sklearn.decomposition.PCA

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
TruncatedSVD for an alternative with sparse data.

Read more in the User Guide.

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 500×500 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.

New in version 0.18.0.

tol

float, default=0.0

Tolerance for singular values computed by svd_solver == ‘arpack’.
Must be of range [0.0, infinity).

New in version 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).

New in version 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
randomized_svd for more details.

New in version 1.1.

power_iteration_normalizer

{‘auto’, ‘QR’, ‘LU’, ‘none’}, default=’auto’

Power iteration normalizer for randomized SVD solver.
Not used by ARPACK. See randomized_svd
for more details.

New in version 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 Glossary.

New in version 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 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.

New in version 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.

New in version 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.

n_features_in_

int

Number of features seen during fit.

New in version 0.24.

feature_names_in_

ndarray of shape (

n_features_in_

,)

Names of features seen during fit. Defined only when X
has feature names that are all strings.

New in version 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

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.
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.
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.

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...]

Methods

fit(X[, y])

Fit the model with X.

fit_transform(X[, y])

Fit the model with X and apply the dimensionality reduction on X.

get_covariance()

Compute data covariance with the generative model.

get_feature_names_out([input_features])

Get output feature names for transformation.

get_params([deep])

Get parameters for this estimator.

get_precision()

Compute data precision matrix with the generative model.

inverse_transform(X)

Transform data back to its original space.

score(X[, y])

Return the average log-likelihood of all samples.

score_samples(X)

Return the log-likelihood of each sample.

set_params(**params)

Set the parameters of this estimator.

transform(X)

Apply dimensionality reduction to X.

fit

(

X

,

y

=

None

)

[source]

¶

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

Ignored.

Returns

:

self

object

Returns the instance itself.

fit_transform

(

X

,

y

=

None

)

[source]

¶

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

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’.

get_covariance

(

)

[source]

¶

Compute data covariance with the generative model.

cov = components_.T * S**2 * components_ + sigma2 * eye(n_features)
where S**2 contains the explained variances, and sigma2 contains the
noise variances.

Returns

:

cov

array of shape=(n_features, n_features)

Estimated covariance of data.

get_feature_names_out

(

input_features

=

None

)

[source]

¶

Get output feature names for transformation.

Parameters

:

input_features

array-like of str or None, default=None

Only used to validate feature names with the names seen in fit.

Returns

:

feature_names_out

ndarray of str objects

Transformed feature names.

get_params

(

deep

=

True

)

[source]

¶

Get parameters for this estimator.

Parameters

:

deep

bool, default=True

If True, will return the parameters for this estimator and
contained subobjects that are estimators.

Returns

:

params

dict

Parameter names mapped to their values.

get_precision

(

)

[source]

¶

Compute data precision matrix with the generative model.

Equals the inverse of the covariance but computed with
the matrix inversion lemma for efficiency.

Returns

:

precision

array, shape=(n_features, n_features)

Estimated precision of data.

inverse_transform

(

X

)

[source]

¶

Transform data back to its original space.

In other words, return an input X_original whose transform would be X.

Parameters

:

X

array-like of shape (n_samples, n_components)

New data, where n_samples is the number of samples
and n_components is the number of components.

Returns

:

X_original array-like of shape (n_samples, n_features)

Original data, where n_samples is the number of samples
and n_features is the number of features.

Notes

If whitening is enabled, inverse_transform will compute the
exact inverse operation, which includes reversing whitening.

score

(

X

,

y

=

None

)

[source]

¶

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.

score_samples

(

X

)

[source]

¶

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.

set_params

(

**

params

)

[source]

¶

Set the parameters of this estimator.

The method works on simple estimators as well as on nested objects
(such as Pipeline). The latter have
parameters of the form <component>__<parameter> so that it’s
possible to update each component of a nested object.

Parameters

:

**params

dict

Estimator parameters.

Returns

:

self

estimator instance

Estimator instance.

transform

(

X

)

[source]

¶

Apply dimensionality reduction to X.

X is projected on the first principal components previously extracted
from a training set.

Parameters

:

X

array-like of shape (n_samples, n_features)

New data, where n_samples is the number of samples
and n_features is the number of features.

Returns

:

X_new

array-like of shape (n_samples, n_components)

Projection of X in the first principal components, where n_samples
is the number of samples and n_components is the number of the components.