""" 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 numbers
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 ._base import _BasePCA
from ..utils import check_random_state
from ..utils._arpack import _init_arpack_v0
from ..utils.extmath import fast_logdet, randomized_svd, svd_flip
from ..utils.extmath import stable_cumsum
from ..utils.validation import check_is_fitted
from ..utils.validation import _deprecate_positional_args
def _assess_dimension(spectrum, rank, n_samples):
"""Compute the log-likelihood of a rank ``rank`` dataset.
The dataset is assumed to be embedded in gaussian noise of shape(n,
dimf) having spectrum ``spectrum``. This implements the method of
T. P. Minka.
Parameters
----------
spectrum : ndarray of shape (n_features,)
Data spectrum.
rank : int
Tested rank value. It should be strictly lower than n_features,
otherwise the method isn't specified (division by zero in equation
(31) from the paper).
n_samples : int
Number of samples.
Returns
-------
ll : float
The log-likelihood.
References
----------
This implements the method of `Thomas P. Minka:
Automatic Choice of Dimensionality for PCA. NIPS 2000: 598-604
<https://proceedings.neurips.cc/paper/2000/file/7503cfacd12053d309b6bed5c89de212-Paper.pdf>`_
"""
n_features = spectrum.shape[0]
if not 1 <= rank < n_features:
raise ValueError("the tested rank should be in [1, n_features - 1]")
eps = 1e-15
if spectrum[rank - 1] < eps:
# When the tested rank is associated with a small eigenvalue, there's
# no point in computing the log-likelihood: it's going to be very
# small and won't be the max anyway. Also, it can lead to numerical
# issues below when computing pa, in particular in log((spectrum[i] -
# spectrum[j]) because this will take the log of something very small.
return -np.inf
pu = -rank * log(2.)
for i in range(1, rank + 1):
pu += (gammaln((n_features - i + 1) / 2.) -
log(np.pi) * (n_features - i + 1) / 2.)
pl = np.sum(np.log(spectrum[:rank]))
pl = -pl * n_samples / 2.
v = max(eps, 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) / 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):
"""Infers the dimension of a dataset with a given spectrum.
The returned value will be in [1, n_features - 1].
"""
ll = np.empty_like(spectrum)
ll[0] = -np.inf # we don't want to return n_components = 0
for rank in range(1, spectrum.shape[0]):
ll[rank] = _assess_dimension(spectrum, rank, n_samples)
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. The input data is centered
but not scaled for each feature before applying the SVD.
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 or 'mle', default=None
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. Use of ``n_components == 'mle'``
will interpret ``svd_solver == 'auto'`` as ``svd_solver == 'full'``.
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.
If ``svd_solver == 'arpack'``, the number of components must be
strictly less than the minimum of n_features and n_samples.
Hence, the None case results in::
n_components == min(n_samples, n_features) - 1
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, 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 : {'auto', 'full', 'arpack', 'randomized'}, default='auto'
If 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.
If full :
run exact full SVD calling the standard LAPACK solver via
`scipy.linalg.svd` and select the components by postprocessing
If arpack :
run SVD truncated to n_components calling ARPACK solver via
`scipy.sparse.linalg.svds`. It requires strictly
0 < n_components < min(X.shape)
If randomized :
run randomized SVD by the method of Halko et al.
.. versionadded:: 0.18.0
tol : float, default=0.0
Tolerance for singular values computed by svd_solver == 'arpack'.
Must be of range [0.0, infinity).
.. versionadded:: 0.18.0
iterated_power : int or 'auto', default='auto'
Number of iterations for the power method computed by
svd_solver == 'randomized'.
Must be of range [0, infinity).
.. versionadded:: 0.18.0
random_state : int, RandomState instance or None, default=None
Used when the 'arpack' or 'randomized' solvers are used. Pass an int
for reproducible results across multiple function calls.
See :term:`Glossary <random_state>`.
