def test_pseudodet_pinv():
# Make sure that pseudo-inverse and pseudo-det agree on cutoff
# Assemble random covariance matrix with large and small eigenvalues
np.random.seed(1234)
n = 7
x = np.random.randn(n, n)
cov = np.dot(x, x.T)
s, u = scipy.linalg.eigh(cov)
s = 0.5 * np.ones(n)
s[0] = 1.0
s[-1] = 1e-7
cov = np.dot(u, np.dot(np.diag(s), u.T))
# Set cond so that the lowest eigenvalue is below the cutoff
cond = 1e-5
psd = _PSD(cov, cond=cond)
psd_pinv = _PSD(psd.pinv, cond=cond)
# Check that the log pseudo-determinant agrees with the sum
# of the logs of all but the smallest eigenvalue
assert_allclose(psd.log_pdet, np.sum(np.log(s[:-1])))
# Check that the pseudo-determinant of the pseudo-inverse
# agrees with 1 / pseudo-determinant
assert_allclose(-psd.log_pdet, psd_pinv.log_pdet)
def logpdf(self, x, loc=None, scale=1, gamma=3):
"""Log of the multivariate EFF probability density function.
Parameters
----------
x : array_like
Points at which to evaluate the log of the probability density
function.
loc : array_like, optional
Centre of the distribution (default zero).
scale : array_like, optional
Positive definite shape matrix. This is not the distribution's
covariance matrix (default one).
gamma : EFF's gamma parameter.
Returns
-------
logpdf : Log of the probability density function evaluated at `x`.
"""
dim, loc, scale, gamma = self._process_parameters(loc, scale, gamma)
x = self._process_quantiles(x, dim)
scale_info = _PSD(scale)
if dim == 3:
return self._logpdf3d(x,loc,scale_info.U,scale_info.log_pdet,gamma,dim)
else:
sys.exit("ERROR DIM NOT IMPLEMENTED")
def pdf(self, x, location=None, scale=1, dof=None, allow_singular=False):
"""
Multivariate Student's t probability density function.
Parameters
----------
x : array_like
Quantiles, with the last axis of `x` denoting the components.
location : ndarray
Location of the distribution
scale : array_like
Scale matrix of the distribution
dof : scalar
Degrees-of-freedom of the distribution
Returns
-------
pdf : ndarray or scalar
Probability density function evaluated at `x`
"""
dim, location, scale, dof = self._process_parameters(
None, location, scale, dof)
x = self._process_quantiles(x, dim)
psd = _PSD(scale, allow_singular=allow_singular)
out = np.exp(
self._logpdf(x, location, psd.U, psd.log_pdet, psd.rank, dof))
return _squeeze_output(out)
def logpdf(self, x, loc=None, scale=1, rt=10):
"""Log of the multivariate King probability density function.
Parameters
----------
x : array_like
Points at which to evaluate the log of the probability density
function.
loc : array_like, optional
Centre of the distribution (default zero).
scale : array_like, optional
Positive definite shape matrix. This is not the distribution's
covariance matrix (default one).
rt : King's tidal radius parameter.
Returns
-------
logpdf : Log of the probability density function evaluated at `x`.
"""
dim, loc, scale, rt = self._process_parameters(loc, scale, rt)
x = self._process_quantiles(x, dim)
scale_info = _PSD(scale)
if dim == 1:
return self._logpdf1d(x,loc,scale_info.U,scale_info.log_pdet,rt)
elif dim == 3:
return self._logpdf3d(x,loc,scale_info.U,scale_info.log_pdet,rt)
else:
sys.exit("Dimension not implemented")
def logpdf(self, x, mean=None, shape=1, df=1):
"""Log of the multivariate t-distribution probability density function.
Parameters
----------
x : array_like
Points at which to evaluate the log of the probability density
function.
mean : array_like, optional
Mean of the distribution (default zero).
shape : array_like, optional
Positive definite shape matrix. This is not the distribution's
covariance matrix (default one).
df : Degrees of freedom.
Returns
-------
logpdf : Log of the probability density function evaluated at `x`.
Examples
--------
FIXME.
