sklearn.decomposition .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.
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.
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 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
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.
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.
run exact full SVD calling the standard LAPACK solver via scipy.linalg.svd and select the components by postprocessing
run SVD truncated to n_components calling ARPACK solver via scipy.sparse.linalg.svds . It requires strictly 0 < n_components < min(X.shape)
run randomized SVD by the method of Halko et al.
Tolerance for singular values computed by svd_solver == ‘arpack’. Must be of range [0.0, infinity).
Number of iterations for the power method computed by svd_solver == ‘randomized’. Must be of range [0, infinity).
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.
Power iteration normalizer for randomized SVD solver. Not used by ARPACK. See randomized_svd for more details.
Used when the ‘arpack’ or ‘randomized’ solvers are used. Pass an int for reproducible results across multiple function calls. See Glossary .
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.
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.
Per-feature empirical mean, estimated from the training set.
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 .
Names of features seen during fit . Defined only when X has feature names that are all strings.
Kernel Principal Component Analysis.
Sparse Principal Component Analysis.
Dimensionality reduction using truncated SVD.
Incremental Principal Component Analysis.
For svd_solver == ‘arpack’, refer to scipy.sparse.linalg.svds .
>>> 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 the model with X and apply the dimensionality reduction on X.
Compute data covariance with the generative model.
Get output feature names for transformation.
Get metadata routing of this object.
Get parameters for this estimator.
Compute data precision matrix with the generative model.
Transform data back to its original space.
Return the average log-likelihood of all samples.
Return the log-likelihood of each sample.
Set the parameters of this estimator.
Apply dimensionality reduction to 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.
Returns : self object
Returns the instance itself.
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.
Returns : X_new ndarray of shape (n_samples, n_components)
This method returns a Fortran-ordered array. To convert it to a C-ordered array, use ‘np.ascontiguousarray’.
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.
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_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 metadata routing of this object.
Please check User Guide on how the routing mechanism works.
Returns : routing MetadataRequest
A MetadataRequest encapsulating routing information.
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.
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.
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.
If whitening is enabled, inverse_transform will compute the exact inverse operation, which includes reversing whitening.
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)
Returns : ll float
Average log-likelihood of the samples under the current model.
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)
Returns : ll ndarray of shape (n_samples,)
Log-likelihood of each sample under the current model.
See Introducing the set_output API for an example on how to use the API.
Parameters : transform , default=None
Configure output of transform and fit_transform .
- «default» : Default output format of a transformer
- «pandas» : DataFrame output
- None : Transform configuration is unchanged
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 __ so that it’s possible to update each component of a nested object.
Parameters : **params dict
Returns : self estimator instance
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.