.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_examples/linear_model/plot_lasso_and_elasticnet.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 Binder .. rst-class:: sphx-glr-example-title .. _sphx_glr_auto_examples_linear_model_plot_lasso_and_elasticnet.py: ================================== L1-based models for Sparse Signals ================================== The present example compares three l1-based regression models on a synthetic signal obtained from sparse and correlated features that are further corrupted with additive gaussian noise: - a :ref:`lasso`; - an :ref:`automatic_relevance_determination`; - an :ref:`elastic_net`. It is known that the Lasso estimates turn to be close to the model selection estimates when the data dimensions grow, given that the irrelevant variables are not too correlated with the relevant ones. In the presence of correlated features, Lasso itself cannot select the correct sparsity pattern [1]_. Here we compare the performance of the three models in terms of the :math:`R^2` score, the fitting time and the sparsity of the estimated coefficients when compared with the ground-truth. .. GENERATED FROM PYTHON SOURCE LINES 23-27 .. code-block:: Python # Authors: The scikit-learn developers # SPDX-License-Identifier: BSD-3-Clause .. GENERATED FROM PYTHON SOURCE LINES 28-43 Generate synthetic dataset -------------------------- We generate a dataset where the number of samples is lower than the total number of features. This leads to an underdetermined system, i.e. the solution is not unique, and thus we cannot apply an :ref:`ordinary_least_squares` by itself. Regularization introduces a penalty term to the objective function, which modifies the optimization problem and can help alleviate the underdetermined nature of the system. The target `y` is a linear combination with alternating signs of sinusoidal signals. Only the 10 lowest out of the 100 frequencies in `X` are used to generate `y`, while the rest of the features are not informative. This results in a high dimensional sparse feature space, where some degree of l1-penalization is necessary. .. GENERATED FROM PYTHON SOURCE LINES 43-60 .. code-block:: Python import numpy as np rng = np.random.RandomState(0) n_samples, n_features, n_informative = 50, 100, 10 time_step = np.linspace(-2, 2, n_samples) freqs = 2 * np.pi * np.sort(rng.rand(n_features)) / 0.01 X = np.zeros((n_samples, n_features)) for i in range(n_features): X[:, i] = np.sin(freqs[i] * time_step) idx = np.arange(n_features) true_coef = (-1) ** idx * np.exp(-idx / 10) true_coef[n_informative:] = 0 # sparsify coef y = np.dot(X, true_coef) .. GENERATED FROM PYTHON SOURCE LINES 61-63 Some of the informative features have close frequencies to induce (anti-)correlations. .. GENERATED FROM PYTHON SOURCE LINES 63-66 .. code-block:: Python freqs[:n_informative] .. rst-class:: sphx-glr-script-out .. code-block:: none array([ 2.9502547 , 11.8059798 , 12.63394388, 12.70359377, 24.62241605, 37.84077985, 40.30506066, 44.63327171, 54.74495357, 59.02456369]) .. GENERATED FROM PYTHON SOURCE LINES 67-70 A random phase is introduced using :func:`numpy.random.random_sample` and some gaussian noise (implemented by :func:`numpy.random.normal`) is added to both the features and the target. .. GENERATED FROM PYTHON SOURCE LINES 70-77 .. code-block:: Python for i in range(n_features): X[:, i] = np.sin(freqs[i] * time_step + 2 * (rng.random_sample() - 0.5)) X[:, i] += 0.2 * rng.normal(0, 1, n_samples) y += 0.2 * rng.normal(0, 1, n_samples) .. GENERATED FROM PYTHON SOURCE LINES 78-82 Such sparse, noisy and correlated features can be obtained, for instance, from sensor nodes monitoring some environmental variables, as they typically register similar values depending on their positions (spatial correlations). We can visualize the target. .. GENERATED FROM PYTHON SOURCE LINES 82-90 .. code-block:: Python import matplotlib.pyplot as plt plt.plot(time_step, y) plt.ylabel("target signal") plt.xlabel("time") _ = plt.title("Superposition of sinusoidal signals") .. image-sg:: /auto_examples/linear_model/images/sphx_glr_plot_lasso_and_elasticnet_001.png :alt: Superposition of sinusoidal signals :srcset: /auto_examples/linear_model/images/sphx_glr_plot_lasso_and_elasticnet_001.