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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.
import numpy as np import matplotlib.pyplot as plt 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 0.8225220806196526
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 0.7436926291700356
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')
Total running time of the script: ( 0 minutes 0.072 seconds)