""" Principal Component Analysis
"""
# Author: Alexandre Gramfort <alexandre.gramfort@inria.fr>
# Olivier Grisel <olivier.grisel@ensta.org>
# Mathieu Blondel <mathieu@mblondel.org>
# Denis A. Engemann <denis-alexander.engemann@inria.fr>
# Michael Eickenberg <michael.eickenberg@inria.fr>
# Giorgio Patrini <giorgio.patrini@anu.edu.au>
#
# License: BSD 3 clause
from math import log, sqrt
import numpy as np
from scipy import linalg
from scipy.special import gammaln
from scipy.sparse import issparse
from scipy.sparse.linalg import svds
from ..externals import six
from .base import _BasePCA
from ..base import BaseEstimator, TransformerMixin
from ..utils import deprecated
from ..utils import check_random_state, as_float_array
from ..utils import check_array
from ..utils.extmath import fast_logdet, randomized_svd, svd_flip
from ..utils.extmath import stable_cumsum
from ..utils.validation import check_is_fitted
def _assess_dimension_(spectrum, rank, n_samples, n_features):
"""Compute the likelihood of a rank ``rank`` dataset
The dataset is assumed to be embedded in gaussian noise of shape(n,
dimf) having spectrum ``spectrum``.
Parameters
----------
spectrum : array of shape (n)
Data spectrum.
rank : int
Tested rank value.
n_samples : int
Number of samples.
n_features : int
Number of features.
Returns
-------
ll : float,
The log-likelihood
Notes
-----
This implements the method of `Thomas P. Minka:
Automatic Choice of Dimensionality for PCA. NIPS 2000: 598-604`
"""
if rank > len(spectrum):
raise ValueError("The tested rank cannot exceed the rank of the"
" dataset")
pu = -rank * log(2.)
for i in range(rank):
pu += (gammaln((n_features - i) / 2.) -
log(np.pi) * (n_features - i) / 2.)
pl = np.sum(np.log(spectrum[:rank]))
pl = -pl * n_samples / 2.
if rank == n_features:
pv = 0
v = 1
else:
v = np.sum(spectrum[rank:]) / (n_features - rank)
pv = -np.log(v) * n_samples * (n_features - rank) / 2.
m = n_features * rank - rank * (rank + 1.) / 2.
pp = log(2. * np.pi) * (m + rank + 1.) / 2.
pa = 0.
spectrum_ = spectrum.copy()
spectrum_[rank:n_features] = v
for i in range(rank):
for j in range(i + 1, len(spectrum)):
pa += log((spectrum[i] - spectrum[j]) *
(1. / spectrum_[j] - 1. / spectrum_[i])) + log(n_samples)
ll = pu + pl + pv + pp - pa / 2. - rank * log(n_samples) / 2.
return ll
def _infer_dimension_(spectrum, n_samples, n_features):
"""Infers the dimension of a dataset of shape (n_samples, n_features)
The dataset is described by its spectrum `spectrum`.
"""
n_spectrum = len(spectrum)
ll = np.empty(n_spectrum)
for rank in range(n_spectrum):
ll[rank] = _assess_dimension_(spectrum, rank, n_samples, n_features)
return ll.argmax()
class PCA(_BasePCA):
"""Principal component analysis (PCA)
Linear dimensionality reduction using Singular Value Decomposition of the
data to project it to a lower dimensional space.
It uses the LAPACK implementation of the full SVD or a randomized truncated
SVD by the method of Halko et al. 2009, depending on the shape of the input
data and the number of components to extract.
It can also use the scipy.sparse.linalg ARPACK implementation of the
truncated SVD.
Notice that this class does not support sparse input. See
:class:`TruncatedSVD` for an alternative with sparse data.
Read more in the :ref:`User Guide <PCA>`.
Parameters
----------
n_components : int, float, None or string
Number of components to keep.
if n_components is not set all components are kept::
n_components == min(n_samples, n_features)
if n_components == 'mle' and svd_solver == 'full', Minka\'s MLE is used
to guess the dimension
if ``0 < n_components < 1`` and svd_solver == 'full', select the number
of components such that the amount of variance that needs to be
explained is greater than the percentage specified by n_components
n_components cannot be equal to n_features for svd_solver == 'arpack'.
copy : bool (default True)
If False, data passed to fit are overwritten and running
fit(X).transform(X) will not yield the expected results,
use fit_transform(X) instead.
whiten : bool, optional (default False)
When True (False by default) the `components_` vectors are multiplied
by the square root of n_samples and then divided by the singular values
to ensure uncorrelated outputs with unit component-wise variances.
