EVOLUTION-MANAGER

Edit File: _multivariate.py

#
# Author: Joris Vankerschaver 2013
#
from __future__ import division, print_function, absolute_import

import math
import numpy as np
from numpy import asarray_chkfinite, asarray
import scipy.linalg
from scipy._lib import doccer
from scipy.special import gammaln, psi, multigammaln, xlogy, entr
from scipy._lib._util import check_random_state
from scipy.linalg.blas import drot
from scipy.linalg.misc import LinAlgError
from scipy.linalg.lapack import get_lapack_funcs

from ._discrete_distns import binom
from . import mvn

__all__ = ['multivariate_normal',
           'matrix_normal',
           'dirichlet',
           'wishart',
           'invwishart',
           'multinomial',
           'special_ortho_group',
           'ortho_group',
           'random_correlation',
           'unitary_group']

_LOG_2PI = np.log(2 * np.pi)
_LOG_2 = np.log(2)
_LOG_PI = np.log(np.pi)

_doc_random_state = """\
random_state : None or int or np.random.RandomState instance, optional
    If int or RandomState, use it for drawing the random variates.
    If None (or np.random), the global np.random state is used.
    Default is None.
"""

def _squeeze_output(out):
    """
    Remove single-dimensional entries from array and convert to scalar,
    if necessary.

"""
    out = out.squeeze()
    if out.ndim == 0:
        out = out[()]
    return out

def _eigvalsh_to_eps(spectrum, cond=None, rcond=None):
    """
    Determine which eigenvalues are "small" given the spectrum.

This is for compatibility across various linear algebra functions
    that should agree about whether or not a Hermitian matrix is numerically
    singular and what is its numerical matrix rank.
    This is designed to be compatible with scipy.linalg.pinvh.

Parameters
    ----------
    spectrum : 1d ndarray
        Array of eigenvalues of a Hermitian matrix.
    cond, rcond : float, optional
        Cutoff for small eigenvalues.
        Singular values smaller than rcond * largest_eigenvalue are
        considered zero.
        If None or -1, suitable machine precision is used.

Returns
    -------
    eps : float
        Magnitude cutoff for numerical negligibility.

"""
    if rcond is not None:
        cond = rcond
    if cond in [None, -1]:
        t = spectrum.dtype.char.lower()
        factor = {'f': 1E3, 'd': 1E6}
        cond = factor[t] * np.finfo(t).eps
    eps = cond * np.max(abs(spectrum))
    return eps

def _pinv_1d(v, eps=1e-5):
    """
    A helper function for computing the pseudoinverse.

Parameters
    ----------
    v : iterable of numbers
        This may be thought of as a vector of eigenvalues or singular values.
    eps : float
        Values with magnitude no greater than eps are considered negligible.

Returns
    -------
    v_pinv : 1d float ndarray
        A vector of pseudo-inverted numbers.

"""
    return np.array([0 if abs(x) <= eps else 1/x for x in v], dtype=float)

class _PSD(object):
    """
    Compute coordinated functions of a symmetric positive semidefinite matrix.

This class addresses two issues.  Firstly it allows the pseudoinverse,
    the logarithm of the pseudo-determinant, and the rank of the matrix
    to be computed using one call to eigh instead of three.
    Secondly it allows these functions to be computed in a way
    that gives mutually compatible results.
    All of the functions are computed with a common understanding as to
    which of the eigenvalues are to be considered negligibly small.
    The functions are designed to coordinate with scipy.linalg.pinvh()
    but not necessarily with np.linalg.det() or with np.linalg.matrix_rank().

Parameters
    ----------
    M : array_like
        Symmetric positive semidefinite matrix (2-D).
    cond, rcond : float, optional
        Cutoff for small eigenvalues.
        Singular values smaller than rcond * largest_eigenvalue are
        considered zero.
        If None or -1, suitable machine precision is used.
    lower : bool, optional
        Whether the pertinent array data is taken from the lower
        or upper triangle of M. (Default: lower)
    check_finite : bool, optional
        Whether to check that the input matrices contain only finite
        numbers. Disabling may give a performance gain, but may result
        in problems (crashes, non-termination) if the inputs do contain
        infinities or NaNs.
    allow_singular : bool, optional
        Whether to allow a singular matrix.  (Default: True)

Notes
    -----
    The arguments are similar to those of scipy.linalg.pinvh().

"""

def __init__(self, M, cond=None, rcond=None, lower=True,
                 check_finite=True, allow_singular=True):
        # Compute the symmetric eigendecomposition.
        # Note that eigh takes care of array conversion, chkfinite,
        # and assertion that the matrix is square.
        s, u = scipy.linalg.eigh(M, lower=lower, check_finite=check_finite)

eps = _eigvalsh_to_eps(s, cond, rcond)
        if np.min(s) < -eps:
            raise ValueError('the input matrix must be positive semidefinite')
        d = s[s > eps]
        if len(d) < len(s) and not allow_singular:
            raise np.linalg.LinAlgError('singular matrix')
        s_pinv = _pinv_1d(s, eps)
        U = np.multiply(u, np.sqrt(s_pinv))

# Initialize the eagerly precomputed attributes.
        self.rank = len(d)
        self.U = U
        self.log_pdet = np.sum(np.log(d))

# Initialize an attribute to be lazily computed.
        self._pinv = None

@property
    def pinv(self):
        if self._pinv is None:
            self._pinv = np.dot(self.U, self.U.T)
        return self._pinv

class multi_rv_generic(object):
    """
    Class which encapsulates common functionality between all multivariate
    distributions.

"""
    def __init__(self, seed=None):
        super(multi_rv_generic, self).__init__()
        self._random_state = check_random_state(seed)

@property
    def random_state(self):
        """ Get or set the RandomState object for generating random variates.

This can be either None or an existing RandomState object.

If None (or np.random), use the RandomState singleton used by np.random.
        If already a RandomState instance, use it.
        If an int, use a new RandomState instance seeded with seed.

"""
        return self._random_state

@random_state.setter
    def random_state(self, seed):
        self._random_state = check_random_state(seed)

def _get_random_state(self, random_state):
        if random_state is not None:
            return check_random_state(random_state)
        else:
            return self._random_state

class multi_rv_frozen(object):
    """
    Class which encapsulates common functionality between all frozen
    multivariate distributions.
    """
    @property
    def random_state(self):
        return self._dist._random_state

@random_state.setter
    def random_state(self, seed):
        self._dist._random_state = check_random_state(seed)

_mvn_doc_default_callparams = """\
mean : array_like, optional
    Mean of the distribution (default zero)
cov : array_like, optional
    Covariance matrix of the distribution (default one)
allow_singular : bool, optional
    Whether to allow a singular covariance matrix.  (Default: False)
"""

_mvn_doc_callparams_note = \
    """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, or a two-dimensional
    array_like.
    """

_mvn_doc_frozen_callparams = ""

_mvn_doc_frozen_callparams_note = \
    """See class definition for a detailed description of parameters."""

mvn_docdict_params = {
    '_mvn_doc_default_callparams': _mvn_doc_default_callparams,
    '_mvn_doc_callparams_note': _mvn_doc_callparams_note,
    '_doc_random_state': _doc_random_state
}

mvn_docdict_noparams = {
    '_mvn_doc_default_callparams': _mvn_doc_frozen_callparams,
    '_mvn_doc_callparams_note': _mvn_doc_frozen_callparams_note,
    '_doc_random_state': _doc_random_state
}

class multivariate_normal_gen(multi_rv_generic):
    r"""
    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)``
        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()``
        Compute the differential entropy of the multivariate normal.

Parameters
    ----------
    x : array_like
        Quantiles, with the last axis of `x` denoting the components.
    %(_mvn_doc_default_callparams)s
    %(_doc_random_state)s

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.

Notes
    -----
    %(_mvn_doc_callparams_note)s

The covariance matrix `cov` must be a (symmetric) positive
    semi-definite matrix. 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,
    and :math:`k` is the dimension of the space where :math:`x` takes values.

.. versionadded:: 0.14.0

Examples
    --------
    >>> 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)

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))

"""

def __init__(self, seed=None):
        super(multivariate_normal_gen, self).__init__(seed)
        self.__doc__ = doccer.docformat(self.__doc__, mvn_docdict_params)

def __call__(self, mean=None, cov=1, allow_singular=False, seed=None):
        """
        Create a frozen multivariate normal distribution.

See `multivariate_normal_frozen` for more information.

"""
        return multivariate_normal_frozen(mean, cov,
                                          allow_singular=allow_singular,
                                          seed=seed)

def _process_parameters(self, dim, mean, cov):
        """
        Infer dimensionality from mean or covariance matrix, ensure that
        mean and covariance are full vector resp. matrix.

"""

# Try to infer dimensionality
        if dim is None:
            if mean is None:
                if cov is None:
                    dim = 1
                else:
                    cov = np.asarray(cov, dtype=float)
                    if cov.ndim < 2:
                        dim = 1
                    else:
                        dim = cov.shape[0]
            else:
                mean = np.asarray(mean, dtype=float)
                dim = mean.size
        else:
            if not np.isscalar(dim):
                raise ValueError("Dimension of random variable must be "
                                 "a scalar.")

# Check input sizes and return full arrays for mean and cov if
        # necessary
        if mean is None:
            mean = np.zeros(dim)
        mean = np.asarray(mean, dtype=float)

if cov is None:
            cov = 1.0
        cov = np.asarray(cov, dtype=float)

if dim == 1:
            mean.shape = (1,)
            cov.shape = (1, 1)

if mean.ndim != 1 or mean.shape[0] != dim:
            raise ValueError("Array 'mean' must be a vector of length %d." %
                             dim)
        if cov.ndim == 0:
            cov = cov * np.eye(dim)
        elif cov.ndim == 1:
            cov = np.diag(cov)
        elif cov.ndim == 2 and cov.shape != (dim, dim):
            rows, cols = cov.shape
            if rows != cols:
                msg = ("Array 'cov' must be square if it is two dimensional,"
                       " but cov.shape = %s." % str(cov.shape))
            else:
                msg = ("Dimension mismatch: array 'cov' is of shape %s,"
                       " but 'mean' is a vector of length %d.")
                msg = msg % (str(cov.shape), len(mean))
            raise ValueError(msg)
        elif cov.ndim > 2:
            raise ValueError("Array 'cov' must be at most two-dimensional,"
                             " but cov.ndim = %d" % cov.ndim)

return dim, mean, cov

def _process_quantiles(self, x, dim):
        """
        Adjust quantiles array so that last axis labels the components of
        each data point.

"""
        x = np.asarray(x, dtype=float)

if x.ndim == 0:
            x = x[np.newaxis]
        elif x.ndim == 1:
            if dim == 1:
                x = x[:, np.newaxis]
            else:
                x = x[np.newaxis, :]

return x

def _logpdf(self, x, mean, prec_U, log_det_cov, rank):
        """
        Parameters
        ----------
        x : ndarray
            Points at which to evaluate the log of the probability
            density function
        mean : ndarray
            Mean of the distribution
        prec_U : ndarray
            A decomposition such that np.dot(prec_U, prec_U.T)
            is the precision matrix, i.e. inverse of the covariance matrix.
        log_det_cov : float
            Logarithm of the determinant of the covariance matrix
        rank : int
            Rank of the covariance matrix.

Notes
        -----
        As this function does no argument checking, it should not be
        called directly; use 'logpdf' instead.

"""
        dev = x - mean
        maha = np.sum(np.square(np.dot(dev, prec_U)), axis=-1)
        return -0.5 * (rank * _LOG_2PI + log_det_cov + maha)

def logpdf(self, x, mean=None, cov=1, allow_singular=False):
        """
        Log of the multivariate normal probability density function.

