content: """
  ====================
  Theil-Sen Regression
  ====================
  
  Computes a Theil-Sen Regression on a synthetic dataset.
  
  See :ref:`theil_sen_regression` for more information on the regressor.
  
  Compared to the OLS (ordinary least squares) estimator, the Theil-Sen
  estimator is robust against outliers. It has a breakdown point of about 29.3%
  in case of a simple linear regression which means that it can tolerate
  arbitrary corrupted data (outliers) of up to 29.3% in the two-dimensional
  case.
  
  The estimation of the model is done by calculating the slopes and intercepts
  of a subpopulation of all possible combinations of p subsample points. If an
  intercept is fitted, p must be greater than or equal to n_features + 1. The
  final slope and intercept is then defined as the spatial median of these
  slopes and intercepts.
  
  In certain cases Theil-Sen performs better than :ref:`RANSAC
  <ransac_regression>` which is also a robust method. This is illustrated in the
  second example below where outliers with respect to the x-axis perturb RANSAC.
  Tuning the ``residual_threshold`` parameter of RANSAC remedies this but in
  general a priori knowledge about the data and the nature of the outliers is
  needed.
  Due to the computational complexity of Theil-Sen it is recommended to use it
  only for small problems in terms of number of samples and features. For larger
  problems the ``max_subpopulation`` parameter restricts the magnitude of all
  possible combinations of p subsample points to a randomly chosen subset and
  therefore also limits the runtime. Therefore, Theil-Sen is applicable to larger
  problems with the drawback of losing some of its mathematical properties since
  it then works on a random subset.
  """
  
  # Author: Florian Wilhelm -- <florian.wilhelm@gmail.com>
  # License: BSD 3 clause
  
  import time
  import numpy as np
  import matplotlib.pyplot as plt
  from sklearn.linear_model import LinearRegression, TheilSenRegressor
  from sklearn.linear_model import RANSACRegressor
  
  print(__doc__)
  
  estimators = [('OLS', LinearRegression()),
                ('Theil-Sen', TheilSenRegressor(random_state=42)),
                ('RANSAC', RANSACRegressor(random_state=42)), ]
  
  ##############################################################################
  # Outliers only in the y direction
  
  np.random.seed(0)
  n_samples = 200
  # Linear model y = 3*x + N(2, 0.1**2)
  x = np.random.randn(n_samples)
  w = 3.
  c = 2.
  noise = 0.1 * np.random.randn(n_samples)
  y = w * x + c + noise
  # 10% outliers
  y[-20:] += -20 * x[-20:]
  X = x[:, np.newaxis]
  
  plt.plot(x, y, 'k+', mew=2, ms=8)
  line_x = np.array([-3, 3])
  for name, estimator in estimators:
      t0 = time.time()
      estimator.fit(X, y)
      elapsed_time = time.time() - t0
      y_pred = estimator.predict(line_x.reshape(2, 1))
      plt.plot(line_x, y_pred,
               label='%s (fit time: %.2fs)' % (name, elapsed_time))
  
  plt.axis('tight')
  plt.legend(loc='upper left')
  
  
  ##############################################################################
  # Outliers in the X direction
  
  np.random.seed(0)
  # Linear model y = 3*x + N(2, 0.1**2)
  x = np.random.randn(n_samples)
  noise = 0.1 * np.random.randn(n_samples)
  y = 3 * x + 2 + noise
  # 10% outliers
  x[-20:] = 9.9
  y[-20:] += 22
  X = x[:, np.newaxis]
  
  plt.figure()
  plt.plot(x, y, 'k+', mew=2, ms=8)
  
  line_x = np.array([-3, 10])
  for name, estimator in estimators:
      t0 = time.time()
      estimator.fit(X, y)
      elapsed_time = time.time() - t0
      y_pred = estimator.predict(line_x.reshape(2, 1))
      plt.plot(line_x, y_pred,
               label='%s (fit time: %.2fs)' % (name, elapsed_time))
  
  plt.axis('tight')
  plt.legend(loc='upper left')
  plt.show()
  