content: """
  ===================================
  Compare cross decomposition methods
  ===================================
  
  Simple usage of various cross decomposition algorithms:
  - PLSCanonical
  - PLSRegression, with multivariate response, a.k.a. PLS2
  - PLSRegression, with univariate response, a.k.a. PLS1
  - CCA
  
  Given 2 multivariate covarying two-dimensional datasets, X, and Y,
  PLS extracts the 'directions of covariance', i.e. the components of each
  datasets that explain the most shared variance between both datasets.
  This is apparent on the **scatterplot matrix** display: components 1 in
  dataset X and dataset Y are maximally correlated (points lie around the
  first diagonal). This is also true for components 2 in both dataset,
  however, the correlation across datasets for different components is
  weak: the point cloud is very spherical.
  """
  print(__doc__)
  
  import numpy as np
  import matplotlib.pyplot as plt
  from sklearn.cross_decomposition import PLSCanonical, PLSRegression, CCA
  
  ###############################################################################
  # Dataset based latent variables model
  
  n = 500
  # 2 latents vars:
  l1 = np.random.normal(size=n)
  l2 = np.random.normal(size=n)
  
  latents = np.array([l1, l1, l2, l2]).T
  X = latents + np.random.normal(size=4 * n).reshape((n, 4))
  Y = latents + np.random.normal(size=4 * n).reshape((n, 4))
  
  X_train = X[:n / 2]
  Y_train = Y[:n / 2]
  X_test = X[n / 2:]
  Y_test = Y[n / 2:]
  
  print("Corr(X)")
  print(np.round(np.corrcoef(X.T), 2))
  print("Corr(Y)")
  print(np.round(np.corrcoef(Y.T), 2))
  
  ###############################################################################
  # Canonical (symmetric) PLS
  
  # Transform data
  # ~~~~~~~~~~~~~~
  plsca = PLSCanonical(n_components=2)
  plsca.fit(X_train, Y_train)
  X_train_r, Y_train_r = plsca.transform(X_train, Y_train)
  X_test_r, Y_test_r = plsca.transform(X_test, Y_test)
  
  # Scatter plot of scores
  # ~~~~~~~~~~~~~~~~~~~~~~
  # 1) On diagonal plot X vs Y scores on each components
  plt.figure(figsize=(12, 8))
  plt.subplot(221)
  plt.plot(X_train_r[:, 0], Y_train_r[:, 0], "ob", label="train")
  plt.plot(X_test_r[:, 0], Y_test_r[:, 0], "or", label="test")
  plt.xlabel("x scores")
  plt.ylabel("y scores")
  plt.title('Comp. 1: X vs Y (test corr = %.2f)' %
            np.corrcoef(X_test_r[:, 0], Y_test_r[:, 0])[0, 1])
  plt.xticks(())
  plt.yticks(())
  plt.legend(loc="best")
  
  plt.subplot(224)
  plt.plot(X_train_r[:, 1], Y_train_r[:, 1], "ob", label="train")
  plt.plot(X_test_r[:, 1], Y_test_r[:, 1], "or", label="test")
  plt.xlabel("x scores")
  plt.ylabel("y scores")
  plt.title('Comp. 2: X vs Y (test corr = %.2f)' %
            np.corrcoef(X_test_r[:, 1], Y_test_r[:, 1])[0, 1])
  plt.xticks(())
  plt.yticks(())
  plt.legend(loc="best")
  
  # 2) Off diagonal plot components 1 vs 2 for X and Y
  plt.subplot(222)
  plt.plot(X_train_r[:, 0], X_train_r[:, 1], "*b", label="train")
  plt.plot(X_test_r[:, 0], X_test_r[:, 1], "*r", label="test")
  plt.xlabel("X comp. 1")
  plt.ylabel("X comp. 2")
  plt.title('X comp. 1 vs X comp. 2 (test corr = %.2f)'
            % np.corrcoef(X_test_r[:, 0], X_test_r[:, 1])[0, 1])
  plt.legend(loc="best")
  plt.xticks(())
  plt.yticks(())
  
  plt.subplot(223)
  plt.plot(Y_train_r[:, 0], Y_train_r[:, 1], "*b", label="train")
  plt.plot(Y_test_r[:, 0], Y_test_r[:, 1], "*r", label="test")
  plt.xlabel("Y comp. 1")
  plt.ylabel("Y comp. 2")
  plt.title('Y comp. 1 vs Y comp. 2 , (test corr = %.2f)'
            % np.corrcoef(Y_test_r[:, 0], Y_test_r[:, 1])[0, 1])
  plt.legend(loc="best")
  plt.xticks(())
  plt.yticks(())
  plt.show()
  
  ###############################################################################
  # PLS regression, with multivariate response, a.k.a. PLS2
  
  n = 1000
  q = 3
  p = 10
  X = np.random.normal(size=n * p).reshape((n, p))
  B = np.array([[1, 2] + [0] * (p - 2)] * q).T
  # each Yj = 1*X1 + 2*X2 + noize
  Y = np.dot(X, B) + np.random.normal(size=n * q).reshape((n, q)) + 5
  
  pls2 = PLSRegression(n_components=3)
  pls2.fit(X, Y)
  print("True B (such that: Y = XB + Err)")
  print(B)
  # compare pls2.coef_ with B
  print("Estimated B")
  print(np.round(pls2.coef_, 1))
  pls2.predict(X)
  
  ###############################################################################
  # PLS regression, with univariate response, a.k.a. PLS1
  
  n = 1000
  p = 10
  X = np.random.normal(size=n * p).reshape((n, p))
  y = X[:, 0] + 2 * X[:, 1] + np.random.normal(size=n * 1) + 5
  pls1 = PLSRegression(n_components=3)
  pls1.fit(X, y)
  # note that the number of components exceeds 1 (the dimension of y)
  print("Estimated betas")
  print(np.round(pls1.coef_, 1))
  
  ###############################################################################
  # CCA (PLS mode B with symmetric deflation)
  
  cca = CCA(n_components=2)
  cca.fit(X_train, Y_train)
  X_train_r, Y_train_r = plsca.transform(X_train, Y_train)
  X_test_r, Y_test_r = plsca.transform(X_test, Y_test)
  