Parameters
        ----------
        x : array_like
            Quantiles, with the last axis of `x` denoting the components.
        %(_mvn_doc_default_callparams)s

Returns
        -------
        pdf : ndarray or scalar
            Log of the probability density function evaluated at `x`

Notes
        -----
        %(_mvn_doc_callparams_note)s

"""
        dim, mean, cov = self._process_parameters(None, mean, cov)
        x = self._process_quantiles(x, dim)
        psd = _PSD(cov, allow_singular=allow_singular)
        out = self._logpdf(x, mean, psd.U, psd.log_pdet, psd.rank)
        return _squeeze_output(out)

def pdf(self, x, mean=None, cov=1, allow_singular=False):
        """
        Multivariate normal probability density function.

Parameters
        ----------
        x : array_like
            Quantiles, with the last axis of `x` denoting the components.
        %(_mvn_doc_default_callparams)s

Returns
        -------
        pdf : ndarray or scalar
            Probability density function evaluated at `x`

Notes
        -----
        %(_mvn_doc_callparams_note)s

"""
        dim, mean, cov = self._process_parameters(None, mean, cov)
        x = self._process_quantiles(x, dim)
        psd = _PSD(cov, allow_singular=allow_singular)
        out = np.exp(self._logpdf(x, mean, psd.U, psd.log_pdet, psd.rank))
        return _squeeze_output(out)

def _cdf(self, x, mean, cov, maxpts, abseps, releps):
        """
        Parameters
        ----------
        x : ndarray
            Points at which to evaluate the cumulative distribution function.
        mean : ndarray
            Mean of the distribution
        cov : array_like
            Covariance matrix of the distribution
        maxpts: integer
            The maximum number of points to use for integration
        abseps: float
            Absolute error tolerance
        releps: float
            Relative error tolerance

Notes
        -----
        As this function does no argument checking, it should not be
        called directly; use 'cdf' instead.

.. versionadded:: 1.0.0

"""
        lower = np.full(mean.shape, -np.inf)
        # mvnun expects 1-d arguments, so process points sequentially
        func1d = lambda x_slice: mvn.mvnun(lower, x_slice, mean, cov,
                                           maxpts, abseps, releps)[0]
        out = np.apply_along_axis(func1d, -1, x)
        return _squeeze_output(out)

def logcdf(self, x, mean=None, cov=1, allow_singular=False, maxpts=None,
               abseps=1e-5, releps=1e-5):
        """
        Log of the multivariate normal cumulative distribution function.

Parameters
        ----------
        x : array_like
            Quantiles, with the last axis of `x` denoting the components.
        %(_mvn_doc_default_callparams)s
        maxpts: integer, optional
            The maximum number of points to use for integration
            (default `1000000*dim`)
        abseps: float, optional
            Absolute error tolerance (default 1e-5)
        releps: float, optional
            Relative error tolerance (default 1e-5)

Returns
        -------
        cdf : ndarray or scalar
            Log of the cumulative distribution function evaluated at `x`

Notes
        -----
        %(_mvn_doc_callparams_note)s

.. versionadded:: 1.0.0

"""
        dim, mean, cov = self._process_parameters(None, mean, cov)
        x = self._process_quantiles(x, dim)
        # Use _PSD to check covariance matrix
        _PSD(cov, allow_singular=allow_singular)
        if not maxpts:
            maxpts = 1000000 * dim
        out = np.log(self._cdf(x, mean, cov, maxpts, abseps, releps))
        return out

def cdf(self, x, mean=None, cov=1, allow_singular=False, maxpts=None,
            abseps=1e-5, releps=1e-5):
        """
        Multivariate normal cumulative distribution function.

Returns
        -------
        cdf : ndarray or scalar
            Cumulative distribution function evaluated at `x`

Notes
        -----
        %(_mvn_doc_callparams_note)s

.. versionadded:: 1.0.0

"""
        dim, mean, cov = self._process_parameters(None, mean, cov)
        x = self._process_quantiles(x, dim)
        # Use _PSD to check covariance matrix
        _PSD(cov, allow_singular=allow_singular)
        if not maxpts:
            maxpts = 1000000 * dim
        out = self._cdf(x, mean, cov, maxpts, abseps, releps)
        return out

def rvs(self, mean=None, cov=1, size=1, random_state=None):
        """
        Draw random samples from a multivariate normal distribution.

Parameters
        ----------
        %(_mvn_doc_default_callparams)s
        size : integer, optional
            Number of samples to draw (default 1).
        %(_doc_random_state)s

Returns
        -------
        rvs : ndarray or scalar
            Random variates of size (`size`, `N`), where `N` is the
            dimension of the random variable.

Notes
        -----
        %(_mvn_doc_callparams_note)s

"""
        dim, mean, cov = self._process_parameters(None, mean, cov)

random_state = self._get_random_state(random_state)
        out = random_state.multivariate_normal(mean, cov, size)
        return _squeeze_output(out)

def entropy(self, mean=None, cov=1):
        """
        Compute the differential entropy of the multivariate normal.

Parameters
        ----------
        %(_mvn_doc_default_callparams)s

Returns
        -------
        h : scalar
            Entropy of the multivariate normal distribution

Notes
        -----
        %(_mvn_doc_callparams_note)s

"""
        dim, mean, cov = self._process_parameters(None, mean, cov)
        _, logdet = np.linalg.slogdet(2 * np.pi * np.e * cov)
        return 0.5 * logdet

multivariate_normal = multivariate_normal_gen()

class multivariate_normal_frozen(multi_rv_frozen):
    def __init__(self, mean=None, cov=1, allow_singular=False, seed=None,
                 maxpts=None, abseps=1e-5, releps=1e-5):
        """
        Create a frozen multivariate normal distribution.

Parameters
        ----------
        mean : array_like, optional
            Mean of the distribution (default zero)
        cov : array_like, optional
            Covariance matrix of the distribution (default one)
        allow_singular : bool, optional
            If this flag is True then tolerate a singular
            covariance matrix (default False).
        seed : None or int or np.random.RandomState instance, optional
            This parameter defines the RandomState object to use for drawing
            random variates.
            If None (or np.random), the global np.random state is used.
            If integer, it is used to seed the local RandomState instance
            Default is None.
        maxpts: integer, optional
            The maximum number of points to use for integration of the
            cumulative distribution function (default `1000000*dim`)
        abseps: float, optional
            Absolute error tolerance for the cumulative distribution function
            (default 1e-5)
        releps: float, optional
            Relative error tolerance for the cumulative distribution function
            (default 1e-5)

Examples
        --------
        When called with the default parameters, this will create a 1D random
        variable with mean 0 and covariance 1:

>>> from scipy.stats import multivariate_normal
        >>> r = multivariate_normal()
        >>> r.mean
        array([ 0.])
        >>> r.cov
        array([[1.]])

"""
        self._dist = multivariate_normal_gen(seed)
        self.dim, self.mean, self.cov = self._dist._process_parameters(
                                                            None, mean, cov)
        self.cov_info = _PSD(self.cov, allow_singular=allow_singular)
        if not maxpts:
            maxpts = 1000000 * self.dim
        self.maxpts = maxpts
        self.abseps = abseps
        self.releps = releps

def logpdf(self, x):
        x = self._dist._process_quantiles(x, self.dim)
        out = self._dist._logpdf(x, self.mean, self.cov_info.U,
                                 self.cov_info.log_pdet, self.cov_info.rank)
        return _squeeze_output(out)

def pdf(self, x):
        return np.exp(self.logpdf(x))

def logcdf(self, x):
        return np.log(self.cdf(x))

def cdf(self, x):
        x = self._dist._process_quantiles(x, self.dim)
        out = self._dist._cdf(x, self.mean, self.cov, self.maxpts, self.abseps,
                              self.releps)
        return _squeeze_output(out)

def rvs(self, size=1, random_state=None):
        return self._dist.rvs(self.mean, self.cov, size, random_state)

def entropy(self):
        """
        Computes the differential entropy of the multivariate normal.

Returns
        -------
        h : scalar
            Entropy of the multivariate normal distribution

"""
        log_pdet = self.cov_info.log_pdet
        rank = self.cov_info.rank
        return 0.5 * (rank * (_LOG_2PI + 1) + log_pdet)

# Set frozen generator docstrings from corresponding docstrings in
# multivariate_normal_gen and fill in default strings in class docstrings
for name in ['logpdf', 'pdf', 'logcdf', 'cdf', 'rvs']:
    method = multivariate_normal_gen.__dict__[name]
    method_frozen = multivariate_normal_frozen.__dict__[name]
    method_frozen.__doc__ = doccer.docformat(method.__doc__,
                                             mvn_docdict_noparams)
    method.__doc__ = doccer.docformat(method.__doc__, mvn_docdict_params)

_matnorm_doc_default_callparams = """\
mean : array_like, optional
    Mean of the distribution (default: `None`)
rowcov : array_like, optional
    Among-row covariance matrix of the distribution (default: `1`)
colcov : array_like, optional
    Among-column covariance matrix of the distribution (default: `1`)
"""

_matnorm_doc_callparams_note = \
    """If `mean` is set to `None` then a matrix of zeros is used for the mean.
    The dimensions of this matrix are inferred from the shape of `rowcov` and
    `colcov`, if these are provided, or set to `1` if ambiguous.

`rowcov` and `colcov` can be two-dimensional array_likes specifying the
    covariance matrices directly. Alternatively, a one-dimensional array will
    be be interpreted as the entries of a diagonal matrix, and a scalar or
    zero-dimensional array will be interpreted as this value times the
    identity matrix.
    """

_matnorm_doc_frozen_callparams = ""

_matnorm_doc_frozen_callparams_note = \
    """See class definition for a detailed description of parameters."""

matnorm_docdict_params = {
    '_matnorm_doc_default_callparams': _matnorm_doc_default_callparams,
    '_matnorm_doc_callparams_note': _matnorm_doc_callparams_note,
    '_doc_random_state': _doc_random_state
}

matnorm_docdict_noparams = {
    '_matnorm_doc_default_callparams': _matnorm_doc_frozen_callparams,
    '_matnorm_doc_callparams_note': _matnorm_doc_frozen_callparams_note,
    '_doc_random_state': _doc_random_state
}

class matrix_normal_gen(multi_rv_generic):
    r"""
    A matrix normal random variable.

The `mean` keyword specifies the mean. The `rowcov` keyword specifies the
    among-row covariance matrix. The 'colcov' keyword specifies the
    among-column covariance matrix.

Methods
    -------
    ``pdf(X, mean=None, rowcov=1, colcov=1)``
        Probability density function.
    ``logpdf(X, mean=None, rowcov=1, colcov=1)``
        Log of the probability density function.
    ``rvs(mean=None, rowcov=1, colcov=1, size=1, random_state=None)``
        Draw random samples.

Parameters
    ----------
    X : array_like
        Quantiles, with the last two axes of `X` denoting the components.
    %(_matnorm_doc_default_callparams)s
    %(_doc_random_state)s

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

rv = matrix_normal(mean=None, rowcov=1, colcov=1)
        - Frozen object with the same methods but holding the given
          mean and covariance fixed.

Notes
    -----
    %(_matnorm_doc_callparams_note)s

The covariance matrices specified by `rowcov` and `colcov` must be
    (symmetric) positive definite. If the samples in `X` are
    :math:`m \times n`, then `rowcov` must be :math:`m \times m` and
    `colcov` must be :math:`n \times n`. `mean` must be the same shape as `X`.

The probability density function for `matrix_normal` is

.. math::

f(X) = (2 \pi)^{-\frac{mn}{2}}|U|^{-\frac{n}{2}} |V|^{-\frac{m}{2}}
               \exp\left( -\frac{1}{2} \mathrm{Tr}\left[ U^{-1} (X-M) V^{-1}
               (X-M)^T \right] \right),

where :math:`M` is the mean, :math:`U` the among-row covariance matrix,
    :math:`V` the among-column covariance matrix.

