1.4 and 2
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ShaaniBel
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1 changed files with 175 additions and 8 deletions
181
A2.py
181
A2.py
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@ -50,6 +50,10 @@ def gaussian_features(x, D, sigma=1.0):
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return np.ones((len(x), 1))
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x_min, x_max = np.min(x), np.max(x)
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if D == 1:
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mu_i = np.array([(x_min + x_max) / 2])
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else:
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mu_i = x_min + (x_max - x_min) / (D - 1) * np.arange(D)
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features = np.ones((len(x), D + 1)) # with bias term
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@ -102,7 +106,8 @@ class GaussianRegression:
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self.D = D
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# create features for training and fit using least squares
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X = gaussian_features(x, D, self.sigma)
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self.w = np.linalg.lstsq(X, y, rcond=None)[0]
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#self.w = np.linalg.lstsq(X, y, rcond=None)[0]
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self.w = np.linalg.pinv(X.T @ X) @ (X.T @ y)
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return self
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@ -159,7 +164,7 @@ plt.show()
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#1.4 Model Selection
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# Split the data into training and validation sets
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x_train, x_val, y_train, y_val = train_test_split(x, y_noisy, test_size=0.2, random_state=42)
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x_train, x_val, y_train, y_val = train_test_split(x, y_noisy, test_size=0.3, random_state=100)
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# range of basis functions to test
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D_values = list(range(0, 46)) # 0 to 45
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@ -175,12 +180,9 @@ for D in D_values:
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model = GaussianRegression(sigma=1.0)
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model.fit(x_train, y_train, D)
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# predict on training then validation
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# predict on training and validation set
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yh_train = model.predict(x_train)
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yh_train = yh_train.flatten() if yh_train.ndim > 1 else yh_train
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yh_val = model.predict(x_val)
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yh_val = yh_val.flatten() if yh_val.ndim > 1 else yh_val
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# compute SSE
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sse_train = np.sum((y_train - yh_train)**2)
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@ -189,7 +191,172 @@ for D in D_values:
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train_sse.append(sse_train)
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val_sse.append(sse_val)
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print(f"D={D:2d}: Train SSE = {sse_train:8.2f}, Val SSE = {sse_val:8.2f}")
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print(f"D={D}: Train SSE={sse_train:.0f}, Val SSE={sse_val:.0f}")
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optimal_D = D_values[int(np.argmin(val_sse))]
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print(f"Optimal D on single split = {optimal_D}")
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#optimal_sse = np.min(val_sse)
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#MAYBE CAN ADD A MANUAL LOWER BOUND
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# Plot training and validation SSE vs D for this single split
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plt.figure(figsize=(12, 6))
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plt.plot(D_values, train_sse, 'b-', label='Train SSE', linewidth=2, marker='o', markersize=4)
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plt.plot(D_values, val_sse, 'r-', label='Validation SSE', linewidth=2, marker='s', markersize=4)
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plt.axvline(x=optimal_D, color='g', linestyle='--', label=f'Optimal D = {optimal_D}')
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#plt.scatter([optimal_D], [val_sse[optimal_D]], label=f"Opt D = {optimal_D}", zorder=5)
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plt.xlabel('Number of Gaussian bases (D)')
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plt.ylabel('Sum of Squared Errors (SSE)')
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plt.title('Train and Validation SSE vs D (single split)')
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plt.legend()
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plt.grid(True, alpha=0.3)
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plt.yscale('log')
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plt.show()
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# plot optimal model fit
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plt.figure(figsize=(10, 4))
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optimal_model = GaussianRegression(sigma=1.0)
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yh_opt = optimal_model.fit(x_train, y_train, optimal_D)
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yh_opt = optimal_model.predict(x_plot)
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plt.plot(x_plot, true_function(x_plot), 'b-', label='True Function', linewidth=2)
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plt.plot(x_train, y_train, 'bo', label='Training Data', alpha=0.6, markersize=4)
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plt.plot(x_val, y_val, 'ro', label='Validation Data', alpha=0.6, markersize=4)
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plt.plot(x_plot, yh_opt, 'g-', label=f'Optimal Model (D={optimal_D})', linewidth=2)
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plt.ylim(-6, 6)
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plt.title(f'Optimal Model with {optimal_D} Gaussian Basis Functions')
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plt.ylabel('y')
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plt.legend()
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plt.grid(True, alpha=0.3)
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plt.tight_layout()
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plt.show()
