""" 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
from numbers import Integral, Real
import numpy as np
from scipy import linalg
from scipy.sparse import issparse
from scipy.sparse.linalg import svds
from scipy.special import gammaln
from ..base import _fit_context
from ..utils import check_random_state
from ..utils._arpack import _init_arpack_v0
from ..utils._array_api import _convert_to_numpy, get_namespace
from ..utils._param_validation import Interval, RealNotInt, StrOptions
from ..utils.extmath import fast_logdet, randomized_svd, stable_cumsum, svd_flip
from ..utils.sparsefuncs import _implicit_column_offset, mean_variance_axis
from ..utils.validation import check_is_fitted
from ._base import _BasePCA
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>`_
"""
xp, _ = get_namespace(spectrum)
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 -xp.inf
pu = -rank * log(2.0)
for i in range(1, rank + 1):
pu += (
gammaln((n_features - i + 1) / 2.0)
- log(xp.pi) * (n_features - i + 1) / 2.0
)
pl = xp.sum(xp.log(spectrum[:rank]))
pl = -pl * n_samples / 2.0
v = max(eps, xp.sum(spectrum[rank:]) / (n_features - rank))
pv = -log(v) * n_samples * (n_features - rank) / 2.0
m = n_features * rank - rank * (rank + 1.0) / 2.0
pp = log(2.0 * xp.pi) * (m + rank) / 2.0
pa = 0.0
spectrum_ = xp.asarray(spectrum, copy=True)
spectrum_[rank:n_features] = v
for i in range(rank):
for j in range(i + 1, spectrum.shape[0]):
pa += log(
(spectrum[i] - spectrum[j]) * (1.0 / spectrum_[j] - 1.0 / spectrum_[i])
) + log(n_samples)
ll = pu + pl + pv + pp - pa / 2.0 - rank * log(n_samples) / 2.0
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].
"""
xp, _ = get_namespace(spectrum)
ll = xp.empty_like(spectrum)
ll[0] = -xp.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 xp.argmax(ll)
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.
For a usage example, see
:ref:`sphx_glr_auto_examples_decomposition_plot_pca_iris.py`
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
n_oversamples : int, default=10
This parameter is only relevant when `svd_solver="randomized"`.
It corresponds to the additional number of random vectors to sample the
range of `X` so as to ensure proper conditioning. See
:func:`~sklearn.utils.extmath.randomized_svd` for more details.
.. versionadded:: 1.1
power_iteration_normalizer : {'auto', 'QR', 'LU', 'none'}, default='auto'
Power iteration normalizer for randomized SVD solver.
Not used by ARPACK. See :func:`~sklearn.utils.extmath.randomized_svd`
for more details.
.. versionadded:: 1.1
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. Equivalently, the right singular
vectors of the centered input data, parallel to its eigenvectors.
The components are sorted by decreasing ``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_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.
n_features_in_ : int
Number of features seen during :term:`fit`.
.. versionadded:: 0.24
feature_names_in_ : ndarray of shape (`n_features_in_`,)
Names of features seen during :term:`fit`. Defined only when `X`
has feature names that are all strings.
.. versionadded:: 1.0
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:
:doi:`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.
<10.1137/090771806>`
and also
:doi:`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.
<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...]
"""
_parameter_constraints: dict = {
"n_components": [
Interval(Integral, 0, None, closed="left"),
Interval(RealNotInt, 0, 1, closed="neither"),
StrOptions({"mle"}),
None,
],
"copy": ["boolean"],
"whiten": ["boolean"],
"svd_solver": [StrOptions({"auto", "full", "arpack", "randomized"})],
"tol": [Interval(Real, 0, None, closed="left")],
"iterated_power": [
StrOptions({"auto"}),
Interval(Integral, 0, None, closed="left"),
],
"n_oversamples": [Interval(Integral, 1, None, closed="left")],
"power_iteration_normalizer": [StrOptions({"auto", "QR", "LU", "none"})],
"random_state": ["random_state"],
}
def __init__(
self,
n_components=None,
*,
copy=True,
whiten=False,
svd_solver="auto",
tol=0.0,
iterated_power="auto",
n_oversamples=10,
power_iteration_normalizer="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.n_oversamples = n_oversamples
self.power_iteration_normalizer = power_iteration_normalizer
self.random_state = random_state
[docs]
@_fit_context(prefer_skip_nested_validation=True)
def fit(self, X, y=None):
"""Fit the model with X.
Parameters
----------
X : {array-like, sparse matrix} 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
Ignored.
Returns
-------
self : object
Returns the instance itself.
"""
self._fit(X)
return self
@_fit_context(prefer_skip_nested_validation=True)
def fit_transform(self, X, y=None):
"""Fit the model with X and apply the dimensionality reduction on X.