The `allow_singular` behaviour of the `multivariate_normal`
    distribution is not currently supported. Covariance matrices must be
    full rank.

The `matrix_normal` distribution is closely related to the
    `multivariate_normal` distribution. Specifically, :math:`\mathrm{Vec}(X)`
    (the vector formed by concatenating the columns  of :math:`X`) has a
    multivariate normal distribution with mean :math:`\mathrm{Vec}(M)`
    and covariance :math:`V \otimes U` (where :math:`\otimes` is the Kronecker
    product). Sampling and pdf evaluation are
    :math:`\mathcal{O}(m^3 + n^3 + m^2 n + m n^2)` for the matrix normal, but
    :math:`\mathcal{O}(m^3 n^3)` for the equivalent multivariate normal,
    making this equivalent form algorithmically inefficient.

.. versionadded:: 0.17.0

Examples
    --------

>>> from scipy.stats import matrix_normal

>>> M = np.arange(6).reshape(3,2); M
    array([[0, 1],
           [2, 3],
           [4, 5]])
    >>> U = np.diag([1,2,3]); U
    array([[1, 0, 0],
           [0, 2, 0],
           [0, 0, 3]])
    >>> V = 0.3*np.identity(2); V
    array([[ 0.3,  0. ],
           [ 0. ,  0.3]])
    >>> X = M + 0.1; X
    array([[ 0.1,  1.1],
           [ 2.1,  3.1],
           [ 4.1,  5.1]])
    >>> matrix_normal.pdf(X, mean=M, rowcov=U, colcov=V)
    0.023410202050005054

>>> # Equivalent multivariate normal
    >>> from scipy.stats import multivariate_normal
    >>> vectorised_X = X.T.flatten()
    >>> equiv_mean = M.T.flatten()
    >>> equiv_cov = np.kron(V,U)
    >>> multivariate_normal.pdf(vectorised_X, mean=equiv_mean, cov=equiv_cov)
    0.023410202050005054
    """

def __init__(self, seed=None):
        super(matrix_normal_gen, self).__init__(seed)
        self.__doc__ = doccer.docformat(self.__doc__, matnorm_docdict_params)

def __call__(self, mean=None, rowcov=1, colcov=1, seed=None):
        """
        Create a frozen matrix normal distribution.

See `matrix_normal_frozen` for more information.

"""
        return matrix_normal_frozen(mean, rowcov, colcov, seed=seed)

def _process_parameters(self, mean, rowcov, colcov):
        """
        Infer dimensionality from mean or covariance matrices. Handle
        defaults. Ensure compatible dimensions.

"""

# Process mean
        if mean is not None:
            mean = np.asarray(mean, dtype=float)
            meanshape = mean.shape
            if len(meanshape) != 2:
                raise ValueError("Array `mean` must be two dimensional.")
            if np.any(meanshape == 0):
                raise ValueError("Array `mean` has invalid shape.")

# Process among-row covariance
        rowcov = np.asarray(rowcov, dtype=float)
        if rowcov.ndim == 0:
            if mean is not None:
                rowcov = rowcov * np.identity(meanshape[0])
            else:
                rowcov = rowcov * np.identity(1)
        elif rowcov.ndim == 1:
            rowcov = np.diag(rowcov)
        rowshape = rowcov.shape
        if len(rowshape) != 2:
            raise ValueError("`rowcov` must be a scalar or a 2D array.")
        if rowshape[0] != rowshape[1]:
            raise ValueError("Array `rowcov` must be square.")
        if rowshape[0] == 0:
            raise ValueError("Array `rowcov` has invalid shape.")
        numrows = rowshape[0]

# Process among-column covariance
        colcov = np.asarray(colcov, dtype=float)
        if colcov.ndim == 0:
            if mean is not None:
                colcov = colcov * np.identity(meanshape[1])
            else:
                colcov = colcov * np.identity(1)
        elif colcov.ndim == 1:
            colcov = np.diag(colcov)
        colshape = colcov.shape
        if len(colshape) != 2:
            raise ValueError("`colcov` must be a scalar or a 2D array.")
        if colshape[0] != colshape[1]:
            raise ValueError("Array `colcov` must be square.")
        if colshape[0] == 0:
            raise ValueError("Array `colcov` has invalid shape.")
        numcols = colshape[0]

# Ensure mean and covariances compatible
        if mean is not None:
            if meanshape[0] != numrows:
                raise ValueError("Arrays `mean` and `rowcov` must have the "
                                 "same number of rows.")
            if meanshape[1] != numcols:
                raise ValueError("Arrays `mean` and `colcov` must have the "
                                 "same number of columns.")
        else:
            mean = np.zeros((numrows, numcols))

dims = (numrows, numcols)

return dims, mean, rowcov, colcov

def _process_quantiles(self, X, dims):
        """
        Adjust quantiles array so that last two axes labels the components of
        each data point.

"""
        X = np.asarray(X, dtype=float)
        if X.ndim == 2:
            X = X[np.newaxis, :]
        if X.shape[-2:] != dims:
            raise ValueError("The shape of array `X` is not compatible "
                             "with the distribution parameters.")
        return X

def _logpdf(self, dims, X, mean, row_prec_rt, log_det_rowcov,
                col_prec_rt, log_det_colcov):
        """
        Parameters
        ----------
        dims : tuple
            Dimensions of the matrix variates
        X : ndarray
            Points at which to evaluate the log of the probability
            density function
        mean : ndarray
            Mean of the distribution
        row_prec_rt : ndarray
            A decomposition such that np.dot(row_prec_rt, row_prec_rt.T)
            is the inverse of the among-row covariance matrix
        log_det_rowcov : float
            Logarithm of the determinant of the among-row covariance matrix
        col_prec_rt : ndarray
            A decomposition such that np.dot(col_prec_rt, col_prec_rt.T)
            is the inverse of the among-column covariance matrix
        log_det_colcov : float
            Logarithm of the determinant of the among-column covariance matrix

Notes
        -----
        As this function does no argument checking, it should not be
        called directly; use 'logpdf' instead.

"""
        numrows, numcols = dims
        roll_dev = np.rollaxis(X-mean, axis=-1, start=0)
        scale_dev = np.tensordot(col_prec_rt.T,
                                 np.dot(roll_dev, row_prec_rt), 1)
        maha = np.sum(np.sum(np.square(scale_dev), axis=-1), axis=0)
        return -0.5 * (numrows*numcols*_LOG_2PI + numcols*log_det_rowcov
                       + numrows*log_det_colcov + maha)

def logpdf(self, X, mean=None, rowcov=1, colcov=1):
        """
        Log of the matrix normal probability density function.

Parameters
        ----------
        X : array_like
            Quantiles, with the last two axes of `X` denoting the components.
        %(_matnorm_doc_default_callparams)s

Returns
        -------
        logpdf : ndarray
            Log of the probability density function evaluated at `X`

Notes
        -----
        %(_matnorm_doc_callparams_note)s

"""
        dims, mean, rowcov, colcov = self._process_parameters(mean, rowcov,
                                                              colcov)
        X = self._process_quantiles(X, dims)
        rowpsd = _PSD(rowcov, allow_singular=False)
        colpsd = _PSD(colcov, allow_singular=False)
        out = self._logpdf(dims, X, mean, rowpsd.U, rowpsd.log_pdet, colpsd.U,
                           colpsd.log_pdet)
        return _squeeze_output(out)

def pdf(self, X, mean=None, rowcov=1, colcov=1):
        """
        Matrix normal probability density function.

Parameters
        ----------
        X : array_like
            Quantiles, with the last two axes of `X` denoting the components.
        %(_matnorm_doc_default_callparams)s

Returns
        -------
        pdf : ndarray
            Probability density function evaluated at `X`

Notes
        -----
        %(_matnorm_doc_callparams_note)s

"""
        return np.exp(self.logpdf(X, mean, rowcov, colcov))

def rvs(self, mean=None, rowcov=1, colcov=1, size=1, random_state=None):
        """
        Draw random samples from a matrix normal distribution.

Parameters
        ----------
        %(_matnorm_doc_default_callparams)s
        size : integer, optional
            Number of samples to draw (default 1).
        %(_doc_random_state)s

Returns
        -------
        rvs : ndarray or scalar
            Random variates of size (`size`, `dims`), where `dims` is the
            dimension of the random matrices.

Notes
        -----
        %(_matnorm_doc_callparams_note)s

"""
        size = int(size)
        dims, mean, rowcov, colcov = self._process_parameters(mean, rowcov,
                                                              colcov)
        rowchol = scipy.linalg.cholesky(rowcov, lower=True)
        colchol = scipy.linalg.cholesky(colcov, lower=True)
        random_state = self._get_random_state(random_state)
        std_norm = random_state.standard_normal(size=(dims[1], size, dims[0]))
        roll_rvs = np.tensordot(colchol, np.dot(std_norm, rowchol.T), 1)
        out = np.rollaxis(roll_rvs.T, axis=1, start=0) + mean[np.newaxis, :, :]
        if size == 1:
            out = out.reshape(mean.shape)
        return out

matrix_normal = matrix_normal_gen()

class matrix_normal_frozen(multi_rv_frozen):
    def __init__(self, mean=None, rowcov=1, colcov=1, seed=None):
        """
        Create a frozen matrix normal distribution.

Parameters
        ----------
        %(_matnorm_doc_default_callparams)s
        seed : None or int or np.random.RandomState instance, optional
            If int or RandomState, use it for drawing the random variates.
            If None (or np.random), the global np.random state is used.
            Default is None.

Examples
        --------
        >>> from scipy.stats import matrix_normal

>>> distn = matrix_normal(mean=np.zeros((3,3)))
        >>> X = distn.rvs(); X
        array([[-0.02976962,  0.93339138, -0.09663178],
               [ 0.67405524,  0.28250467, -0.93308929],
               [-0.31144782,  0.74535536,  1.30412916]])
        >>> distn.pdf(X)
        2.5160642368346784e-05
        >>> distn.logpdf(X)
        -10.590229595124615
        """
        self._dist = matrix_normal_gen(seed)
        self.dims, self.mean, self.rowcov, self.colcov = \
            self._dist._process_parameters(mean, rowcov, colcov)
        self.rowpsd = _PSD(self.rowcov, allow_singular=False)
        self.colpsd = _PSD(self.colcov, allow_singular=False)

def logpdf(self, X):
        X = self._dist._process_quantiles(X, self.dims)
        out = self._dist._logpdf(self.dims, X, self.mean, self.rowpsd.U,
                                 self.rowpsd.log_pdet, self.colpsd.U,
                                 self.colpsd.log_pdet)
        return _squeeze_output(out)

def pdf(self, X):
        return np.exp(self.logpdf(X))

def rvs(self, size=1, random_state=None):
        return self._dist.rvs(self.mean, self.rowcov, self.colcov, size,
                              random_state)

# Set frozen generator docstrings from corresponding docstrings in
# matrix_normal_gen and fill in default strings in class docstrings
for name in ['logpdf', 'pdf', 'rvs']:
    method = matrix_normal_gen.__dict__[name]
    method_frozen = matrix_normal_frozen.__dict__[name]
    method_frozen.__doc__ = doccer.docformat(method.__doc__,
                                             matnorm_docdict_noparams)
    method.__doc__ = doccer.docformat(method.__doc__, matnorm_docdict_params)

_dirichlet_doc_default_callparams = """\
alpha : array_like
    The concentration parameters. The number of entries determines the
    dimensionality of the distribution.
"""
_dirichlet_doc_frozen_callparams = ""

_dirichlet_doc_frozen_callparams_note = \
    """See class definition for a detailed description of parameters."""

dirichlet_docdict_params = {
    '_dirichlet_doc_default_callparams': _dirichlet_doc_default_callparams,
    '_doc_random_state': _doc_random_state
}

dirichlet_docdict_noparams = {
    '_dirichlet_doc_default_callparams': _dirichlet_doc_frozen_callparams,
    '_doc_random_state': _doc_random_state
}

def _dirichlet_check_parameters(alpha):
    alpha = np.asarray(alpha)
    if np.min(alpha) <= 0:
        raise ValueError("All parameters must be greater than 0")
    elif alpha.ndim != 1:
        raise ValueError("Parameter vector 'a' must be one dimensional, "
                         "but a.shape = %s." % (alpha.shape, ))
    return alpha

def _dirichlet_check_input(alpha, x):
    x = np.asarray(x)

if x.shape[0] + 1 != alpha.shape[0] and x.shape[0] != alpha.shape[0]:
        raise ValueError("Vector 'x' must have either the same number "
                         "of entries as, or one entry fewer than, "
                         "parameter vector 'a', but alpha.shape = %s "
                         "and x.shape = %s." % (alpha.shape, x.shape))

if x.shape[0] != alpha.shape[0]:
        xk = np.array([1 - np.sum(x, 0)])
        if xk.ndim == 1:
            x = np.append(x, xk)
        elif xk.ndim == 2:
            x = np.vstack((x, xk))
        else:
            raise ValueError("The input must be one dimensional or a two "
                             "dimensional matrix containing the entries.")

if np.min(x) < 0:
        raise ValueError("Each entry in 'x' must be greater than or equal "
                         "to zero.")

if np.max(x) > 1:
        raise ValueError("Each entry in 'x' must be smaller or equal one.")

