Plot NonnegativeRidge coefficients as a function of regularization

Note

This example is a copy of plot_ridge_path.py by Fabian Pedregosa in the package Scikit-learn, using NonnegativeRidge.

NonnegativeRidge Regression is the estimator used in this example. Each color represents a different feature of the coefficient vector, and this is displayed as a function of the regularization parameter.

This example also shows the usefulness of applying Ridge regression to highly ill-conditioned matrices. For such matrices, a slight change in the target variable can cause huge variances in the calculated weights. In such cases, it is useful to set a certain regularization (alpha) to reduce this variation (noise).

When alpha is very large, the regularization effect dominates the squared loss function and the coefficients tend to zero. At the end of the path, as alpha tends toward zero and the solution tends towards the ordinary least squares, coefficients exhibit big oscillations. In practise it is necessary to tune alpha in such a way that a balance is maintained between both.

Out:

# Authors: Joseph Knox <josephk@alleninstitute.org>
# License: Allen Institute Software License

# NOTE: modified from plot_ridge_path.py by Fabian Pedregosa
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from __future__ import division, print_function

import numpy as np
import matplotlib.pyplot as plt

from mcmodels.regressors import NonnegativeRidge


print(__doc__)

# X is the n  x n Hilbert matrix
n = 6
X = 1. / (np.arange(1, n + 1) + np.arange(n)[:, np.newaxis])
y = np.ones(n)

# #############################################################################
# Compute paths

n_alphas = 41
alphas = np.logspace(-2, 2, n_alphas)

coefs = []
for a in alphas:
    ridge = NonnegativeRidge(alpha=a)
    ridge.fit(X, y)
    coefs.append(ridge.coef_)

# #############################################################################
# Display results

fig, axes = plt.subplots(1, 2, figsize=(8, 3))

for ax in axes:
    ax.plot(alphas, coefs, lw=2)
    ax.set_xscale('log')
    ax.set_xlabel('alpha')
    ax.set_ylabel('weights')

# trim subplots
axes[0].set_xlim(1e-2, 1e2)
axes[1].set_xlim(1e-1, 1e1)

axes[0].set_ylim(0, 8)
axes[1].set_ylim(0, 1)

plt.suptitle('Ridge coefficients as a function of the regularization')
plt.show()