Vous êtes un professionnel et vous avez besoin d'une formation ? Calcul scientifique
avec Python Voir le programme détaillé

Module « scipy.stats »

Fonction lognorm - module scipy.stats

Signature de la fonction lognorm

def lognorm(*args, **kwds) 

Description

help(scipy.stats.lognorm)

A lognormal continuous random variable.

As an instance of the `rv_continuous` class, `lognorm` 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(s, loc=0, scale=1, size=1, random_state=None)
    Random variates.
pdf(x, s, loc=0, scale=1)
    Probability density function.
logpdf(x, s, loc=0, scale=1)
    Log of the probability density function.
cdf(x, s, loc=0, scale=1)
    Cumulative distribution function.
logcdf(x, s, loc=0, scale=1)
    Log of the cumulative distribution function.
sf(x, s, loc=0, scale=1)
    Survival function  (also defined as ``1 - cdf``, but `sf` is sometimes more accurate).
logsf(x, s, loc=0, scale=1)
    Log of the survival function.
ppf(q, s, loc=0, scale=1)
    Percent point function (inverse of ``cdf`` --- percentiles).
isf(q, s, loc=0, scale=1)
    Inverse survival function (inverse of ``sf``).
moment(order, s, loc=0, scale=1)
    Non-central moment of the specified order.
stats(s, loc=0, scale=1, moments='mv')
    Mean('m'), variance('v'), skew('s'), and/or kurtosis('k').
entropy(s, 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=(s,), 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(s, loc=0, scale=1)
    Median of the distribution.
mean(s, loc=0, scale=1)
    Mean of the distribution.
var(s, loc=0, scale=1)
    Variance of the distribution.
std(s, loc=0, scale=1)
    Standard deviation of the distribution.
interval(confidence, s, loc=0, scale=1)
    Confidence interval with equal areas around the median.

Notes
-----
The probability density function for `lognorm` is:

.. math::

    f(x, s) = \frac{1}{s x \sqrt{2\pi}}
              \exp\left(-\frac{\log^2(x)}{2s^2}\right)

for :math:`x > 0`, :math:`s > 0`.

`lognorm` takes ``s`` as a shape parameter for :math:`s`.

The probability density above is defined in the "standardized" form. To shift
and/or scale the distribution use the ``loc`` and ``scale`` parameters.
Specifically, ``lognorm.pdf(x, s, loc, scale)`` is identically
equivalent to ``lognorm.pdf(y, s) / 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.

Suppose a normally distributed random variable ``X`` has  mean ``mu`` and
standard deviation ``sigma``. Then ``Y = exp(X)`` is lognormally
distributed with ``s = sigma`` and ``scale = exp(mu)``.

Examples
--------
>>> import numpy as np
>>> from scipy.stats import lognorm
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots(1, 1)

Calculate the first four moments:

>>> s = 0.954
>>> mean, var, skew, kurt = lognorm.stats(s, moments='mvsk')

Display the probability density function (``pdf``):

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

Check accuracy of ``cdf`` and ``ppf``:

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

Generate random numbers:

>>> r = lognorm.rvs(s, 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()


The logarithm of a log-normally distributed random variable is
normally distributed:

>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy import stats
>>> fig, ax = plt.subplots(1, 1)
>>> mu, sigma = 2, 0.5
>>> X = stats.norm(loc=mu, scale=sigma)
>>> Y = stats.lognorm(s=sigma, scale=np.exp(mu))
>>> x = np.linspace(*X.interval(0.999))
>>> y = Y.rvs(size=10000)
>>> ax.plot(x, X.pdf(x), label='X (pdf)')
>>> ax.hist(np.log(y), density=True, bins=x, label='log(Y) (histogram)')
>>> ax.legend()
>>> plt.show()

Vous êtes un professionnel et vous avez besoin d'une formation ? Programmation Python
Les compléments Voir le programme détaillé