Whitening will remove some information from the transformed signal
(the relative variance scales of the components) but can sometime
improve the predictive accuracy of the downstream estimators by
making their data respect some hard-wired assumptions.
svd_solver : string {'auto', 'full', 'arpack', 'randomized'}
auto :
the solver is selected by a default policy based on `X.shape` and
`n_components`: if the input data is larger than 500x500 and the
number of components to extract is lower than 80% of the smallest
dimension of the data, then the more efficient 'randomized'
method is enabled. Otherwise the exact full SVD is computed and
optionally truncated afterwards.
full :
run exact full SVD calling the standard LAPACK solver via
`scipy.linalg.svd` and select the components by postprocessing
arpack :
run SVD truncated to n_components calling ARPACK solver via
`scipy.sparse.linalg.svds`. It requires strictly
0 < n_components < X.shape[1]
randomized :
run randomized SVD by the method of Halko et al.
.. versionadded:: 0.18.0
tol : float >= 0, optional (default .0)
Tolerance for singular values computed by svd_solver == 'arpack'.
.. versionadded:: 0.18.0
iterated_power : int >= 0, or 'auto', (default 'auto')
Number of iterations for the power method computed by
svd_solver == 'randomized'.
.. versionadded:: 0.18.0
random_state : int, RandomState instance or None, optional (default None)
If int, random_state is the seed used by the random number generator;
If RandomState instance, random_state is the random number generator;
If None, the random number generator is the RandomState instance used
by `np.random`. Used when ``svd_solver`` == 'arpack' or 'randomized'.
.. versionadded:: 0.18.0
Attributes
----------
components_ : array, shape (n_components, n_features)
Principal axes in feature space, representing the directions of
maximum variance in the data. The components are sorted by
``explained_variance_``.
explained_variance_ : array, shape (n_components,)
The amount of variance explained by each of the selected components.
Equal to n_components largest eigenvalues
of the covariance matrix of X.
.. versionadded:: 0.18
explained_variance_ratio_ : array, shape (n_components,)
Percentage of variance explained by each of the selected components.
If ``n_components`` is not set then all components are stored and the
sum of explained variances is equal to 1.0.
singular_values_ : array, shape (n_components,)
The singular values corresponding to each of the selected components.
The singular values are equal to the 2-norms of the ``n_components``
variables in the lower-dimensional space.
mean_ : array, shape (n_features,)
Per-feature empirical mean, estimated from the training set.
Equal to `X.mean(axis=0)`.
n_components_ : int
The estimated number of components. When n_components is set
to 'mle' or a number between 0 and 1 (with svd_solver == 'full') this
number is estimated from input data. Otherwise it equals the parameter
n_components, or n_features if n_components is None.
noise_variance_ : float
The estimated noise covariance following the Probabilistic PCA model
from Tipping and Bishop 1999. See "Pattern Recognition and
Machine Learning" by C. Bishop, 12.2.1 p. 574 or
http://www.miketipping.com/papers/met-mppca.pdf. It is required to
computed the estimated data covariance and score samples.
Equal to the average of (min(n_features, n_samples) - n_components)
smallest eigenvalues of the covariance matrix of X.
References
----------
For n_components == 'mle', this class uses the method of `Thomas P. Minka:
Automatic Choice of Dimensionality for PCA. NIPS 2000: 598-604`
Implements the probabilistic PCA model from:
M. Tipping and C. Bishop, Probabilistic Principal Component Analysis,
Journal of the Royal Statistical Society, Series B, 61, Part 3, pp. 611-622
via the score and score_samples methods.
See http://www.miketipping.com/papers/met-mppca.pdf
For svd_solver == 'arpack', refer to `scipy.sparse.linalg.svds`.
For svd_solver == 'randomized', see:
`Finding structure with randomness: Stochastic algorithms
for constructing approximate matrix decompositions Halko, et al., 2009
(arXiv:909)`
`A randomized algorithm for the decomposition of matrices
Per-Gunnar Martinsson, Vladimir Rokhlin and Mark Tygert`
Examples
--------
>>> import numpy as np
>>> from sklearn.decomposition import PCA
>>> X = np.array([[-1, -1], [-2, -1], [-3, -2], [1, 1], [2, 1], [3, 2]])
>>> pca = PCA(n_components=2)
>>> pca.fit(X)
PCA(copy=True, iterated_power='auto', n_components=2, random_state=None,
svd_solver='auto', tol=0.0, whiten=False)
>>> print(pca.explained_variance_ratio_) # doctest: +ELLIPSIS
[ 0.99244... 0.00755...]
