content: """
  ==========
  Kernel PCA
  ==========
  
  This example shows that Kernel PCA is able to find a projection of the data
  that makes data linearly separable.
  """
  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()
  