From a28a335a1cc88f8f6cd9ae3058469e774b004e84 Mon Sep 17 00:00:00 2001 From: ShaaniBel Date: Wed, 22 Oct 2025 19:07:16 -0400 Subject: [PATCH] final version --- results/task1-data-generation.png | Bin 31346 -> 29687 bytes results/task1-model-fitting.png | Bin 160494 -> 221273 bytes .../task1-model-selection-optimal-model.png | Bin 47000 -> 42188 bytes .../task1-model-selection-single-split.png | Bin 47722 -> 35616 bytes results/task1-non-linear-basis-functions.png | Bin 320667 -> 320721 bytes results/task2-plotting-multiple-fits.png | Bin 619916 -> 611658 bytes .../task2-plotting-train-and-test-errors.png | Bin 46340 -> 41111 bytes tasks-1-&-2.py | 109 ++++++++---------- 8 files changed, 50 insertions(+), 59 deletions(-) diff --git a/results/task1-data-generation.png b/results/task1-data-generation.png index 83e3ccc625c742bfefe668bb29e9a85a8ebe7d84..ab8615648a8eff72f0898a781b55687d41bfec8e 100644 GIT binary patch literal 29687 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#__________________________________________________________________________________ +def true_function(x): + #return (np.sin(x) + 0.5 * np.cos(3*x) + 0.25 * np.sin(7*x)) + #return np.exp(-0.1 * (x - 5)**2) * np.sin(4 * x) + #return np.piecewise(x,[x < 4, x >= 4], [lambda x: np.sin(2*x), lambda x: 0.5 * x-3]) + return np.log(x + 1) * np.cos(x) + np.sin(2*x) + + + #Task 1 #1.1 def generate_data(n_samples=100, noise_std=1.0): @@ -16,7 +24,8 @@ def generate_data(n_samples=100, noise_std=1.0): x = np.linspace(0, 10, n_samples) #y values without noise - y_clean = np.log(x + 1) * np.cos(x) + np.sin(2*x) + #y_clean = np.log(x + 1) * np.cos(x) + np.sin(2*x) + y_clean = true_function(x) #noise noise = np.random.normal(0, noise_std, n_samples) @@ -28,11 +37,12 @@ def generate_data(n_samples=100, noise_std=1.0): 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.figure(figsize=(12, 3)) +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.title('Clean vs noisy data') plt.legend() plt.grid(True, alpha=0.3) plt.savefig('results/task1-data-generation.png') @@ -82,9 +92,9 @@ for i, D in enumerate(D_values_to_plot, 1): phi = gaussian_basis(x_plot, mu) plt.plot(x_plot, phi, alpha=0.7) - plt.title(f'Gaussian Basis Functions (D={D})') + plt.title(f'Gaussian basis functions (D={D})') plt.xlabel('x') - plt.ylabel(r'$\phi(x)$') + plt.ylabel('$\phi(x)$') plt.grid(True, alpha=0.3) plt.tight_layout() @@ -119,10 +129,8 @@ class GaussianRegression: return yh - -def true_function(x): - return np.log(x + 1) * np.cos(x) + np.sin(2*x) + # fit models with different numbers of basis functions and plot D_values = [0, 5, 10, 13, 15, 17, 20, 25, 30, 45] @@ -142,19 +150,20 @@ for i, D in enumerate(D_values): y_hat = y_hat.flatten() if y_hat.ndim > 1 else y_hat # Plot - plt.plot(x_plot, true_function(x_plot), 'b-', label='True Function', linewidth=2, alpha=0.7) - plt.plot(x, y_noisy, 'ro', label='Noisy Data', alpha=0.4, markersize=3) + plt.plot(x_plot, true_function(x_plot), 'b-', label='True function', linewidth=2, alpha=0.7) + plt.plot(x, y_noisy, 'ro', label='Noisy data', alpha=0.4, markersize=3) plt.plot(x_plot, y_hat, 'g-', label=f'Fitted (D={D})', linewidth=2) - plt.ylim(-6, 6) + plt.ylim(-4.2, 4.2) plt.title(f'D = {D}') plt.grid(True, alpha=0.3) - plt.legend(fontsize=8) + if D == 0: + plt.legend(fontsize=8) # x and y labels - if i % 3 == 0: + if i % 5 == 0: plt.ylabel('y') - if i >= 9: + if i >= 5: plt.xlabel('x') plt.tight_layout() @@ -201,14 +210,14 @@ print(f"Optimal D on single split = {optimal_D}") # Plot training and validation SSE vs D for this single split -plt.figure(figsize=(12, 6)) +plt.figure(figsize=(12, 3)) plt.plot(D_values, train_sse, 'b-', label='Train SSE', linewidth=2, marker='o', markersize=4) plt.plot(D_values, val_sse, 'r-', label='Validation SSE', linewidth=2, marker='s', markersize=4) plt.axvline(x=optimal_D, color='g', linestyle='--', label=f'Optimal D = {optimal_D}') #plt.scatter([optimal_D], [val_sse[optimal_D]], label=f"Opt