>>> print(pca.singular_values_) # doctest: +ELLIPSIS
[ 6.30061... 0.54980...]
>>> pca = PCA(n_components=2, svd_solver='full')
>>> pca.fit(X) # doctest: +ELLIPSIS +NORMALIZE_WHITESPACE
PCA(copy=True, iterated_power='auto', n_components=2, random_state=None,
svd_solver='full', tol=0.0, whiten=False)
>>> print(pca.explained_variance_ratio_) # doctest: +ELLIPSIS
[ 0.99244... 0.00755...]
>>> print(pca.singular_values_) # doctest: +ELLIPSIS
[ 6.30061... 0.54980...]
>>> pca = PCA(n_components=1, svd_solver='arpack')
>>> pca.fit(X)
PCA(copy=True, iterated_power='auto', n_components=1, random_state=None,
svd_solver='arpack', tol=0.0, whiten=False)
>>> print(pca.explained_variance_ratio_) # doctest: +ELLIPSIS
[ 0.99244...]
>>> print(pca.singular_values_) # doctest: +ELLIPSIS
[ 6.30061...]
See also
--------
KernelPCA
SparsePCA
TruncatedSVD
IncrementalPCA
"""
def __init__(self, n_components=None, copy=True, whiten=False,
svd_solver='auto', tol=0.0, iterated_power='auto',
random_state=None):
self.n_components = n_components
self.copy = copy
self.whiten = whiten
self.svd_solver = svd_solver
self.tol = tol
self.iterated_power = iterated_power
self.random_state = random_state
[docs] def fit(self, X, y=None):
"""Fit the model with X.
Parameters
----------
X : array-like, shape (n_samples, n_features)
Training data, where n_samples in the number of samples
and n_features is the number of features.
y : Ignored.
Returns
-------
self : object
Returns the instance itself.
"""
self._fit(X)
return self
def _fit(self, X):
"""Dispatch to the right submethod depending on the chosen solver."""
# Raise an error for sparse input.
# This is more informative than the generic one raised by check_array.
if issparse(X):
raise TypeError('PCA does not support sparse input. See '
'TruncatedSVD for a possible alternative.')
X = check_array(X, dtype=[np.float64, np.float32], ensure_2d=True,
copy=self.copy)
# Handle n_components==None
if self.n_components is None:
n_components = X.shape[1]
else:
n_components = self.n_components
# Handle svd_solver
svd_solver = self.svd_solver
if svd_solver == 'auto':
# Small problem, just call full PCA
if max(X.shape) <= 500:
svd_solver = 'full'
elif n_components >= 1 and n_components < .8 * min(X.shape):
svd_solver = 'randomized'
# This is also the case of n_components in (0,1)
else:
svd_solver = 'full'
# Call different fits for either full or truncated SVD
if svd_solver == 'full':
return self._fit_full(X, n_components)
elif svd_solver in ['arpack', 'randomized']:
return self._fit_truncated(X, n_components, svd_solver)
else:
raise ValueError("Unrecognized svd_solver='{0}'"
"".format(svd_solver))
def _fit_full(self, X, n_components):
"""Fit the model by computing full SVD on X"""
n_samples, n_features = X.shape
if n_components == 'mle':
if n_samples < n_features:
raise ValueError("n_components='mle' is only supported "
"if n_samples >= n_features")
elif not 0 <= n_components <= n_features:
raise ValueError("n_components=%r must be between 0 and "
"n_features=%r with svd_solver='full'"
% (n_components, n_features))
# Center data
self.mean_ = np.mean(X, axis=0)
X -= self.mean_
U, S, V = linalg.svd(X, full_matrices=False)
# flip eigenvectors' sign to enforce deterministic output
U, V = svd_flip(U, V)
components_ = V
# Get variance explained by singular values
explained_variance_ = (S ** 2) / (n_samples - 1)
total_var = explained_variance_.sum()
explained_variance_ratio_ = explained_variance_ / total_var
singular_values_ = S.copy() # Store the singular values.
