text: """
  ==================================================================
  Principal Component Regression vs Partial Least Squares Regression
  ==================================================================
  
  This example compares `Principal Component Regression
  <https://en.wikipedia.org/wiki/Principal_component_regression>`_ (PCR) and
  `Partial Least Squares Regression
  <https://en.wikipedia.org/wiki/Partial_least_squares_regression>`_ (PLS) on a
  toy dataset. Our goal is to illustrate how PLS can outperform PCR when the
  target is strongly correlated with some directions in the data that have a
  low variance.
  
  PCR is a regressor composed of two steps: first,
  :class:`~sklearn.decomposition.PCA` is applied to the training data, possibly
  performing dimensionality reduction; then, a regressor (e.g. a linear
  regressor) is trained on the transformed samples. In
  :class:`~sklearn.decomposition.PCA`, the transformation is purely
  unsupervised, meaning that no information about the targets is used. As a
  result, PCR may perform poorly in some datasets where the target is strongly
  correlated with *directions* that have low variance. Indeed, the
  dimensionality reduction of PCA projects the data into a lower dimensional
  space where the variance of the projected data is greedily maximized along
  each axis. Despite them having the most predictive power on the target, the
  directions with a lower variance will be dropped, and the final regressor
  will not be able to leverage them.
  
  PLS is both a transformer and a regressor, and it is quite similar to PCR: it
  also applies a dimensionality reduction to the samples before applying a
  linear regressor to the transformed data. The main difference with PCR is
  that the PLS transformation is supervised. Therefore, as we will see in this
  example, it does not suffer from the issue we just mentioned.
  """
  
  print(__doc__)
  
  # %%
  # The data
  # --------
  #
  # We start by creating a simple dataset with two features. Before we even dive
  # into PCR and PLS, we fit a PCA estimator to display the two principal
  # components of this dataset, i.e. the two directions that explain the most
  # variance in the data.
  import numpy as np
  import matplotlib.pyplot as plt
  from sklearn.decomposition import PCA
  
  rng = np.random.RandomState(0)
  n_samples = 500
  cov = [[3, 3],
         [3, 4]]
  X = rng.multivariate_normal(mean=[0, 0], cov=cov, size=n_samples)
  pca = PCA(n_components=2).fit(X)
  
  
  plt.scatter(X[:, 0], X[:, 1], alpha=.3, label='samples')
  for i, (comp, var) in enumerate(zip(pca.components_, pca.explained_variance_)):
      comp = comp * var  # scale component by its variance explanation power
      plt.plot([0, comp[0]], [0, comp[1]], label=f"Component {i}", linewidth=5,
               color=f"C{i + 2}")
  plt.gca().set(aspect='equal',
                title="2-dimensional dataset with principal components",
                xlabel='first feature', ylabel='second feature')
  plt.legend()
  #plt.show()
  
  # %%
  # For the purpose of this example, we now define the target `y` such that it is
  # strongly correlated with a direction that has a small variance. To this end,
  # we will project `X` onto the second component, and add some noise to it.
  
  y = X.dot(pca.components_[1]) + rng.normal(size=n_samples) / 2
  
  fig, axes = plt.subplots(1, 2, figsize=(10, 3))
  
  axes[0].scatter(X.dot(pca.components_[0]), y, alpha=.3)
  axes[0].set(xlabel='Projected data onto first PCA component', ylabel='y')
  axes[1].scatter(X.dot(pca.components_[1]), y, alpha=.3)
  axes[1].set(xlabel='Projected data onto second PCA component', ylabel='y')
  plt.tight_layout()
  #plt.show()
  
  # %%
  # Projection on one component and predictive power
  # ------------------------------------------------
  #
  # We now create two regressors: PCR and PLS, and for our illustration purposes
  # we set the number of components to 1. Before feeding the data to the PCA step
  # of PCR, we first standardize it, as recommended by good practice. The PLS
  # estimator has built-in scaling capabilities.
  #
  # For both models, we plot the projected data onto the first component against
  # the target. In both cases, this projected data is what the regressors will
  # use as training data.
  from sklearn.model_selection import train_test_split
  from sklearn.pipeline import make_pipeline
  from sklearn.linear_model import LinearRegression
  from sklearn.preprocessing import StandardScaler
  from sklearn.decomposition import PCA
  from sklearn.cross_decomposition import PLSRegression
  
  X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=rng)
  
  pcr = make_pipeline(StandardScaler(), PCA(n_components=1), LinearRegression())
  pcr.fit(X_train, y_train)
  pca = pcr.named_steps['pca']  # retrieve the PCA step of the pipeline
  
  pls = PLSRegression(n_components=1)
  pls.fit(X_train, y_train)
  
  fig, axes = plt.subplots(1, 2, figsize=(10, 3))
  axes[0].scatter(pca.transform(X_test), y_test, alpha=.3, label='ground truth')
  axes[0].scatter(pca.transform(X_test), pcr.predict(X_test), alpha=.3,
                  label='predictions')
  axes[0].set(xlabel='Projected data onto first PCA component',
              ylabel='y', title='PCR / PCA')
  axes[0].legend()
  axes[1].scatter(pls.transform(X_test), y_test, alpha=.3, label='ground truth')
  axes[1].scatter(pls.transform(X_test), pls.predict(X_test), alpha=.3,
                  label='predictions')
  axes[1].set(xlabel='Projected data onto first PLS component',
              ylabel='y', title='PLS')
  axes[1].legend()
  plt.tight_layout()
  #plt.show()
  
  # %%
  # As expected, the unsupervised PCA transformation of PCR has dropped the
  # second component, i.e. the direction with the lowest variance, despite
  # it being the most predictive direction. This is because PCA is a completely
  # unsupervised transformation, and results in the projected data having a low
  # predictive power on the target.
  #
  # On the other hand, the PLS regressor manages to capture the effect of the
  # direction with the lowest variance, thanks to its use of target information
  # during the transformation: it can recogize that this direction is actually
  # the most predictive. We note that the first PLS component is negatively
  # correlated with the target, which comes from the fact that the signs of
  # eigenvectors are arbitrary.
  #
  # We also print the R-squared scores of both estimators, which further confirms
  # that PLS is a better alternative than PCR in this case. A negative R-squared
  # indicates that PCR performs worse than a regressor that would simply predict
  # the mean of the target.
  
  print(f"PCR r-squared {pcr.score(X_test, y_test):.3f}")
  print(f"PLS r-squared {pls.score(X_test, y_test):.3f}")
  
  # %%
  # As a final remark, we note that PCR with 2 components performs as well as
  # PLS: this is because in this case, PCR was able to leverage the second
  # component which has the most preditive power on the target.
  
  pca_2 = make_pipeline(PCA(n_components=2), LinearRegression())
  pca_2.fit(X_train, y_train)
  print(f"PCR r-squared with 2 components {pca_2.score(X_test, y_test):.3f}")
  