chatgpt is bit worse today. We dont know what it went through. We can only hope for the best.
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import matplotlib.pyplot as plt
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import numpy as np
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def plot_cost_function(costs, iterations):
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"""Plot the cost function over iterations."""
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plt.figure(figsize=(8, 6))
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plt.plot(range(iterations), costs, label="Cost Function")
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plt.xlabel("Iterations")
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plt.ylabel("Cost")
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plt.title("Cost Function over Iterations")
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plt.legend()
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plt.grid(True)
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plt.show()
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def plot_data_with_regression_line(X, y, theta):
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"""Plot the data points and the regression line."""
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plt.figure(figsize=(8, 6))
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plt.scatter(X[:, 1], y, color="blue", label="Data Points")
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regression_line = np.dot(X, theta)
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plt.plot(X[:, 1], regression_line, color="red", label="Regression Line")
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plt.xlabel("Feature")
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plt.ylabel("Target")
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plt.title("Data and Regression Line")
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plt.legend()
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plt.grid(True)
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plt.show()
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def plot_predictions_vs_actual(y_actual, y_predicted):
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"""Plot predicted values against actual values."""
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plt.figure(figsize=(8, 6))
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plt.scatter(y_actual, y_predicted, color="green", label="Predictions vs Actual")
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plt.plot(
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[min(y_actual), max(y_actual)],
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[min(y_actual), max(y_actual)],
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color="red",
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label="Ideal Fit",
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)
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plt.xlabel("Actual Values")
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plt.ylabel("Predicted Values")
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plt.title("Predictions vs Actual Values")
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plt.legend()
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plt.grid(True)
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plt.show()
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def plot_residuals(y_actual, y_predicted):
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"""Plot residuals (errors) to evaluate model fit."""
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residuals = y_actual - y_predicted
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plt.figure(figsize=(8, 6))
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plt.scatter(y_predicted, residuals, color="purple", label="Residuals")
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plt.axhline(0, color="red", linestyle="--", label="Zero Error Line")
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plt.xlabel("Predicted Values")
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plt.ylabel("Residuals")
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plt.title("Residual Plot")
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plt.legend()
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plt.grid(True)
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plt.show()
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def plot_cost_comparison(costs_dict, iterations):
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"""Plot cost functions for multiple methods on the same graph."""
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plt.figure(figsize=(8, 6))
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for method, costs in costs_dict.items():
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plt.plot(range(iterations), costs, label=f"{method} Cost")
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plt.xlabel("Iterations")
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plt.ylabel("Cost")
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plt.title("Cost Function Comparison")
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plt.legend()
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plt.grid(True)
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plt.show()
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@@ -0,0 +1,36 @@
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import pandas as pd
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import numpy as np
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def load_data(filename, verbose=False, drop_unnamed=True):
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try:
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# Load the dataset
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df = pd.read_csv(
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filename, dtype=np.float64
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) # Force float64 for higher precision
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if verbose:
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print("Data loaded successfully.")
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print("Data types:\n", df.dtypes)
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# Drop unnamed/index columns if specified
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if drop_unnamed:
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df = df.loc[:, ~df.columns.str.contains("^Unnamed")]
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if verbose:
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print("Dropped unnamed columns.")
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# Features (X) = all but last column, Target (y) = last column
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X = df.iloc[:, :-1].values
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y = df.iloc[:, -1].values
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if verbose:
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print("Shape of X:", X.shape)
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print("Shape of y:", y.shape)
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return X, y
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except FileNotFoundError:
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print(f"Error: File '{filename}' not found.")
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raise
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except Exception as e:
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print(f"An error occurred: {e}")
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raise
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+125
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import numpy as np
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# Hypothesis function
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def hypothesis(X, theta):
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"""
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Compute the hypothesis (predictions) for the given input and parameters.
