Automatic Relevance Determination Regression (ARD)¶

Fit regression model with Bayesian Ridge Regression.

Compared to the OLS (ordinary least squares) estimator, the coefficient weights are slightly shifted toward zeros, wich 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.

Python source code: plot_ard.py

print __doc__

import numpy as np
import pylab as pl
from scipy import stats

from sklearn.linear_model import ARDRegression, LinearRegression

###############################################################################
# Generating simulated data with Gaussian weigthts

# 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 weigts 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 noite 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
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
pl.figure(figsize=(6, 5))
pl.title("Weights of the model")
pl.plot(clf.coef_, 'b-', label="ARD estimate")
pl.plot(ols.coef_, 'r--', label="OLS estimate")
pl.plot(w, 'g-', label="Ground truth")
pl.xlabel("Features")
pl.ylabel("Values of the weights")
pl.legend(loc=1)

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

pl.figure(figsize=(6, 5))
pl.title("Marginal log-likelihood")
pl.plot(clf.scores_)
pl.ylabel("Score")
pl.xlabel("Iterations")
pl.show()

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Automatic Relevance Determination Regression (ARD)¶