.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_examples/model_selection/plot_train_error_vs_test_error.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. or to run this example in your browser via JupyterLite or Binder .. rst-class:: sphx-glr-example-title .. _sphx_glr_auto_examples_model_selection_plot_train_error_vs_test_error.py: ========================================================= Effect of model regularization on training and test error ========================================================= In this example, we evaluate the impact of the regularization parameter in a linear model called :class:`~sklearn.linear_model.ElasticNet`. To carry out this evaluation, we use a validation curve using :class:`~sklearn.model_selection.ValidationCurveDisplay`. This curve shows the training and test scores of the model for different values of the regularization parameter. Once we identify the optimal regularization parameter, we compare the true and estimated coefficients of the model to determine if the model is able to recover the coefficients from the noisy input data. .. GENERATED FROM PYTHON SOURCE LINES 17-21 .. code-block:: Python # Authors: The scikit-learn developers # SPDX-License-Identifier: BSD-3-Clause .. GENERATED FROM PYTHON SOURCE LINES 22-29 Generate sample data -------------------- We generate a regression dataset that contains many features relative to the number of samples. However, only 10% of the features are informative. In this context, linear models exposing L1 penalization are commonly used to recover a sparse set of coefficients. .. GENERATED FROM PYTHON SOURCE LINES 29-46 .. code-block:: Python from sklearn.datasets import make_regression from sklearn.model_selection import train_test_split n_samples_train, n_samples_test, n_features = 150, 300, 500 X, y, true_coef = make_regression( n_samples=n_samples_train + n_samples_test, n_features=n_features, n_informative=50, shuffle=False, noise=1.0, coef=True, random_state=42, ) X_train, X_test, y_train, y_test = train_test_split( X, y, train_size=n_samples_train, test_size=n_samples_test, shuffle=False ) .. GENERATED FROM PYTHON SOURCE LINES 47-61 Model definition ---------------- Here, we do not use a model that only exposes an L1 penalty. Instead, we use an :class:`~sklearn.linear_model.ElasticNet` model that exposes both L1 and L2 penalties. We fix the `l1_ratio` parameter such that the solution found by the model is still sparse. Therefore, this type of model tries to find a sparse solution but at the same time also tries to shrink all coefficients towards zero. In addition, we force the coefficients of the model to be positive since we know that `make_regression` generates a response with a positive signal. So we use this pre-knowledge to get a better model. .. GENERATED FROM PYTHON SOURCE LINES 61-67 .. code-block:: Python from sklearn.linear_model import ElasticNet enet = ElasticNet(l1_ratio=0.9, positive=True, max_iter=10_000) .. GENERATED FROM PYTHON SOURCE LINES 68-88 Evaluate the impact of the regularization parameter --------------------------------------------------- To evaluate the impact of the regularization parameter, we use a validation curve. This curve shows the training and test scores of the model for different values of the regularization parameter. The regularization `alpha` is a parameter applied to the coefficients of the model: when it tends to zero, no regularization is applied and the model tries to fit the training data with the least amount of error. However, it leads to overfitting when features are noisy. When `alpha` increases, the model coefficients are constrained, and thus the model cannot fit the training data as closely, avoiding overfitting. However, if too much regularization is applied, the model underfits the data and is not able to properly capture the signal. The validation curve helps in finding a good trade-off between both extremes: the model is not regularized and thus flexible enough to fit the signal, but not too flexible to overfit. The :class:`~sklearn.model_selection.ValidationCurveDisplay` allows us to display the training and validation scores across a range of alpha values. .. GENERATED FROM PYTHON SOURCE LINES 88-122 .. code-block:: Python import numpy as np from sklearn.model_selection import ValidationCurveDisplay