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import numpy as np import matplotlib.pyplot as plt def weight(x, xi, tau): return np.exp(-np.sum((x - xi) ** 2) / (2 * tau ** 2)) def predict(x, X, y, tau): W = np.diag([weight(x, xi, tau) for xi in X]) theta = np.linalg.pinv(X.T @ W @ X) @ X.T @ W @ y return x @ theta # Generate training data X = np.linspace(0, 2 * np.pi, 100) y = np.sin(X) + 0.1 * np.random.randn(100) X_ = np.c_[np.ones(X.shape), X] # Add bias term # Generate test data x_test = np.linspace(0, 2 * np.pi, 200) x_test_ = np.c_[np.ones(x_test.shape), x_test] # Predict tau = 0.5 y_pred = [predict(xi, X_, y, tau) for xi in x_test_] # Plot plt.scatter(X, y, alpha=0.6, label="Data") plt.plot(x_test, y_pred, color="red", label="LWR Prediction") plt.legend() plt.title("Locally Weighted Regression") plt.show()

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