content: #!/usr/bin/env python
  """
  =============================================
  Joint feature selection with multi-task Lasso
  =============================================
  
  The multi-task lasso allows to fit multiple regression problems
  jointly enforcing the selected features to be the same across
  tasks. This example simulates sequential measurements, each task
  is a time instant, and the relevant features vary in amplitude
  over time while being the same. The multi-task lasso imposes that
  features that are selected at one time point are select for all time
  point. This makes feature selection by the Lasso more stable.
  
  """
  print(__doc__)
  
  # Author: Alexandre Gramfort <alexandre.gramfort@inria.fr>
  # License: BSD 3 clause
  
  import matplotlib.pyplot as plt
  import numpy as np
  
  from sklearn.linear_model import MultiTaskLasso, Lasso
  
  rng = np.random.RandomState(42)
  
  # Generate some 2D coefficients with sine waves with random frequency and phase
  n_samples, n_features, n_tasks = 100, 30, 40
  n_relevant_features = 5
  coef = np.zeros((n_tasks, n_features))
  times = np.linspace(0, 2 * np.pi, n_tasks)
  for k in range(n_relevant_features):
      coef[:, k] = np.sin((1. + rng.randn(1)) * times + 3 * rng.randn(1))
  
  X = rng.randn(n_samples, n_features)
  Y = np.dot(X, coef.T) + rng.randn(n_samples, n_tasks)
  
  coef_lasso_ = np.array([Lasso(alpha=0.5).fit(X, y).coef_ for y in Y.T])
  coef_multi_task_lasso_ = MultiTaskLasso(alpha=1.).fit(X, Y).coef_
  
  ###############################################################################
  # Plot support and time series
  fig = plt.figure(figsize=(8, 5))
  plt.subplot(1, 2, 1)
  plt.spy(coef_lasso_)
  plt.xlabel('Feature')
  plt.ylabel('Time (or Task)')
  plt.text(10, 5, 'Lasso')
  plt.subplot(1, 2, 2)
  plt.spy(coef_multi_task_lasso_)
  plt.xlabel('Feature')
  plt.ylabel('Time (or Task)')
  plt.text(10, 5, 'MultiTaskLasso')
  fig.suptitle('Coefficient non-zero location')
  
  feature_to_plot = 0
  plt.figure()
  lw = 2
  plt.plot(coef[:, feature_to_plot], color='seagreen', linewidth=lw,
           label='Ground truth')
  plt.plot(coef_lasso_[:, feature_to_plot], color='cornflowerblue', linewidth=lw,
           label='Lasso')
  plt.plot(coef_multi_task_lasso_[:, feature_to_plot], color='gold', linewidth=lw,
           label='MultiTaskLasso')
  plt.legend(loc='upper center')
  plt.axis('tight')
  plt.ylim([-1.1, 1.1])
  plt.show()
  