PCA#

class sklearn.decomposition.PCA(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)[source]#

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.

With sparse inputs, the ARPACK implementation of the truncated SVD can be used (i.e. through scipy.sparse.linalg.svds). Alternatively, one may consider TruncatedSVD where the data are not centered.

Notice that this class only supports sparse inputs for some solvers such as “arpack” and “covariance_eigh”. See TruncatedSVD for an alternative with sparse data.

For a usage example, see Principal Component Analysis (PCA) on Iris Dataset

Read more in the User Guide.

Parameters:
n_componentsint, 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
copybool, 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.

whitenbool, 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’, ‘covariance_eigh’, ‘arpack’, ‘randomized’}, default=’auto’
“auto” :

The solver is selected by a default ‘auto’ policy is based on X.shape and n_components: if the input data has fewer than 1000 features and more than 10 times as many samples, then the “covariance_eigh” solver is used. Otherwise, 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 selected. 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

“covariance_eigh” :

Precompute the covariance matrix (on centered data), run a classical eigenvalue decomposition on the covariance matrix typically using LAPACK and select the components by postprocessing. This solver is very efficient for n_samples >> n_features and small n_features. It is, however, not tractable otherwise for large n_features (large memory footprint required to materialize the covariance matrix). Also note that compared to the “full” solver, this solver effectively doubles the condition number and is therefore less numerical stable (e.g. on input data with a large range of singular values).

“arpack” :

Run SVD truncated to n_components calling ARPACK solver via scipy.sparse.linalg.svds. It requires strictly 0 < n_components < min(X.shape)

“randomized” :

Run randomized SVD by the method of Halko et al.

Added in version 0.18.0.

Changed in version 1.5: Added the ‘covariance_eigh’ solver.

tolfloat, default=0.0

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

Added in version 0.18.0.

iterated_powerint or ‘auto’, default=’auto’

Number of iterations for the power method computed by svd_solver == ‘randomized’. Must be of range [0, infinity).

Added in version 0.18.0.

n_oversamplesint, 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.

Added 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.

Added in version 1.1.

random_stateint, 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.

Added 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 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.

Added 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.

Added 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_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.

Added 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.

Added 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...]
fit(X, y=None)[source]#

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.

yIgnored

Ignored.

Returns:
selfobject

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, 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.

yIgnored

Ignored.

Returns:
X_newndarray 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:
covarray of shape=(n_features, n_features)

Estimated covariance of data.

get_feature_names_out(input_features=None)[source]#

Get output feature names for transformation.

The feature names out will prefixed by the lowercased class name. For example, if the transformer outputs 3 features, then the feature names out are: ["class_name0", "class_name1", "class_name2"].

Parameters:
input_featuresarray-like of str or None, default=None

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

Returns:
feature_names_outndarray of str objects

Transformed feature names.

get_metadata_routing()[source]#

Get metadata routing of this object.

Please check User Guide on how the routing mechanism works.

Returns:
routingMetadataRequest

A MetadataRequest encapsulating routing information.

get_params(deep=True)[source]#

Get parameters for this estimator.

Parameters:
deepbool, default=True

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

Returns:
paramsdict

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:
precisionarray, 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:
Xarray-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:
Xarray-like of shape (n_samples, n_features)

The data.

yIgnored

Ignored.

Returns:
llfloat

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:
Xarray-like of shape (n_samples, n_features)

The data.

Returns:
llndarray of shape (n_samples,)

Log-likelihood of each sample under the current model.

set_output(*, transform=None)[source]#

Set output container.

See Introducing the set_output API for an example on how to use the API.

Parameters:
transform{“default”, “pandas”, “polars”}, default=None

Configure output of transform and fit_transform.

  • "default": Default output format of a transformer

  • "pandas": DataFrame output

  • "polars": Polars output

  • None: Transform configuration is unchanged

Added in version 1.4: "polars" option was added.

Returns:
selfestimator instance

Estimator instance.

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:
**paramsdict

Estimator parameters.

Returns:
selfestimator 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, sparse matrix} 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_newarray-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.