.. _`Principal Component Analysis (PCA)`:

.. _`org.sysess.sympathy.machinelearning.pca`:

Principal Component Analysis (PCA)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. image:: PCA.svg
   :width: 48


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

:Configuration:


  - *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
    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
    n_components cannot be equal to n_features for svd_solver == 'arpack'.


  - *svd_solver*

    auto :
        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.
    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 < X.shape
    randomized :
        run randomized SVD by the method of Halko et al.

    .. versionadded:: 0.18.0


  - *tol*

    Tolerance for singular values computed by svd_solver == 'arpack'.

    .. versionadded:: 0.18.0


  - *iterated_power*

    Number of iterations for the power method computed by
    svd_solver == 'randomized'.

    .. versionadded:: 0.18.0


  - *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.




:Attributes:


  - *components_*

    Principal axes in feature space, representing the directions of
    maximum variance in the data. The components are sorted by
    ``explained_variance_``.


  - *explained_variance_*

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

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

    .. versionadded:: 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 explained variances 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 n_features 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
    computed 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.




:Inputs:


:Outputs:
    **model** : model
        Model

*Ports*:

    **Outputs**:

        :model: model

            Model

*Configuration*:

    **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
        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
        n_components cannot be equal to n_features for svd_solver == 'arpack'.
    **svd_solver**
        auto :
            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.
        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 < X.shape
        randomized :
            run randomized SVD by the method of Halko et al.

        .. versionadded:: 0.18.0
    **tol**
        Tolerance for singular values computed by svd_solver == 'arpack'.

        .. versionadded:: 0.18.0
    **iterated_power**
        Number of iterations for the power method computed by
        svd_solver == 'randomized'.

        .. versionadded:: 0.18.0
    **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.

.. automodule:: node_decomposition

.. class:: PrincipalComponentAnalysis