"""
dim, mean, shape, df = self._process_parameters(mean, shape, df)
x = self._process_quantiles(x, dim)
shape_info = _PSD(shape)
return self._logpdf(x, mean, shape_info.U, shape_info.log_pdet, df,
dim)
def test_large_pseudo_determinant():
# Check that large pseudo-determinants are handled appropriately.
# Construct a singular diagonal covariance matrix
# whose pseudo determinant overflows double precision.
large_total_log = 1000.0
npos = 100
nzero = 2
large_entry = np.exp(large_total_log / npos)
n = npos + nzero
cov = np.zeros((n, n), dtype=float)
np.fill_diagonal(cov, large_entry)
cov[-nzero:, -nzero:] = 0
# Check some determinants.
assert_equal(scipy.linalg.det(cov), 0)
assert_equal(scipy.linalg.det(cov[:npos, :npos]), np.inf)
# np.linalg.slogdet is only available in numpy 1.6+
# but scipy currently supports numpy 1.5.1.
# assert_allclose(np.linalg.slogdet(cov[:npos, :npos]), (1, large_total_log))
# Check the pseudo-determinant.
psd = _PSD(cov)
assert_allclose(psd.log_pdet, large_total_log)
def pdf(self, x, mean=None, cov=1, allow_singular=False):
"""
Multivariate laplace probability density function.
Parameters
----------
x : array_like
Quantiles, with the last axis of `x` denoting the components.
%(_mvl_doc_default_callparams)s
Returns
-------
pdf : ndarray or scalar
Probability density function evaluated at `x`
Notes
-----
%(_mvl_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=None, cov=1, allow_singular=False):
"""
Multivariate laplace cumulative distribution function.
Parameters
----------
x : array_like
Quantiles, with the last axis of `x` denoting the components.
%(_mvl_doc_default_callparams)s
Returns
-------
cdf : ndarray or scalar
Cumulative distribution function evaluated at `x`
Notes
-----
%(_mvl_doc_callparams_note)s
.. versionadded:: 1.0.0
"""
dim, mean, cov = self._process_parameters(None, mean, cov)
x = self._process_quantiles(x, dim)
psd = _PSD(cov, allow_singular=allow_singular)
out = self._cdf(x, mean, psd.U)
return _squeeze_output(out)
def _e_step(self, X):
''' E-step of the algorithm
Computes the conditional probability of the model
Parameters
----------
X: array-like, shape (n, p)
data
Returns
----------
cond_prob_matrix: array-like, shape (n, K)
(cond_prob_matrix)_ik = P(Z_i=k|X_i=x_i)
'''
n, p = X.shape
K = len(self.alpha_)
cond_prob_matrix = np.zeros((n,K))
for k in range(K):
psd = _PSD(self.Sigma_[k])
prec_U, logdet = psd.U, psd.log_pdet
diff = X - self.mu_[k]
logdensity = -0.5 * (p * np.log(2 * np.pi) + p * np.log(self.tau_[:, k]) + logdet + p)
cond_prob_matrix[:, k] = np.exp(logdensity) * self.alpha_[k]
sum_row = np.sum(cond_prob_matrix, axis = 1)
bool_sum_zero = (sum_row == 0)
cond_prob_matrix[bool_sum_zero, :] = self.alpha_
cond_prob_matrix /= cond_prob_matrix.sum(axis=1)[:,np.newaxis]
return cond_prob_matrix
def __init__(self, mean=None, shape=1, df=1, seed=None):
"""Create a frozen multivariate t-distribution.
See `MultivariateTGenerator` for parameters.
"""
self._dist = MultivariateTGenerator(seed)
mean, shape, df = self._dist._process_parameters(mean, shape, df)
self.shape_info = _PSD(shape)
self.mean, self.shape, self.df = mean, shape, df
def logcdf(self,
x,
location=None,
scale=1,
dof=None,
allow_singular=False,
maxpts=None,
abseps=1e-5,
releps=1e-5):
"""
Log of the multivariate Student's t cumulative distribution function.
