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.


The probability density function for vonmises and vonmises_line is:

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

for \(-\pi \le x \le \pi\), \(\kappa > 0\). \(I_0\) 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.

vonmises and vonmises_line take kappa as a shape parameter.

The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the loc and scale parameters. Specifically, vonmises.pdf(x, kappa, loc, scale) is identically equivalent to vonmises.pdf(y, kappa) / scale with y = (x - loc) / scale. Note that shifting the location of a distribution does not make it a “noncentral” distribution; noncentral generalizations of some distributions are available in separate classes.


>>> from scipy.stats import vonmises
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots(1, 1)

Calculate the first four moments:

>>> kappa = 3.99
>>> mean, var, skew, kurt = vonmises.stats(kappa, moments='mvsk')

Display the probability density function (pdf):

>>> x = np.linspace(vonmises.ppf(0.01, kappa),
...                 vonmises.ppf(0.99, kappa), 100)
>>> ax.plot(x, vonmises.pdf(x, kappa),
...        'r-', lw=5, alpha=0.6, label='vonmises pdf')

Alternatively, the distribution object can be called (as a function) to fix the shape, location and scale parameters. This returns a “frozen” RV object holding the given parameters fixed.

Freeze the distribution and display the frozen pdf:

>>> rv = vonmises(kappa)
>>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

Check accuracy of cdf and ppf:

>>> vals = vonmises.ppf([0.001, 0.5, 0.999], kappa)
>>> np.allclose([0.001, 0.5, 0.999], vonmises.cdf(vals, kappa))

Generate random numbers:

>>> r = vonmises.rvs(kappa, size=1000)

And compare the histogram:

>>> ax.hist(r, density=True, histtype='stepfilled', alpha=0.2)
>>> ax.legend(loc='best', frameon=False)


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

Non-central moment of order n

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

Endpoints of the range that contains fraction alpha [0, 1] of the distribution