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# 

# Author: Joris Vankerschaver 2013 

# 

from __future__ import division, print_function, absolute_import 

 

import math 

import numpy as np 

import scipy.linalg 

from scipy.misc 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 ._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. 

 

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 

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 = np.squeeze(out, axis=0) 

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. 

 

""" 

 

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. 

 

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. 

 

""" 

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) 

 

 

from numpy import asarray_chkfinite, asarray 

from scipy.linalg.misc import LinAlgError 

from scipy.linalg.lapack import get_lapack_funcs 

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,)) 

 

tril_idx = np.tril_indices(a.shape[-2], k=-1) 

triu_idx = 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_idx] = a1[index][tril_idx] 

 

return a1 

 

 

class invwishart_gen(wishart_gen): 

r""" 

An inverse 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 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 an inverse 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" 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]) 

#scale_x_inv = np.zeros(x.shape) 

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())) 

 

#scale_x_inv[:,:,i] = scipy.linalg.cho_solve((C, True), scale).T 

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. 

 

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) 

_, 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 

 

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) if out is not None else out 

 

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. 

 

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) 

 

# 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.multinomial : Sampling from the multinomial distribution. 

""" 

 

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. 

Each quantile must be a symmetric positive definite matrix. 

%(_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. 

Each quantile must be a symmetric positive definite matrix. 

%(_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. 

 

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 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 

http://en.wikipedia.org/wiki/Orthogonal_matrix#Randomization 

 

See also the similar `ortho_group`. 

 

Examples 

-------- 

>>> from scipy.stats import special_ortho_group 

>>> x = special_ortho_group.rvs(3) 

 

>>> np.dot(x, x.T) 

array([[ 1.00000000e+00, 1.13231364e-17, -2.86852790e-16], 

[ 1.13231364e-17, 1.00000000e+00, -1.46845020e-16], 

[ -2.86852790e-16, -1.46845020e-16, 1.00000000e+00]]) 

 

>>> import scipy.linalg 

>>> scipy.linalg.det(x) 

1.0 

 

This generates one random matrix from SO(3). It is orthogonal and 

has a determinant of 1. 

 

""" 

 

def __init__(self, seed=None): 

super(special_ortho_group_gen, self).__init__(seed) 

self.__doc__ = doccer.docformat(self.__doc__) 

 

def __call__(self, dim=None, seed=None): 

""" 

Create a frozen SO(N) distribution. 

 

See `special_ortho_group_frozen` for more information. 

 

""" 

return special_ortho_group_frozen(dim, seed=seed) 

 

def _process_parameters(self, dim): 

""" 

Dimension N must be specified; it cannot be inferred. 

""" 

 

if dim is None or not np.isscalar(dim) or dim <= 1 or dim != int(dim): 

raise ValueError("""Dimension of rotation must be specified, 

and must be a scalar greater than 1.""") 

 

return dim 

 

def rvs(self, dim, size=1, random_state=None): 

""" 

Draw random samples from SO(N). 

 

Parameters 

---------- 

dim : integer 

Dimension of rotation space (N). 

size : integer, optional 

Number of samples to draw (default 1). 

 

Returns 

------- 

rvs : ndarray or scalar 

Random size N-dimensional matrices, dimension (size, dim, dim) 

 

""" 

random_state = self._get_random_state(random_state) 

 

size = int(size) 

if size > 1: 

return np.array([self.rvs(dim, size=1, random_state=random_state) 

for i in range(size)]) 

 

dim = self._process_parameters(dim) 

 

H = np.eye(dim) 

D = np.empty((dim,)) 

for n in range(dim-1): 

x = random_state.normal(size=(dim-n,)) 

D[n] = np.sign(x[0]) if x[0] != 0 else 1 

x[0] += D[n]*np.sqrt((x*x).sum()) 

# Householder transformation 

Hx = (np.eye(dim-n) 

- 2.*np.outer(x, x)/(x*x).sum()) 

mat = np.eye(dim) 

mat[n:, n:] = Hx 

H = np.dot(H, mat) 

D[-1] = (-1)**(dim-1)*D[:-1].prod() 

# Equivalent to np.dot(np.diag(D), H) but faster, apparently 

H = (D*H.T).T 

return H 

 

 

special_ortho_group = special_ortho_group_gen() 

 

class special_ortho_group_frozen(multi_rv_frozen): 

def __init__(self, dim=None, seed=None): 

""" 

Create a frozen SO(N) distribution. 