Parameters
----------
x : array_like
Quantiles, with the last axis of `x` denoting the components.
location : ndarray
Location of the distribution
scale : array_like
Scale matrix of the distribution
dof : scalar
Degrees-of-freedom of the distribution
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`
"""
dim, location, scale, dof = self._process_parameters(
None, location, scale, dof)
x = self._process_quantiles(x, dim)
# Use _PSD to check covariance matrix
_PSD(scale, allow_singular=allow_singular)
if not maxpts:
maxpts = 1000000 * dim
out = np.log(self._cdf(x, location, scale, dof, maxpts, abseps,
releps))
return out
def cdf(self,
x,
df,
mean=None,
cov=1,
allow_singular=False,
maxpts=None,
abseps=1e-5,
releps=1e-5):
"""
Multivariate Student's T cumulative distribution function.
Parameters
----------
x : array_like
Quantiles, with the last axis of `x` denoting the components.
%(_mvt_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
-----
%(_mvt_doc_callparams_note)s
.. versionadded:: 1.0.0
"""
dim, df, mean, cov = self._process_parameters(None, df, 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, df, mean, cov, maxpts, abseps, releps)
return out
def __init__(self, mean=None, cov=1, allow_singular=False, seed=None,
maxpts=None, abseps=1e-5, releps=1e-5):
"""
Create a frozen multivariate laplace 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, int, `~np.random.RandomState`, `~np.random.Generator`}, optional
This parameter defines the object to use for drawing random
variates.
If `seed` is `None` the `~np.random.RandomState` singleton is used.
If `seed` is an int, a new ``RandomState`` instance is used, seeded
with seed.
If `seed` is already a ``RandomState`` or ``Generator`` instance,
then that object is used.
Default is None.
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_laplace
>>> r = multivariate_laplace()
>>> r.mean
array([ 0.])
>>> r.cov
array([[1.]])
"""
self._dist = multivariate_laplace_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(x,
mean=None,
cov=None,
allow_singular=False,
coef=1,
psd=None,
return_psd=False):
if mean is None:
mean = np.zeros(x.shape[-1], dtype=np.float64)
if cov is None:
cov = np.eye(x.shape[-1], dtype=np.float64)
if psd is None:
psd = _PSD(cov, allow_singular=allow_singular)
out = _logpdf(x, mean, psd.U, psd.log_pdet, psd.rank, coef)
return (_squeeze_output(out), psd) if return_psd else _squeeze_output(out)
def rvs(self, loc=None, scale=1, rt=10, size=1, random_state=None,
min_u=1e-5):
"""Draw random samples from a multivariate King distribution.
Parameters
----------
x : array_like
Points at which to evaluate the log of the probability density
function.
loc : array_like, optional
Mean of the distribution (default zero).
scale : array_like, optional
Positive definite shape matrix. This is not the distribution's
covariance matrix (default one).
rt : King's tidal radius parameter.
Returns
-------
"""
if random_state is not None:
rng = check_random_state(random_state)
else:
rng = self._random_state
dim, loc, scale, rt = self._process_parameters(loc, scale, rt)
scale_info = _PSD(scale)
# rt = rt*np.exp(0.5*scale_info.log_pdet)
#------ Take samples from the distance -------------
if dim == 1 :
rho = self._rvs_r1d(rt=rt, size=size,min_u=min_u)
elif dim == 3:
rho = self._rvs_r3d(rt=rt, size=size,min_u=min_u)
else:
sys.exit("Dimension {0} not implemented".format(dim))
#------ Samples from the angles -------
samples = toCartesian(rho,dim,random_state=rng).reshape(size,dim)
if dim > 1:
chol = np.linalg.cholesky(scale)
samples = np.dot(samples,chol)
else:
samples = scale*samples
return loc + samples
def rvs(self, loc=None, scale=1, gamma=3, size=1, random_state=None):
"""Draw random samples from a multivariate EFF distribution.
Parameters
----------
x : array_like
Points at which to evaluate the log of the probability density
function.
loc : array_like, optional
Mean of the distribution (default zero).
scale : array_like, optional
Positive definite shape matrix. This is not the distribution's
covariance matrix (default one).
gamma : EFF gamma parameter.