# Check x_i > 0 or alpha_i > 1
    xeq0 = (x == 0)
    alphalt1 = (alpha < 1)
    if x.shape != alpha.shape:
        alphalt1 = np.repeat(alphalt1, x.shape[-1], axis=-1).reshape(x.shape)
    chk = np.logical_and(xeq0, alphalt1)

if np.sum(chk):
        raise ValueError("Each entry in 'x' must be greater than zero if its "
                         "alpha is less than one.")

if (np.abs(np.sum(x, 0) - 1.0) > 10e-10).any():
        raise ValueError("The input vector 'x' must lie within the normal "
                         "simplex. but np.sum(x, 0) = %s." % np.sum(x, 0))

return x

def _lnB(alpha):
    r"""
    Internal helper function to compute the log of the useful quotient

.. math::

B(\alpha) = \frac{\prod_{i=1}{K}\Gamma(\alpha_i)}
                         {\Gamma\left(\sum_{i=1}^{K} \alpha_i \right)}

Parameters
    ----------
    %(_dirichlet_doc_default_callparams)s

Returns
    -------
    B : scalar
        Helper quotient, internal use only

"""
    return np.sum(gammaln(alpha)) - gammaln(np.sum(alpha))

class dirichlet_gen(multi_rv_generic):
    r"""
    A Dirichlet random variable.

The `alpha` keyword specifies the concentration parameters of the
    distribution.

.. versionadded:: 0.15.0

Methods
    -------
    ``pdf(x, alpha)``
        Probability density function.
    ``logpdf(x, alpha)``
        Log of the probability density function.
    ``rvs(alpha, size=1, random_state=None)``
        Draw random samples from a Dirichlet distribution.
    ``mean(alpha)``
        The mean of the Dirichlet distribution
    ``var(alpha)``
        The variance of the Dirichlet distribution
    ``entropy(alpha)``
        Compute the differential entropy of the Dirichlet distribution.

Parameters
    ----------
    x : array_like
        Quantiles, with the last axis of `x` denoting the components.
    %(_dirichlet_doc_default_callparams)s
    %(_doc_random_state)s

Alternatively, the object may be called (as a function) to fix
    concentration parameters, returning a "frozen" Dirichlet
    random variable:

rv = dirichlet(alpha)
        - Frozen object with the same methods but holding the given
          concentration parameters fixed.

Notes
    -----
    Each :math:`\alpha` entry must be positive. The distribution has only
    support on the simplex defined by

.. math::
        \sum_{i=1}^{K} x_i \le 1

The probability density function for `dirichlet` is

.. math::

f(x) = \frac{1}{\mathrm{B}(\boldsymbol\alpha)} \prod_{i=1}^K x_i^{\alpha_i - 1}

where

.. math::

\mathrm{B}(\boldsymbol\alpha) = \frac{\prod_{i=1}^K \Gamma(\alpha_i)}
                                     {\Gamma\bigl(\sum_{i=1}^K \alpha_i\bigr)}

and :math:`\boldsymbol\alpha=(\alpha_1,\ldots,\alpha_K)`, the
    concentration parameters and :math:`K` is the dimension of the space
    where :math:`x` takes values.

Note that the dirichlet interface is somewhat inconsistent.
    The array returned by the rvs function is transposed
    with respect to the format expected by the pdf and logpdf.

Examples
    --------
    >>> from scipy.stats import dirichlet

Generate a dirichlet random variable

>>> quantiles = np.array([0.2, 0.2, 0.6])  # specify quantiles
    >>> alpha = np.array([0.4, 5, 15])  # specify concentration parameters
    >>> dirichlet.pdf(quantiles, alpha)
    0.2843831684937255

The same PDF but following a log scale

>>> dirichlet.logpdf(quantiles, alpha)
    -1.2574327653159187

Once we specify the dirichlet distribution
    we can then calculate quantities of interest

>>> dirichlet.mean(alpha)  # get the mean of the distribution
    array([0.01960784, 0.24509804, 0.73529412])
    >>> dirichlet.var(alpha) # get variance
    array([0.00089829, 0.00864603, 0.00909517])
    >>> dirichlet.entropy(alpha)  # calculate the differential entropy
    -4.3280162474082715

We can also return random samples from the distribution

>>> dirichlet.rvs(alpha, size=1, random_state=1)
    array([[0.00766178, 0.24670518, 0.74563305]])
    >>> dirichlet.rvs(alpha, size=2, random_state=2)
    array([[0.01639427, 0.1292273 , 0.85437844],
           [0.00156917, 0.19033695, 0.80809388]])

"""

def __init__(self, seed=None):
        super(dirichlet_gen, self).__init__(seed)
        self.__doc__ = doccer.docformat(self.__doc__, dirichlet_docdict_params)

def __call__(self, alpha, seed=None):
        return dirichlet_frozen(alpha, seed=seed)

def _logpdf(self, x, alpha):
        """
        Parameters
        ----------
        x : ndarray
            Points at which to evaluate the log of the probability
            density function
        %(_dirichlet_doc_default_callparams)s

Notes
        -----
        As this function does no argument checking, it should not be
        called directly; use 'logpdf' instead.

"""
        lnB = _lnB(alpha)
        return - lnB + np.sum((xlogy(alpha - 1, x.T)).T, 0)

def logpdf(self, x, alpha):
        """
        Log of the Dirichlet probability density function.

Parameters
        ----------
        x : array_like
            Quantiles, with the last axis of `x` denoting the components.
        %(_dirichlet_doc_default_callparams)s

Returns
        -------
        pdf : ndarray or scalar
            Log of the probability density function evaluated at `x`.

"""
        alpha = _dirichlet_check_parameters(alpha)
        x = _dirichlet_check_input(alpha, x)

out = self._logpdf(x, alpha)
        return _squeeze_output(out)

def pdf(self, x, alpha):
        """
        The Dirichlet probability density function.

Parameters
        ----------
        x : array_like
            Quantiles, with the last axis of `x` denoting the components.
        %(_dirichlet_doc_default_callparams)s

Returns
        -------
        pdf : ndarray or scalar
            The probability density function evaluated at `x`.

"""
        alpha = _dirichlet_check_parameters(alpha)
        x = _dirichlet_check_input(alpha, x)

out = np.exp(self._logpdf(x, alpha))
        return _squeeze_output(out)

def mean(self, alpha):
        """
        Compute the mean of the dirichlet distribution.

Parameters
        ----------
        %(_dirichlet_doc_default_callparams)s

Returns
        -------
        mu : ndarray or scalar
            Mean of the Dirichlet distribution.

"""
        alpha = _dirichlet_check_parameters(alpha)

out = alpha / (np.sum(alpha))
        return _squeeze_output(out)

def var(self, alpha):
        """
        Compute the variance of the dirichlet distribution.

Parameters
        ----------
        %(_dirichlet_doc_default_callparams)s

Returns
        -------
        v : ndarray or scalar
            Variance of the Dirichlet distribution.

"""

alpha = _dirichlet_check_parameters(alpha)

alpha0 = np.sum(alpha)
        out = (alpha * (alpha0 - alpha)) / ((alpha0 * alpha0) * (alpha0 + 1))
        return _squeeze_output(out)

def entropy(self, alpha):
        """
        Compute the differential entropy of the dirichlet distribution.

Parameters
        ----------
        %(_dirichlet_doc_default_callparams)s

Returns
        -------
        h : scalar
            Entropy of the Dirichlet distribution

"""

alpha = _dirichlet_check_parameters(alpha)

alpha0 = np.sum(alpha)
        lnB = _lnB(alpha)
        K = alpha.shape[0]

out = lnB + (alpha0 - K) * scipy.special.psi(alpha0) - np.sum(
            (alpha - 1) * scipy.special.psi(alpha))
        return _squeeze_output(out)

def rvs(self, alpha, size=1, random_state=None):
        """
        Draw random samples from a Dirichlet distribution.

Parameters
        ----------
        %(_dirichlet_doc_default_callparams)s
        size : int, optional
            Number of samples to draw (default 1).
        %(_doc_random_state)s

Returns
        -------
        rvs : ndarray or scalar
            Random variates of size (`size`, `N`), where `N` is the
            dimension of the random variable.

"""
        alpha = _dirichlet_check_parameters(alpha)
        random_state = self._get_random_state(random_state)
        return random_state.dirichlet(alpha, size=size)

dirichlet = dirichlet_gen()

class dirichlet_frozen(multi_rv_frozen):
    def __init__(self, alpha, seed=None):
        self.alpha = _dirichlet_check_parameters(alpha)
        self._dist = dirichlet_gen(seed)

def logpdf(self, x):
        return self._dist.logpdf(x, self.alpha)

def pdf(self, x):
        return self._dist.pdf(x, self.alpha)

def mean(self):
        return self._dist.mean(self.alpha)

def var(self):
        return self._dist.var(self.alpha)

def entropy(self):
        return self._dist.entropy(self.alpha)

def rvs(self, size=1, random_state=None):
        return self._dist.rvs(self.alpha, size, random_state)

# Set frozen generator docstrings from corresponding docstrings in
# multivariate_normal_gen and fill in default strings in class docstrings
for name in ['logpdf', 'pdf', 'rvs', 'mean', 'var', 'entropy']:
    method = dirichlet_gen.__dict__[name]
    method_frozen = dirichlet_frozen.__dict__[name]
    method_frozen.__doc__ = doccer.docformat(
        method.__doc__, dirichlet_docdict_noparams)
    method.__doc__ = doccer.docformat(method.__doc__, dirichlet_docdict_params)

_wishart_doc_default_callparams = """\
df : int
    Degrees of freedom, must be greater than or equal to dimension of the
    scale matrix
scale : array_like
    Symmetric positive definite scale matrix of the distribution
"""

_wishart_doc_callparams_note = ""

_wishart_doc_frozen_callparams = ""

_wishart_doc_frozen_callparams_note = \
    """See class definition for a detailed description of parameters."""

wishart_docdict_params = {
    '_doc_default_callparams': _wishart_doc_default_callparams,
    '_doc_callparams_note': _wishart_doc_callparams_note,
    '_doc_random_state': _doc_random_state
}

wishart_docdict_noparams = {
    '_doc_default_callparams': _wishart_doc_frozen_callparams,
    '_doc_callparams_note': _wishart_doc_frozen_callparams_note,
    '_doc_random_state': _doc_random_state
}

class wishart_gen(multi_rv_generic):
    r"""
    A Wishart random variable.

