scipy.stats.landau#

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.

Methods

rvs(loc=0, scale=1, size=1, random_state=None)	Random variates.
pdf(x, loc=0, scale=1)	Probability density function.
logpdf(x, loc=0, scale=1)	Log of the probability density function.
cdf(x, loc=0, scale=1)	Cumulative distribution function.
logcdf(x, loc=0, scale=1)	Log of the cumulative distribution function.
sf(x, loc=0, scale=1)	Survival function (also defined as `1 - cdf`, but sf is sometimes more accurate).
logsf(x, loc=0, scale=1)	Log of the survival function.
ppf(q, loc=0, scale=1)	Percent point function (inverse of `cdf` — percentiles).
isf(q, loc=0, scale=1)	Inverse survival function (inverse of `sf`).
moment(order, loc=0, scale=1)	Non-central moment of the specified order.
stats(loc=0, scale=1, moments=’mv’)	Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).
entropy(loc=0, scale=1)	(Differential) entropy of the RV.
fit(data)	Parameter estimates for generic data. See scipy.stats.rv_continuous.fit for detailed documentation of the keyword arguments.
expect(func, args=(), 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(loc=0, scale=1)	Median of the distribution.
mean(loc=0, scale=1)	Mean of the distribution.
var(loc=0, scale=1)	Variance of the distribution.
std(loc=0, scale=1)	Standard deviation of the distribution.
interval(confidence, loc=0, scale=1)	Confidence interval with equal areas around the median.

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 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.

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 isf methods. [1]

References

[1] (1,2)

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

[2] (1,2)

“Landau Distribution”, Wikipedia, https://en.wikipedia.org/wiki/Landau_distribution

[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.

[4]

The Boost Developers. “Boost C++ Libraries”. https://www.boost.org/.

[5]

Yoshimura, T. “Numerical Evaluation and High Precision Approximation Formula for Landau Distribution”. DOI:10.36227/techrxiv.171822215.53612870/v2

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()