content: """
  =============================================
  Comparison of kernel ridge regression and SVR
  =============================================
  
  Both kernel ridge regression (KRR) and SVR learn a non-linear function by
  employing the kernel trick, i.e., they learn a linear function in the space
  induced by the respective kernel which corresponds to a non-linear function in
  the original space. They differ in the loss functions (ridge versus
  epsilon-insensitive loss). In contrast to SVR, fitting a KRR can be done in
  closed-form and is typically faster for medium-sized datasets. On the other
  hand, the learned model is non-sparse and thus slower than SVR at
  prediction-time.
  
  This example illustrates both methods on an artificial dataset, which
  consists of a sinusoidal target function and strong noise added to every fifth
  datapoint. The first figure compares the learned model of KRR and SVR when both
  complexity/regularization and bandwidth of the RBF kernel are optimized using
  grid-search. The learned functions are very similar; however, fitting KRR is
  approx. seven times faster than fitting SVR (both with grid-search). However,
  prediction of 100000 target values is more than tree times faster with SVR
  since it has learned a sparse model using only approx. 1/3 of the 100 training
  datapoints as support vectors.
  
  The next figure compares the time for fitting and prediction of KRR and SVR for
  different sizes of the training set. Fitting KRR is faster than SVR for medium-
  sized training sets (less than 1000 samples); however, for larger training sets
  SVR scales better. With regard to prediction time, SVR is faster than
  KRR for all sizes of the training set because of the learned sparse
  solution. Note that the degree of sparsity and thus the prediction time depends
  on the parameters epsilon and C of the SVR.
  """
  
  # Authors: Jan Hendrik Metzen <jhm@informatik.uni-bremen.de>
  # License: BSD 3 clause
  
  
  from __future__ import division
  import time
  
  import numpy as np
  
  from sklearn.svm import SVR
  from sklearn.grid_search import GridSearchCV
  from sklearn.learning_curve import learning_curve
  from sklearn.kernel_ridge import KernelRidge
  import matplotlib.pyplot as plt
  
  rng = np.random.RandomState(0)
  
  #############################################################################
  # Generate sample data
  X = 5 * rng.rand(10000, 1)
  y = np.sin(X).ravel()
  
  # Add noise to targets
  y[::5] += 3 * (0.5 - rng.rand(X.shape[0]/5))
  
  X_plot = np.linspace(0, 5, 100000)[:, None]
  
  #############################################################################
  # Fit regression model
  train_size = 100
  svr = GridSearchCV(SVR(kernel='rbf', gamma=0.1), cv=5,
                     param_grid={"C": [1e0, 1e1, 1e2, 1e3],
                                 "gamma": np.logspace(-2, 2, 5)})
  
  kr = GridSearchCV(KernelRidge(kernel='rbf', gamma=0.1), cv=5,
                    param_grid={"alpha": [1e0, 0.1, 1e-2, 1e-3],
                                "gamma": np.logspace(-2, 2, 5)})
  
  t0 = time.time()
  svr.fit(X[:train_size], y[:train_size])
  svr_fit = time.time() - t0
  print("SVR complexity and bandwidth selected and model fitted in %.3f s"
        % svr_fit)
  
  t0 = time.time()
  kr.fit(X[:train_size], y[:train_size])
  kr_fit = time.time() - t0
  print("KRR complexity and bandwidth selected and model fitted in %.3f s"
        % kr_fit)
  
  sv_ratio = svr.best_estimator_.support_.shape[0] / train_size
  print("Support vector ratio: %.3f" % sv_ratio)
  
  t0 = time.time()
  y_svr = svr.predict(X_plot)
  svr_predict = time.time() - t0
  print("SVR prediction for %d inputs in %.3f s"
        % (X_plot.shape[0], svr_predict))
  
  t0 = time.time()
  y_kr = kr.predict(X_plot)
  kr_predict = time.time() - t0
  print("KRR prediction for %d inputs in %.3f s"
        % (X_plot.shape[0], kr_predict))
  
  
  #############################################################################
  # look at the results
  sv_ind = svr.best_estimator_.support_
  plt.scatter(X[sv_ind], y[sv_ind], c='r', s=50, label='SVR support vectors')
  plt.scatter(X[:100], y[:100], c='k', label='data')
  plt.hold('on')
  plt.plot(X_plot, y_svr, c='r',
           label='SVR (fit: %.3fs, predict: %.3fs)' % (svr_fit, svr_predict))
  plt.plot(X_plot, y_kr, c='g',
           label='KRR (fit: %.3fs, predict: %.3fs)' % (kr_fit, kr_predict))
  plt.xlabel('data')
  plt.ylabel('target')
  plt.title('SVR versus Kernel Ridge')
  plt.legend()
  
  # Visualize training and prediction time
  plt.figure()
  
  # Generate sample data
  X = 5 * rng.rand(10000, 1)
  y = np.sin(X).ravel()
  y[::5] += 3 * (0.5 - rng.rand(X.shape[0]/5))
  sizes = np.logspace(1, 4, 7)
  for name, estimator in {"KRR": KernelRidge(kernel='rbf', alpha=0.1,
                                             gamma=10),
                          "SVR": SVR(kernel='rbf', C=1e1, gamma=10)}.items():
      train_time = []
      test_time = []
      for train_test_size in sizes:
          t0 = time.time()
          estimator.fit(X[:train_test_size], y[:train_test_size])
          train_time.append(time.time() - t0)
  
          t0 = time.time()
          estimator.predict(X_plot[:1000])
          test_time.append(time.time() - t0)
  
      plt.plot(sizes, train_time, 'o-', color="r" if name == "SVR" else "g",
               label="%s (train)" % name)
      plt.plot(sizes, test_time, 'o--', color="r" if name == "SVR" else "g",
               label="%s (test)" % name)
  
  plt.xscale("log")
  plt.yscale("log")
  plt.xlabel("Train size")
  plt.ylabel("Time (seconds)")
  plt.title('Execution Time')
  plt.legend(loc="best")
  
  # Visualize learning curves
  plt.figure()
  
  svr = SVR(kernel='rbf', C=1e1, gamma=0.1)
  kr = KernelRidge(kernel='rbf', alpha=0.1, gamma=0.1)
  train_sizes, train_scores_svr, test_scores_svr = \
      learning_curve(svr, X[:100], y[:100], train_sizes=np.linspace(0.1, 1, 10),
                     scoring="mean_squared_error", cv=10)
  train_sizes_abs, train_scores_kr, test_scores_kr = \
      learning_curve(kr, X[:100], y[:100], train_sizes=np.linspace(0.1, 1, 10),
                     scoring="mean_squared_error", cv=10)
  
  plt.plot(train_sizes, test_scores_svr.mean(1), 'o-', color="r",
           label="SVR")
  plt.plot(train_sizes, test_scores_kr.mean(1), 'o-', color="g",
           label="KRR")
  plt.xlabel("Train size")
  plt.ylabel("Mean Squared Error")
  plt.title('Learning curves')
  plt.legend(loc="best")
  
  plt.show()
  