NonnegativeLinear Least Squares and NonnegativeRidge Regression Variance

Note

This example is a copy of plot_ols_ridge_variance.py by Gael Varoquaux and Jaques Grobler in the package Scikit-learn, using NonnegativeLinear and NonnegativeRidge.

Due to the few points in each dimension and the straight line that linear regression uses to follow these points as well as it can, noise on the observations will cause great variance as shown in the first plot. Every line’s slope can vary quite a bit for each prediction due to the noise induced in the observations.

Ridge regression is basically minimizing a penalised version of the least-squared function. The penalising shrinks the value of the regression coefficients. Despite the few data points in each dimension, the slope of the prediction is much more stable and the variance in the line itself is greatly reduced, in comparison to that of the standard linear regression

Out:

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

# NOTE: modified from plot_ols_ridge_variance.py by Gael Varoquaux and Jaques Grobler
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from __future__ import print_function

import numpy as np
import matplotlib.pyplot as plt

from mcmodels.regressors import NonnegativeLinear, NonnegativeRidge


print(__doc__)

n_datasets = 5

X_train = np.c_[.5, 1].T
y_train = [.5, 1]
X_test = np.c_[0, 2].T

np.random.seed(0)

regressors = dict(NonnegativeLinear=NonnegativeLinear(),
                  NonnegativeRidge=NonnegativeRidge(alpha=0.1))

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

for ax, (name, reg) in zip(axes, regressors.items()):

    for _ in range(n_datasets):
        this_X = .15 * np.random.normal(size=(2, 1)) + X_train
        reg.fit(this_X, y_train)

        ax.plot(X_test, reg.predict(X_test), color='.5')
        ax.scatter(this_X, y_train, s=3, c='.5', marker='o', zorder=10)

    reg.fit(X_train, y_train)
    ax.plot(X_test, reg.predict(X_test), linewidth=2, color='blue')
    ax.scatter(X_train, y_train, s=30, c='r', marker='+', zorder=10)

    ax.set_xlim(0, 2)
    ax.set_ylim((0, 1.6))

    ax.set_xlabel('X')
    ax.set_ylabel('y')

    ax.set_title(name)

plt.show()