Added new Directory Ml and implemented basic algos

Author: lck6055

Python/machine_learning/gradient_descent.py

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+import numpy as np
+import matplotlib.pyplot as plt
+
+# Linear Regression using Gradient Descent (Full-Batch and SGD) 
+
+# Dataset
+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])
+
+# Gradient Descent Function
+def gradient_descent(x, y, lr=0.01, epochs=10000, tolerance=1e-6, stochastic=False):
+    b0, b1 = 0, 0
+    n = len(x)
+
+    for epoch in range(epochs):
+        if stochastic:
+            # Stochastic (single 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
+            y_pred = b0 + b1 * x
+            b0_grad = -np.sum(y - y_pred) / n
+            b1_grad = -np.sum((y - y_pred) * x) / n
+
+        # Update parameters
+        b0_new = b0 - lr * b0_grad
+        b1_new = b1 - lr * b1_grad
+
+        # Check convergence
+        if abs(b0_new - b0) < tolerance and abs(b1_new - b1) < tolerance:
+            break
+
+        b0, b1 = b0_new, b1_new
+
+    return b0, b1
+
+
+# Full-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
+
+# Compute 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("=== Gradient Descent (Full-Batch) ===")
+print(f"Intercept (b0): {b0_gd:.4f}, Slope (b1): {b1_gd:.4f}")
+print(f"SSE: {SSE_gd:.4f}, R²: {R2_gd:.4f}")
+
+print("\n=== Gradient Descent (Stochastic) ===")
+print(f"Intercept (b0): {b0_sgd:.4f}, Slope (b1): {b1_sgd:.4f}")
+print(f"SSE: {SSE_sgd:.4f}, R²: {R2_sgd:.4f}")
+
+# Plot comparison
+plt.scatter(x, y, color="blue", label="Data")
+plt.plot(x, y_pred_gd, "g--", label="Batch Gradient Descent")
+plt.plot(x, y_pred_sgd, "m:", label="Stochastic GD")
+plt.xlabel("x")
+plt.ylabel("y")
+plt.legend()
+plt.title("Linear Regression using Gradient Descent (Batch & Stochastic)")
+plt.show()

Python/machine_learning/linear_regression.py

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+import numpy as np
+import matplotlib.pyplot as plt
+
+# --- Linear Regression using Analytical (Closed-form) Solution ---
+
+# Dataset
+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])
+
+# Mean values
+x_mean, y_mean = np.mean(x), np.mean(y)
+
+# Compute coefficients (Closed-form)
+b1 = np.sum((x - x_mean) * (y - y_mean)) / np.sum((x - x_mean)**2)
+b0 = y_mean - b1 * x_mean
+
+# Predictions
+y_pred = b0 + b1 * x
+
+# Compute SSE and R²
+SSE = np.sum((y - y_pred)**2)
+SST = np.sum((y - y_mean)**2)
+R2 = 1 - (SSE / SST)
+
+# Print results
+print("=== Linear Regression (Analytical Solution) ===")
+print(f"Intercept (b0): {b0:.4f}")
+print(f"Slope (b1): {b1:.4f}")
+print(f"SSE: {SSE:.4f}")
+print(f"R²: {R2:.4f}")
+
+# Plot results
+plt.scatter(x, y, color="blue", label="Data")
+plt.plot(x, y_pred, "r-", label="Analytical Solution")
+plt.xlabel("x")
+plt.ylabel("y")
+plt.legend()
+plt.title("Linear Regression - Analytical Solution")
+plt.show()