The `df` keyword specifies the degrees of freedom. The `scale` keyword
    specifies the scale matrix, which must be symmetric and positive definite.
    In this context, the scale matrix is often interpreted in terms of a
    multivariate normal precision matrix (the inverse of the covariance
    matrix).

Methods
    -------
    ``pdf(x, df, scale)``
        Probability density function.
    ``logpdf(x, df, scale)``
        Log of the probability density function.
    ``rvs(df, scale, size=1, random_state=None)``
        Draw random samples from a Wishart distribution.
    ``entropy()``
        Compute the differential entropy of the Wishart distribution.

Parameters
    ----------
    x : array_like
        Quantiles, with the last axis of `x` denoting the components.
    %(_doc_default_callparams)s
    %(_doc_random_state)s

Alternatively, the object may be called (as a function) to fix the degrees
    of freedom and scale parameters, returning a "frozen" Wishart random
    variable:

rv = wishart(df=1, scale=1)
        - Frozen object with the same methods but holding the given
          degrees of freedom and scale fixed.

See Also
    --------
    invwishart, chi2

Notes
    -----
    %(_doc_callparams_note)s

The scale matrix `scale` must be a symmetric positive definite
    matrix. Singular matrices, including the symmetric positive semi-definite
    case, are not supported.

The Wishart distribution is often denoted

.. math::

W_p(\nu, \Sigma)

where :math:`\nu` is the degrees of freedom and :math:`\Sigma` is the
    :math:`p \times p` scale matrix.

The probability density function for `wishart` has support over positive
    definite matrices :math:`S`; if :math:`S \sim W_p(\nu, \Sigma)`, then
    its PDF is given by:

.. math::

f(S) = \frac{|S|^{\frac{\nu - p - 1}{2}}}{2^{ \frac{\nu p}{2} }
               |\Sigma|^\frac{\nu}{2} \Gamma_p \left ( \frac{\nu}{2} \right )}
               \exp\left( -tr(\Sigma^{-1} S) / 2 \right)

If :math:`S \sim W_p(\nu, \Sigma)` (Wishart) then
    :math:`S^{-1} \sim W_p^{-1}(\nu, \Sigma^{-1})` (inverse Wishart).

If the scale matrix is 1-dimensional and equal to one, then the Wishart
    distribution :math:`W_1(\nu, 1)` collapses to the :math:`\chi^2(\nu)`
    distribution.

.. versionadded:: 0.16.0

References
    ----------
    .. [1] M.L. Eaton, "Multivariate Statistics: A Vector Space Approach",
           Wiley, 1983.
    .. [2] W.B. Smith and R.R. Hocking, "Algorithm AS 53: Wishart Variate
           Generator", Applied Statistics, vol. 21, pp. 341-345, 1972.

Examples
    --------
    >>> import matplotlib.pyplot as plt
    >>> from scipy.stats import wishart, chi2
    >>> x = np.linspace(1e-5, 8, 100)
    >>> w = wishart.pdf(x, df=3, scale=1); w[:5]
    array([ 0.00126156,  0.10892176,  0.14793434,  0.17400548,  0.1929669 ])
    >>> c = chi2.pdf(x, 3); c[:5]
    array([ 0.00126156,  0.10892176,  0.14793434,  0.17400548,  0.1929669 ])
    >>> plt.plot(x, w)

The input quantiles can be any shape of array, as long as the last
    axis labels the components.

"""

def __init__(self, seed=None):
        super(wishart_gen, self).__init__(seed)
        self.__doc__ = doccer.docformat(self.__doc__, wishart_docdict_params)

def __call__(self, df=None, scale=None, seed=None):
        """
        Create a frozen Wishart distribution.

See `wishart_frozen` for more information.

"""
        return wishart_frozen(df, scale, seed)

def _process_parameters(self, df, scale):
        if scale is None:
            scale = 1.0
        scale = np.asarray(scale, dtype=float)

if scale.ndim == 0:
            scale = scale[np.newaxis, np.newaxis]
        elif scale.ndim == 1:
            scale = np.diag(scale)
        elif scale.ndim == 2 and not scale.shape[0] == scale.shape[1]:
            raise ValueError("Array 'scale' must be square if it is two"
                             " dimensional, but scale.scale = %s."
                             % str(scale.shape))
        elif scale.ndim > 2:
            raise ValueError("Array 'scale' must be at most two-dimensional,"
                             " but scale.ndim = %d" % scale.ndim)

dim = scale.shape[0]

if df is None:
            df = dim
        elif not np.isscalar(df):
            raise ValueError("Degrees of freedom must be a scalar.")
        elif df < dim:
            raise ValueError("Degrees of freedom cannot be less than dimension"
                             " of scale matrix, but df = %d" % df)

return dim, df, scale

def _process_quantiles(self, x, dim):
        """
        Adjust quantiles array so that last axis labels the components of
        each data point.
        """
        x = np.asarray(x, dtype=float)

if x.ndim == 0:
            x = x * np.eye(dim)[:, :, np.newaxis]
        if x.ndim == 1:
            if dim == 1:
                x = x[np.newaxis, np.newaxis, :]
            else:
                x = np.diag(x)[:, :, np.newaxis]
        elif x.ndim == 2:
            if not x.shape[0] == x.shape[1]:
                raise ValueError("Quantiles must be square if they are two"
                                 " dimensional, but x.shape = %s."
                                 % str(x.shape))
            x = x[:, :, np.newaxis]
        elif x.ndim == 3:
            if not x.shape[0] == x.shape[1]:
                raise ValueError("Quantiles must be square in the first two"
                                 " dimensions if they are three dimensional"
                                 ", but x.shape = %s." % str(x.shape))
        elif x.ndim > 3:
            raise ValueError("Quantiles must be at most two-dimensional with"
                             " an additional dimension for multiple"
                             "components, but x.ndim = %d" % x.ndim)

# Now we have 3-dim array; should have shape [dim, dim, *]
        if not x.shape[0:2] == (dim, dim):
            raise ValueError('Quantiles have incompatible dimensions: should'
                             ' be %s, got %s.' % ((dim, dim), x.shape[0:2]))

return x

def _process_size(self, size):
        size = np.asarray(size)

if size.ndim == 0:
            size = size[np.newaxis]
        elif size.ndim > 1:
            raise ValueError('Size must be an integer or tuple of integers;'
                             ' thus must have dimension <= 1.'
                             ' Got size.ndim = %s' % str(tuple(size)))
        n = size.prod()
        shape = tuple(size)

return n, shape

def _logpdf(self, x, dim, df, scale, log_det_scale, C):
        """
        Parameters
        ----------
        x : ndarray
            Points at which to evaluate the log of the probability
            density function
        dim : int
            Dimension of the scale matrix
        df : int
            Degrees of freedom
        scale : ndarray
            Scale matrix
        log_det_scale : float
            Logarithm of the determinant of the scale matrix
        C : ndarray
            Cholesky factorization of the scale matrix, lower triagular.

Notes
        -----
        As this function does no argument checking, it should not be
        called directly; use 'logpdf' instead.

"""
        # log determinant of x
        # Note: x has components along the last axis, so that x.T has
        # components alone the 0-th axis. Then since det(A) = det(A'), this
        # gives us a 1-dim vector of determinants

# Retrieve tr(scale^{-1} x)
        log_det_x = np.zeros(x.shape[-1])
        scale_inv_x = np.zeros(x.shape)
        tr_scale_inv_x = np.zeros(x.shape[-1])
        for i in range(x.shape[-1]):
            _, log_det_x[i] = self._cholesky_logdet(x[:, :, i])
            scale_inv_x[:, :, i] = scipy.linalg.cho_solve((C, True), x[:, :, i])
            tr_scale_inv_x[i] = scale_inv_x[:, :, i].trace()

# Log PDF
        out = ((0.5 * (df - dim - 1) * log_det_x - 0.5 * tr_scale_inv_x) -
               (0.5 * df * dim * _LOG_2 + 0.5 * df * log_det_scale +
                multigammaln(0.5*df, dim)))

return out

def logpdf(self, x, df, scale):
        """
        Log of the Wishart probability density function.

Parameters
        ----------
        x : array_like
            Quantiles, with the last axis of `x` denoting the components.
            Each quantile must be a symmetric positive definite matrix.
        %(_doc_default_callparams)s

Returns
        -------
        pdf : ndarray
            Log of the probability density function evaluated at `x`

Notes
        -----
        %(_doc_callparams_note)s

"""
        dim, df, scale = self._process_parameters(df, scale)
        x = self._process_quantiles(x, dim)

# Cholesky decomposition of scale, get log(det(scale))
        C, log_det_scale = self._cholesky_logdet(scale)

out = self._logpdf(x, dim, df, scale, log_det_scale, C)
        return _squeeze_output(out)

def pdf(self, x, df, scale):
        """
        Wishart probability density function.

Returns
        -------
        pdf : ndarray
            Probability density function evaluated at `x`

Notes
        -----
        %(_doc_callparams_note)s

"""
        return np.exp(self.logpdf(x, df, scale))

def _mean(self, dim, df, scale):
        """
        Parameters
        ----------
        dim : int
            Dimension of the scale matrix
        %(_doc_default_callparams)s

Notes
        -----
        As this function does no argument checking, it should not be
        called directly; use 'mean' instead.

"""
        return df * scale

def mean(self, df, scale):
        """
        Mean of the Wishart distribution

Parameters
        ----------
        %(_doc_default_callparams)s

Returns
        -------
        mean : float
            The mean of the distribution
        """
        dim, df, scale = self._process_parameters(df, scale)
        out = self._mean(dim, df, scale)
        return _squeeze_output(out)

def _mode(self, dim, df, scale):
        """
        Parameters
        ----------
        dim : int
            Dimension of the scale matrix
        %(_doc_default_callparams)s

Notes
        -----
        As this function does no argument checking, it should not be
        called directly; use 'mode' instead.

"""
        if df >= dim + 1:
            out = (df-dim-1) * scale
        else:
            out = None
        return out

def mode(self, df, scale):
        """
        Mode of the Wishart distribution

Only valid if the degrees of freedom are greater than the dimension of
        the scale matrix.

Parameters
        ----------
        %(_doc_default_callparams)s

Returns
        -------
        mode : float or None
            The Mode of the distribution
        """
        dim, df, scale = self._process_parameters(df, scale)
        out = self._mode(dim, df, scale)
        return _squeeze_output(out) if out is not None else out

def _var(self, dim, df, scale):
        """
        Parameters
        ----------
        dim : int
            Dimension of the scale matrix
        %(_doc_default_callparams)s

Notes
        -----
        As this function does no argument checking, it should not be
        called directly; use 'var' instead.

"""
        var = scale**2
        diag = scale.diagonal()  # 1 x dim array
        var += np.outer(diag, diag)
        var *= df
        return var

def var(self, df, scale):
        """
        Variance of the Wishart distribution

Parameters
        ----------
        %(_doc_default_callparams)s

Returns
        -------
        var : float
            The variance of the distribution
        """
        dim, df, scale = self._process_parameters(df, scale)
        out = self._var(dim, df, scale)
        return _squeeze_output(out)

def _standard_rvs(self, n, shape, dim, df, random_state):
        """
        Parameters
        ----------
        n : integer
            Number of variates to generate
        shape : iterable
            Shape of the variates to generate
        dim : int
            Dimension of the scale matrix
        df : int
            Degrees of freedom
        random_state : np.random.RandomState instance
            RandomState used for drawing the random variates.

Notes
        -----
        As this function does no argument checking, it should not be
        called directly; use 'rvs' instead.