# Postprocess the number of components required
if n_components == 'mle':
n_components = \
_infer_dimension_(explained_variance_, n_samples, n_features)
elif 0 < n_components < 1.0:
# number of components for which the cumulated explained
# variance percentage is superior to the desired threshold
ratio_cumsum = stable_cumsum(explained_variance_ratio_)
n_components = np.searchsorted(ratio_cumsum, n_components) + 1
# Compute noise covariance using Probabilistic PCA model
# The sigma2 maximum likelihood (cf. eq. 12.46)
if n_components < min(n_features, n_samples):
self.noise_variance_ = explained_variance_[n_components:].mean()
else:
self.noise_variance_ = 0.
self.n_samples_, self.n_features_ = n_samples, n_features
self.components_ = components_[:n_components]
self.n_components_ = n_components
self.explained_variance_ = explained_variance_[:n_components]
self.explained_variance_ratio_ = \
explained_variance_ratio_[:n_components]
self.singular_values_ = singular_values_[:n_components]
return U, S, V
def _fit_truncated(self, X, n_components, svd_solver):
"""Fit the model by computing truncated SVD (by ARPACK or randomized)
on X
"""
n_samples, n_features = X.shape
if isinstance(n_components, six.string_types):
raise ValueError("n_components=%r cannot be a string "
"with svd_solver='%s'"
% (n_components, svd_solver))
elif not 1 <= n_components <= n_features:
raise ValueError("n_components=%r must be between 1 and "
"n_features=%r with svd_solver='%s'"
% (n_components, n_features, svd_solver))
elif svd_solver == 'arpack' and n_components == n_features:
raise ValueError("n_components=%r must be stricly less than "
"n_features=%r with svd_solver='%s'"
% (n_components, n_features, svd_solver))
random_state = check_random_state(self.random_state)
# Center data
self.mean_ = np.mean(X, axis=0)
X -= self.mean_
if svd_solver == 'arpack':
# random init solution, as ARPACK does it internally
v0 = random_state.uniform(-1, 1, size=min(X.shape))
U, S, V = svds(X, k=n_components, tol=self.tol, v0=v0)
# svds doesn't abide by scipy.linalg.svd/randomized_svd
# conventions, so reverse its outputs.
S = S[::-1]
# flip eigenvectors' sign to enforce deterministic output
U, V = svd_flip(U[:, ::-1], V[::-1])
elif svd_solver == 'randomized':
# sign flipping is done inside
U, S, V = randomized_svd(X, n_components=n_components,
n_iter=self.iterated_power,
flip_sign=True,
random_state=random_state)
self.n_samples_, self.n_features_ = n_samples, n_features
self.components_ = V
self.n_components_ = n_components
# Get variance explained by singular values
self.explained_variance_ = (S ** 2) / (n_samples - 1)
total_var = np.var(X, ddof=1, axis=0)
self.explained_variance_ratio_ = \
self.explained_variance_ / total_var.sum()
self.singular_values_ = S.copy() # Store the singular values.
if self.n_components_ < min(n_features, n_samples):
self.noise_variance_ = (total_var.sum() -
self.explained_variance_.sum())
self.noise_variance_ /= min(n_features, n_samples) - n_components
else:
self.noise_variance_ = 0.
return U, S, V
[docs] def score_samples(self, X):
"""Return the log-likelihood of each sample.
See. "Pattern Recognition and Machine Learning"
by C. Bishop, 12.2.1 p. 574
or http://www.miketipping.com/papers/met-mppca.pdf
Parameters
----------
X : array, shape(n_samples, n_features)
The data.
Returns
-------
ll : array, shape (n_samples,)
Log-likelihood of each sample under the current model
"""
check_is_fitted(self, 'mean_')
X = check_array(X)
Xr = X - self.mean_
n_features = X.shape[1]
log_like = np.zeros(X.shape[0])
precision = self.get_precision()
log_like = -.5 * (Xr * (np.dot(Xr, precision))).sum(axis=1)
log_like -= .5 * (n_features * log(2. * np.pi) -
fast_logdet(precision))
return log_like
[docs] def score(self, X, y=None):
"""Return the average log-likelihood of all samples.
See. "Pattern Recognition and Machine Learning"
by C. Bishop, 12.2.1 p. 574
or http://www.miketipping.com/papers/met-mppca.pdf
Parameters
----------
X : array, shape(n_samples, n_features)
The data.
y : Ignored.
Returns
-------
ll : float
Average log-likelihood of the samples under the current model
"""
return np.mean(self.score_samples(X))
@deprecated("RandomizedPCA was deprecated in 0.18 and will be removed in "
"0.20. "
"Use PCA(svd_solver='randomized') instead. The new implementation "
"DOES NOT store whiten ``components_``. Apply transform to get "
"them.")
class RandomizedPCA(BaseEstimator, TransformerMixin):
"""Principal component analysis (PCA) using randomized SVD
.. deprecated:: 0.18
This class will be removed in 0.20.
Use :class:`PCA` with parameter svd_solver 'randomized' instead.
The new implementation DOES NOT store whiten ``components_``.
