.. _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.



.. image:: /auto_examples/decomposition/images/sphx_glr_plot_kernel_pca_001.png
    :align: center





.. 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.plot(X[reds, 0], X[reds, 1], "ro")
    plt.plot(X[blues, 0], X[blues, 1], "bo")
    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.plot(X_pca[reds, 0], X_pca[reds, 1], "ro")
    plt.plot(X_pca[blues, 0], X_pca[blues, 1], "bo")
    plt.title("Projection by PCA")
    plt.xlabel("1st principal component")
    plt.ylabel("2nd component")

    plt.subplot(2, 2, 3, aspect='equal')
    plt.plot(X_kpca[reds, 0], X_kpca[reds, 1], "ro")
    plt.plot(X_kpca[blues, 0], X_kpca[blues, 1], "bo")
    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.plot(X_back[reds, 0], X_back[reds, 1], "ro")
    plt.plot(X_back[blues, 0], X_back[blues, 1], "bo")
    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 0.499 seconds)



.. container:: sphx-glr-download

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


.. container:: sphx-glr-download

    **Download IPython notebook:** :download:`plot_kernel_pca.ipynb <plot_kernel_pca.ipynb>`