"""
        # Random normal variates for off-diagonal elements
        n_tril = dim * (dim-1) // 2
        covariances = random_state.normal(
            size=n*n_tril).reshape(shape+(n_tril,))

# Random chi-square variates for diagonal elements
        variances = (np.r_[[random_state.chisquare(df-(i+1)+1, size=n)**0.5
                            for i in range(dim)]].reshape((dim,) +
                                                          shape[::-1]).T)

# Create the A matri(ces) - lower triangular
        A = np.zeros(shape + (dim, dim))

# Input the covariances
        size_idx = tuple([slice(None, None, None)]*len(shape))
        tril_idx = np.tril_indices(dim, k=-1)
        A[size_idx + tril_idx] = covariances

# Input the variances
        diag_idx = np.diag_indices(dim)
        A[size_idx + diag_idx] = variances

return A

def _rvs(self, n, shape, dim, df, C, random_state):
        """
        Parameters
        ----------
        n : integer
            Number of variates to generate
        shape : iterable
            Shape of the variates to generate
        dim : int
            Dimension of the scale matrix
        df : int
            Degrees of freedom
        scale : ndarray
            Scale matrix
        C : ndarray
            Cholesky factorization of the scale matrix, lower triangular.
        %(_doc_random_state)s

Notes
        -----
        As this function does no argument checking, it should not be
        called directly; use 'rvs' instead.

"""
        random_state = self._get_random_state(random_state)
        # Calculate the matrices A, which are actually lower triangular
        # Cholesky factorizations of a matrix B such that B ~ W(df, I)
        A = self._standard_rvs(n, shape, dim, df, random_state)

# Calculate SA = C A A' C', where SA ~ W(df, scale)
        # Note: this is the product of a (lower) (lower) (lower)' (lower)'
        #       or, denoting B = AA', it is C B C' where C is the lower
        #       triangular Cholesky factorization of the scale matrix.
        #       this appears to conflict with the instructions in [1]_, which
        #       suggest that it should be D' B D where D is the lower
        #       triangular factorization of the scale matrix. However, it is
        #       meant to refer to the Bartlett (1933) representation of a
        #       Wishart random variate as L A A' L' where L is lower triangular
        #       so it appears that understanding D' to be upper triangular
        #       is either a typo in or misreading of [1]_.
        for index in np.ndindex(shape):
            CA = np.dot(C, A[index])
            A[index] = np.dot(CA, CA.T)

return A

def rvs(self, df, scale, size=1, random_state=None):
        """
        Draw random samples from a Wishart distribution.

Parameters
        ----------
        %(_doc_default_callparams)s
        size : integer or iterable of integers, optional
            Number of samples to draw (default 1).
        %(_doc_random_state)s

Returns
        -------
        rvs : ndarray
            Random variates of shape (`size`) + (`dim`, `dim), where `dim` is
            the dimension of the scale matrix.

Notes
        -----
        %(_doc_callparams_note)s

"""
        n, shape = self._process_size(size)
        dim, df, scale = self._process_parameters(df, scale)

# Cholesky decomposition of scale
        C = scipy.linalg.cholesky(scale, lower=True)

out = self._rvs(n, shape, dim, df, C, random_state)

return _squeeze_output(out)

def _entropy(self, dim, df, log_det_scale):
        """
        Parameters
        ----------
        dim : int
            Dimension of the scale matrix
        df : int
            Degrees of freedom
        log_det_scale : float
            Logarithm of the determinant of the scale matrix

Notes
        -----
        As this function does no argument checking, it should not be
        called directly; use 'entropy' instead.

"""
        return (
            0.5 * (dim+1) * log_det_scale +
            0.5 * dim * (dim+1) * _LOG_2 +
            multigammaln(0.5*df, dim) -
            0.5 * (df - dim - 1) * np.sum(
                [psi(0.5*(df + 1 - (i+1))) for i in range(dim)]
            ) +
            0.5 * df * dim
        )

def entropy(self, df, scale):
        """
        Compute the differential entropy of the Wishart.

Parameters
        ----------
        %(_doc_default_callparams)s

Returns
        -------
        h : scalar
            Entropy of the Wishart distribution

Notes
        -----
        %(_doc_callparams_note)s

"""
        dim, df, scale = self._process_parameters(df, scale)
        _, log_det_scale = self._cholesky_logdet(scale)
        return self._entropy(dim, df, log_det_scale)

def _cholesky_logdet(self, scale):
        """
        Compute Cholesky decomposition and determine (log(det(scale)).

Parameters
        ----------
        scale : ndarray
            Scale matrix.

Returns
        -------
        c_decomp : ndarray
            The Cholesky decomposition of `scale`.
        logdet : scalar
            The log of the determinant of `scale`.

Notes
        -----
        This computation of ``logdet`` is equivalent to
        ``np.linalg.slogdet(scale)``.  It is ~2x faster though.

"""
        c_decomp = scipy.linalg.cholesky(scale, lower=True)
        logdet = 2 * np.sum(np.log(c_decomp.diagonal()))
        return c_decomp, logdet

wishart = wishart_gen()

class wishart_frozen(multi_rv_frozen):
    """
    Create a frozen Wishart distribution.

Parameters
    ----------
    df : array_like
        Degrees of freedom of the distribution
    scale : array_like
        Scale matrix of the distribution
    seed : None or int or np.random.RandomState instance, optional
        This parameter defines the RandomState object to use for drawing
        random variates.
        If None (or np.random), the global np.random state is used.
        If integer, it is used to seed the local RandomState instance
        Default is None.

"""
    def __init__(self, df, scale, seed=None):
        self._dist = wishart_gen(seed)
        self.dim, self.df, self.scale = self._dist._process_parameters(
            df, scale)
        self.C, self.log_det_scale = self._dist._cholesky_logdet(self.scale)

def logpdf(self, x):
        x = self._dist._process_quantiles(x, self.dim)

out = self._dist._logpdf(x, self.dim, self.df, self.scale,
                                 self.log_det_scale, self.C)
        return _squeeze_output(out)

def pdf(self, x):
        return np.exp(self.logpdf(x))

def mean(self):
        out = self._dist._mean(self.dim, self.df, self.scale)
        return _squeeze_output(out)

def mode(self):
        out = self._dist._mode(self.dim, self.df, self.scale)
        return _squeeze_output(out) if out is not None else out

def var(self):
        out = self._dist._var(self.dim, self.df, self.scale)
        return _squeeze_output(out)

def rvs(self, size=1, random_state=None):
        n, shape = self._dist._process_size(size)
        out = self._dist._rvs(n, shape, self.dim, self.df,
                              self.C, random_state)
        return _squeeze_output(out)

def entropy(self):
        return self._dist._entropy(self.dim, self.df, self.log_det_scale)

# Set frozen generator docstrings from corresponding docstrings in
# Wishart and fill in default strings in class docstrings
for name in ['logpdf', 'pdf', 'mean', 'mode', 'var', 'rvs', 'entropy']:
    method = wishart_gen.__dict__[name]
    method_frozen = wishart_frozen.__dict__[name]
    method_frozen.__doc__ = doccer.docformat(
        method.__doc__, wishart_docdict_noparams)
    method.__doc__ = doccer.docformat(method.__doc__, wishart_docdict_params)

def _cho_inv_batch(a, check_finite=True):
    """
    Invert the matrices a_i, using a Cholesky factorization of A, where
    a_i resides in the last two dimensions of a and the other indices describe
    the index i.

Overwrites the data in a.

Parameters
    ----------
    a : array
        Array of matrices to invert, where the matrices themselves are stored
        in the last two dimensions.
    check_finite : bool, optional
        Whether to check that the input matrices contain only finite numbers.
        Disabling may give a performance gain, but may result in problems
        (crashes, non-termination) if the inputs do contain infinities or NaNs.

Returns
    -------
    x : array
        Array of inverses of the matrices ``a_i``.

See also
    --------
    scipy.linalg.cholesky : Cholesky factorization of a matrix

"""
    if check_finite:
        a1 = asarray_chkfinite(a)
    else:
        a1 = asarray(a)
    if len(a1.shape) < 2 or a1.shape[-2] != a1.shape[-1]:
        raise ValueError('expected square matrix in last two dimensions')

potrf, potri = get_lapack_funcs(('potrf', 'potri'), (a1,))

triu_rows, triu_cols = np.triu_indices(a.shape[-2], k=1)
    for index in np.ndindex(a1.shape[:-2]):

# Cholesky decomposition
        a1[index], info = potrf(a1[index], lower=True, overwrite_a=False,
                                clean=False)
        if info > 0:
            raise LinAlgError("%d-th leading minor not positive definite"
                              % info)
        if info < 0:
            raise ValueError('illegal value in %d-th argument of internal'
                             ' potrf' % -info)
        # Inversion
        a1[index], info = potri(a1[index], lower=True, overwrite_c=False)
        if info > 0:
            raise LinAlgError("the inverse could not be computed")
        if info < 0:
            raise ValueError('illegal value in %d-th argument of internal'
                             ' potrf' % -info)

# Make symmetric (dpotri only fills in the lower triangle)
        a1[index][triu_rows, triu_cols] = a1[index][triu_cols, triu_rows]

return a1

class invwishart_gen(wishart_gen):
    r"""
    An inverse Wishart random variable.

Parameters
    ----------
    x : array_like
        Quantiles, with the last axis of `x` denoting the components.
    %(_doc_default_callparams)s
    %(_doc_random_state)s

Alternatively, the object may be called (as a function) to fix the degrees
    of freedom and scale parameters, returning a "frozen" inverse Wishart
    random variable:

rv = invwishart(df=1, scale=1)
        - Frozen object with the same methods but holding the given
          degrees of freedom and scale fixed.

See Also
    --------
    wishart

Notes
    -----
    %(_doc_callparams_note)s

The scale matrix `scale` must be a symmetric positive definite
    matrix. Singular matrices, including the symmetric positive semi-definite
    case, are not supported.

The inverse Wishart distribution is often denoted

.. math::

W_p^{-1}(\nu, \Psi)

where :math:`\nu` is the degrees of freedom and :math:`\Psi` is the
    :math:`p \times p` scale matrix.

The probability density function for `invwishart` has support over positive
    definite matrices :math:`S`; if :math:`S \sim W^{-1}_p(\nu, \Sigma)`,
    then its PDF is given by:

.. math::

f(S) = \frac{|\Sigma|^\frac{\nu}{2}}{2^{ \frac{\nu p}{2} }
               |S|^{\frac{\nu + p + 1}{2}} \Gamma_p \left(\frac{\nu}{2} \right)}
               \exp\left( -tr(\Sigma S^{-1}) / 2 \right)

If :math:`S \sim W_p^{-1}(\nu, \Psi)` (inverse Wishart) then
    :math:`S^{-1} \sim W_p(\nu, \Psi^{-1})` (Wishart).

If the scale matrix is 1-dimensional and equal to one, then the inverse
    Wishart distribution :math:`W_1(\nu, 1)` collapses to the
    inverse Gamma distribution with parameters shape = :math:`\frac{\nu}{2}`
    and scale = :math:`\frac{1}{2}`.

.. versionadded:: 0.16.0

References
    ----------
    .. [1] M.L. Eaton, "Multivariate Statistics: A Vector Space Approach",
           Wiley, 1983.
    .. [2] M.C. Jones, "Generating Inverse Wishart Matrices", Communications
           in Statistics - Simulation and Computation, vol. 14.2, pp.511-514,
           1985.

Examples
    --------
    >>> import matplotlib.pyplot as plt
    >>> from scipy.stats import invwishart, invgamma
    >>> x = np.linspace(0.01, 1, 100)
    >>> iw = invwishart.pdf(x, df=6, scale=1)
    >>> iw[:3]
    array([  1.20546865e-15,   5.42497807e-06,   4.45813929e-03])
    >>> ig = invgamma.pdf(x, 6/2., scale=1./2)
    >>> ig[:3]
    array([  1.20546865e-15,   5.42497807e-06,   4.45813929e-03])
    >>> plt.plot(x, iw)

The input quantiles can be any shape of array, as long as the last
    axis labels the components.