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"""
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result = np.dot(X, theta).reshape(-1, 1)
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# print("Shape of X:", X.shape)
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# print("Shape of theta:", theta.shape)
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# print("Shape of hypothesis output:", result.shape)
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return result
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# Cost function (Mean Squared Error)
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def compute_cost(X, y, theta):
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"""
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Compute the cost (Mean Squared Error) for the given input, target, and parameters.
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"""
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m = len(y)
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predictions = hypothesis(X, theta)
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errors = predictions - y
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cost = (1 / (2 * m)) * np.sum(errors**2)
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# # print(f"Max error: {np.max(errors):.4f}, Cost: {cost:.4f}") # Debugging
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return cost
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# Compute gradients
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def compute_gradients(theta, X, y):
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"""
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Compute gradients for gradient descent.
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Parameters:
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theta (numpy.ndarray): Model parameters (n,).
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X (numpy.ndarray): Feature matrix (m, n).
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y (numpy.ndarray): Target values (m, 1).
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Returns:
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numpy.ndarray: Gradients (n,).
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"""
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# print("Shape of theta, X, y:", theta.shape, X.shape, y.shape)
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m = len(y)
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errors = hypothesis(X, theta) - y # Compute prediction errors
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# print("Shape of errors:", errors.shape) # Debugging
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gradients = (1 / m) * np.dot(X.T, errors) # Compute gradients
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# print("Shape of gradients:", gradients.shape) # Debugging
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return gradients.flatten()
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# Gradient Descent function (Batch)
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def gradient_descent(X, y, theta, learning_rate, iterations, return_cost: bool = False):
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"""
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Perform batch gradient descent to optimize theta.
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"""
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if return_cost:
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costs = []
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for _i in range(iterations):
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# print("Shape of X:", X.shape) # Should be (200, 4)
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# print("Shape of y:", y.shape) # Should be (200, 1)
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# print("Shape of theta:", theta.shape) # Should be (4,)
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gradients = compute_gradients(theta, X, y) # Compute gradients
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# print(f"Iteration {i + 1}:")
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# print(f" Shape of theta: {theta.shape}")
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# print(f" Shape of gradients: {gradients.shape}")
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theta -= learning_rate * gradients # Update theta
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if return_cost:
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cost = compute_cost(X, y, theta)
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costs.append(cost) # Track cost
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# print(f" Cost: {cost:.4f}")
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return (theta, costs) if return_cost else theta
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# Mini-batch Gradient Descent function
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def mini_batch_gradient_descent(
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X, y, theta, learning_rate, iterations, batch_size, return_cost: bool = False
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):
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"""
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Perform mini-batch gradient descent to optimize theta.
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"""
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m = len(y)
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if return_cost:
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costs = []
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for _i in range(iterations):
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shuffle_indices = np.random.permutation(m)
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X_shuffled = X[shuffle_indices]
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y_shuffled = y[shuffle_indices]
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for j in range(0, m, batch_size):
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X_batch = X_shuffled[j : j + batch_size]
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y_batch = y_shuffled[j : j + batch_size]
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gradients = compute_gradients(theta, X_batch, y_batch) # Compute gradients
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theta -= learning_rate * gradients # Update theta
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if return_cost:
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cost = compute_cost(X, y, theta)
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costs.append(cost) # Track cost
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# print(f"Iteration {i + 1}/{iterations}, Cost: {cost:.4f}") # Debugging
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return (theta, costs) if return_cost else theta
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# Stochastic Gradient Descent function
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def stochastic_gradient_descent(
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X, y, theta, learning_rate, iterations, return_cost: bool = False
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):
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"""
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Perform stochastic gradient descent to optimize theta.
