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_continuousclass,- landauobject 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 - locand- scaleparameters. Specifically,- landau.pdf(x, loc, scale)is identically equivalent to- landau.pdf(y) / scalewith- 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 - muand- care 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,- sfand- 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. [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) - Get the support: - >>> lb, ub = landau.support() - 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 - cdfand- 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() 