"""

def __init__(self, seed=None):
        super(invwishart_gen, self).__init__(seed)
        self.__doc__ = doccer.docformat(self.__doc__, wishart_docdict_params)

def __call__(self, df=None, scale=None, seed=None):
        """
        Create a frozen inverse Wishart distribution.

See `invwishart_frozen` for more information.

"""
        return invwishart_frozen(df, scale, seed)

def _logpdf(self, x, dim, df, scale, log_det_scale):
        """
        Parameters
        ----------
        x : ndarray
            Points at which to evaluate the log of the probability
            density function.
        dim : int
            Dimension of the scale matrix
        df : int
            Degrees of freedom
        scale : ndarray
            Scale matrix
        log_det_scale : float
            Logarithm of the determinant of the scale matrix

Notes
        -----
        As this function does no argument checking, it should not be
        called directly; use 'logpdf' instead.

"""
        log_det_x = np.zeros(x.shape[-1])
        x_inv = np.copy(x).T
        if dim > 1:
            _cho_inv_batch(x_inv)  # works in-place
        else:
            x_inv = 1./x_inv
        tr_scale_x_inv = np.zeros(x.shape[-1])

for i in range(x.shape[-1]):
            C, lower = scipy.linalg.cho_factor(x[:, :, i], lower=True)

log_det_x[i] = 2 * np.sum(np.log(C.diagonal()))

tr_scale_x_inv[i] = np.dot(scale, x_inv[i]).trace()

# Log PDF
        out = ((0.5 * df * log_det_scale - 0.5 * tr_scale_x_inv) -
               (0.5 * df * dim * _LOG_2 + 0.5 * (df + dim + 1) * log_det_x) -
               multigammaln(0.5*df, dim))

return out

def logpdf(self, x, df, scale):
        """
        Log of the inverse Wishart probability density function.

Returns
        -------
        pdf : ndarray
            Log of the probability density function evaluated at `x`

Notes
        -----
        %(_doc_callparams_note)s

"""
        dim, df, scale = self._process_parameters(df, scale)
        x = self._process_quantiles(x, dim)
        _, log_det_scale = self._cholesky_logdet(scale)
        out = self._logpdf(x, dim, df, scale, log_det_scale)
        return _squeeze_output(out)

def pdf(self, x, df, scale):
        """
        Inverse Wishart probability density function.

Parameters
        ----------
        x : array_like
            Quantiles, with the last axis of `x` denoting the components.
            Each quantile must be a symmetric positive definite matrix.

%(_doc_default_callparams)s

Returns
        -------
        pdf : ndarray
            Probability density function evaluated at `x`

Notes
        -----
        %(_doc_callparams_note)s

"""
        return np.exp(self.logpdf(x, df, scale))

def _mean(self, dim, df, scale):
        """
        Parameters
        ----------
        dim : int
            Dimension of the scale matrix
        %(_doc_default_callparams)s

Notes
        -----
        As this function does no argument checking, it should not be
        called directly; use 'mean' instead.

"""
        if df > dim + 1:
            out = scale / (df - dim - 1)
        else:
            out = None
        return out

def mean(self, df, scale):
        """
        Mean of the inverse Wishart distribution

Only valid if the degrees of freedom are greater than the dimension of
        the scale matrix plus one.

Parameters
        ----------
        %(_doc_default_callparams)s

Returns
        -------
        mean : float or None
            The mean of the distribution

"""
        dim, df, scale = self._process_parameters(df, scale)
        out = self._mean(dim, df, scale)
        return _squeeze_output(out) if out is not None else out

def _mode(self, dim, df, scale):
        """
        Parameters
        ----------
        dim : int
            Dimension of the scale matrix
        %(_doc_default_callparams)s

Notes
        -----
        As this function does no argument checking, it should not be
        called directly; use 'mode' instead.

"""
        return scale / (df + dim + 1)

def mode(self, df, scale):
        """
        Mode of the inverse Wishart distribution

Parameters
        ----------
        %(_doc_default_callparams)s

Returns
        -------
        mode : float
            The Mode of the distribution

"""
        dim, df, scale = self._process_parameters(df, scale)
        out = self._mode(dim, df, scale)
        return _squeeze_output(out)

def _var(self, dim, df, scale):
        """
        Parameters
        ----------
        dim : int
            Dimension of the scale matrix
        %(_doc_default_callparams)s

Notes
        -----
        As this function does no argument checking, it should not be
        called directly; use 'var' instead.

"""
        if df > dim + 3:
            var = (df - dim + 1) * scale**2
            diag = scale.diagonal()  # 1 x dim array
            var += (df - dim - 1) * np.outer(diag, diag)
            var /= (df - dim) * (df - dim - 1)**2 * (df - dim - 3)
        else:
            var = None
        return var

def var(self, df, scale):
        """
        Variance of the inverse Wishart distribution

Only valid if the degrees of freedom are greater than the dimension of
        the scale matrix plus three.

Parameters
        ----------
        %(_doc_default_callparams)s

def _rvs(self, n, shape, dim, df, C, random_state):
        """
        Parameters
        ----------
        n : integer
            Number of variates to generate
        shape : iterable
            Shape of the variates to generate
        dim : int
            Dimension of the scale matrix
        df : int
            Degrees of freedom
        C : ndarray
            Cholesky factorization of the scale matrix, lower triagular.
        %(_doc_random_state)s

Notes
        -----
        As this function does no argument checking, it should not be
        called directly; use 'rvs' instead.

"""
        random_state = self._get_random_state(random_state)
        # Get random draws A such that A ~ W(df, I)
        A = super(invwishart_gen, self)._standard_rvs(n, shape, dim,
                                                      df, random_state)

# Calculate SA = (CA)'^{-1} (CA)^{-1} ~ iW(df, scale)
        eye = np.eye(dim)
        trtrs = get_lapack_funcs(('trtrs'), (A,))

for index in np.ndindex(A.shape[:-2]):
            # Calculate CA
            CA = np.dot(C, A[index])
            # Get (C A)^{-1} via triangular solver
            if dim > 1:
                CA, info = trtrs(CA, eye, lower=True)
                if info > 0:
                    raise LinAlgError("Singular matrix.")
                if info < 0:
                    raise ValueError('Illegal value in %d-th argument of'
                                     ' internal trtrs' % -info)
            else:
                CA = 1. / CA
            # Get SA
            A[index] = np.dot(CA.T, CA)

return A

def rvs(self, df, scale, size=1, random_state=None):
        """
        Draw random samples from an inverse Wishart distribution.

Returns
        -------
        rvs : ndarray
            Random variates of shape (`size`) + (`dim`, `dim), where `dim` is
            the dimension of the scale matrix.

Notes
        -----
        %(_doc_callparams_note)s

"""
        n, shape = self._process_size(size)
        dim, df, scale = self._process_parameters(df, scale)

# Invert the scale
        eye = np.eye(dim)
        L, lower = scipy.linalg.cho_factor(scale, lower=True)
        inv_scale = scipy.linalg.cho_solve((L, lower), eye)
        # Cholesky decomposition of inverted scale
        C = scipy.linalg.cholesky(inv_scale, lower=True)

out = self._rvs(n, shape, dim, df, C, random_state)

return _squeeze_output(out)

def entropy(self):
        # Need to find reference for inverse Wishart entropy
        raise AttributeError

invwishart = invwishart_gen()

class invwishart_frozen(multi_rv_frozen):
    def __init__(self, df, scale, seed=None):
        """
        Create a frozen inverse Wishart distribution.

Parameters
        ----------
        df : array_like
            Degrees of freedom of the distribution
        scale : array_like
            Scale matrix of the distribution
        seed : None or int or np.random.RandomState instance, optional
            This parameter defines the RandomState object to use for drawing
            random variates.
            If None (or np.random), the global np.random state is used.
            If integer, it is used to seed the local RandomState instance
            Default is None.

"""
        self._dist = invwishart_gen(seed)
        self.dim, self.df, self.scale = self._dist._process_parameters(
            df, scale
        )

# Get the determinant via Cholesky factorization
        C, lower = scipy.linalg.cho_factor(self.scale, lower=True)
        self.log_det_scale = 2 * np.sum(np.log(C.diagonal()))

# Get the inverse using the Cholesky factorization
        eye = np.eye(self.dim)
        self.inv_scale = scipy.linalg.cho_solve((C, lower), eye)

# Get the Cholesky factorization of the inverse scale
        self.C = scipy.linalg.cholesky(self.inv_scale, lower=True)

def logpdf(self, x):
        x = self._dist._process_quantiles(x, self.dim)
        out = self._dist._logpdf(x, self.dim, self.df, self.scale,
                                 self.log_det_scale)
        return _squeeze_output(out)

def pdf(self, x):
        return np.exp(self.logpdf(x))

def mean(self):
        out = self._dist._mean(self.dim, self.df, self.scale)
        return _squeeze_output(out) if out is not None else out

def mode(self):
        out = self._dist._mode(self.dim, self.df, self.scale)
        return _squeeze_output(out)

def var(self):
        out = self._dist._var(self.dim, self.df, self.scale)
        return _squeeze_output(out) if out is not None else out

def rvs(self, size=1, random_state=None):
        n, shape = self._dist._process_size(size)

out = self._dist._rvs(n, shape, self.dim, self.df,
                              self.C, random_state)

return _squeeze_output(out)

def entropy(self):
        # Need to find reference for inverse Wishart entropy
        raise AttributeError

# Set frozen generator docstrings from corresponding docstrings in
# inverse Wishart and fill in default strings in class docstrings
for name in ['logpdf', 'pdf', 'mean', 'mode', 'var', 'rvs']:
    method = invwishart_gen.__dict__[name]
    method_frozen = wishart_frozen.__dict__[name]
    method_frozen.__doc__ = doccer.docformat(
        method.__doc__, wishart_docdict_noparams)
    method.__doc__ = doccer.docformat(method.__doc__, wishart_docdict_params)

_multinomial_doc_default_callparams = """\
n : int
    Number of trials
p : array_like
    Probability of a trial falling into each category; should sum to 1
"""

_multinomial_doc_callparams_note = \
"""`n` should be a positive integer. Each element of `p` should be in the
interval :math:`[0,1]` and the elements should sum to 1. If they do not sum to
1, the last element of the `p` array is not used and is replaced with the
remaining probability left over from the earlier elements.
"""

_multinomial_doc_frozen_callparams = ""

_multinomial_doc_frozen_callparams_note = \
    """See class definition for a detailed description of parameters."""

multinomial_docdict_params = {
    '_doc_default_callparams': _multinomial_doc_default_callparams,
    '_doc_callparams_note': _multinomial_doc_callparams_note,
    '_doc_random_state': _doc_random_state
}

multinomial_docdict_noparams = {
    '_doc_default_callparams': _multinomial_doc_frozen_callparams,
    '_doc_callparams_note': _multinomial_doc_frozen_callparams_note,
    '_doc_random_state': _doc_random_state
}

class multinomial_gen(multi_rv_generic):
    r"""
    A multinomial random variable.

Methods
    -------
    ``pmf(x, n, p)``
        Probability mass function.
    ``logpmf(x, n, p)``
        Log of the probability mass function.
    ``rvs(n, p, size=1, random_state=None)``
        Draw random samples from a multinomial distribution.
    ``entropy(n, p)``
        Compute the entropy of the multinomial distribution.
    ``cov(n, p)``
        Compute the covariance matrix of the multinomial distribution.