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"""
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m = len(y)
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if return_cost:
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costs = []
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for _i in range(iterations):
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for j in range(m):
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X_sample = X[j : j + 1]
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y_sample = y[j : j + 1]
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gradients = compute_gradients(
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theta, X_sample, y_sample
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) # Compute gradients
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theta -= learning_rate * gradients # Update theta
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if return_cost:
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cost = compute_cost(X, y, theta)
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costs.append(cost) # Track cost
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# print(f"Iteration {i + 1}/{iterations}, Cost: {cost:.4f}") # Debugging
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return (theta, costs) if return_cost else theta
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+299
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{
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"cells": [
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{
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"cell_type": "markdown",
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"id": "ee8e9389",
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"metadata": {},
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"source": [
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"# no way\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "fa275d88",
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"metadata": {},
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"outputs": [],
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"source": [
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"import warnings\n",
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"\n",
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"warnings.filterwarnings(\"error\")"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "0d63f004",
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"Data loaded successfully.\n",
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"Data types:\n",
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" Unnamed: 0 float64\n",
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"TV float64\n",
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"radio float64\n",
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"newspaper float64\n",
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"sales float64\n",
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"dtype: object\n",
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"Dropped unnamed columns.\n",
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"Shape of X: (200, 3)\n",
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"Shape of y: (200,)\n",
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"Shape of y before reshaping: (200,)\n",
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"Shape of y after reshaping: (200, 1)\n",
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"Shape of X after adding bias term: (200, 4)\n"
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]
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}
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],
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"source": [
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"# Load the data\n",
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"import file_reader\n",
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"\n",
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"filename = \"Advertising.csv\"\n",
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"X, y = file_reader.load_data(filename, verbose=True)\n",
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"\n",
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"# print(\"Shape of X:\", X.shape)\n",
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"# print(\"Shape of y:\", y.shape)\n",
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"\n",
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"print(\"Shape of y before reshaping:\", y.shape)\n",
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"y = y.reshape(-1, 1)\n",
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"print(\"Shape of y after reshaping:\", y.shape)\n",
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"# Add a bias term to X (for the intercept)\n",
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"import numpy as np\n",
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"\n",
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"\n",
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"import regulations\n",
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"\n",
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"X_scaled = regulations.standardize(X)\n",
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"## Add bias term (column of 1s)\n",
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"# X = np.hstack([np.ones((X_scaled.shape[0], 1)), X_scaled])\n",
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"X = np.c_[np.ones(X_scaled.shape[0]), X_scaled]\n",
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"\n",
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"\n",
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"print(\"Shape of X after adding bias term:\", X.shape)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"id": "0fa06674",
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"Shape of theta: (4,)\n"
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]
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}
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],
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"source": [
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"# Initialize parameters\n",
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"theta = np.zeros(X.shape[1]) # Initialize theta with zeros\n",
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"# Reshape theta to ensure proper alignment\n",
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"# theta = theta.reshape(-1, 1)\n",
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"print(\"Shape of theta:\", theta.shape)\n",
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"\n",
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"learning_rate = 0.001\n",
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"iterations = 1000\n",
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"batch_size = 32 # For mini-batch gradient descent"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 8,
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"id": "8d5bf6a6",
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"Theta batch: [ 1.40204919e+01 3.91562216e+00 2.79885945e+00 -1.09326135e-02]\n",
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"Theta (mini_batch): [ 1.40204919e+01 3.91562216e+00 2.79885945e+00 -1.09326135e-02]\n",
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"Theta (stochastic): [ 1.40204919e+01 3.91562216e+00 2.79885945e+00 -1.09326135e-02]\n"