.. versionadded:: 0.18.0
Attributes
----------
components_ : ndarray of 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_ : ndarray of shape (n_components,)
The amount of variance explained by each of the selected components.
The variance estimation uses `n_samples - 1` degrees of freedom.
Equal to n_components largest eigenvalues
of the covariance matrix of X.
.. versionadded:: 0.18
explained_variance_ratio_ : ndarray of 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 the ratios is equal to 1.0.
singular_values_ : ndarray of 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.
.. versionadded:: 0.19
mean_ : ndarray of 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 the lesser value of n_features and n_samples
if n_components is None.
n_features_ : int
Number of features in the training data.
n_samples_ : int
Number of samples in the training data.
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
compute 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.
See Also
--------
KernelPCA : Kernel Principal Component Analysis.
SparsePCA : Sparse Principal Component Analysis.
TruncatedSVD : Dimensionality reduction using truncated SVD.
IncrementalPCA : Incremental Principal Component Analysis.
References
----------
For n_components == 'mle', this class uses the method from:
`Minka, T. P.. "Automatic choice of dimensionality for PCA".
In NIPS, pp. 598-604 <https://tminka.github.io/papers/pca/minka-pca.pdf>`_
Implements the probabilistic PCA model from:
`Tipping, M. E., and Bishop, C. M. (1999). "Probabilistic principal
component analysis". Journal of the Royal Statistical Society:
Series B (Statistical Methodology), 61(3), 611-622.
<http://www.miketipping.com/papers/met-mppca.pdf>`_
via the score and score_samples methods.
For svd_solver == 'arpack', refer to `scipy.sparse.linalg.svds`.
For svd_solver == 'randomized', see:
`Halko, N., Martinsson, P. G., and Tropp, J. A. (2011).
"Finding structure with randomness: Probabilistic algorithms for
constructing approximate matrix decompositions".
SIAM review, 53(2), 217-288.
<https://doi.org/10.1137/090771806>`_
and also
`Martinsson, P. G., Rokhlin, V., and Tygert, M. (2011).
"A randomized algorithm for the decomposition of matrices".
Applied and Computational Harmonic Analysis, 30(1), 47-68
<https://doi.org/10.1016/j.acha.2010.02.003>`_.
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(n_components=2)
>>> print(pca.explained_variance_ratio_)
[0.9924... 0.0075...]
>>> print(pca.singular_values_)
[6.30061... 0.54980...]
>>> pca = PCA(n_components=2, svd_solver='full')
>>> pca.fit(X)
PCA(n_components=2, svd_solver='full')
>>> print(pca.explained_variance_ratio_)
[0.9924... 0.00755...]
>>> print(pca.singular_values_)
[6.30061... 0.54980...]
>>> pca = PCA(n_components=1, svd_solver='arpack')
>>> pca.fit(X)
PCA(n_components=1, svd_solver='arpack')
>>> print(pca.explained_variance_ratio_)
[0.99244...]
>>> print(pca.singular_values_)
[6.30061...]
"""
@_deprecate_positional_args
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 of shape (n_samples, n_features)
Training data, where n_samples is 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_transform(self, X, y=None):
"""Fit the model with X and apply the dimensionality reduction on X.
Parameters
----------
X : array-like of shape (n_samples, n_features)
Training data, where n_samples is the number of samples
and n_features is the number of features.
y : Ignored
Returns
-------
X_new : ndarray of shape (n_samples, n_components)
Transformed values.
Notes
-----
This method returns a Fortran-ordered array. To convert it to a
C-ordered array, use 'np.ascontiguousarray'.
