.. _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  .. container:: sphx-glr-download **Download IPython notebook:** :download:plot_kernel_pca.ipynb