.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "11_demos\python_packages\scikit-learn\demo_componentRegression.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:`Go to the end ` to download the full example code. .. rst-class:: sphx-glr-example-title .. _sphx_glr_11_demos_python_packages_scikit-learn_demo_componentRegression.py: Principal Component Regression vs Partial Least Squares Regression ================================================================== An example of ``scikit-learn``, copied from `scikit-learn.org `_. This example compares `Principal Component Regression `_ (PCR) and `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. .. GENERATED FROM PYTHON SOURCE LINES 38-45 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. .. GENERATED FROM PYTHON SOURCE LINES 45-75 .. code-block:: Python 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=0.3, 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() .. image-sg:: /11_demos/python_packages/scikit-learn/images/sphx_glr_demo_componentRegression_001.png :alt: 2-dimensional dataset with principal components :srcset: /11_demos/python_packages/scikit-learn/images/sphx_glr_demo_componentRegression_001.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 76-79 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. .. GENERATED FROM PYTHON SOURCE LINES 79-91 .. code-block:: Python 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=0.3) axes[0].set(xlabel="Projected data onto first PCA component", ylabel="y") axes[1].scatter(X.dot(pca.components_[1]), y, alpha=0.3) axes[1].set(xlabel="Projected data onto second PCA component", ylabel="y") plt.tight_layout() plt.show() .. image-sg:: /11_demos/python_packages/scikit-learn/images/sphx_glr_demo_componentRegression_002.png :alt: demo componentRegression :srcset: /11_demos/python_packages/scikit-learn/images/sphx_glr_demo_componentRegression_002.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 92-103 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. .. GENERATED FROM PYTHON SOURCE LINES 103-137 .. code-block:: Python 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=0.3, label="ground truth") axes[0].scatter( pca.transform(X_test), pcr.predict(X_test), alpha=0.3, 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=0.3, label="ground truth") axes[1].scatter( pls.transform(X_test), pls.predict(X_test), alpha=0.3, label="predictions" ) axes[1].set(xlabel="Projected data onto first PLS component", ylabel="y", title="PLS") axes[1].legend() plt.tight_layout() plt.show() .. image-sg:: /11_demos/python_packages/scikit-learn/images/sphx_glr_demo_componentRegression_003.png :alt: PCR / PCA, PLS :srcset: /11_demos/python_packages/scikit-learn/images/sphx_glr_demo_componentRegression_003.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 138-155 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 recognize 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. .. GENERATED FROM PYTHON SOURCE LINES 155-159 .. code-block:: Python print(f"PCR r-squared {pcr.score(X_test, y_test):.3f}") print(f"PLS r-squared {pls.score(X_test, y_test):.3f}") .. rst-class:: sphx-glr-script-out .. code-block:: none PCR r-squared -0.026 PLS r-squared 0.658 .. GENERATED FROM PYTHON SOURCE LINES 160-163 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. .. GENERATED FROM PYTHON SOURCE LINES 163-167 .. code-block:: Python 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}") .. rst-class:: sphx-glr-script-out .. code-block:: none PCR r-squared with 2 components 0.673 .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 12.061 seconds) .. _sphx_glr_download_11_demos_python_packages_scikit-learn_demo_componentRegression.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: demo_componentRegression.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: demo_componentRegression.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: demo_componentRegression.zip `