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Showing content from https://numpy.org/doc/stable/reference/random/generated/numpy.random.vonmises.html below:

numpy.random.vonmises — NumPy v2.3 Manual

random.vonmises(mu, kappa, size=None)#

Draw samples from a von Mises distribution.

Samples are drawn from a von Mises distribution with specified mode (mu) and concentration (kappa), on the interval [-pi, pi].

The von Mises distribution (also known as the circular normal distribution) is a continuous probability distribution on the unit circle. It may be thought of as the circular analogue of the normal distribution.

Parameters:

mufloat or array_like of floats

Mode (“center”) of the distribution.

kappafloat or array_like of floats

Concentration of the distribution, has to be >=0.

sizeint or tuple of ints, optional

Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if mu and kappa are both scalars. Otherwise, np.broadcast(mu, kappa).size samples are drawn.

Returns:

outndarray or scalar

Drawn samples from the parameterized von Mises distribution.

Notes

The probability density for the von Mises distribution is

\[p(x) = \frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)},\]

where \(\mu\) is the mode and \(\kappa\) the concentration, and \(I_0(\kappa)\) is the modified Bessel function of order 0.

The von Mises is named for Richard Edler von Mises, who was born in Austria-Hungary, in what is now the Ukraine. He fled to the United States in 1939 and became a professor at Harvard. He worked in probability theory, aerodynamics, fluid mechanics, and philosophy of science.

References

[1]

Abramowitz, M. and Stegun, I. A. (Eds.). “Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing,” New York: Dover, 1972.

[2]

von Mises, R., “Mathematical Theory of Probability and Statistics”, New York: Academic Press, 1964.

Examples

Draw samples from the distribution:

>>> mu, kappa = 0.0, 4.0 # mean and concentration
>>> s = np.random.vonmises(mu, kappa, 1000)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> from scipy.special import i0  
>>> plt.hist(s, 50, density=True)
>>> x = np.linspace(-np.pi, np.pi, num=51)
>>> y = np.exp(kappa*np.cos(x-mu))/(2*np.pi*i0(kappa))  
>>> plt.plot(x, y, linewidth=2, color='r')  
>>> plt.show()

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