.. _`Kernel Principal Component Analysis (KPCA)`:

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

Kernel Principal Component Analysis (KPCA)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

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


Non-linear dimensionality reduction through the use of kernels

*Configuration*:


  - *n_components*

    Number of components. If None, all non-zero components are kept.


  - *kernel*

    Kernel. Default="linear".


  - *degree*

    Degree for poly kernels. Ignored by other kernels.


  - *gamma*

    Kernel coefficient for rbf, poly and sigmoid kernels. Ignored by other
    kernels.


  - *coef0*

    Independent term in poly and sigmoid kernels.
    Ignored by other kernels.


  - *alpha*

    Hyperparameter of the ridge regression that learns the
    inverse transform (when fit_inverse_transform=True).


  - *fit_inverse_transform*

    Learn the inverse transform for non-precomputed kernels.
    (i.e. learn to find the pre-image of a point)


  - *remove_zero_eig*

    If True, then all components with zero eigenvalues are removed, so
    that the number of components in the output may be < n_components
    (and sometimes even zero due to numerical instability).
    When n_components is None, this parameter is ignored and components
    with zero eigenvalues are removed regardless.


  - *eigen_solver*

    Select eigensolver to use. If n_components is much less than
    the number of training samples, arpack may be more efficient
    than the dense eigensolver.


  - *tol*

    Convergence tolerance for arpack.
    If 0, optimal value will be chosen by arpack.


  - *max_iter*

    Maximum number of iterations for arpack.
    If None, optimal value will be chosen by arpack.


  - *random_state*

    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 ``eigen_solver`` == 'arpack'.

    .. versionadded:: 0.18


  - *n_jobs*

    The number of parallel jobs to run.
    ``None`` means 1 unless in a :obj:`joblib.parallel_backend` context.
    ``-1`` means using all processors. See n_jobs
    for more details.

    .. versionadded:: 0.18




*Attributes*:


  - *lambdas_*

    Eigenvalues of the centered kernel matrix in decreasing order.
    If `n_components` and `remove_zero_eig` are not set,
    then all values are stored.


  - *alphas_*

    Eigenvectors of the centered kernel matrix. If `n_components` and
    `remove_zero_eig` are not set, then all components are stored.


  - *dual_coef_*

    Inverse transform matrix. Only available when
    ``fit_inverse_transform`` is True.


  - *X_transformed_fit_*

    Projection of the fitted data on the kernel principal components.
    Only available when ``fit_inverse_transform`` is True.


  - *X_fit_*

    The data used to fit the model. If `copy_X=False`, then `X_fit_` is
    a reference. This attribute is used for the calls to transform.




*Input ports*:


*Output ports*:
    **model** : model
        Model



**n_components** (n_components)
    Number of components. If None, all non-zero components are kept.
**kernel** (kernel)
    Kernel. Default="linear".
**degree** (degree)
    Degree for poly kernels. Ignored by other kernels.
**gamma** (gamma)
    Kernel coefficient for rbf, poly and sigmoid kernels. Ignored by other
    kernels.
**coef0** (coef0)
    Independent term in poly and sigmoid kernels.
    Ignored by other kernels.
**alpha** (alpha)
    Hyperparameter of the ridge regression that learns the
    inverse transform (when fit_inverse_transform=True).
**fit_inverse_transform** (fit_inverse_transform)
    Learn the inverse transform for non-precomputed kernels.
    (i.e. learn to find the pre-image of a point)
**remove_zero_eig** (remove_zero_eig)
    If True, then all components with zero eigenvalues are removed, so
    that the number of components in the output may be < n_components
    (and sometimes even zero due to numerical instability).
    When n_components is None, this parameter is ignored and components
    with zero eigenvalues are removed regardless.
**eigen_solver** (eigen_solver)
    Select eigensolver to use. If n_components is much less than
    the number of training samples, arpack may be more efficient
    than the dense eigensolver.
**tol** (tol)
    Convergence tolerance for arpack.
    If 0, optimal value will be chosen by arpack.
**max_iter** (max_iter)
    Maximum number of iterations for arpack.
    If None, optimal value will be chosen by arpack.
**random_state** (random_state)
    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 ``eigen_solver`` == 'arpack'.

    .. versionadded:: 0.18
**n_jobs** (n_jobs)
    The number of parallel jobs to run.
    ``None`` means 1 unless in a :obj:`joblib.parallel_backend` context.
    ``-1`` means using all processors. See n_jobs
    for more details.

    .. versionadded:: 0.18

.. automodule:: node_decomposition

.. class:: KernelPCA