.. only:: html .. note:: :class: sphx-glr-download-link-note Click :ref:`here ` 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_bayesian_ridge.py: ========================= Bayesian Ridge Regression ========================= Computes a Bayesian Ridge Regression on a synthetic dataset. See :ref:`bayesian_ridge_regression` for more information on the regressor. Compared to the OLS (ordinary least squares) estimator, the coefficient weights are slightly shifted toward zeros, which stabilises them. As the prior on the weights is a Gaussian prior, the histogram of the estimated weights is Gaussian. The estimation of the model is done by iteratively maximizing the marginal log-likelihood of the observations. We also plot predictions and uncertainties for Bayesian Ridge Regression for one dimensional regression using polynomial feature expansion. Note the uncertainty starts going up on the right side of the plot. This is because these test samples are outside of the range of the training samples. .. rst-class:: sphx-glr-horizontal * .. image:: /auto_examples/linear_model/images/sphx_glr_plot_bayesian_ridge_001.png :alt: Weights of the model :class: sphx-glr-multi-img * .. image:: /auto_examples/linear_model/images/sphx_glr_plot_bayesian_ridge_002.png :alt: Histogram of the weights :class: sphx-glr-multi-img * .. image:: /auto_examples/linear_model/images/sphx_glr_plot_bayesian_ridge_003.png :alt: Marginal log-likelihood :class: sphx-glr-multi-img * .. image:: /auto_examples/linear_model/images/sphx_glr_plot_bayesian_ridge_004.png :alt: plot bayesian ridge :class: sphx-glr-multi-img .. code-block:: default print(__doc__) import numpy as np import matplotlib.pyplot as plt from scipy import stats from sklearn.linear_model import BayesianRidge, LinearRegression # ############################################################################# # Generating simulated data with Gaussian weights np.random.seed(0) n_samples, n_features = 100, 100 X = np.random.randn(n_samples, n_features) # Create Gaussian data # Create weights with a precision lambda_ of 4. lambda_ = 4. w = np.zeros(n_features) # Only keep 10 weights of interest relevant_features = np.random.randint(0, n_features, 10) for i in relevant_features: w[i] = stats.norm.rvs(loc=0, scale=1. / np.sqrt(lambda_)) # Create noise with a precision alpha of 50. alpha_ = 50. noise = stats.norm.rvs(loc=0, scale=1. / np.sqrt(alpha_), size=n_samples) # Create the target y = np.dot(X, w) + noise # ############################################################################# # Fit the Bayesian Ridge Regression and an OLS for comparison clf = BayesianRidge(compute_score=True) clf.fit(X, y) ols = LinearRegression() ols.fit(X, y) # ############################################################################# # Plot true weights, estimated weights, histogram of the weights, and # predictions with standard deviations lw = 2 plt.figure(figsize=(6, 5)) plt.title("Weights of the model") plt.plot(clf.coef_, color='lightgreen', linewidth=lw, label="Bayesian Ridge estimate") plt.plot(w, color='gold', linewidth=lw, label="Ground truth") plt.plot(ols.coef_, color='navy', linestyle='--', label="OLS estimate") plt.xlabel("Features") plt.ylabel("Values of the weights") plt.legend(loc="best", prop=dict(size=12)) plt.figure(figsize=(6, 5)) plt.title("Histogram of the weights") plt.hist(clf.coef_, bins=n_features, color='gold', log=True, edgecolor='black') plt.scatter(clf.coef_[relevant_features], np.full(len(relevant_features), 5.), color='navy', label="Relevant features") plt.ylabel("Features") plt.xlabel("Values of the weights") plt.legend(loc="upper left") plt.figure(figsize=(6, 5)) plt.title("Marginal log-likelihood") plt.plot(clf.scores_, color='navy', linewidth=lw) plt.ylabel("Score") plt.xlabel("Iterations") # Plotting some predictions for polynomial regression def f(x, noise_amount): y = np.sqrt(x) * np.sin(x) noise = np.random.normal(0, 1, len(x)) return y + noise_amount * noise degree = 10 X = np.linspace(0, 10, 100) y = f(X, noise_amount=0.1) clf_poly = BayesianRidge() clf_poly.fit(np.vander(X, degree), y) X_plot = np.linspace(0, 11, 25) y_plot = f(X_plot, noise_amount=0) y_mean, y_std = clf_poly.predict(np.vander(X_plot, degree), return_std=True) plt.figure(figsize=(6, 5)) plt.errorbar(X_plot, y_mean, y_std, color='navy', label="Polynomial Bayesian Ridge Regression", linewidth=lw) plt.plot(X_plot, y_plot, color='gold', linewidth=lw, label="Ground Truth") plt.ylabel("Output y") plt.xlabel("Feature X") plt.legend(loc="lower left") plt.show() .. rst-class:: sphx-glr-timing **Total running time of the script:** ( 0 minutes 0.557 seconds) .. _sphx_glr_download_auto_examples_linear_model_plot_bayesian_ridge.py: .. only :: html .. container:: sphx-glr-footer :class: sphx-glr-footer-example .. container:: binder-badge .. image:: images/binder_badge_logo.svg :target: https://mybinder.org/v2/gh/scikit-learn/scikit-learn/master?urlpath=lab/tree/notebooks/auto_examples/linear_model/plot_bayesian_ridge.ipynb :alt: Launch binder :width: 150 px .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_bayesian_ridge.py ` .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_bayesian_ridge.ipynb ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_