In [ ]:
%matplotlib inline

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.

In [ ]:
import numpy as np
import matplotlib.pyplot as plt
from sklearn.metrics import r2_score

Generate some random data

In [ ]:

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 =, true_coef)

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

Split the data in train set and test set

In [ ]:
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.

In [ ]:
from sklearn.linear_model import LinearRegression

reg_nnls = LinearRegression(positive=True)
y_pred_nnls =, 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.

In [ ]:
reg_ols = LinearRegression()
y_pred_ols =, 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.

In [ ]:
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=.5)
ax.set_xlabel("OLS regression coefficients", fontweight="bold")
ax.set_ylabel("NNLS regression coefficients", fontweight="bold")