This example shows that Kernel PCA is able to find a projection of the data that makes data linearly separable.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 | print(__doc__)# Authors: Mathieu Blondel# Andreas Mueller# License: BSD 3 clauseimport numpy as npimport matplotlib.pyplot as pltfrom sklearn.decomposition import PCA, KernelPCAfrom sklearn.datasets import make_circlesnp.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 resultsplt.figure()plt.subplot(2, 2, 1, aspect='equal')plt.title("Original space")reds = y == 0blues = y == 1plt.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 0.494 seconds)
Download Python source code:
plot_kernel_pca.py
Download IPython notebook:
plot_kernel_pca.ipynb
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