6

 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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