Apply transform to get them.
Linear dimensionality reduction using approximated Singular Value
Decomposition of the data and keeping only the most significant
singular vectors to project the data to a lower dimensional space.
Read more in the :ref:`User Guide <RandomizedPCA>`.
Parameters
----------
n_components : int, optional
Maximum number of components to keep. When not given or None, this
is set to n_features (the second dimension of the training data).
copy : bool
If False, data passed to fit are overwritten and running
fit(X).transform(X) will not yield the expected results,
use fit_transform(X) instead.
iterated_power : int, default=2
Number of iterations for the power method.
.. versionchanged:: 0.18
whiten : bool, optional
When True (False by default) the `components_` vectors are multiplied
by the square root of (n_samples) and divided by the singular values to
ensure uncorrelated outputs with unit component-wise variances.
Whitening will remove some information from the transformed signal
(the relative variance scales of the components) but can sometime
improve the predictive accuracy of the downstream estimators by
making their data respect some hard-wired assumptions.
random_state : int, RandomState instance or None, optional, default=None
If int, random_state is the seed used by the random number generator;
If RandomState instance, random_state is the random number generator;
If None, the random number generator is the RandomState instance used
by `np.random`.
Attributes
----------
components_ : array, shape (n_components, n_features)
Components with maximum variance.
explained_variance_ratio_ : array, shape (n_components,)
Percentage of variance explained by each of the selected components.
If k is not set then all components are stored and the sum of explained
variances is equal to 1.0.
singular_values_ : array, shape (n_components,)
The singular values corresponding to each of the selected components.
The singular values are equal to the 2-norms of the ``n_components``
variables in the lower-dimensional space.
mean_ : array, shape (n_features,)
Per-feature empirical mean, estimated from the training set.
Examples
--------
>>> import numpy as np
>>> from sklearn.decomposition import RandomizedPCA
>>> X = np.array([[-1, -1], [-2, -1], [-3, -2], [1, 1], [2, 1], [3, 2]])
>>> pca = RandomizedPCA(n_components=2)
>>> pca.fit(X) # doctest: +ELLIPSIS +NORMALIZE_WHITESPACE
RandomizedPCA(copy=True, iterated_power=2, n_components=2,
random_state=None, whiten=False)
>>> print(pca.explained_variance_ratio_) # doctest: +ELLIPSIS
[ 0.99244... 0.00755...]
>>> print(pca.singular_values_) # doctest: +ELLIPSIS
[ 6.30061... 0.54980...]
See also
--------
PCA
TruncatedSVD
References
----------
.. [Halko2009] `Finding structure with randomness: Stochastic algorithms
for constructing approximate matrix decompositions Halko, et al., 2009
(arXiv:909)`
.. [MRT] `A randomized algorithm for the decomposition of matrices
Per-Gunnar Martinsson, Vladimir Rokhlin and Mark Tygert`
"""
def __init__(self, n_components=None, copy=True, iterated_power=2,
whiten=False, random_state=None):
self.n_components = n_components
self.copy = copy
self.iterated_power = iterated_power
self.whiten = whiten
self.random_state = random_state
[docs] def fit(self, X, y=None):
"""Fit the model with X by extracting the first principal components.
Parameters
----------
X : array-like, shape (n_samples, n_features)
Training data, where n_samples in the number of samples
and n_features is the number of features.
y : Ignored.
Returns
-------
self : object
Returns the instance itself.
"""
self._fit(check_array(X))
return self
def _fit(self, X):
"""Fit the model to the data X.
Parameters
----------
X : array-like, shape (n_samples, n_features)
Training vector, where n_samples in the number of samples and
n_features is the number of features.
Returns
-------
X : ndarray, shape (n_samples, n_features)
The input data, copied, centered and whitened when requested.
"""
random_state = check_random_state(self.random_state)
X = np.atleast_2d(as_float_array(X, copy=self.copy))
n_samples = X.shape[0]
# Center data
self.mean_ = np.mean(X, axis=0)
X -= self.mean_
if self.n_components is None:
n_components = X.shape[1]
else:
n_components = self.n_components
U, S, V = randomized_svd(X, n_components,
n_iter=self.iterated_power,
random_state=random_state)
self.explained_variance_ = exp_var = (S ** 2) / (n_samples - 1)
full_var = np.var(X, ddof=1, axis=0).sum()
self.explained_variance_ratio_ = exp_var / full_var
self.singular_values_ = S # Store the singular values.
if self.whiten:
self.components_ = V / S[:, np.newaxis] * sqrt(n_samples)
else:
self.components_ = V
return X
