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Module « scipy.stats »

Fonction multivariate_normal - module scipy.stats

Signature de la fonction multivariate_normal

def multivariate_normal(mean=None, cov=1, allow_singular=False, seed=None) 

Description

help(scipy.stats.multivariate_normal)

A multivariate normal random variable.

The `mean` keyword specifies the mean. The `cov` keyword specifies the
covariance matrix.

Methods
-------
pdf(x, mean=None, cov=1, allow_singular=False)
    Probability density function.
logpdf(x, mean=None, cov=1, allow_singular=False)
    Log of the probability density function.
cdf(x, mean=None, cov=1, allow_singular=False, maxpts=1000000*dim, abseps=1e-5, releps=1e-5, lower_limit=None)
    Cumulative distribution function.
logcdf(x, mean=None, cov=1, allow_singular=False, maxpts=1000000*dim, abseps=1e-5, releps=1e-5)
    Log of the cumulative distribution function.
rvs(mean=None, cov=1, size=1, random_state=None)
    Draw random samples from a multivariate normal distribution.
entropy(mean=None, cov=1)
    Compute the differential entropy of the multivariate normal.
fit(x, fix_mean=None, fix_cov=None)
    Fit a multivariate normal distribution to data.

Parameters
----------
mean : array_like, default: ``[0]``
    Mean of the distribution.
cov : array_like or `Covariance`, default: ``[1]``
    Symmetric positive (semi)definite covariance matrix of the distribution.
allow_singular : bool, default: ``False``
    Whether to allow a singular covariance matrix. This is ignored if `cov` is
    a `Covariance` object.
seed : {None, int, np.random.RandomState, np.random.Generator}, optional
    Used for drawing random variates.
    If `seed` is `None`, the `~np.random.RandomState` singleton is used.
    If `seed` is an int, a new ``RandomState`` instance is used, seeded
    with seed.
    If `seed` is already a ``RandomState`` or ``Generator`` instance,
    then that object is used.
    Default is `None`.

Notes
-----
Setting the parameter `mean` to `None` is equivalent to having `mean`
be the zero-vector. The parameter `cov` can be a scalar, in which case
the covariance matrix is the identity times that value, a vector of
diagonal entries for the covariance matrix, a two-dimensional array_like,
or a `Covariance` object.

The covariance matrix `cov` may be an instance of a subclass of
`Covariance`, e.g. `scipy.stats.CovViaPrecision`. If so, `allow_singular`
is ignored.

Otherwise, `cov` must be a symmetric positive semidefinite
matrix when `allow_singular` is True; it must be (strictly) positive
definite when `allow_singular` is False.
Symmetry is not checked; only the lower triangular portion is used.
The determinant and inverse of `cov` are computed
as the pseudo-determinant and pseudo-inverse, respectively, so
that `cov` does not need to have full rank.

The probability density function for `multivariate_normal` is

.. math::

    f(x) = \frac{1}{\sqrt{(2 \pi)^k \det \Sigma}}
           \exp\left( -\frac{1}{2} (x - \mu)^T \Sigma^{-1} (x - \mu) \right),

where :math:`\mu` is the mean, :math:`\Sigma` the covariance matrix,
:math:`k` the rank of :math:`\Sigma`. In case of singular :math:`\Sigma`,
SciPy extends this definition according to [1]_.

.. versionadded:: 0.14.0

References
----------
.. [1] Multivariate Normal Distribution - Degenerate Case, Wikipedia,
       https://en.wikipedia.org/wiki/Multivariate_normal_distribution#Degenerate_case

Examples
--------
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy.stats import multivariate_normal

>>> x = np.linspace(0, 5, 10, endpoint=False)
>>> y = multivariate_normal.pdf(x, mean=2.5, cov=0.5); y
array([ 0.00108914,  0.01033349,  0.05946514,  0.20755375,  0.43939129,
        0.56418958,  0.43939129,  0.20755375,  0.05946514,  0.01033349])
>>> fig1 = plt.figure()
>>> ax = fig1.add_subplot(111)
>>> ax.plot(x, y)
>>> plt.show()

Alternatively, the object may be called (as a function) to fix the mean
and covariance parameters, returning a "frozen" multivariate normal
random variable:

>>> rv = multivariate_normal(mean=None, cov=1, allow_singular=False)
>>> # Frozen object with the same methods but holding the given
>>> # mean and covariance fixed.

The input quantiles can be any shape of array, as long as the last
axis labels the components.  This allows us for instance to
display the frozen pdf for a non-isotropic random variable in 2D as
follows:

>>> x, y = np.mgrid[-1:1:.01, -1:1:.01]
>>> pos = np.dstack((x, y))
>>> rv = multivariate_normal([0.5, -0.2], [[2.0, 0.3], [0.3, 0.5]])
>>> fig2 = plt.figure()
>>> ax2 = fig2.add_subplot(111)
>>> ax2.contourf(x, y, rv.pdf(pos))

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