1.1 and 1.2 data generation

A2.py

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+import numpy as np
+import matplotlib.pyplot as plt
+import warnings
+warnings.filterwarnings('ignore')
+
+#reproducibility
+np.random.seed(1)
+
+#__________________________________________________________________________________
+#Task 1
+#1.1
+def generate_data(n_samples=100, noise_std=1.0):
+    """Generates synthetic data with noise"""
+    # generate x values uniformly in [0, 10]
+    x = np.linspace(0, 10, n_samples)
+    
+    #y values without noise
+    y_clean = (np.log(x + 1e-10) + 1) * np.cos(x) + np.sin(2*x)
+    
+    #noise
+    noise = np.random.normal(0, noise_std, n_samples)
+    y_noisy = y_clean + noise
+    
+    return x, y_clean, y_noisy
+
+# generate data
+x, y_clean, y_noisy = generate_data(100)
+
+# Plot clean and noisy data
+plt.plot(x, y_clean, 'b-', label='Clean Data', linewidth=2)
+plt.plot(x, y_noisy, 'ro', label='Noisy Data', alpha=0.6, markersize=4)
+plt.xlabel('x')
+plt.ylabel('y')
+plt.title('Clean vs Noisy Data')
+plt.legend()
+plt.grid(True, alpha=0.3)
+plt.show()
+
+
+#__________________________________________________________________________________
+#1.2
+def gaussian_basis(x, mu, sigma=1.0):
+    """Gaussian basis function"""
+    return np.exp(-(x - mu)**2 / sigma**2)
+
+def gaussian_features(x, D, sigma=1.0):
+    """Create Gaussian basis features"""
+    if D == 0:
+        return np.ones((len(x), 1))  
+    
+    x_min, x_max = np.min(x), np.max(x)
+    mu_i = x_min + (x_max - x_min) / (D - 1) * np.arange(D)
+    
+    features = np.ones((len(x), D + 1))  # with bias term
+    
+    for i, mu in enumerate(mu_i):
+        features[:, i+1] = gaussian_basis(x, mu, sigma)
+    
+    return features
+
+
+# Plot Gaussian basis functions for different D values
+D_values_to_plot = [5, 15, 30,45]
+x_plot = np.linspace(0, 10, 200)
+
+plt.figure(figsize=(15, 4))
+
+for i, D in enumerate(D_values_to_plot, 1):
+    plt.subplot(1, 4, i)
+    
+    # Calculate means
+    x_min, x_max = np.min(x_plot), np.max(x_plot)
+    mu_i = x_min + (x_max - x_min) / (D - 1) * np.arange(D)
+    
+    # Plot each Gaussian basis
+    for mu in mu_i:
+        phi = gaussian_basis(x_plot, mu)
+        plt.plot(x_plot, phi, alpha=0.7)
+    
+    plt.title(f'Gaussian Basis Functions (D={D})')
+    plt.xlabel('x')
+    plt.ylabel('$\phi(x)$')
+    plt.grid(True, alpha=0.3)
+
+plt.tight_layout()
+plt.show()
+
+#__________________________________________________________________________________