.. 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_curvefit.py: ============================================ Curve Fitting with Bayesian Ridge Regression ============================================ Computes a Bayesian Ridge Regression of Sinusoids. See :ref:bayesian_ridge_regression for more information on the regressor. In general, when fitting a curve with a polynomial by Bayesian ridge regression, the selection of initial values of the regularization parameters (alpha, lambda) may be important. This is because the regularization parameters are determined by an iterative procedure that depends on initial values. In this example, the sinusoid is approximated by a polynomial using different pairs of initial values. When starting from the default values (alpha_init = 1.90, lambda_init = 1.), the bias of the resulting curve is large, and the variance is small. So, lambda_init should be relatively small (1.e-3) so as to reduce the bias. Also, by evaluating log marginal likelihood (L) of these models, we can determine which one is better. It can be concluded that the model with larger L is more likely. .. image:: /auto_examples/linear_model/images/sphx_glr_plot_bayesian_ridge_curvefit_001.png :class: sphx-glr-single-img .. code-block:: default print(__doc__) # Author: Yoshihiro Uchida import numpy as np import matplotlib.pyplot as plt from sklearn.linear_model import BayesianRidge def func(x): return np.sin(2*np.pi*x) # ############################################################################# # Generate sinusoidal data with noise size = 25 rng = np.random.RandomState(1234) x_train = rng.uniform(0., 1., size) y_train = func(x_train) + rng.normal(scale=0.1, size=size) x_test = np.linspace(0., 1., 100) # ############################################################################# # Fit by cubic polynomial n_order = 3 X_train = np.vander(x_train, n_order + 1, increasing=True) X_test = np.vander(x_test, n_order + 1, increasing=True) # ############################################################################# # Plot the true and predicted curves with log marginal likelihood (L) reg = BayesianRidge(tol=1e-6, fit_intercept=False, compute_score=True) fig, axes = plt.subplots(1, 2, figsize=(8, 4)) for i, ax in enumerate(axes): # Bayesian ridge regression with different initial value pairs if i == 0: init = [1 / np.var(y_train), 1.] # Default values elif i == 1: init = [1., 1e-3] reg.set_params(alpha_init=init[0], lambda_init=init[1]) reg.fit(X_train, y_train) ymean, ystd = reg.predict(X_test, return_std=True) ax.plot(x_test, func(x_test), color="blue", label="sin($2\\pi x$)") ax.scatter(x_train, y_train, s=50, alpha=0.5, label="observation") ax.plot(x_test, ymean, color="red", label="predict mean") ax.fill_between(x_test, ymean-ystd, ymean+ystd, color="pink", alpha=0.5, label="predict std") ax.set_ylim(-1.3, 1.3) ax.legend() title = "$\\alpha$_init$={:.2f},\\ \\lambda$_init$={}$".format( init[0], init[1]) if i == 0: title += " (Default)" ax.set_title(title, fontsize=12) text = "$\\alpha={:.1f}$\n$\\lambda={:.3f}$\n$L={:.1f}$".format( reg.alpha_, reg.lambda_, reg.scores_[-1]) ax.text(0.05, -1.0, text, fontsize=12) plt.tight_layout() plt.show() .. rst-class:: sphx-glr-timing **Total running time of the script:** ( 0 minutes 0.201 seconds) .. _sphx_glr_download_auto_examples_linear_model_plot_bayesian_ridge_curvefit.py: .. only :: html .. container:: sphx-glr-footer :class: sphx-glr-footer-example .. container:: binder-badge .. image:: https://mybinder.org/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_curvefit.ipynb :width: 150 px .. container:: sphx-glr-download sphx-glr-download-python :download:Download Python source code: plot_bayesian_ridge_curvefit.py  .. container:: sphx-glr-download sphx-glr-download-jupyter :download:Download Jupyter notebook: plot_bayesian_ridge_curvefit.ipynb  .. only:: html .. rst-class:: sphx-glr-signature Gallery generated by Sphinx-Gallery _