scipy.stats.landau — SciPy v1.15.2 Manual (original) (raw)

scipy.stats.landau = <scipy.stats._continuous_distns.landau_gen object>[source]#

A Landau continuous random variable.

As an instance of the rv_continuous class, landau 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.

Notes

The probability density function for landau ([1], [2]) is:

\[f(x) = \frac{1}{\pi}\int_0^\infty \exp(-t \log t - xt)\sin(\pi t) dt\]

for a real number \(x\).

The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the loc and scale parameters. Specifically, landau.pdf(x, loc, scale) is identically equivalent to landau.pdf(y) / scale withy = (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.

Often (e.g. [2]), the Landau distribution is parameterized in terms of a location parameter \(\mu\) and scale parameter \(c\), the latter of which also introduces a location shift. If mu and c are used to represent these parameters, this corresponds with SciPy’s parameterization with loc = mu + 2*c / np.pi * np.log(c) and scale = c.

This distribution uses routines from the Boost Math C++ library for the computation of the pdf, cdf, ppf, sf and isfmethods. [1]

References

[1] (1,2)

Landau, L. (1944). “On the energy loss of fast particles by ionization”. J. Phys. (USSR). 8: 201.

[3]

Chambers, J. M., Mallows, C. L., & Stuck, B. (1976). “A method for simulating stable random variables.” Journal of the American Statistical Association, 71(354), 340-344.

Examples

import numpy as np from scipy.stats import landau import matplotlib.pyplot as plt fig, ax = plt.subplots(1, 1)

Calculate the first four moments:

mean, var, skew, kurt = landau.stats(moments='mvsk')

Display the probability density function (pdf):

x = np.linspace(landau.ppf(0.01), ... landau.ppf(0.99), 100) ax.plot(x, landau.pdf(x), ... 'r-', lw=5, alpha=0.6, label='landau 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 = landau() ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

Check accuracy of cdf and ppf:

vals = landau.ppf([0.001, 0.5, 0.999]) np.allclose([0.001, 0.5, 0.999], landau.cdf(vals)) True

Generate random numbers:

r = landau.rvs(size=1000)

And compare the histogram:

ax.hist(r, density=True, bins='auto', histtype='stepfilled', alpha=0.2) ax.set_xlim([x[0], x[-1]]) ax.legend(loc='best', frameon=False) plt.show()

Methods