Principal Component Analysis (PCA)

Linear dimensionality reduction using Singular Value Decomposition of the data to project it to a lower dimensional space.

Documentation

Attributes

components_

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_

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_

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.

mean_

Per-feature empirical mean, estimated from the training set.

Equal to X.mean(axis=0).

n_components_

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.

noise_variance_

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.

Definition

Output ports

model model

Model

Configuration

N. of iteratins (for randomized solver) (iterated_power)

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

Added in version 0.18.0.

Number of components to keep (n_components)

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

Solver (svd_solver)

“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.

Tolerance for singular values (tol)

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

Added in version 0.18.0.

Whiten (whiten)

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.

Implementation

class node_decomposition.PrincipalComponentAnalysis