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#__________________________________________________________________________________
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#2. Bias-Variance Tradeoff with Multiple Fits
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#sigma = (x.max() - x.min()) / D
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n_repetitions = 10
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D_values = [0, 5, 7, 10, 12, 15, 20, 25, 30, 45]
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x = np.linspace(0, 10, 300)
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# Initialize arrays to store results
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train_errs = np.zeros((n_repetitions, len(D_values)))
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test_errs = np.zeros((n_repetitions, len(D_values)))
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predictions = np.zeros((n_repetitions, len(D_values), len(x)))
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for rep in range(n_repetitions):
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#create new dataset
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x_data, y_true, y_data = generate_data(100, noise_std=1.0)
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# split into train and test
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x_train, x_test, y_train, y_test = train_test_split(x_data, y_data, test_size=0.3, random_state=rep)
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for D_i, D in enumerate(D_values):
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# fit model
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model = GaussianRegression(sigma=1.0)
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model.fit(x_train, y_train, D)
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# predict on both sets
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yh_train = model.predict(x_train)
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yh_test = model.predict(x_test)
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# compute and store errors (MSE)
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train_err = np.mean((y_train - yh_train)**2)
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test_err = np.mean((y_test - yh_test)**2)
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train_errs[rep, D_i] = train_err
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test_errs[rep, D_i] = test_err
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# predict for visualization
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yh_cont = model.predict(x)
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predictions[rep, D_i, :] = yh_cont
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# Plot 1: fitted models on the same plot, bias-variance tradeoff visualization
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fig, axes = plt.subplots(2, 5, figsize=(25, 10))
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axes = axes.flatten()
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for D_i, D in enumerate(D_values):
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ax = axes[D_i]
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# plot individual fits
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for rep in range(n_repetitions):
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if train_errs[rep, D_i] != np.inf:
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ax.plot(x, predictions[rep, D_i, :], color='green', alpha=0.3, linewidth=1)
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# plot true function
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ax.plot(x, true_function(x), 'b-', linewidth=3, label='True Function')
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#plot average prediction
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valid_predictions = [predictions[rep, D_i, :]
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for rep in range(n_repetitions)
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if train_errs[rep, D_i] != np.inf]
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if valid_predictions:
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avg_prediction = np.mean(valid_predictions, axis=0)
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ax.plot(x, avg_prediction, 'r-', linewidth=2, label='Average Prediction')
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ax.set_title(f'D = {D} Gaussian Bases')
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ax.set_xlabel('x')
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ax.set_ylabel('y')
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ax.set_ylim(-4, 4)
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ax.grid(True, alpha=0.3)
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if D_i == 0:
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ax.legend()
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plt.tight_layout()
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plt.suptitle('Bias-Variance tradeoff with 10 different fits',
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fontsize=16, y=1.02)
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plt.show()
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# Plot 2: average training and test errors
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plt.figure(figsize=(12, 6))
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# Compute mean and std
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avg_train_errors = np.mean(train_errs, axis=0)
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avg_test_errors = np.mean(test_errs, axis=0)
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std_train_errors = np.std(train_errs, axis=0)
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std_test_errors = np.std(test_errs, axis=0)
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# Plot with error bars
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plt.errorbar(D_values, avg_train_errors, yerr=std_train_errors, label='Average Training Error', marker='o', capsize=5, linewidth=2)
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plt.errorbar(D_values, avg_test_errors, yerr=std_test_errors, label='Average Test Error', marker='s', capsize=5, linewidth=2)
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plt.xlabel('Number of Gaussian Basis Functions (D)')
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plt.ylabel('Mean Squared Error')
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plt.title('Average Training and Test Errors Across 10 Repetitions')
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plt.legend()
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plt.grid(True, alpha=0.3)
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plt.yscale('log')
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plt.xticks(D_values)
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plt.show()
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