new_contents: """
  ==========
  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
  # License: BSD
  
  import numpy as np
  import pylab as pl
  
  from scikits.learn.pca import PCA, KernelPCA
  
  np.random.seed(0)
  
  def genenerate_rings(n_samples=200):
      x_red = np.random.random((n_samples,)) * 2 - 1
      signs_red = np.sign(np.random.random(x_red.shape) - 0.5)
      y_red = np.sqrt(np.abs(x_red ** 2 - 1)) * signs_red
  
      x_blue = np.random.random((n_samples,)) * 6 - 3
      signs_blue = np.sign(np.random.random(x_blue.shape) - 0.5)
      y_blue = np.sqrt(np.abs(x_blue ** 2 - 9)) * signs_blue
  
      return np.hstack(([x_red, y_red], [x_blue, y_blue])).T
  
  def generate_clusters(n_samples=200):
      mean1 = np.array([0, 2])
      mean2 = np.array([2, 0])
      cov = np.array([[2.0, 1.0], [1.0, 2.0]])
      X_red = np.random.multivariate_normal(mean1, cov, n_samples)
      X_blue = np.random.multivariate_normal(mean2, cov, n_samples)
      return np.vstack((X_red, X_blue))
  
  X = genenerate_rings()
  #X = generate_clusters()
  
  kpca = KernelPCA(kernel="rbf", fit_inverse_transform=True)
  X_kpca = kpca.fit_transform(X)
  X_back = kpca.inverse_transform(X_kpca)
  pca = PCA()
  X_pca = pca.fit_transform(X)
  
  pl.subplot(4, 1, 1)
  pl.title("Original space")
  pl.plot(X[:200, 0], X[:200, 1], "ro")
  pl.plot(X[200:, 0], X[200:, 1], "bo")
  pl.xlabel("$x_1$")
  pl.ylabel("$x_2$")
  
  X1, X2 = np.meshgrid(np.linspace(-6,6,50), np.linspace(-6,6,50))
  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)
  pl.contour(X1, X2, Z_grid, colors='grey', linewidths=1, origin='lower')
  
  pl.subplot(4, 1, 2)
  pl.plot(X_kpca[:200, 0], X_pca[:200, 1], "ro")
  pl.plot(X_pca[200:, 0], X_pca[200:, 1], "bo")
  pl.title("Projection by PCA")
  pl.xlabel("1st principal component")
  pl.ylabel("2nd component")
  
  pl.subplot(4, 1, 3)
  pl.plot(X_kpca[:200, 0], X_kpca[:200, 1], "ro")
  pl.plot(X_kpca[200:, 0], X_kpca[200:, 1], "bo")
  pl.title("Projection by KPCA")
  pl.xlabel("1st principal component in space induced by $\phi$")
  pl.ylabel("2nd component")
  
  pl.subplot(4, 1, 4)
  pl.plot(X_back[:200, 0], X_back[:200, 1], "ro")
  pl.plot(X_back[200:, 0], X_back[200:, 1], "bo")
  pl.title("Original space after inverse transform")
  pl.xlabel("$x_1$")
  pl.ylabel("$x_2$")
  
  pl.subplots_adjust(hspace=1.5)
  
  pl.show()
  
subject: Add kernel PCA example.