Plot Nnls¶
========================== 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.
Imports for Non-Negative Least Squares (NNLS) Regression¶
Non-negative least squares constrains all regression coefficients to be zero or positive, which is essential in domains where negative contributions are physically impossible – for example, concentrations in chemical mixtures, pixel intensities in image decomposition, or topic weights in document analysis. Scikit-learn implements this via LinearRegression(positive=True), which solves the same least squares objective as OLS but subject to the constraint w >= 0.
Effect on the solution: The non-negativity constraint naturally induces sparsity because coefficients that OLS would estimate as negative are forced to zero. This means NNLS performs a form of implicit feature selection without requiring an explicit regularization penalty. Comparing NNLS coefficients against unconstrained OLS coefficients reveals which features the unconstrained model relied on with negative weights – these are the features whose contribution is “corrected” by the constraint.
# 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)
# %%
# 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)
# %%
# 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")