Parameters
----------
X : {array-like, sparse matrix} 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
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."""
xp, is_array_api_compliant = get_namespace(X)
# Raise an error for sparse input and unsupported svd_solver
if issparse(X) and self.svd_solver != "arpack":
raise TypeError(
'PCA only support sparse inputs with the "arpack" solver, while '
f'"{self.svd_solver}" was passed. See TruncatedSVD for a possible'
" alternative."
)
# Raise an error for non-Numpy input and arpack solver.
if self.svd_solver == "arpack" and is_array_api_compliant:
raise ValueError(
"PCA with svd_solver='arpack' is not supported for Array API inputs."
)
X = self._validate_data(
X,
dtype=[xp.float64, xp.float32],
accept_sparse=("csr", "csc"),
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 1 <= n_components < 0.8 * 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)
def _fit_full(self, X, n_components):
"""Fit the model by computing full SVD on X."""
xp, is_array_api_compliant = get_namespace(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))
)
# Center data
self.mean_ = xp.mean(X, axis=0)
X -= self.mean_
if not is_array_api_compliant:
# Use scipy.linalg with NumPy/SciPy inputs for the sake of not
# introducing unanticipated behavior changes. In the long run we
# could instead decide to always use xp.linalg.svd for all inputs,
# but that would make this code rely on numpy's SVD instead of
# scipy's. It's not 100% clear whether they use the same LAPACK
# solver by default though (assuming both are built against the
# same BLAS).
U, S, Vt = linalg.svd(X, full_matrices=False)
else:
U, S, Vt = xp.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 = xp.sum(explained_variance_)
explained_variance_ratio_ = explained_variance_ / total_var
singular_values_ = xp.asarray(S, copy=True) # 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
if is_array_api_compliant:
# Convert to numpy as xp.cumsum and xp.searchsorted are not
# part of the Array API standard yet:
#
# https://github.com/data-apis/array-api/issues/597
# https://github.com/data-apis/array-api/issues/688
#
# Furthermore, it's not always safe to call them for namespaces
# that already implement them: for instance as
# cupy.searchsorted does not accept a float as second argument.
explained_variance_ratio_np = _convert_to_numpy(
explained_variance_ratio_, xp=xp
)
else:
explained_variance_ratio_np = explained_variance_ratio_
ratio_cumsum = stable_cumsum(explained_variance_ratio_np)
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_ = xp.mean(explained_variance_[n_components:])
else:
self.noise_variance_ = 0.0
self.n_samples_ = n_samples
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.
"""
xp, _ = get_namespace(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 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
total_var = None
if issparse(X):
self.mean_, var = mean_variance_axis(X, axis=0)
total_var = var.sum() * n_samples / (n_samples - 1) # ddof=1
X = _implicit_column_offset(X, self.mean_)
else:
self.mean_ = xp.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_oversamples=self.n_oversamples,
n_iter=self.iterated_power,
power_iteration_normalizer=self.power_iteration_normalizer,
flip_sign=True,
random_state=random_state,
)
self.n_samples_ = n_samples
self.components_ = Vt
self.n_components_ = n_components
# Get variance explained by singular values
self.explained_variance_ = (S**2) / (n_samples - 1)
# Workaround in-place variance calculation since at the time numpy
# did not have a way to calculate variance in-place.
#
# TODO: update this code to either:
# * Use the array-api variance calculation, unless memory usage suffers
# * Update sklearn.utils.extmath._incremental_mean_and_var to support array-api
# See: https://github.com/scikit-learn/scikit-learn/pull/18689#discussion_r1335540991
if total_var is None:
N = X.shape[0] - 1
X **= 2
total_var = xp.sum(X) / N
self.explained_variance_ratio_ = self.explained_variance_ / total_var
self.singular_values_ = xp.asarray(S, copy=True) # Store the singular values.
if self.n_components_ < min(n_features, n_samples):
self.noise_variance_ = total_var - xp.sum(self.explained_variance_)
self.noise_variance_ /= min(n_features, n_samples) - n_components
else:
self.noise_variance_ = 0.0
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)
xp, _ = get_namespace(X)
X = self._validate_data(X, dtype=[xp.float64, xp.float32], reset=False)
Xr = X - self.mean_
n_features = X.shape[1]
precision = self.get_precision()
log_like = -0.5 * xp.sum(Xr * (Xr @ precision), axis=1)
log_like -= 0.5 * (n_features * log(2.0 * 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
Ignored.
Returns
-------
ll : float
Average log-likelihood of the samples under the current model.
"""
xp, _ = get_namespace(X)
return float(xp.mean(self.score_samples(X)))
def _more_tags(self):
return {"preserves_dtype": [np.float64, np.float32], "array_api_support": True}