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Module « scipy.stats »

Fonction landau - module scipy.stats

Signature de la fonction landau

def landau(*args, **kwds) 

Description

help(scipy.stats.landau)

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 <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.rv_continuous.fit.html#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:

.. math::

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

for a real number :math:`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 :math:`\mu` and scale parameter :math:`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] Landau, L. (1944). "On the energy loss of fast particles by
       ionization". J. Phys. (USSR). 8: 201.
.. [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()

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