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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.

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

Probability type

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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