scipy.stats.vonmises = <scipy.stats._continuous_distns.vonmises_gen object>[source]#

A Von Mises continuous random variable.

As an instance of the rv_continuous class, vonmises object inherits from it a collection of generic methods (see below for the full list), and completes them with details specific for this particular distribution.

See also


Von-Mises Fisher distribution on a hypersphere


The probability density function for vonmises and vonmises_line is:

\[f(x, \kappa) = \frac{ \exp(\kappa \cos(x)) }{ 2 \pi I_0(\kappa) }\]

for \(-\pi \le x \le \pi\), \(\kappa \ge 0\). \(I_0\) is the modified Bessel function of order zero (scipy.special.i0).

vonmises is a circular distribution which does not restrict the distribution to a fixed interval. Currently, there is no circular distribution framework in SciPy. The cdf is implemented such that cdf(x + 2*np.pi) == cdf(x) + 1.

vonmises_line is the same distribution, defined on \([-\pi, \pi]\) on the real line. This is a regular (i.e. non-circular) distribution.

Note about distribution parameters: vonmises and vonmises_line take kappa as a shape parameter (concentration) and loc as the location (circular mean). A scale parameter is accepted but does not have any effect.


Import the necessary modules.

>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy.stats import vonmises

Define distribution parameters.

>>> loc = 0.5 * np.pi  # circular mean
>>> kappa = 1  # concentration

Compute the probability density at x=0 via the pdf method.

>>> vonmises.pdf(0, loc=loc, kappa=kappa)

Verify that the percentile function ppf inverts the cumulative distribution function cdf up to floating point accuracy.

>>> x = 1
>>> cdf_value = vonmises.cdf(x, loc=loc, kappa=kappa)
>>> ppf_value = vonmises.ppf(cdf_value, loc=loc, kappa=kappa)
>>> x, cdf_value, ppf_value
(1, 0.31489339900904967, 1.0000000000000004)

Draw 1000 random variates by calling the rvs method.

>>> sample_size = 1000
>>> sample = vonmises(loc=loc, kappa=kappa).rvs(sample_size)

Plot the von Mises density on a Cartesian and polar grid to emphasize that it is a circular distribution.

>>> fig = plt.figure(figsize=(12, 6))
>>> left = plt.subplot(121)
>>> right = plt.subplot(122, projection='polar')
>>> x = np.linspace(-np.pi, np.pi, 500)
>>> vonmises_pdf = vonmises.pdf(x, loc=loc, kappa=kappa)
>>> ticks = [0, 0.15, 0.3]

The left image contains the Cartesian plot.

>>> left.plot(x, vonmises_pdf)
>>> left.set_yticks(ticks)
>>> number_of_bins = int(np.sqrt(sample_size))
>>> left.hist(sample, density=True, bins=number_of_bins)
>>> left.set_title("Cartesian plot")
>>> left.set_xlim(-np.pi, np.pi)
>>> left.grid(True)

The right image contains the polar plot.

>>> right.plot(x, vonmises_pdf, label="PDF")
>>> right.set_yticks(ticks)
>>> right.hist(sample, density=True, bins=number_of_bins,
...            label="Histogram")
>>> right.set_title("Polar plot")
>>> right.legend(bbox_to_anchor=(0.15, 1.06))


rvs(kappa, loc=0, scale=1, size=1, random_state=None)

Random variates.

pdf(x, kappa, loc=0, scale=1)

Probability density function.

logpdf(x, kappa, loc=0, scale=1)

Log of the probability density function.

cdf(x, kappa, loc=0, scale=1)

Cumulative distribution function.

logcdf(x, kappa, loc=0, scale=1)

Log of the cumulative distribution function.

sf(x, kappa, loc=0, scale=1)

Survival function (also defined as 1 - cdf, but sf is sometimes more accurate).

logsf(x, kappa, loc=0, scale=1)

Log of the survival function.

ppf(q, kappa, loc=0, scale=1)

Percent point function (inverse of cdf — percentiles).

isf(q, kappa, loc=0, scale=1)

Inverse survival function (inverse of sf).

moment(order, kappa, loc=0, scale=1)

Non-central moment of the specified order.

stats(kappa, loc=0, scale=1, moments=’mv’)

Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).

entropy(kappa, loc=0, scale=1)

(Differential) entropy of the RV.


Parameter estimates for generic data. See for detailed documentation of the keyword arguments.

expect(func, args=(kappa,), loc=0, scale=1, lb=None, ub=None, conditional=False, **kwds)

Expected value of a function (of one argument) with respect to the distribution.

median(kappa, loc=0, scale=1)

Median of the distribution.

mean(kappa, loc=0, scale=1)

Mean of the distribution.

var(kappa, loc=0, scale=1)

Variance of the distribution.

std(kappa, loc=0, scale=1)

Standard deviation of the distribution.

interval(confidence, kappa, loc=0, scale=1)

Confidence interval with equal areas around the median.