 

Parameters 

---------- 

dim : scalar 

Dimension of matrices 

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. 

 

Examples 

-------- 

>>> from scipy.stats import special_ortho_group 

>>> g = special_ortho_group(5) 

>>> x = g.rvs() 

 

""" 

self._dist = special_ortho_group_gen(seed) 

self.dim = self._dist._process_parameters(dim) 

 

def rvs(self, size=1, random_state=None): 

return self._dist.rvs(self.dim, size, random_state) 

 

class ortho_group_gen(multi_rv_generic): 

r""" 

A matrix-valued O(N) random variable. 

 

Return a random orthogonal matrix, drawn from the O(N) Haar 

distribution (the only uniform distribution on O(N)). 

 

The `dim` keyword specifies the dimension N. 

 

Methods 

------- 

``rvs(dim=None, size=1, random_state=None)`` 

Draw random samples from O(N). 

 

Parameters 

---------- 

dim : scalar 

Dimension of matrices 

 

Notes 

---------- 

This class is closely related to `special_ortho_group`. 

 

Some care is taken to avoid numerical error, as per the paper by Mezzadri. 

 

References 

---------- 

.. [1] F. Mezzadri, "How to generate random matrices from the classical 

compact groups", :arXiv:`math-ph/0609050v2`. 

 

Examples 

-------- 

>>> from scipy.stats import ortho_group 

>>> x = ortho_group.rvs(3) 

 

>>> np.dot(x, x.T) 

array([[ 1.00000000e+00, 1.13231364e-17, -2.86852790e-16], 

[ 1.13231364e-17, 1.00000000e+00, -1.46845020e-16], 

[ -2.86852790e-16, -1.46845020e-16, 1.00000000e+00]]) 

 

>>> import scipy.linalg 

>>> np.fabs(scipy.linalg.det(x)) 

1.0 

 

This generates one random matrix from O(3). It is orthogonal and 

has a determinant of +1 or -1. 

 

""" 

 

def __init__(self, seed=None): 

super(ortho_group_gen, self).__init__(seed) 

self.__doc__ = doccer.docformat(self.__doc__) 

 

def _process_parameters(self, dim): 

""" 

Dimension N must be specified; it cannot be inferred. 

""" 

 

if dim is None or not np.isscalar(dim) or dim <= 1 or dim != int(dim): 

raise ValueError("Dimension of rotation must be specified," 

"and must be a scalar greater than 1.") 

 

return dim 

 

def rvs(self, dim, size=1, random_state=None): 

""" 

Draw random samples from O(N). 

 

Parameters 

---------- 

dim : integer 

Dimension of rotation space (N). 

size : integer, optional 

Number of samples to draw (default 1). 

 

Returns 

------- 

rvs : ndarray or scalar 

Random size N-dimensional matrices, dimension (size, dim, dim) 

 

""" 

random_state = self._get_random_state(random_state) 

 

size = int(size) 

if size > 1: 

return np.array([self.rvs(dim, size=1, random_state=random_state) 

for i in range(size)]) 

 

dim = self._process_parameters(dim) 

 

H = np.eye(dim) 

for n in range(dim): 

x = random_state.normal(size=(dim-n,)) 

# random sign, 50/50, but chosen carefully to avoid roundoff error 

D = np.sign(x[0]) if x[0] != 0 else 1 

x[0] += D*np.sqrt((x*x).sum()) 

# Householder transformation 

Hx = -D*(np.eye(dim-n) 

- 2.*np.outer(x, x)/(x*x).sum()) 

mat = np.eye(dim) 

mat[n:, n:] = Hx 

H = np.dot(H, mat) 

return H 

 

 

ortho_group = ortho_group_gen() 

 

class random_correlation_gen(multi_rv_generic): 

r""" 

A random correlation matrix. 

 

Return a random correlation matrix, given a vector of eigenvalues. 

 

The `eigs` keyword specifies the eigenvalues of the correlation matrix, 

and implies the dimension. 

 

Methods 

------- 

``rvs(eigs=None, random_state=None)`` 

Draw random correlation matrices, all with eigenvalues eigs. 