D = {optimal_D}", zorder=5) -plt.xlabel('Number of Gaussian bases (D)') +plt.xlabel('Number of gaussian bases (D)') plt.ylabel('Sum of Squared Errors (SSE)') -plt.title('Train and Validation SSE vs D (single split)') +plt.title('Train and validation SSE vs D (single split)') plt.legend() plt.grid(True, alpha=0.3) plt.yscale('log') @@ -216,18 +225,19 @@ plt.savefig('results/task1-model-selection-single-split.png') # plot optimal model fit -plt.figure(figsize=(10, 4)) +plt.figure(figsize=(10, 3)) optimal_model = GaussianRegression(sigma=1.0) yh_opt = optimal_model.fit(x_train, y_train, optimal_D) yh_opt = optimal_model.predict(x_plot) -plt.plot(x_plot, true_function(x_plot), 'b-', label='True Function', linewidth=2) -plt.plot(x_train, y_train, 'bo', label='Training Data', alpha=0.6, markersize=4) -plt.plot(x_val, y_val, 'ro', label='Validation Data', alpha=0.6, markersize=4) +plt.plot(x_plot, true_function(x_plot), 'b-', label='true function', linewidth=2) +plt.plot(x_train, y_train, 'bo', label='Training data', alpha=0.6, markersize=4) +plt.plot(x_val, y_val, 'ro', label='validation data', alpha=0.6, markersize=4) plt.plot(x_plot, yh_opt, 'g-', label=f'Optimal Model (D={optimal_D})', linewidth=2) -plt.ylim(-6, 6) -plt.title(f'Optimal Model with {optimal_D} Gaussian Basis Functions') +plt.ylim(-5, 5) +plt.title(f'Optimal model with {optimal_D} gaussian basis functions') plt.ylabel('y') +plt.xlabel('x') plt.legend() plt.grid(True, alpha=0.3) @@ -256,7 +266,15 @@ for rep in range(n_repetitions): for D_i, D in enumerate(D_values): # fit model - model = GaussianRegression(sigma=1.0) + ''' + if D > 10: + sigma = (x.max() - x.min()) / D + else: + sigma = 1 + ''' + sigma = 1 + model = GaussianRegression(sigma) + #model = GaussianRegression(sigma=1.0) model.fit(x_train, y_train, D) # predict on both sets @@ -288,7 +306,7 @@ for D_i, D in enumerate(D_values): ax.plot(x, predictions[rep, D_i, :], color='green', alpha=0.3, linewidth=1) # plot true function - ax.plot(x, true_function(x), 'b-', linewidth=3, label='True Function') + ax.plot(x, true_function(x), 'b-', linewidth=3, label='true function') #plot average prediction valid_predictions = [predictions[rep, D_i, :] @@ -297,9 +315,9 @@ for D_i, D in enumerate(D_values): if valid_predictions: avg_prediction = np.mean(valid_predictions, axis=0) - ax.plot(x, avg_prediction, 'r-', linewidth=2, label='Average Prediction') + ax.plot(x, avg_prediction, 'r-', linewidth=2, label='Average prediction') - ax.set_title(f'D = {D} Gaussian Bases') + ax.set_title(f'D = {D}') ax.set_xlabel('x') ax.set_ylabel('y') ax.set_ylim(-4, 4) @@ -309,13 +327,11 @@ for D_i, D in enumerate(D_values): ax.legend() plt.tight_layout() -plt.suptitle('Bias-Variance tradeoff with 10 different fits', - fontsize=16, y=1.02) +plt.suptitle('Bias-Variance tradeoff with 10 different fits', fontsize=16, y=1.02) plt.savefig('results/task2-plotting-multiple-fits.png') - # Plot 2: average training and test errors -plt.figure(figsize=(12, 6)) +plt.figure(figsize=(12, 4)) # Compute mean and std avg_train_errors = np.mean(train_errs, axis=0) @@ -328,36 +344,11 @@ std_test_errors = np.std(test_errs, axis=0) plt.errorbar(D_values, avg_train_errors, yerr=std_train_errors, label='Average Training Error', marker='o', capsize=5, linewidth=2) plt.errorbar(D_values, avg_test_errors, yerr=std_test_errors, label='Average Test Error', marker='s', capsize=5, linewidth=2) -plt.xlabel('Number of Gaussian Basis Functions (D)') +plt.xlabel('number of gaussian basis functions (D)') plt.ylabel('Mean Squared Error') -plt.title('Average Training and Test Errors Across 10 Repetitions') +plt.title('Average training and test errors across 10 repetitions') plt.legend() plt.grid(True, alpha=0.3) plt.yscale('log') plt.xticks(D_values) plt.savefig('results/task2-plotting-train-and-test-errors.png') - - - - - - - - - - - - - - - - - - - - - - - - -

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