decomposition.PCA

class pai4sk.decomposition.PCA(n_components=None, copy=True, whiten=False, svd_solver='auto', tol=0.0, iterated_power='auto', random_state=None, use_gpu=True)

Principal component analysis (PCA)

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

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.

If cuml is installed and input data is cudf dataframe, then pai4sk will try to use the accelerated PCA algorithm from cuML. Otherwise, scikit-learn’s PCA algorithm will be used.

cuML in pai4sk is currently supported only | (a) with python 3.6 and | (b) without MPI. | If PCA from cuML is run, then the return values from the APIs will be cudf dataframe and cudf Series objects instead of the return types of scikit-learn API.

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, None or string) –

    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, optional (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 (string {'auto', 'full', 'arpack', 'randomized', 'cuml', 'jacobi'}) –
    auto :
    when cuml is not used, 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 cuml is used, then the default algorithm ‘full’ will be used when the svd_solver is ‘auto’ or ‘cuml’.
    full :
    run exact full SVD calling the standard LAPACK solver via scipy.linalg.svd and select the components by postprocessing
    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.

    New in version 0.18.0.

  • tol (float >= 0, optional (default .0)) –

    Tolerance for singular values computed by svd_solver == ‘arpack’.

    New in version 0.18.0.

  • iterated_power (int >= 0, or 'auto', (default 'auto')) –

    Number of iterations for the power method computed by svd_solver == ‘randomized’. Note : cuML for pai4sk only supports integer values for this parameter.

    New in version 0.18.0.

  • random_state (int, RandomState instance or None, optional (default None)) –

    If int, random_state is the seed used by the random number generator; If RandomState instance, random_state is the random number generator; If None, the random number generator is the RandomState instance used by np.random. Used when svd_solver == ‘arpack’ or ‘randomized’.

    New in version 0.18.0.

  • use_gpu (boolean, Default is True) – If True, cuML will use GPU 0. Applicable only for cuML.
Variables:
  • components (array of shape (n_components, n_features) or cudf dataframe) – Principal axes in feature space, representing the directions of maximum variance in the data. The components are sorted by explained_variance_.
  • explained_variance (array of shape (n_components,) or cudf Series) –

    The amount of variance explained by each of the selected components.

    Equal to n_components largest eigenvalues of the covariance matrix of X.

    New in version 0.18.

  • explained_variance_ratio (array of shape (n_components,) or cudf Series) –

    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 (array of shape (n_components,) or cudf Series) – 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.
  • mean (array, 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.
  • noise_variance (float or cudf Series) –

    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.

References

For n_components == ‘mle’, this class uses the method of 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. See http://www.miketipping.com/papers/met-mppca.pdf

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 pai4sk.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(copy=True, iterated_power='auto', n_components=2, random_state=None,
  svd_solver='auto', tol=0.0, whiten=False)
>>> print(pca.explained_variance_ratio_)  # doctest: +ELLIPSIS
[0.9924... 0.0075...]
>>> print(pca.singular_values_)  # doctest: +ELLIPSIS
[6.30061... 0.54980...]

>>> pca = PCA(n_components=2, svd_solver='full')
>>> pca.fit(X)                 # doctest: +ELLIPSIS +NORMALIZE_WHITESPACE
PCA(copy=True, iterated_power='auto', n_components=2, random_state=None,
  svd_solver='full', tol=0.0, whiten=False)
>>> print(pca.explained_variance_ratio_)  # doctest: +ELLIPSIS
[0.9924... 0.00755...]
>>> print(pca.singular_values_)  # doctest: +ELLIPSIS
[6.30061... 0.54980...]

>>> pca = PCA(n_components=1, svd_solver='arpack')
>>> pca.fit(X)
PCA(copy=True, iterated_power='auto', n_components=1, random_state=None,
  svd_solver='arpack', tol=0.0, whiten=False)
>>> print(pca.explained_variance_ratio_)  # doctest: +ELLIPSIS
[0.99244...]
>>> print(pca.singular_values_)  # doctest: +ELLIPSIS
[6.30061...]

See also

KernelPCA, SparsePCA, TruncatedSVD, IncrementalPCA

fit(X, y=None, _transform=True)

Fit the model with X.

Parameters:
  • X (array-like of shape (n_samples, n_features) or cudf dataframe) – Training data, where n_samples is the number of samples and n_features is the number of features.
  • y (Ignored) –
Returns:

self – Returns the instance itself. If PCA from cuML is run, then this fit method saves the computed values as cudf dataframes and cudf Series objects instead of the results’ types seen from scikit-learn’s fit method.
Return type:

object
fit_transform(X, y=None)

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

Parameters:
  • X (array-like of shape (n_samples, n_features) or cudf dataframe) – Training data, where n_samples is the number of samples and n_features is the number of features.
  • y (Ignored) –
Returns:

X_new – If PCA from cuML is run, then this method saves the computed values as cudf dataframes and cudf Series objects instead of the results’ types seen from scikit-learn’s fit_transform method.
Return type:

array-like of shape (n_samples, n_components) or cudf dataframe
inverse_transform(X)

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) or cudf dataframe) – New data, where n_samples is the number of samples and n_components is the number of components.
Returns:X_original – If PCA from cuML is run, then this method returns cudf dataframe instead of the results’ types seen from scikit-learn’s inverse_transform method.
Return type:array-like of shape (n_samples, n_features) or cudf dataframe

Notes

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

score(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, shape(n_samples, n_features)) – The data.
  • y (Ignored) –
Returns:

ll – Average log-likelihood of the samples under the current model
Return type:

float
transform(X)

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) or cudf dataframe) – New data, where n_samples is the number of samples and n_features is the number of features.
Returns:X_new – If PCA from cuML is run, then this method saves the computed values as cudf dataframe instead of the results’ types seen from scikit-learn’s transform method.
Return type:array-like of shape (n_samples, n_components) or cudf dataframe

Examples

>>> import numpy as np
>>> from pai4sk.decomposition import IncrementalPCA
>>> X = np.array([[-1, -1], [-2, -1], [-3, -2], [1, 1], [2, 1], [3, 2]])
>>> ipca = IncrementalPCA(n_components=2, batch_size=3)
>>> ipca.fit(X)
IncrementalPCA(batch_size=3, copy=True, n_components=2, whiten=False)
>>> ipca.transform(X) # doctest: +SKIP