updated linear_regression

Author: lck6055

Python/machine_learning/linear_regression.py

@@ -2,34 +2,54 @@
 Algorithm: Linear Regression (Analytical / Closed-form Solution)
 
 Description:
-This script implements Linear Regression using the closed-form solution (Analytical method).
-It calculates the best-fit line for a dataset by directly computing the slope (b1) and intercept (b0)
-using the formulas derived from minimizing the Mean Squared Error (MSE).
+This script implements Linear Regression using the closed-form solution.
+It calculates the best-fit line by directly computing the slope (b1) and intercept (b0)
+using formulas derived from minimizing the Mean Squared Error (MSE).
 
-Mathematical Formulation:
-    y_pred = b0 + b1 * x
-    b1 = Σ((x - mean(x)) * (y - mean(y))) / Σ((x - mean(x))^2)
-    b0 = mean(y) - b1 * mean(x)
-
-Time Complexity: O(n)        # Single pass through the data
+Time Complexity: O(n)        # Single pass through data
 Space Complexity: O(1)       # Only a few variables stored
 """
 
 import numpy as np
 import matplotlib.pyplot as plt
 
-# Function for Analytical Linear Regression
+# --- Function for Analytical Linear Regression ---
 def linear_regression_analytical(x, y):
+    """
+    Computes linear regression coefficients using the analytical method.
+    
+    Parameters:
+        x (np.ndarray): Feature values
+        y (np.ndarray): Target values
+        
+    Returns:
+        tuple: b0 (intercept), b1 (slope), y_pred (predictions), SSE, R²
+    """
+    # Compute mean of x and y
     x_mean, y_mean = np.mean(x), np.mean(y)
+
+    # Compute slope (b1) using formula
     b1 = np.sum((x - x_mean) * (y - y_mean)) / np.sum((x - x_mean)**2)
+
+    # Compute intercept (b0) using formula
     b0 = y_mean - b1 * x_mean
+
+    # Compute predicted y values
     y_pred = b0 + b1 * x
+
+    # Compute Sum of Squared Errors (SSE)
     SSE = np.sum((y - y_pred)**2)
+
+    # Compute Total Sum of Squares (SST) for R²
     SST = np.sum((y - y_mean)**2)
+
+    # Compute R² score (coefficient of determination)
     R2 = 1 - (SSE / SST)
+
     return b0, b1, y_pred, SSE, R2
 
-# Test Cases
+
+# --- Test Cases ---
 test_cases = {
     "Simple Linear": {
         "x": np.array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9]),
@@ -49,19 +69,24 @@ def linear_regression_analytical(x, y):
     }
 }
 
+# Loop through each test case
 for name, data in test_cases.items():
     print(f"\n=== {name} ===")
     x, y = data["x"], data["y"]
+
+    # Compute linear regression using analytical method
     b0, b1, y_pred, SSE, R2 = linear_regression_analytical(x, y)
+
+    # Print coefficients and metrics
     print(f"Intercept (b0): {b0:.4f}, Slope (b1): {b1:.4f}")
     print(f"SSE: {SSE:.4f}, R²: {R2:.4f}")
 
-    # Plot
+    # Plot data points and fitted line
     plt.figure(figsize=(6, 4))
-    plt.scatter(x, y, color="blue", label="Data Points")
-    plt.plot(x, y_pred, "r-", label="Analytical Solution")
-    plt.xlabel("x")
-    plt.ylabel("y")
+    plt.scatter(x, y, color="blue", label="Data Points")           # Original data points
+    plt.plot(x, y_pred, "r-", label="Analytical Solution")        # Fitted line
+    plt.xlabel("x")                                               # x-axis label
+    plt.ylabel("y")                                               # y-axis label
     plt.legend()
-    plt.title(f"Linear Regression - {name}")
+    plt.title(f"Linear Regression - {name}")                      # Plot title
     plt.show()