Von Mises Distribution Percent Point
Find the angle percent point (quantile) of the von Mises distribution from probability p, mean direction μ, and concentration κ.
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Input
Enter probability p, mean direction μ in radians, and concentration κ to compute the percent point of the von Mises distribution, the corresponding angle.
A value greater than 0 and less than 1
Choose lower-tail or upper-tail probability.
The central angle around which the distribution concentrates
A value greater than 0. Larger values are more sharply concentrated.
Result
Percent point at lower probability 0.95
1.41796946 rad
= 81.2436655 degrees
Mean direction μ
0
Concentration κ
2
Lower probability F
0.95000023
Upper probability
0.04999977
Density f
0.09466553
Circular variance
0.30222534
Probability density function
Cumulative distribution function
How it works
- The von Mises density is f(θ) = exp(κ cos(θ − μ)) / (2π I0(κ)), where I0 is the modified Bessel function of the first kind, implemented here from its series expansion.
- The cumulative probability is obtained by trapezoidal numerical integration over the interval from μ − π to θ, normalized by the integral over the full period.
- The percent point is found by inverting the cumulative probability with bisection. The median (p = 0.5) equals the mean direction μ.
- When the upper tail is selected, the probability is converted to the lower probability 1 − p before solving for the angle.
- Larger concentration κ makes the distribution sharply peaked around the mean direction μ, while κ near 0 approaches a uniform distribution on the circle.
- Circular variance is 1 − I1(κ)/I0(κ); values near 0 indicate that directions are tightly clustered.
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Von Mises Distribution Percent Point