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]
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}
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],
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"source": [
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"# Import gradient functions\n",
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"import gradient\n",
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"\n",
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"# Run batch gradient descent and track costs\n",
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"theta_batch, costs_batch = gradient.gradient_descent(\n",
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" X, y, theta, learning_rate, iterations, return_cost=True\n",
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")\n",
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"\n",
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"# Run mini-batch gradient descent and track costs\n",
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"theta_mini_batch, costs_mini_batch = gradient.mini_batch_gradient_descent(\n",
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" X, y, theta, learning_rate, iterations, batch_size, return_cost=True\n",
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")\n",
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"\n",
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"# Run stochastic gradient descent and track costs\n",
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"theta_stochastic, costs_stochastic = gradient.stochastic_gradient_descent(\n",
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" X, y, theta, learning_rate, iterations, return_cost=True\n",
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")\n",
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"print(\"Theta batch:\", theta_batch)\n",
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"print(\"Theta (mini_batch):\", theta_mini_batch)\n",
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"print(\"Theta (stochastic):\", theta_stochastic)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"id": "f9d6d9bf",
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"Theta (Normal Equation): [[14.0225 ]\n",
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" [ 3.91925365]\n",
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" [ 2.79206274]\n",
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" [-0.02253861]]\n"
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]
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}
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],
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"source": [
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"# Solve using the improved normal equation\n",
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"import normal\n",
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"\n",
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"theta_normal = normal.normal_equation(X, y)\n",
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"print(\"Theta (Normal Equation):\", theta_normal)"
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]
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},
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{
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||||
"cell_type": "code",
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||||
"execution_count": 6,
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||||
"id": "5becf82a",
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||||
"metadata": {
|
||||
"id": "7cf220ac",
|
||||
"language": "python"
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||||
},
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"outputs": [
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||||
{
|
||||
"data": {
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||||
"image/png": 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|
||||
"text/plain": [
|
||||
"<Figure size 800x600 with 1 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
"output_type": "display_data"
|
||||
},
|
||||
{
|
||||
"data": {
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||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 800x600 with 1 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
"output_type": "display_data"
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 800x600 with 1 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
"output_type": "display_data"
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 800x600 with 1 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
"output_type": "display_data"
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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|
||||
"text/plain": [
|
||||
"<Figure size 800x600 with 1 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
"output_type": "display_data"
|
||||
},
|
||||
{
|
||||
"data": {
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||||
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|
||||
"text/plain": [
|
||||
"<Figure size 800x600 with 1 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
"output_type": "display_data"
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": "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 truncated
|
||||
"text/plain": [
|
||||
"<Figure size 800x600 with 1 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
"output_type": "display_data"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"# Import display functions\n",
|
||||
"import display\n",
|
||||
"%matplotlib inline\n",
|
||||
"\n",
|
||||
"# Plot cost functions for each gradient descent method\n",
|
||||
"display.plot_cost_function(costs_batch, iterations)\n",
|
||||
"display.plot_cost_function(costs_mini_batch, iterations)\n",
|
||||
"display.plot_cost_function(costs_stochastic, iterations)\n",
|
||||
"\n",
|
||||
"# Compare cost functions across methods\n",
|
||||
"costs_dict = {\n",
|
||||
" 'Batch': costs_batch,\n",
|
||||
" 'Mini-batch': costs_mini_batch,\n",
|
||||
" 'Stochastic': costs_stochastic\n",
|
||||
"}\n",
|
||||
"display.plot_cost_comparison(costs_dict, iterations)\n",
|
||||
"\n",
|
||||
"# Plot data with regression line (Batch Gradient Descent)\n",
|
||||
"display.plot_data_with_regression_line(X, y, theta_batch)\n",
|
||||
"\n",
|
||||
"# Compare predictions vs actual values (Batch Gradient Descent)\n",
|
||||
"y_pred_batch = np.dot(X, theta_batch).reshape(-1, 1)\n",
|
||||
"display.plot_predictions_vs_actual(y, y_pred_batch)\n",
|
||||
"\n",
|
||||
"# Plot residuals (Batch Gradient Descent)\n",
|
||||
"display.plot_residuals(y, y_pred_batch)"
|
||||
]
|
||||
}
|
||||
],
|
||||
"metadata": {
|
||||
"kernelspec": {
|
||||
"display_name": "base",
|
||||
"language": "python",
|
||||
"name": "python3"
|
||||
},
|
||||
"language_info": {
|
||||
"codemirror_mode": {
|
||||
"name": "ipython",
|
||||
"version": 3
|
||||
},
|
||||
"file_extension": ".py",
|
||||
"mimetype": "text/x-python",
|
||||
"name": "python",
|
||||
"nbconvert_exporter": "python",
|
||||
"pygments_lexer": "ipython3",
|
||||
"version": "3.12.9"
|
||||
}
|
||||
},
|
||||
"nbformat": 4,
|
||||
"nbformat_minor": 5
|
||||
}
|
||||
+12
@@ -0,0 +1,12 @@
|
||||
import numpy as np
|
||||
|
||||
|
||||
def normal_equation(X, y):
|
||||
"""
|
||||
Solve linear regression using the normal equation:
|
||||
θ = (X^T X)^(-1) X^T y
|
||||
Efficiently implemented using np.linalg.solve to avoid explicit matrix inversion.
