content: """
  =============================
  Ledoit-Wolf vs OAS estimation
  =============================
  
  The usual covariance maximum likelihood estimate can be regularized
  using shrinkage. Ledoit and Wolf proposed a close formula to compute
  the asymptotical optimal shrinkage parameter (minimizing a MSE
  criterion), yielding the Ledoit-Wolf covariance estimate.
  
  Chen et al. proposed an improvement of the Ledoit-Wolf shrinkage
  parameter, the OAS coefficient, whose convergence is significantly
  better under the assumption that the data are gaussian.
  
  This example, inspired from Chen's publication [1], shows a comparison
  of the estimated MSE of the LW and OAS methods, using gaussian
  distributed data.
  
  [1] "Shrinkage Algorithms for MMSE Covariance Estimation"
  Chen et al., IEEE Trans. on Sign. Proc., Volume 58, Issue 10, October 2010.
  
  """
  print __doc__
  
  import numpy as np
  import pylab as pl
  from scipy.linalg import toeplitz, cholesky
  
  from sklearn.covariance import LedoitWolf, OAS
  
  np.random.seed(0)
  ###############################################################################
  n_features = 100
  # simulation covariance matrix (AR(1) process)
  r = 0.1
  real_cov = toeplitz(r ** np.arange(n_features))
  coloring_matrix = cholesky(real_cov)
  
  n_samples_range = np.arange(6, 31, 1)
  repeat = 100
  lw_mse = np.zeros((n_samples_range.size, repeat))
  oa_mse = np.zeros((n_samples_range.size, repeat))
  lw_shrinkage = np.zeros((n_samples_range.size, repeat))
  oa_shrinkage = np.zeros((n_samples_range.size, repeat))
  for i, n_samples in enumerate(n_samples_range):
      for j in range(repeat):
          X = np.dot(
              np.random.normal(size=(n_samples, n_features)), coloring_matrix.T)
  
          lw = LedoitWolf(store_precision=False, assume_centered=True)
          lw.fit(X)
          lw_mse[i, j] = lw.error_norm(real_cov, scaling=False)
          lw_shrinkage[i, j] = lw.shrinkage_
  
          oa = OAS(store_precision=False, assume_centered=True)
          oa.fit(X)
          oa_mse[i, j] = oa.error_norm(real_cov, scaling=False)
          oa_shrinkage[i, j] = oa.shrinkage_
  
  # plot MSE
  pl.subplot(2, 1, 1)
  pl.errorbar(n_samples_range, lw_mse.mean(1), yerr=lw_mse.std(1),
              label='Ledoit-Wolf', color='g')
  pl.errorbar(n_samples_range, oa_mse.mean(1), yerr=oa_mse.std(1),
              label='OAS', color='r')
  pl.ylabel("Squared error")
  pl.legend(loc="upper right")
  pl.title("Comparison of covariance estimators")
  pl.xlim(5, 31)
  
  # plot shrinkage coefficient
  pl.subplot(2, 1, 2)
  pl.errorbar(n_samples_range, lw_shrinkage.mean(1), yerr=lw_shrinkage.std(1),
              label='Ledoit-Wolf', color='g')
  pl.errorbar(n_samples_range, oa_shrinkage.mean(1), yerr=oa_shrinkage.std(1),
              label='OAS', color='r')
  pl.xlabel("n_samples")
  pl.ylabel("Shrinkage")
  pl.legend(loc="lower right")
  pl.ylim(pl.ylim()[0], 1. + (pl.ylim()[1] - pl.ylim()[0]) / 10.)
  pl.xlim(5, 31)
  
  pl.show()
  