Returns
-------
"""
if random_state is not None:
rng = check_random_state(random_state)
else:
rng = self._random_state
dim, loc, scale, gamma = self._process_parameters(loc, scale, gamma)
scale_info = _PSD(scale)
# rc = np.exp(log_pdet)
#------ Take samples from the distance -------------
if dim == 1 :
rho = eff.rvs(gamma=gamma,size=size)
elif dim == 3:
rho = self._rvs_r3d(rc=1.0, gamma=gamma, size=size)
else:
sys.exit("Dimension {0} not implemented".format(dim))
#------ Samples from the angles -------
samples = toCartesian(rho,dim,random_state=rng).reshape(size,dim)
if dim > 1:
chol = np.linalg.cholesky(scale)
samples = np.dot(samples,chol)
else:
samples = samples*scale
return loc + samples
def predict(self, Xnew, thres=None):
n, p = Xnew.shape
cond_prob_matrix = np.zeros((n, self.K))
for k in range(self.K):
psd = _PSD(self.Sigma_[k])
prec_U, logdet = psd.U, psd.log_pdet
diff = Xnew - self.mu_[k]
sig = np.mean(diff * diff)
maha = (np.dot(diff, np.linalg.inv(self.Sigma_[k])) * diff).sum(1)
logdensity = -0.5 * (logdet + maha)
cond_prob_matrix[:, k] = np.exp(logdensity) * self.alpha_[k]
sum_row = np.sum(cond_prob_matrix, axis=1)
bool_sum_zero = (sum_row == 0)
cond_prob_matrix[bool_sum_zero, :] = self.alpha_
cond_prob_matrix /= cond_prob_matrix.sum(axis=1)[:, np.newaxis]
new_labels = np.array([i for i in np.argmax(cond_prob_matrix, axis=1)])
outlierness = np.zeros((n, )).astype(bool)
if thres is None:
thres = self.thres
thres = chi2.ppf(1 - thres, p)
for k in range(self.K):
data_cluster = Xnew[new_labels == k, :]
diff_cluster = data_cluster - self.mu_[k]
sig_cluster = np.mean(diff_cluster * diff_cluster)
maha_cluster = (np.dot(diff_cluster, np.linalg.inv(self.Sigma_[k]))
* diff_cluster).sum(1) / sig_cluster
outlierness[new_labels == k] = (maha_cluster > thres)
new_labels[outlierness] = -1
new_labels = new_labels.astype(str)
return (new_labels)
def __init__(self, mu=None, sigma=None, df=None):
if sigma is not None:
sigma = np.asarray(sigma)
self.mu = np.zeros(sigma.shape[0]) if mu is None else np.asarray(mu)
if sigma is None:
sigma = np.ones(len(mu))
if len(sigma.shape) == 1:
sigma = np.diag(sigma)
self.dim = len(self.mu)
self.sigma = sigma
self.df = df
# Use scipy stats to compute |Sigma| and (x-mu)^T Sigma^{-1} (x - mu),
# and to estimate dimension p from rank. Formula for pdf from wikipedia
# https://en.wikipedia.org/wiki/Multivariate_t-distribution
from scipy.stats._multivariate import _PSD
self._psd = _PSD(self.sigma)
nu, p = self.df, self._psd.rank
self._log_norm = (gammaln((nu + p) / 2) - gammaln(nu / 2) -
p / 2 * log(pi * nu) - self._psd.log_pdet / 2)
def __init__(self, loc=None, scale=1, rt=10, seed=None): """ Create a frozen multivariate King frozen distribution. Parameters ---------- x : array_like Points at which to evaluate the log of the probability density function. loc : array_like, optional Mean of the distribution (default zero). scale : array_like, optional Positive definite shape matrix. This is not the distribution's covariance matrix (default one). rt : King's tidal radius parameter. """ self._dist = multivariate_king_gen(seed) self.dim, self.loc, self.scale, self.rt = self._dist._process_parameters(loc, scale, rt) self.scale_info = _PSD(self.scale)
def __init__(self, loc=None, scale=1, gamma=2, seed=None): """ Create a frozen multivariate EFF frozen distribution. Parameters ---------- x : array_like Points at which to evaluate the log of the probability density function. loc : array_like, optional Mean of the distribution (default zero). scale : array_like, optional Positive definite shape matrix. This is not the distribution's covariance matrix (default one). gamma : EFF gamma parameter. """ self._dist = multivariate_eff_gen(seed) self.dim, self.loc, self.scale, self.gamma = self._dist._process_parameters(loc, scale, gamma) self.scale_info = _PSD(self.scale)
def __init__(self, mean=None, shape=1, df=1, seed=None):
"""
Create a frozen multivariate normal distribution.
