Added new Directory Ml and implemented basic algos

Python/machine_learning/gradient_descent.py

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+"""
+Algorithm: Linear Regression using Gradient Descent (Batch & Stochastic)
+
+Description:
+This script implements Linear Regression using both Batch Gradient Descent (BGD) 
+and Stochastic Gradient Descent (SGD).
+
+Linear Regression fits a straight line to data points (x, y) by minimizing 
+the Mean Squared Error (MSE) between the predicted and actual values.
+
+Gradient Descent iteratively adjusts parameters (b0 - intercept, b1 - slope)
+in the opposite direction of the gradient of the loss function.
+
+Mathematical Formulation:
+    y_pred = b0 + b1 * x
+    Loss (MSE) = (1/n) * Σ(y - y_pred)²
+    Gradients:
+        ∂L/∂b0 = -(2/n) * Σ(y - y_pred)
+        ∂L/∂b1 = -(2/n) * Σ(x * (y - y_pred))
+
+Variants:
+- Batch Gradient Descent: Uses all samples per iteration (stable but slower)
+- Stochastic Gradient Descent: Uses one random sample per iteration (faster but noisier)
+
+Time Complexity:
+    - Batch Gradient Descent: O(n * epochs)
+    - Stochastic Gradient Descent: O(epochs)
+Space Complexity: O(1)
+"""
+
+import numpy as np
+import matplotlib.pyplot as plt
+
+
+def gradient_descent(x, y, lr=0.01, epochs=10000, tolerance=1e-6, stochastic=False):
+    """
+    Performs linear regression using Gradient Descent.
+
+    Parameters:
+        x (np.ndarray): Feature values
+        y (np.ndarray): Target values
+        lr (float): Learning rate
+        epochs (int): Maximum iterations
+        tolerance (float): Convergence threshold
+        stochastic (bool): If True, performs Stochastic Gradient Descent
+
+    Returns:
+        tuple: (b0, b1) - Intercept and Slope
+    """
+    b0, b1 = 0, 0
+    n = len(x)
+
+    for epoch in range(epochs):
+        if stochastic:
+            # Stochastic Gradient Descent (single random sample)
+            i = np.random.randint(0, n)
+            xi, yi = x[i], y[i]
+            y_pred = b0 + b1 * xi
+            b0_grad = -(yi - y_pred)
+            b1_grad = -(yi - y_pred) * xi
+        else:
+            # Batch Gradient Descent (uses full dataset)
+            y_pred = b0 + b1 * x
+            b0_grad = -np.sum(y - y_pred) / n
+            b1_grad = -np.sum((y - y_pred) * x) / n
+
+        # Parameter updates
+        b0_new = b0 - lr * b0_grad
+        b1_new = b1 - lr * b1_grad
+
+        # Convergence check
+        if abs(b0_new - b0) < tolerance and abs(b1_new - b1) < tolerance:
+            print(f"✅ Converged after {epoch} epochs")
+            break
+
+        b0, b1 = b0_new, b1_new
+
+    return b0, b1
+
+
+# Example Test Cases
+if __name__ == "__main__":
+    test_cases = {
+        "Case 1: Simple Linear": {
+            "x": np.array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9]),
+            "y": np.array([1, 3, 2, 5, 7, 8, 8, 9, 10, 12])
+        },
+        "Case 2: Perfectly Linear": {
+            "x": np.arange(0, 10),
+            "y": 3 * np.arange(0, 10) + 2  # y = 3x + 2
+        },
+        "Case 3: Negative Slope": {
+            "x": np.arange(0, 10),
+            "y": 20 - 2 * np.arange(0, 10)  # y = -2x + 20
+        },
+        "Case 4: Random Noise Added": {
+            "x": np.arange(0, 10),
+            "y": 5 * np.arange(0, 10) + np.random.normal(0, 3, 10)  # y = 5x + noise
+        }
+    }
+
+    for name, data in test_cases.items():
+        print(f"\n=== {name} ===")
+        x, y = data["x"], data["y"]
+
+        # Batch Gradient Descent
+        b0_gd, b1_gd = gradient_descent(x, y, lr=0.01, epochs=10000, stochastic=False)
+        y_pred_gd = b0_gd + b1_gd * x
+
+        # Stochastic Gradient Descent
+        b0_sgd, b1_sgd = gradient_descent(x, y, lr=0.01, epochs=10000, stochastic=True)
+        y_pred_sgd = b0_sgd + b1_sgd * x
+
+        # Metrics
+        SST = np.sum((y - np.mean(y))**2)
+        SSE_gd = np.sum((y - y_pred_gd)**2)
+        SSE_sgd = np.sum((y - y_pred_sgd)**2)
+        R2_gd = 1 - (SSE_gd / SST)
+        R2_sgd = 1 - (SSE_sgd / SST)
+
+        # Print results
+        print(f"Batch GD: Intercept={b0_gd:.4f}, Slope={b1_gd:.4f}, R²={R2_gd:.4f}")
+        print(f"SGD:      Intercept={b0_sgd:.4f}, Slope={b1_sgd:.4f}, R²={R2_sgd:.4f}")
+
+        # Visualization
+        plt.figure(figsize=(6, 4))
+        plt.scatter(x, y, color="blue", label="Data Points")
+        plt.plot(x, y_pred_gd, "g--", label="Batch GD")
+        plt.plot(x, y_pred_sgd, "m:", label="Stochastic GD")
+        plt.xlabel("x")
+        plt.ylabel("y")
+        plt.legend()
+        plt.title(f"{name}")
+        plt.show()

Python/machine_learning/linear_regression.py

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+"""
+Algorithm: Linear Regression (Analytical / Closed-form Solution)
+
+Description:
+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).
+
+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 ---
+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 = {
+    "Simple Linear": {
+        "x": np.array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9]),
+        "y": np.array([1, 3, 2, 5, 7, 8, 8, 9, 10, 12])
+    },
+    "Perfectly Linear": {
+        "x": np.arange(0, 10),
+        "y": 3 * np.arange(0, 10) + 2  # y = 3x + 2
+    },
+    "Negative Slope": {
+        "x": np.arange(0, 10),
+        "y": 20 - 2 * np.arange(0, 10)  # y = -2x + 20
+    },
+    "Noisy Data": {
+        "x": np.arange(0, 10),
+        "y": 5 * np.arange(0, 10) + np.random.normal(0, 3, 10)  # y = 5x + noise
+    }
+}
+
+# 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 data points and fitted line
+    plt.figure(figsize=(6, 4))
+    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}")                      # Plot title
+    plt.show()