alphas = np.logspace(-5, 1, 60) disp = ValidationCurveDisplay.from_estimator( enet, X_train, y_train, param_name="alpha", param_range=alphas, scoring="r2", n_jobs=2, score_type="both", ) disp.ax_.set( title=r"Validation Curve for ElasticNet (R$^2$ Score)", xlabel=r"alpha (regularization strength)", ylabel="R$^2$ Score", ) test_scores_mean = disp.test_scores.mean(axis=1) idx_avg_max_test_score = np.argmax(test_scores_mean) disp.ax_.vlines( alphas[idx_avg_max_test_score], disp.ax_.get_ylim()[0], test_scores_mean[idx_avg_max_test_score], color="k", linewidth=2, linestyle="--", label=f"Optimum on test\n$\\alpha$ = {alphas[idx_avg_max_test_score]:.2e}", ) _ = disp.ax_.legend(loc="lower right") .. image-sg:: /auto_examples/model_selection/images/sphx_glr_plot_train_error_vs_test_error_001.png :alt: Validation Curve for ElasticNet (R$^2$ Score) :srcset: /auto_examples/model_selection/images/sphx_glr_plot_train_error_vs_test_error_001.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 123-134 To find the optimal regularization parameter, we can select the value of `alpha` that maximizes the validation score. Coefficients comparison ----------------------- Now that we have identified the optimal regularization parameter, we can compare the true coefficients and the estimated coefficients. First, let's set the regularization parameter to the optimal value and fit the model on the training data. In addition, we'll show the test score for this model. .. GENERATED FROM PYTHON SOURCE LINES 134-139 .. code-block:: Python enet.set_params(alpha=alphas[idx_avg_max_test_score]).fit(X_train, y_train) print( f"Test score: {enet.score(X_test, y_test):.3f}", ) .. rst-class:: sphx-glr-script-out .. code-block:: none Test score: 0.884 .. GENERATED FROM PYTHON SOURCE LINES 140-141 Now, we plot the true coefficients and the estimated coefficients. .. GENERATED FROM PYTHON SOURCE LINES 141-158 .. code-block:: Python import matplotlib.pyplot as plt fig, axs = plt.subplots(ncols=2, figsize=(12, 6), sharex=True, sharey=True) for ax, coef, title in zip(axs, [true_coef, enet.coef_], ["True", "Model"]): ax.stem(coef) ax.set( title=f"{title} Coefficients", xlabel="Feature Index", ylabel="Coefficient Value", ) fig.suptitle( "Comparison of the coefficients of the true generative model and \n" "the estimated elastic net coefficients" ) plt.show() .. image-sg:: /auto_examples/model_selection/images/sphx_glr_plot_train_error_vs_test_error_002.png :alt: Comparison of the coefficients of the true generative model and the estimated elastic net coefficients, True Coefficients, Model Coefficients :srcset: /auto_examples/model_selection/images/sphx_glr_plot_train_error_vs_test_error_002.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 159-167 While the original coefficients are sparse, the estimated coefficients are not as sparse. The reason is that we fixed the `l1_ratio` parameter to 0.9. We could force the model to get a sparser solution by increasing the `l1_ratio` parameter. However, we observed that for the estimated coefficients that are close to zero in the true generative model, our model shrinks them towards zero. So we don't recover the true coefficients, but we get a sensible outcome in line with the performance obtained on the test set. .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 5.138 seconds) .. _sphx_glr_download_auto_examples_model_selection_plot_train_error_vs_test_error.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: binder-badge .. image:: images/binder_badge_logo.svg :target: https://mybinder.org/v2/gh/scikit-learn/scikit-learn/main?urlpath=lab/tree/notebooks/auto_examples/model_selection/plot_train_error_vs_test_error.ipynb :alt: Launch binder :width: 150 px .. container:: lite-badge .. image:: images/jupyterlite_badge_logo.svg :target: ../../lite/lab/index.html?path=auto_examples/model_selection/plot_train_error_vs_test_error.ipynb :alt: Launch JupyterLite :width: 150 px .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_train_error_vs_test_error.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_train_error_vs_test_error.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: plot_train_error_vs_test_error.zip ` .. include:: plot_train_error_vs_test_error.recommendations .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_