149 lines
No EOL
5.8 KiB
Python
149 lines
No EOL
5.8 KiB
Python
import numpy as np
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import matplotlib.pyplot as plt
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from numpy.linalg import inv
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# Linear Regression Data Generation
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def generate_data(N, noise_var):
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x = np.random.uniform(0, 1, N)
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noise = np.random.normal(0, np.sqrt(noise_var), N)
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y = np.sin(2 * np.pi * x) + noise
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return x.reshape(-1, 1), y
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# Gaussian Basis is used for the linear regression
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def gaussian_basis(x, D):
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mus = np.linspace(0, 1, D) # means initialized.
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s = 0.1 # standard deviation
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basis = np.exp(-(x - mus)**2 / (2 * s**2)) # Gaussian Formula
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return basis
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# Ridge Regression Formula
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def l2_ridge_regression(x, y, lam):
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I = np.eye(x.shape[1]) # regularization term
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return inv(x.T @ x + lam * I) @ x.T @ y # (X^T*X+λ*I)^−1*X^T*y
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# Lasso Regression Formula
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def l1_lasso_regression(x, y, lam, lr=0.01, iterations=1000):
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w = np.zeros(x.shape[1]) # w initialized the weight
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for i in range(iterations): # iterations
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grad = -x.T @ (y - x @ w) # gradient formula
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w -= lr * grad # gradient descent
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w = np.sign(w) * np.maximum(0, np.abs(w) - lr * lam) # soft thresholding
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return w
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# Cross Validation
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def cross_validate(x, y, lam_values, reg_type, k):
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fold_size = len(x) // k # fold size is defined as 10
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train_errors, val_errors = [], []
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for lam in lam_values:
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train_mse, val_mse = [], []
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for i in range(k): # trained for 9 folds and validated in the remaining with each λ.
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val_idx = np.arange(i * fold_size, (i + 1) * fold_size)
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train_idx = np.setdiff1d(np.arange(len(x)), val_idx)
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x_train, y_train = x[train_idx], y[train_idx]
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x_val, y_val = x[val_idx], y[val_idx]
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# apply the regression
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if reg_type == 'l2':
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w = l2_ridge_regression(x_train, y_train, lam)
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else:
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w = l1_lasso_regression(x_train, y_train, lam)
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train_pred = x_train @ w
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val_pred = x_val @ w
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train_mse.append(np.mean((y_train - train_pred)**2)) # train mse added for each fold
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val_mse.append(np.mean((y_val - val_pred)**2)) # validation mse added for each fold
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train_errors.append(np.mean(train_mse)) # train error added for each λ
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val_errors.append(np.mean(val_mse)) # validation error added for each λ
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return train_errors, val_errors
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# Train and Validation Errors
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def train_validation_err(reg_type, lam_values, num_datasets, N, D):
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if lam_values is None:
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lam_values = np.logspace(-3, 1, 10) # λ is defined
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train_all, val_all = [], []
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for i in range(num_datasets): # inside given 50 datasets
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x, y = generate_data(N, 1.0) # data generated
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Phi = gaussian_basis(x, D) # linear regression with gaussian basis
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train_err, val_err = cross_validate(Phi, y, lam_values, reg_type, 10) # cross-validation training
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train_all.append(train_err)
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val_all.append(val_err)
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mean_train = np.mean(train_all, 0) # mean training
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mean_val = np.mean(val_all, 0) # mean validation
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# plotting the figure
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plt.figure()
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plt.plot(lam_values, mean_train, label='Train Error')
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plt.plot(lam_values, mean_val, label='Validation Error')
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plt.xscale('log')
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plt.xlabel("λ")
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plt.ylabel("MSE")
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plt.title(f"{reg_type.upper()} Regularization")
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plt.legend()
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plt.grid(True)
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plt.savefig('results/task3-train-validation-errors-' + reg_type + '.png')
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return mean_train, mean_val, lam_values
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# Bias-Variance Decomposition
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def bias_variance_decomp(reg_type, lam_values, num_datasets, N, D):
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if lam_values is None:
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lam_values = np.logspace(-3, 1, 10) # λ is defined
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x_test = np.linspace(0, 1, 100).reshape(-1, 1) # x test value defined
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Phi_test = gaussian_basis(x_test, D) # linear regression with gaussian basis
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y_true = np.sin(2 * np.pi * x_test).ravel() # sin(2*π*x)
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biases, variances, total_mse = [], [], []
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for lam in lam_values:
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preds = []
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for i in range(num_datasets):
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x_train, y_train = generate_data(N, 1.0) # data generated
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Phi_train = gaussian_basis(x_train, D) # linear regression with gaussian basis
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# apply the regression
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if reg_type == 'l2':
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w = l2_ridge_regression(Phi_train, y_train, lam)
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else:
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w = l1_lasso_regression(Phi_train, y_train, lam)
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preds.append(Phi_test @ w)
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preds = np.array(preds)
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mean_pred = np.mean(preds, 0) # mean of predictions
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bias2 = np.mean((mean_pred - y_true) ** 2) # squared bias formula
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# bias is defined as mean of difference between predicted mean and true y value.
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var = np.mean(np.var(preds, 0)) # mean variance of predictions
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mse = bias2 + var + 1 # noise variance
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biases.append(bias2) # add bias for each λ
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variances.append(var) # add variance for each λ
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total_mse.append(mse) # add total mse for each λ
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# plotting the figure
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plt.figure()
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plt.plot(lam_values, biases, label='Bias^2')
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plt.plot(lam_values, variances, label='Variance')
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plt.plot(lam_values, np.array(biases) + np.array(variances), label='Bias^2 + Var')
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plt.plot(lam_values, total_mse, label='Bias^2 + Var + Noise')
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plt.xscale('log')
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plt.xlabel("λ")
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plt.ylabel("Error")
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plt.title(f"{reg_type.upper()} Bias-Variance Decomposition")
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plt.legend()
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plt.grid(True)
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plt.savefig('results/task3-bias-decomposition-' + reg_type + '.png')
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if __name__ == "__main__":
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print("Running Task 3: Regularization with Cross-Validation")
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lam_values = np.logspace(-3, 1, 10) # initial λ values -3, 1, 10
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train_validation_err('l2', lam_values, 50, 20, 45)
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train_validation_err('l1', lam_values, 50, 20, 45)
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bias_variance_decomp('l2', lam_values, 50, 20, 45)
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bias_variance_decomp('l1', lam_values, 50, 20, 45) |