Parameters
    ----------
    x : array_like
        Quantiles, with the last axis of `x` denoting the components.
    %(_doc_default_callparams)s
    %(_doc_random_state)s

Notes
    -----
    %(_doc_callparams_note)s

Alternatively, the object may be called (as a function) to fix the `n` and
    `p` parameters, returning a "frozen" multinomial random variable:

The probability mass function for `multinomial` is

.. math::

f(x) = \frac{n!}{x_1! \cdots x_k!} p_1^{x_1} \cdots p_k^{x_k},

supported on :math:`x=(x_1, \ldots, x_k)` where each :math:`x_i` is a
    nonnegative integer and their sum is :math:`n`.

.. versionadded:: 0.19.0

Examples
    --------

>>> from scipy.stats import multinomial
    >>> rv = multinomial(8, [0.3, 0.2, 0.5])
    >>> rv.pmf([1, 3, 4])
    0.042000000000000072

The multinomial distribution for :math:`k=2` is identical to the
    corresponding binomial distribution (tiny numerical differences
    notwithstanding):

>>> from scipy.stats import binom
    >>> multinomial.pmf([3, 4], n=7, p=[0.4, 0.6])
    0.29030399999999973
    >>> binom.pmf(3, 7, 0.4)
    0.29030400000000012

The functions ``pmf``, ``logpmf``, ``entropy``, and ``cov`` support
    broadcasting, under the convention that the vector parameters (``x`` and
    ``p``) are interpreted as if each row along the last axis is a single
    object. For instance:

>>> multinomial.pmf([[3, 4], [3, 5]], n=[7, 8], p=[.3, .7])
    array([0.2268945,  0.25412184])

Here, ``x.shape == (2, 2)``, ``n.shape == (2,)``, and ``p.shape == (2,)``,
    but following the rules mentioned above they behave as if the rows
    ``[3, 4]`` and ``[3, 5]`` in ``x`` and ``[.3, .7]`` in ``p`` were a single
    object, and as if we had ``x.shape = (2,)``, ``n.shape = (2,)``, and
    ``p.shape = ()``. To obtain the individual elements without broadcasting,
    we would do this:

>>> multinomial.pmf([3, 4], n=7, p=[.3, .7])
    0.2268945
    >>> multinomial.pmf([3, 5], 8, p=[.3, .7])
    0.25412184

This broadcasting also works for ``cov``, where the output objects are
    square matrices of size ``p.shape[-1]``. For example:

>>> multinomial.cov([4, 5], [[.3, .7], [.4, .6]])
    array([[[ 0.84, -0.84],
            [-0.84,  0.84]],
           [[ 1.2 , -1.2 ],
            [-1.2 ,  1.2 ]]])

In this example, ``n.shape == (2,)`` and ``p.shape == (2, 2)``, and
    following the rules above, these broadcast as if ``p.shape == (2,)``.
    Thus the result should also be of shape ``(2,)``, but since each output is
    a :math:`2 \times 2` matrix, the result in fact has shape ``(2, 2, 2)``,
    where ``result[0]`` is equal to ``multinomial.cov(n=4, p=[.3, .7])`` and
    ``result[1]`` is equal to ``multinomial.cov(n=5, p=[.4, .6])``.

See also
    --------
    scipy.stats.binom : The binomial distribution.
    numpy.random.Generator.multinomial : Sampling from the multinomial distribution.
    """  # noqa: E501

def __init__(self, seed=None):
        super(multinomial_gen, self).__init__(seed)
        self.__doc__ = \
            doccer.docformat(self.__doc__, multinomial_docdict_params)

def __call__(self, n, p, seed=None):
        """
        Create a frozen multinomial distribution.

See `multinomial_frozen` for more information.
        """
        return multinomial_frozen(n, p, seed)

def _process_parameters(self, n, p):
        """
        Return: n_, p_, npcond.

n_ and p_ are arrays of the correct shape; npcond is a boolean array
        flagging values out of the domain.
        """
        p = np.array(p, dtype=np.float64, copy=True)
        p[..., -1] = 1. - p[..., :-1].sum(axis=-1)

# true for bad p
        pcond = np.any(p < 0, axis=-1)
        pcond |= np.any(p > 1, axis=-1)

n = np.array(n, dtype=np.int, copy=True)

# true for bad n
        ncond = n <= 0

return n, p, ncond | pcond

def _process_quantiles(self, x, n, p):
        """
        Return: x_, xcond.

x_ is an int array; xcond is a boolean array flagging values out of the
        domain.
        """
        xx = np.asarray(x, dtype=np.int)

if xx.ndim == 0:
            raise ValueError("x must be an array.")

if xx.size != 0 and not xx.shape[-1] == p.shape[-1]:
            raise ValueError("Size of each quantile should be size of p: "
                             "received %d, but expected %d." %
                             (xx.shape[-1], p.shape[-1]))

# true for x out of the domain
        cond = np.any(xx != x, axis=-1)
        cond |= np.any(xx < 0, axis=-1)
        cond = cond | (np.sum(xx, axis=-1) != n)

return xx, cond

def _checkresult(self, result, cond, bad_value):
        result = np.asarray(result)

if cond.ndim != 0:
            result[cond] = bad_value
        elif cond:
            if result.ndim == 0:
                return bad_value
            result[...] = bad_value
        return result

def _logpmf(self, x, n, p):
        return gammaln(n+1) + np.sum(xlogy(x, p) - gammaln(x+1), axis=-1)

def logpmf(self, x, n, p):
        """
        Log of the Multinomial probability mass function.

Parameters
        ----------
        x : array_like
            Quantiles, with the last axis of `x` denoting the components.
        %(_doc_default_callparams)s

Returns
        -------
        logpmf : ndarray or scalar
            Log of the probability mass function evaluated at `x`

Notes
        -----
        %(_doc_callparams_note)s
        """
        n, p, npcond = self._process_parameters(n, p)
        x, xcond = self._process_quantiles(x, n, p)

result = self._logpmf(x, n, p)

# replace values for which x was out of the domain; broadcast
        # xcond to the right shape
        xcond_ = xcond | np.zeros(npcond.shape, dtype=np.bool_)
        result = self._checkresult(result, xcond_, np.NINF)

# replace values bad for n or p; broadcast npcond to the right shape
        npcond_ = npcond | np.zeros(xcond.shape, dtype=np.bool_)
        return self._checkresult(result, npcond_, np.NAN)

def pmf(self, x, n, p):
        """
        Multinomial probability mass function.

Parameters
        ----------
        x : array_like
            Quantiles, with the last axis of `x` denoting the components.
        %(_doc_default_callparams)s

Returns
        -------
        pmf : ndarray or scalar
            Probability density function evaluated at `x`

Notes
        -----
        %(_doc_callparams_note)s
        """
        return np.exp(self.logpmf(x, n, p))

def mean(self, n, p):
        """
        Mean of the Multinomial distribution

Parameters
        ----------
        %(_doc_default_callparams)s

Returns
        -------
        mean : float
            The mean of the distribution
        """
        n, p, npcond = self._process_parameters(n, p)
        result = n[..., np.newaxis]*p
        return self._checkresult(result, npcond, np.NAN)

def cov(self, n, p):
        """
        Covariance matrix of the multinomial distribution.

Parameters
        ----------
        %(_doc_default_callparams)s

Returns
        -------
        cov : ndarray
            The covariance matrix of the distribution
        """
        n, p, npcond = self._process_parameters(n, p)

nn = n[..., np.newaxis, np.newaxis]
        result = nn * np.einsum('...j,...k->...jk', -p, p)

# change the diagonal
        for i in range(p.shape[-1]):
            result[..., i, i] += n*p[..., i]

return self._checkresult(result, npcond, np.nan)

def entropy(self, n, p):
        r"""
        Compute the entropy of the multinomial distribution.

The entropy is computed using this expression:

.. math::

f(x) = - \log n! - n\sum_{i=1}^k p_i \log p_i +
            \sum_{i=1}^k \sum_{x=0}^n \binom n x p_i^x(1-p_i)^{n-x} \log x!

Parameters
        ----------
        %(_doc_default_callparams)s

Returns
        -------
        h : scalar
            Entropy of the Multinomial distribution

Notes
        -----
        %(_doc_callparams_note)s
        """
        n, p, npcond = self._process_parameters(n, p)

x = np.r_[1:np.max(n)+1]

term1 = n*np.sum(entr(p), axis=-1)
        term1 -= gammaln(n+1)

n = n[..., np.newaxis]
        new_axes_needed = max(p.ndim, n.ndim) - x.ndim + 1
        x.shape += (1,)*new_axes_needed

term2 = np.sum(binom.pmf(x, n, p)*gammaln(x+1),
                       axis=(-1, -1-new_axes_needed))

return self._checkresult(term1 + term2, npcond, np.nan)

def rvs(self, n, p, size=None, random_state=None):
        """
        Draw random samples from a Multinomial distribution.

Returns
        -------
        rvs : ndarray or scalar
            Random variates of shape (`size`, `len(p)`)

Notes
        -----
        %(_doc_callparams_note)s
        """
        n, p, npcond = self._process_parameters(n, p)
        random_state = self._get_random_state(random_state)
        return random_state.multinomial(n, p, size)

multinomial = multinomial_gen()

class multinomial_frozen(multi_rv_frozen):
    r"""
    Create a frozen Multinomial distribution.

Parameters
    ----------
    n : int
        number of trials
    p: array_like
        probability of a trial falling into each category; should sum to 1
    seed : None or int or np.random.RandomState instance, optional
        This parameter defines the RandomState object to use for drawing
        random variates.
        If None (or np.random), the global np.random state is used.
        If integer, it is used to seed the local RandomState instance
        Default is None.
    """
    def __init__(self, n, p, seed=None):
        self._dist = multinomial_gen(seed)
        self.n, self.p, self.npcond = self._dist._process_parameters(n, p)

# monkey patch self._dist
        def _process_parameters(n, p):
            return self.n, self.p, self.npcond

self._dist._process_parameters = _process_parameters

def logpmf(self, x):
        return self._dist.logpmf(x, self.n, self.p)

def pmf(self, x):
        return self._dist.pmf(x, self.n, self.p)

def mean(self):
        return self._dist.mean(self.n, self.p)

def cov(self):
        return self._dist.cov(self.n, self.p)

def entropy(self):
        return self._dist.entropy(self.n, self.p)

def rvs(self, size=1, random_state=None):
        return self._dist.rvs(self.n, self.p, size, random_state)

# Set frozen generator docstrings from corresponding docstrings in
# multinomial and fill in default strings in class docstrings
for name in ['logpmf', 'pmf', 'mean', 'cov', 'rvs']:
    method = multinomial_gen.__dict__[name]
    method_frozen = multinomial_frozen.__dict__[name]
    method_frozen.__doc__ = doccer.docformat(
        method.__doc__, multinomial_docdict_noparams)
    method.__doc__ = doccer.docformat(method.__doc__,
                                      multinomial_docdict_params)

class special_ortho_group_gen(multi_rv_generic):
    r"""
    A matrix-valued SO(N) random variable.

Return a random rotation matrix, drawn from the Haar distribution
    (the only uniform distribution on SO(n)).

The `dim` keyword specifies the dimension N.

Methods
    -------
    ``rvs(dim=None, size=1, random_state=None)``
        Draw random samples from SO(N).

Parameters
    ----------
    dim : scalar
        Dimension of matrices

Notes
    ----------
    This class is wrapping the random_rot code from the MDP Toolkit,
    https://github.com/mdp-toolkit/mdp-toolkit

Return a random rotation matrix, drawn from the Haar distribution
    (the only uniform distribution on SO(n)).
    The algorithm is described in the paper
    Stewart, G.W., "The efficient generation of random orthogonal
    matrices with an application to condition estimators", SIAM Journal
    on Numerical Analysis, 17(3), pp. 403-409, 1980.
    For more information see
    https://en.wikipedia.org/wiki/Orthogonal_matrix#Randomization