Parameters
----------
x : array_like
Points at which to evaluate the log of the probability density
function.
mean : array_like, optional
Mean of the distribution (default zero).
shape : array_like, optional
Positive definite shape matrix. This is not the distribution's
covariance matrix (default one).
df : Degrees of freedom.
Examples
--------
FIXME.
"""
self._dist = multivariate_t_gen(seed)
dim, mean, shape, df = self._dist._process_parameters(mean, shape, df)
self.dim, self.mean, self.shape, self.df = dim, mean, shape, df
self.shape_info = _PSD(shape)
def __init__(self, mu=None, sigma=None, df=None):
if sigma is not None:
sigma = np.asarray(sigma)
self.mu = np.zeros(sigma.shape[0]) if mu is None else np.asarray(mu)
if sigma is None:
sigma = np.ones(len(mu))
if len(sigma.shape) == 1:
sigma = np.diag(sigma)
self.dim = len(self.mu)
self.sigma = sigma
self.df = df
# Use scipy stats to compute |Sigma| and (x-mu)^T Sigma^{-1} (x - mu),
# and to estimate dimension p from rank. Formula for pdf from wikipedia
# https://en.wikipedia.org/wiki/Multivariate_t-distribution
from scipy.stats._multivariate import _PSD
self._psd = _PSD(self.sigma)
nu, p = self.df, self._psd.rank
self._log_norm = (gammaln((nu + p)/2)
- gammaln(nu/2)
- p/2*log(pi*nu)
- self._psd.log_pdet/2
)
def logpdf(self, x, mean=None, shape=1, df=1):
"""Log of the multivariate t-distribution probability density function.
Parameters
----------
x : p-dimensional ndarray or n-by-p matrix.
mean : p-dimensional ndarray.
shape : p-by-p positive definite shape matrix. This is not the
distribution's covariance matrix.
df : Degrees of freedom.
Returns
-------
logpdf : Log of the probability density function evaluated at `x`.
"""
mean, shape, df = self._process_parameters(mean, shape, df)
shape_info = _PSD(shape)
return self._logpdf(x, mean, shape_info.U, shape_info.log_pdet, df)
def pdf(x, mean, cov, out=None, x_copy=None):
psd = mv._PSD(cov, allow_singular=False)
y = log_pdf(x, mean, psd.U, psd.log_pdet, psd.rank, out, x_copy)
return np.exp(y, out=y)
def __init__(self,
location=None,
scale=1,
dof=None,
allow_singular=False,
seed=None,
maxpts=None,
abseps=1e-5,
releps=1e-5):
"""
Create a frozen multivariate Student's t-distribution.
Parameters
----------
location : array_like, optional
Location of the distribution (default zero)
scale : array_like, optional
Scale matrix of the distribution (default one)
dof : None or scalar, optional
Degrees-of-freedom of the distribution (default numpy.inf)
allow_singular : bool, optional
If this flag is True then tolerate a singular
scale 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 student_mixture import multivariate_t
>>> r = multivariate_t()
>>> r.location
array([ 0.])
>>> r.scale
array([[1.]])
>>> r.dof
inf
"""
self._dist = multivariate_t_gen(seed)
self.dim, self.location, self.scale, self.dof = self._dist._process_parameters(
None, location, scale, dof)
self.scale_info = _PSD(self.scale, allow_singular=allow_singular)
if not maxpts:
maxpts = 1000000 * self.dim
self.maxpts = maxpts
self.abseps = abseps
self.releps = releps
def __init__(self,
df,
mean=None,
cov=1,
allow_singular=False,
seed=None,
maxpts=None,
abseps=1e-5,
releps=1e-5):
"""
Create a frozen multivariate Student's T distribution.
Parameters
----------
df : float
Degrees of freedom of the distribution
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_t
>>> r = multivariate_t(3)
>>> r.df
3.0
>>> r.mean
array([ 0.])
>>> r.cov
array([[1.]])
"""
self._dist = multivariate_t_gen(seed)
self.dim, self.df, self.mean, self.cov = self._dist._process_parameters(
None, df, 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 __init__(self, mean, cov):
self.mean = mean
self.cov = cov
self.psd = mv._PSD(cov, allow_singular=False)