.. _sphx_glr_auto_examples_linear_model_plot_ard.py:


==================================================
Automatic Relevance Determination Regression (ARD)
==================================================

Fit regression model with Bayesian Ridge Regression.

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.

The histogram of the estimated weights is very peaked, as a sparsity-inducing
prior is implied on the weights.

The estimation of the model is done by iteratively maximizing the
marginal log-likelihood of the observations.


.. code-block:: python

    print(__doc__)

    import numpy as np
    import matplotlib.pyplot as plt
    from scipy import stats

    from sklearn.linear_model import ARDRegression, LinearRegression







Generating simulated data with Gaussian weights


.. code-block:: python


    # Parameters of the example
    np.random.seed(0)
    n_samples, n_features = 100, 100
    # Create Gaussian data
    X = np.random.randn(n_samples, n_features)
    # 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 ARD Regression


.. code-block:: python

    clf = ARDRegression(compute_score=True)
    clf.fit(X, y)

    ols = LinearRegression()
    ols.fit(X, y)







Plot the true weights, the estimated weights and the histogram of the
weights


.. code-block:: python

    plt.figure(figsize=(6, 5))
    plt.title("Weights of the model")
    plt.plot(clf.coef_, color='darkblue', linestyle='-', linewidth=2,
             label="ARD estimate")
    plt.plot(ols.coef_, color='yellowgreen', linestyle=':', linewidth=2,
             label="OLS estimate")
    plt.plot(w, color='orange', linestyle='-', linewidth=2, label="Ground truth")
    plt.xlabel("Features")
    plt.ylabel("Values of the weights")
    plt.legend(loc=1)

    plt.figure(figsize=(6, 5))
    plt.title("Histogram of the weights")
    plt.hist(clf.coef_, bins=n_features, color='navy', log=True)
    plt.scatter(clf.coef_[relevant_features], 5 * np.ones(len(relevant_features)),
                color='gold', marker='o', label="Relevant features")
    plt.ylabel("Features")
    plt.xlabel("Values of the weights")
    plt.legend(loc=1)

    plt.figure(figsize=(6, 5))
    plt.title("Marginal log-likelihood")
    plt.plot(clf.scores_, color='navy', linewidth=2)
    plt.ylabel("Score")
    plt.xlabel("Iterations")
    plt.show()



.. rst-class:: sphx-glr-horizontal


    *

      .. image:: /auto_examples/linear_model/images/sphx_glr_plot_ard_001.png
            :scale: 47

    *

      .. image:: /auto_examples/linear_model/images/sphx_glr_plot_ard_002.png
            :scale: 47

    *

      .. image:: /auto_examples/linear_model/images/sphx_glr_plot_ard_003.png
            :scale: 47




**Total running time of the script:**
(0 minutes 0.406 seconds)



.. container:: sphx-glr-download

    **Download Python source code:** :download:`plot_ard.py <plot_ard.py>`


.. container:: sphx-glr-download

    **Download IPython notebook:** :download:`plot_ard.ipynb <plot_ard.ipynb>`