 

Parameters 

---------- 

eigs : 1d ndarray 

Eigenvalues of correlation matrix. 

 

Notes 

---------- 

 

Generates a random correlation matrix following a numerically stable 

algorithm spelled out by Davies & Higham. This algorithm uses a single O(N) 

similarity transformation to construct a symmetric positive semi-definite 

matrix, and applies a series of Givens rotations to scale it to have ones 

on the diagonal. 

 

References 

---------- 

 

.. [1] Davies, Philip I; Higham, Nicholas J; "Numerically stable generation 

of correlation matrices and their factors", BIT 2000, Vol. 40, 

No. 4, pp. 640 651 

 

Examples 

-------- 

>>> from scipy.stats import random_correlation 

>>> np.random.seed(514) 

>>> x = random_correlation.rvs((.5, .8, 1.2, 1.5)) 

>>> x 

array([[ 1. , -0.20387311, 0.18366501, -0.04953711], 

[-0.20387311, 1. , -0.24351129, 0.06703474], 

[ 0.18366501, -0.24351129, 1. , 0.38530195], 

[-0.04953711, 0.06703474, 0.38530195, 1. ]]) 

>>> import scipy.linalg 

>>> e, v = scipy.linalg.eigh(x) 

>>> e 

array([ 0.5, 0.8, 1.2, 1.5]) 

 

""" 

 

def __init__(self, seed=None): 

super(random_correlation_gen, self).__init__(seed) 

self.__doc__ = doccer.docformat(self.__doc__) 

 

def _process_parameters(self, eigs, tol): 

eigs = np.asarray(eigs, dtype=float) 

dim = eigs.size 

 

if eigs.ndim != 1 or eigs.shape[0] != dim or dim <= 1: 

raise ValueError("Array 'eigs' must be a vector of length greater than 1.") 

 

if np.fabs(np.sum(eigs) - dim) > tol: 

raise ValueError("Sum of eigenvalues must equal dimensionality.") 

 

for x in eigs: 

if x < -tol: 

raise ValueError("All eigenvalues must be non-negative.") 

 

return dim, eigs 

 

def _givens_to_1(self, aii, ajj, aij): 

"""Computes a 2x2 Givens matrix to put 1's on the diagonal for the input matrix. 

 

The input matrix is a 2x2 symmetric matrix M = [ aii aij ; aij ajj ]. 

 

The output matrix g is a 2x2 anti-symmetric matrix of the form [ c s ; -s c ]; 

the elements c and s are returned. 

 

Applying the output matrix to the input matrix (as b=g.T M g) 

results in a matrix with bii=1, provided tr(M) - det(M) >= 1 

and floating point issues do not occur. Otherwise, some other 

valid rotation is returned. When tr(M)==2, also bjj=1. 

 

""" 

aiid = aii - 1. 

ajjd = ajj - 1. 

 

if ajjd == 0: 

# ajj==1, so swap aii and ajj to avoid division by zero 

return 0., 1. 

 

dd = math.sqrt(max(aij**2 - aiid*ajjd, 0)) 

 

# The choice of t should be chosen to avoid cancellation [1] 

t = (aij + math.copysign(dd, aij)) / ajjd 

c = 1. / math.sqrt(1. + t*t) 

if c == 0: 

# Underflow 

s = 1.0 

else: 

s = c*t 

return c, s 

 

def _to_corr(self, m): 

""" 

Given a psd matrix m, rotate to put one's on the diagonal, turning it 

into a correlation matrix. This also requires the trace equal the 

dimensionality. Note: modifies input matrix 

""" 

# Check requirements for in-place Givens 

if not (m.flags.c_contiguous and m.dtype == np.float64 and m.shape[0] == m.shape[1]): 

raise ValueError() 

 

d = m.shape[0] 

for i in range(d-1): 

if m[i,i] == 1: 

continue 

elif m[i, i] > 1: 

for j in range(i+1, d): 

if m[j, j] < 1: 

break 

else: 

for j in range(i+1, d): 

if m[j, j] > 1: 

break 

 

c, s = self._givens_to_1(m[i,i], m[j,j], m[i,j]) 

 

