.. note::
    :class: sphx-glr-download-link-note

    Click :ref:`here <sphx_glr_download_auto_examples_decomposition_plot_kernel_pca.py>` to download the full example code
.. rst-class:: sphx-glr-example-title

.. _sphx_glr_auto_examples_decomposition_plot_kernel_pca.py:


==========
Kernel PCA
==========

This example shows that Kernel PCA is able to find a projection of the data
that makes data linearly separable.



.. code-block:: python

    print(__doc__)

    # Authors: Mathieu Blondel
    #          Andreas Mueller
    # License: BSD 3 clause

    import numpy as np
    import matplotlib.pyplot as plt

    from sklearn.decomposition import PCA, KernelPCA
    from sklearn.datasets import make_circles

    np.random.seed(0)

    X, y = make_circles(n_samples=400, factor=.3, noise=.05)

    kpca = KernelPCA(kernel="rbf", fit_inverse_transform=True, gamma=10)
    X_kpca = kpca.fit_transform(X)
    X_back = kpca.inverse_transform(X_kpca)
    pca = PCA()
    X_pca = pca.fit_transform(X)

    # Plot results

    plt.figure()
    plt.subplot(2, 2, 1, aspect='equal')
    plt.title("Original space")
    reds = y == 0
    blues = y == 1

    plt.scatter(X[reds, 0], X[reds, 1], c="red",
                s=20, edgecolor='k')
    plt.scatter(X[blues, 0], X[blues, 1], c="blue",
                s=20, edgecolor='k')
    plt.xlabel("$x_1$")
    plt.ylabel("$x_2$")

    X1, X2 = np.meshgrid(np.linspace(-1.5, 1.5, 50), np.linspace(-1.5, 1.5, 50))
    X_grid = np.array([np.ravel(X1), np.ravel(X2)]).T
    # projection on the first principal component (in the phi space)
    Z_grid = kpca.transform(X_grid)[:, 0].reshape(X1.shape)
    plt.contour(X1, X2, Z_grid, colors='grey', linewidths=1, origin='lower')

    plt.subplot(2, 2, 2, aspect='equal')
    plt.scatter(X_pca[reds, 0], X_pca[reds, 1], c="red",
                s=20, edgecolor='k')
    plt.scatter(X_pca[blues, 0], X_pca[blues, 1], c="blue",
                s=20, edgecolor='k')
    plt.title("Projection by PCA")
    plt.xlabel("1st principal component")
    plt.ylabel("2nd component")

    plt.subplot(2, 2, 3, aspect='equal')
    plt.scatter(X_kpca[reds, 0], X_kpca[reds, 1], c="red",
                s=20, edgecolor='k')
    plt.scatter(X_kpca[blues, 0], X_kpca[blues, 1], c="blue",
                s=20, edgecolor='k')
    plt.title("Projection by KPCA")
    plt.xlabel("1st principal component in space induced by $\phi$")
    plt.ylabel("2nd component")

    plt.subplot(2, 2, 4, aspect='equal')
    plt.scatter(X_back[reds, 0], X_back[reds, 1], c="red",
                s=20, edgecolor='k')
    plt.scatter(X_back[blues, 0], X_back[blues, 1], c="blue",
                s=20, edgecolor='k')
    plt.title("Original space after inverse transform")
    plt.xlabel("$x_1$")
    plt.ylabel("$x_2$")

    plt.subplots_adjust(0.02, 0.10, 0.98, 0.94, 0.04, 0.35)

    plt.show()

**Total running time of the script:** ( 0 minutes  0.000 seconds)


.. _sphx_glr_download_auto_examples_decomposition_plot_kernel_pca.py:


.. only :: html

 .. container:: sphx-glr-footer
    :class: sphx-glr-footer-example



  .. container:: sphx-glr-download

     :download:`Download Python source code: plot_kernel_pca.py <plot_kernel_pca.py>`



  .. container:: sphx-glr-download

     :download:`Download Jupyter notebook: plot_kernel_pca.ipynb <plot_kernel_pca.ipynb>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.readthedocs.io>`_
