Non-negative least squares

In this example, we fit a linear model with positive constraints on the regression coefficients and compare the estimated coefficients to a classic linear regression.

# Authors: The scikit-learn developers
# SPDX-License-Identifier: BSD-3-Clause

import matplotlib.pyplot as plt
import numpy as np

from sklearn.metrics import r2_score

Generate some random data

np.random.seed(42)

n_samples, n_features = 200, 50
X = np.random.randn(n_samples, n_features)
true_coef = 3 * np.random.randn(n_features)
# Threshold coefficients to render them non-negative
true_coef[true_coef < 0] = 0
y = np.dot(X, true_coef)

# Add some noise
y += 5 * np.random.normal(size=(n_samples,))

Split the data in train set and test set

from sklearn.model_selection import train_test_split

X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.5)

Fit the Non-Negative least squares.

from sklearn.linear_model import LinearRegression

reg_nnls = LinearRegression(positive=True)
y_pred_nnls = reg_nnls.fit(X_train, y_train).predict(X_test)
r2_score_nnls = r2_score(y_test, y_pred_nnls)
print("NNLS R2 score", r2_score_nnls)

NNLS R2 score 0.8225220806196525

Fit an OLS.

reg_ols = LinearRegression()
y_pred_ols = reg_ols.fit(X_train, y_train).predict(X_test)
r2_score_ols = r2_score(y_test, y_pred_ols)
print("OLS R2 score", r2_score_ols)

OLS R2 score 0.7436926291700353

Comparing the regression coefficients between OLS and NNLS, we can observe they are highly correlated (the dashed line is the identity relation), but the non-negative constraint shrinks some to 0. The Non-Negative Least squares inherently yield sparse results.

fig, ax = plt.subplots()
ax.plot(reg_ols.coef_, reg_nnls.coef_, linewidth=0, marker=".")

low_x, high_x = ax.get_xlim()
low_y, high_y = ax.get_ylim()
low = max(low_x, low_y)
high = min(high_x, high_y)
ax.plot([low, high], [low, high], ls="--", c=".3", alpha=0.5)
ax.set_xlabel("OLS regression coefficients", fontweight="bold")
ax.set_ylabel("NNLS regression coefficients", fontweight="bold")

Text(55.847222222222214, 0.5, 'NNLS regression coefficients')

Related examples

Effect of model regularization on training and test error

Effect of model regularization on training and test error

Logistic function

Logistic function

Failure of Machine Learning to infer causal effects

Failure of Machine Learning to infer causal effects

L1-based models for Sparse Signals

L1-based models for Sparse Signals