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 91-96 We split the data into train and test sets for simplicity. In practice one should use a :class:`~sklearn.model_selection.TimeSeriesSplit` cross-validation to estimate the variance of the test score. Here we set `shuffle="False"` as we must not use training data that succeed the testing data when dealing with data that have a temporal relationship. .. GENERATED FROM PYTHON SOURCE LINES 96-101 .. code-block:: Python from sklearn.model_selection import train_test_split X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.5, shuffle=False) .. GENERATED FROM PYTHON SOURCE LINES 102-116 In the following, we compute the performance of three l1-based models in terms of the goodness of fit :math:`R^2` score and the fitting time. Then we make a plot to compare the sparsity of the estimated coefficients with respect to the ground-truth coefficients and finally we analyze the previous results. Lasso ----- In this example, we demo a :class:`~sklearn.linear_model.Lasso` with a fixed value of the regularization parameter `alpha`. In practice, the optimal parameter `alpha` should be selected by passing a :class:`~sklearn.model_selection.TimeSeriesSplit` cross-validation strategy to a :class:`~sklearn.linear_model.LassoCV`. To keep the example simple and fast to execute, we directly set the optimal value for alpha here. .. GENERATED FROM PYTHON SOURCE LINES 116-129 .. code-block:: Python from time import time from sklearn.linear_model import Lasso from sklearn.metrics import r2_score t0 = time() lasso = Lasso(alpha=0.14).fit(X_train, y_train) print(f"Lasso fit done in {(time() - t0):.3f}s") y_pred_lasso = lasso.predict(X_test) r2_score_lasso = r2_score(y_test, y_pred_lasso) print(f"Lasso r^2 on test data : {r2_score_lasso:.3f}") .. rst-class:: sphx-glr-script-out .. code-block:: none Lasso fit done in 0.001s Lasso r^2 on test data : 0.480 .. GENERATED FROM PYTHON SOURCE LINES 130-139 Automatic Relevance Determination (ARD) --------------------------------------- An ARD regression is the bayesian version of the Lasso. It can produce interval estimates for all of the parameters, including the error variance, if required. It is a suitable option when the signals have gaussian noise. See the example :ref:`sphx_glr_auto_examples_linear_model_plot_ard.py` for a comparison of :class:`~sklearn.linear_model.ARDRegression` and :class:`~sklearn.linear_model.BayesianRidge` regressors. .. GENERATED FROM PYTHON SOURCE LINES 139-150 .. code-block:: Python from sklearn.linear_model import ARDRegression t0 = time() ard = ARDRegression().fit(X_train, y_train) print(f"ARD fit done in {(time() - t0):.3f}s") y_pred_ard = ard.predict(X_test) r2_score_ard = r2_score(y_test, y_pred_ard) print(f"ARD r^2 on test data : {r2_score_ard:.3f}") .. rst-class:: sphx-glr-script-out .. code-block:: none ARD fit done in 0.010s ARD r^2 on test data : 0.543 .. GENERATED FROM PYTHON SOURCE LINES 151-167 ElasticNet ---------- :class:`~sklearn.linear_model.ElasticNet` is a middle ground between :class:`~sklearn.linear_model.Lasso` and :class:`~sklearn.linear_model.Ridge`, as it combines a L1 and a L2-penalty. The amount of regularization is controlled by the two hyperparameters `l1_ratio` and `alpha`. For `l1_ratio = 0` the penalty is pure L2 and the model is equivalent to a :class:`~sklearn.linear_model.Ridge`. Similarly, `l1_ratio = 1` is a pure L1 penalty and the model is equivalent to a :class:`~sklearn.linear_model.Lasso`. For `0 < l1_ratio < 1`, the penalty is a combination of L1 and L2. As done before, we train the model with fix values for `alpha` and `l1_ratio`. To select their optimal value we used an :class:`~sklearn.linear_model.ElasticNetCV`, not shown here to keep the example simple. .. GENERATED