|
||||
"""
|
||||
X_transpose = np.transpose(X)
|
||||
theta = np.linalg.solve(X_transpose @ X, X_transpose @ y)
|
||||
return theta
|
||||
+67
-8
@@ -13,29 +13,35 @@
|
||||
3. Redo exercise 1 with regularization
|
||||
4. Redo exercise 2 with regularization
|
||||
|
||||
## plan
|
||||
## plan for 1
|
||||
|
||||
what the fuck is a Linear regression?
|
||||
|
||||
it is to my knowledge: fit a thing to a linear equation, with the Minimal Error.
|
||||
|
||||
### the data
|
||||
|
||||
lets say the data has m rows. and its linear regression. pretty cool how you can train on all the data at the same time.
|
||||
|
||||
this is still basic ml after all
|
||||
|
||||
### error function, cost function: the thig we want reduced
|
||||
|
||||
this is the cost function, also loss function:
|
||||
|
||||
$$
|
||||
E = \frac{1}{2m} \sum^m_{i = 1}\left(h(x_{(i)})- y_{(i)}\right)^2
|
||||
E = \frac{1}{2m} \sum^m_{i = 1}\left(h(x^{(i)})- y^{(i)}\right)^2
|
||||
$$
|
||||
|
||||
with $h(x)$ being our model (we are trying to find a and b and make our guess $h(x)$ as close to $y$ as possible)
|
||||
|
||||
this comes from the error function here:
|
||||
|
||||
$$e = \frac{1}{2}\left(h(x_{(i)})-y_{(i)}\right)^2$$
|
||||
$$e = \frac{1}{2}\left(h(x^{(i)})-y^{(i)}\right)^2$$
|
||||
|
||||
and the cost function from the error function:
|
||||
|
||||
$$E = \frac{1}{m}\sum^m_{i=1}e_{(i)}$$
|
||||
$$E = \frac{1}{m}\sum^m_{i=1}e^{(i)}$$
|
||||
|
||||
as were dealing with linear fitting ($h(x) = ax + b$) the thing turns to this:
|
||||
|
||||
@@ -64,13 +70,66 @@ however they did the calculations:
|
||||
for our case $h(x) = ax + b$ we have
|
||||
|
||||
$$
|
||||
\frac{\partial}{\partial a}{E(a, b)}
|
||||
\text{ or } \\
|
||||
\frac{\partial E}{\partial a} = \frac{1}{m}\sum^m_{i=1}(ax_{(i)}+b - y_{(i)})\cdot x_{(i)}
|
||||
\begin{align*}
|
||||
&\frac{\partial}{\partial a}{E(a, b)} &&= \\
|
||||
&\frac{\partial E}{\partial a} &&= \frac{1}{m}\sum^m_{i=1}(ax^{(i)}+b - y^{(i)})\cdot x^{(i)}
|
||||
\end{align*}
|
||||
$$
|
||||
|
||||
yeah also for the case of b:
|
||||
|
||||
$$b:= b - \alpha \frac{\partial E}{\partial b}$$
|
||||
|
||||
$$\frac{\partial E}{\partial b} = \frac{1}{m}\sum^m_{i=1}(ax_{(i)}+b - y_{(i)})$$
|
||||
$$\frac{\partial E}{\partial b} = \frac{1}{m}\sum^m_{i=1}(ax^{(i)}+b - y^{(i)})$$
|
||||
|
||||
note that that $\alpha$ is important. its called learning rate.