# Use BLAS to apply Givens rotations in-place. Equivalent to: 

# g = np.eye(d) 

# g[i, i] = g[j,j] = c 

# g[j, i] = -s; g[i, j] = s 

# m = np.dot(g.T, np.dot(m, g)) 

mv = m.ravel() 

drot(mv, mv, c, -s, n=d, 

offx=i*d, incx=1, offy=j*d, incy=1, 

overwrite_x=True, overwrite_y=True) 

drot(mv, mv, c, -s, n=d, 

offx=i, incx=d, offy=j, incy=d, 

overwrite_x=True, overwrite_y=True) 

 

return m 

 

def rvs(self, eigs, random_state=None, tol=1e-13, diag_tol=1e-7): 

""" 

Draw random correlation matrices 

 

Parameters 

---------- 

eigs : 1d ndarray 

Eigenvalues of correlation matrix 

tol : float, optional 

Tolerance for input parameter checks 

diag_tol : float, optional 

Tolerance for deviation of the diagonal of the resulting 

matrix. Default: 1e-7 

 

Raises 

------ 

RuntimeError 

Floating point error prevented generating a valid correlation 

matrix. 

 

Returns 

------- 

rvs : ndarray or scalar 

Random size N-dimensional matrices, dimension (size, dim, dim), 

each having eigenvalues eigs. 

 

""" 

dim, eigs = self._process_parameters(eigs, tol=tol) 

 

random_state = self._get_random_state(random_state) 

 

m = ortho_group.rvs(dim, random_state=random_state) 

m = np.dot(np.dot(m, np.diag(eigs)), m.T) # Set the trace of m 

m = self._to_corr(m) # Carefully rotate to unit diagonal 

 

# Check diagonal 

if abs(m.diagonal() - 1).max() > diag_tol: 

raise RuntimeError("Failed to generate a valid correlation matrix") 

 

return m 

 

 

random_correlation = random_correlation_gen() 

 

class unitary_group_gen(multi_rv_generic): 

r""" 

A matrix-valued U(N) random variable. 

 

Return a random unitary matrix. 

 

The `dim` keyword specifies the dimension N. 

 

Methods 

------- 

``rvs(dim=None, size=1, random_state=None)`` 

Draw random samples from U(N). 

 

Parameters 

---------- 

dim : scalar 

Dimension of matrices 

 

Notes 

---------- 

This class is similar to `ortho_group`. 

 

References 

---------- 

.. [1] F. Mezzadri, "How to generate random matrices from the classical 

compact groups", arXiv:math-ph/0609050v2. 

 

Examples 

-------- 

>>> from scipy.stats import unitary_group 

>>> x = unitary_group.rvs(3) 

 

>>> np.dot(x, x.conj().T) 

array([[ 1.00000000e+00, 1.13231364e-17, -2.86852790e-16], 

[ 1.13231364e-17, 1.00000000e+00, -1.46845020e-16], 

[ -2.86852790e-16, -1.46845020e-16, 1.00000000e+00]]) 

 

This generates one random matrix from U(3). The dot product confirms that it is unitary up to machine precision. 

 

""" 

 

def __init__(self, seed=None): 

super(unitary_group_gen, self).__init__(seed) 

self.__doc__ = doccer.docformat(self.__doc__) 

 

def _process_parameters(self, dim): 

""" 

Dimension N must be specified; it cannot be inferred. 

""" 

 

if dim is None or not np.isscalar(dim) or dim <= 1 or dim != int(dim): 

raise ValueError("Dimension of rotation must be specified," 

"and must be a scalar greater than 1.") 

 

return dim 

 

def rvs(self, dim, size=1, random_state=None): 

""" 

Draw random samples from U(N). 

 

Parameters 

---------- 

dim : integer 

Dimension of space (N). 

size : integer, optional 

Number of samples to draw (default 1). 

 

Returns 

------- 

rvs : ndarray or scalar 

Random size N-dimensional matrices, dimension (size, dim, dim) 

 

""" 

random_state = self._get_random_state(random_state) 

 

size = int(size) 

if size > 1: 

return np.array([self.rvs(dim, size=1, random_state=random_state) 

for i in range(size)]) 

 

dim = self._process_parameters(dim) 

 

z = 1/math.sqrt(2)*(random_state.normal(size=(dim,dim)) + 

1j*random_state.normal(size=(dim,dim))) 

q, r = scipy.linalg.qr(z) 

d = r.diagonal() 

q *= d/abs(d) 

return q 

 

 

unitary_group = unitary_group_gen()