FROM PYTHON SOURCE LINES 167-178 .. code-block:: Python from sklearn.linear_model import ElasticNet t0 = time() enet = ElasticNet(alpha=0.08, l1_ratio=0.5).fit(X_train, y_train) print(f"ElasticNet fit done in {(time() - t0):.3f}s") y_pred_enet = enet.predict(X_test) r2_score_enet = r2_score(y_test, y_pred_enet) print(f"ElasticNet r^2 on test data : {r2_score_enet:.3f}") .. rst-class:: sphx-glr-script-out .. code-block:: none ElasticNet fit done in 0.001s ElasticNet r^2 on test data : 0.636 .. GENERATED FROM PYTHON SOURCE LINES 179-184 Plot and analysis of the results -------------------------------- In this section, we use a heatmap to visualize the sparsity of the true and estimated coefficients of the respective linear models. .. GENERATED FROM PYTHON SOURCE LINES 184-215 .. code-block:: Python import matplotlib.pyplot as plt import pandas as pd import seaborn as sns from matplotlib.colors import SymLogNorm df = pd.DataFrame( { "True coefficients": true_coef, "Lasso": lasso.coef_, "ARDRegression": ard.coef_, "ElasticNet": enet.coef_, } ) plt.figure(figsize=(10, 6)) ax = sns.heatmap( df.T, norm=SymLogNorm(linthresh=10e-4, vmin=-1, vmax=1), cbar_kws={"label": "coefficients' values"}, cmap="seismic_r", ) plt.ylabel("linear model") plt.xlabel("coefficients") plt.title( f"Models' coefficients\nLasso $R^2$: {r2_score_lasso:.3f}, " f"ARD $R^2$: {r2_score_ard:.3f}, " f"ElasticNet $R^2$: {r2_score_enet:.3f}" ) plt.tight_layout() .. image-sg:: /auto_examples/linear_model/images/sphx_glr_plot_lasso_and_elasticnet_002.png :alt: Models' coefficients Lasso $R^2$: 0.480, ARD $R^2$: 0.543, ElasticNet $R^2$: 0.636 :srcset: /auto_examples/linear_model/images/sphx_glr_plot_lasso_and_elasticnet_002.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 216-250 In the present example :class:`~sklearn.linear_model.ElasticNet` yields the best score and captures the most of the predictive features, yet still fails at finding all the true components. Notice that both :class:`~sklearn.linear_model.ElasticNet` and :class:`~sklearn.linear_model.ARDRegression` result in a less sparse model than a :class:`~sklearn.linear_model.Lasso`. Conclusions ----------- :class:`~sklearn.linear_model.Lasso` is known to recover sparse data effectively but does not perform well with highly correlated features. Indeed, if several correlated features contribute to the target, :class:`~sklearn.linear_model.Lasso` would end up selecting a single one of them. In the case of sparse yet non-correlated features, a :class:`~sklearn.linear_model.Lasso` model would be more suitable. :class:`~sklearn.linear_model.ElasticNet` introduces some sparsity on the coefficients and shrinks their values to zero. Thus, in the presence of correlated features that contribute to the target, the model is still able to reduce their weights without setting them exactly to zero. This results in a less sparse model than a pure :class:`~sklearn.linear_model.Lasso` and may capture non-predictive features as well. :class:`~sklearn.linear_model.ARDRegression` is better when handling gaussian noise, but is still unable to handle correlated features and requires a larger amount of time due to fitting a prior. References ---------- .. [1] :doi:`"Lasso-type recovery of sparse representations for high-dimensional data" N. Meinshausen, B. Yu - The Annals of Statistics 2009, Vol. 37, No. 1, 246-270 <10.1214/07-AOS582>` .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 0.369 seconds) .. _sphx_glr_download_auto_examples_linear_model_plot_lasso_and_elasticnet.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/linear_model/plot_lasso_and_elasticnet.ipynb :alt: Launch binder :width: 150 px .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_lasso_and_elasticnet.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_lasso_and_elasticnet.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: plot_lasso_and_elasticnet.zip ` .. include:: plot_lasso_and_elasticnet.recommendations .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_