|
||||
|
||||
### matrix rep. we go big
|
||||
|
||||
but now when the columns are many and the rows are also many, what do we do?
|
||||
|
||||
it is very convenient that the linear can be represented by a vector of x times a vector of coefficients (weights):
|
||||
|
||||
$$
|
||||
\theta = \begin{vmatrix}\theta_0 \\ \theta_1 \\ \cdots \\ \theta_n \end{vmatrix} \text{ and }x = \begin{vmatrix}1 \\ x_1 \\ \cdots \\ x_n \end{vmatrix}\\
|
||||
\implies h(x) = \theta^Tx
|
||||
$$
|
||||
|
||||
cost fucntion here:
|
||||
|
||||
$$
|
||||
E = \frac{1}{2m} \sum^m_{i = 1}\left(\theta^Tx^{(i)}- y^{(i)}\right)^2
|
||||
$$
|
||||
|
||||
and a very generalized gradient descent!!!
|
||||
|
||||
$$
|
||||
\theta_j:= \theta_j - \alpha \frac{\partial E}{\partial \theta_j}\\
|
||||
\frac{\partial E}{\partial \theta_j} = \frac{1}{m}\sum^m_{i=1}(\theta^Tx^{(i)} - y^{(i)})\cdot x^{(i)}_j
|
||||
$$
|
||||
|
||||
thats insane! also very codable! we already has dot product of 2 vectors `np.dot`, and that is for $\theta^Tx^{(i)}$. the entire table of x across m versions is just a 2d table.
|
||||
|
||||
so we could plug this into a
|
||||
|
||||
into a chatgpt LMAO aint no way doing this by hand
|
||||
|
||||
### ok when are we done?
|
||||
|
||||
so its an iterative method where you find a and b and all the other things at the same time using that. and again. and again.
|
||||
|
||||
the way we could turn off is jus check if E or e is small enough or we are looping too much
|
||||
|
||||
### any way to not ruin our computers?
|
||||
|
||||
this does NOT harm your computers. it requires NOTHING to run.
|
||||
|
||||
but see this. the m can be anything. it can be one. 64. the whole table.
|
||||
|
||||
that is Stochastic, Mini-batch and Batch. crazy. think about it.
|
||||
|
||||
## plan for 2
|
||||
|
||||
when i calculate $\frac{\partial E}{\partial \theta} = 0$ directly
|
||||
$$ \theta = (X^T X)^{-1} X^T Y$$
|
||||
???
|
||||
@@ -0,0 +1,29 @@
|
||||
def standardize_df(df):
|
||||
"""
|
||||
Standardize the input DataFrame.
|
||||
|
||||
Parameters:
|
||||
df (pandas.DataFrame): The input data.
|
||||
|
||||
Returns:
|
||||
pandas.DataFrame: The standardized data.
|
||||
"""
|
||||
return (df - df.mean()) / df.std()
|
||||
|
||||
|
||||
import numpy as np
|
||||
|
||||
|
||||
def standardize(X):
|
||||
"""
|
||||
Standardize the input features.
|
||||
|
||||
Parameters:
|
||||
X (numpy.ndarray): The input data matrix (m x n).
|
||||
|
||||
Returns:
|
||||
numpy.ndarray: The standardized data matrix.
|
||||
"""
|
||||
mean = np.mean(X, axis=0)
|
||||
std = np.std(X, axis=0)
|
||||
return (X - mean) / std
|
||||
Reference in new issue
Block a user