a circular covariate

Joint circular models connecting direction and a linear response through shared latent structure.

Explore the research

03 / STAN IMPLEMENTATIONS

The methods, worked through in Stan.

All three Stan tutorials
01
Priors / Stan IMPLEMENTATION

Penalized Complexity Priors for Circular Distributions

Construct and calibrate PC priors for von Mises concentration, with circular-uniform and point-mass base models.

02
Circular response / Stan IMPLEMENTATION

Circular Distributions for Regression Models

Explore link-adjusted von Mises regression, from fixed effects to temporal structure and wind-direction data.

03
Circular covariate / Stan IMPLEMENTATION

Joint circular models

Explore regression models with circular covariates and learn how to extend them to joint modelling.