"""
U, S, Vt = self._fit(X)
U = U[:, :self.n_components_]
if self.whiten:
# X_new = X * V / S * sqrt(n_samples) = U * sqrt(n_samples)
U *= sqrt(X.shape[0] - 1)
else:
# X_new = X * V = U * S * Vt * V = U * S
U *= S[:self.n_components_]
return U
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 = self._validate_data(X, dtype=[np.float64, np.float32],
ensure_2d=True, copy=self.copy)
# Handle n_components==None
if self.n_components is None:
if self.svd_solver != 'arpack':
n_components = min(X.shape)
else:
n_components = min(X.shape) - 1
else:
n_components = self.n_components
# Handle svd_solver
self._fit_svd_solver = self.svd_solver
if self._fit_svd_solver == 'auto':
# Small problem or n_components == 'mle', just call full PCA
if max(X.shape) <= 500 or n_components == 'mle':
self._fit_svd_solver = 'full'
elif n_components >= 1 and n_components < .8 * min(X.shape):
self._fit_svd_solver = 'randomized'
# This is also the case of n_components in (0,1)
else:
self._fit_svd_solver = 'full'
# Call different fits for either full or truncated SVD
if self._fit_svd_solver == 'full':
return self._fit_full(X, n_components)
elif self._fit_svd_solver in ['arpack', 'randomized']:
return self._fit_truncated(X, n_components, self._fit_svd_solver)
else:
raise ValueError("Unrecognized svd_solver='{0}'"
"".format(self._fit_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 <= min(n_samples, n_features):
raise ValueError("n_components=%r must be between 0 and "
"min(n_samples, n_features)=%r with "
"svd_solver='full'"
% (n_components, min(n_samples, n_features)))
elif n_components >= 1:
if not isinstance(n_components, numbers.Integral):
raise ValueError("n_components=%r must be of type int "
"when greater than or equal to 1, "
"was of type=%r"
% (n_components, type(n_components)))
# Center data
self.mean_ = np.mean(X, axis=0)
X -= self.mean_
U, S, Vt = linalg.svd(X, full_matrices=False)
# flip eigenvectors' sign to enforce deterministic output
U, Vt = svd_flip(U, Vt)
components_ = Vt
# 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)
elif 0 < n_components < 1.0:
# number of components for which the cumulated explained
# variance percentage is superior to the desired threshold
# side='right' ensures that number of features selected
# their variance is always greater than n_components float
# passed. More discussion in issue: #15669
ratio_cumsum = stable_cumsum(explained_variance_ratio_)
n_components = np.searchsorted(ratio_cumsum, n_components,
side='right') + 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, Vt
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, str):
raise ValueError("n_components=%r cannot be a string "
"with svd_solver='%s'"
% (n_components, svd_solver))
elif not 1 <= n_components <= min(n_samples, n_features):
raise ValueError("n_components=%r must be between 1 and "
"min(n_samples, n_features)=%r with "
"svd_solver='%s'"
% (n_components, min(n_samples, n_features),
svd_solver))
elif not isinstance(n_components, numbers.Integral):
raise ValueError("n_components=%r must be of type int "
"when greater than or equal to 1, was of type=%r"
% (n_components, type(n_components)))
elif svd_solver == 'arpack' and n_components == min(n_samples,
n_features):
raise ValueError("n_components=%r must be strictly less than "
"min(n_samples, n_features)=%r with "
"svd_solver='%s'"
% (n_components, min(n_samples, 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':
v0 = _init_arpack_v0(min(X.shape), random_state)
U, S, Vt = 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, Vt = svd_flip(U[:, ::-1], Vt[::-1])
elif svd_solver == 'randomized':
# sign flipping is done inside
U, S, Vt = 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_ = Vt
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, Vt
[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-like of shape (n_samples, n_features)
The data.
Returns
-------
ll : ndarray of shape (n_samples,)
Log-likelihood of each sample under the current model.
"""
check_is_fitted(self)
X = self._validate_data(X, dtype=[np.float64, np.float32], reset=False)
Xr = X - self.mean_
n_features = X.shape[1]
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-like of 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))
def _more_tags(self):
return {'preserves_dtype': [np.float64, np.float32]}