ass2(non-linear regression)

 import pandas as pd 

import numpy as np 

import matplotlib.pyplot as plt 

 

df = pd.read_csv('china_gdp.csv') 

print(df.head()) 

 

# Extract data 

x_data = df["Year"].values 

y_data = df["Value"].values 

 

# Normalize 

x = x_data / max(x_data) 

y = y_data / max(y_data) 

# Logistic function 

def sigmoid(x, L, k, x0): 

    return L / (1 + np.exp(-k * (x - x0))) 

 

# Gradient descent for non-linear model 

def gradient_descent(x, y, L, k, x0, alpha, iterations): 

    m = len(x) 

    cost_history = [] 

    for i in range(iterations): 

        y_pred = sigmoid(x, L, k, x0) 

        error = y_pred - y 

        cost = np.sum(error ** 2) / (2 * m) 

        cost_history.append(cost) 

 

        # Compute gradients numerically (simplified) 

        dL = np.sum(error * (1 / (1 + np.exp(-k * (x - x0))))) / m 

        dk = np.sum(error * L * (x - x0) * np.exp(-k * (x - x0)) / ((1 + np.exp(-k * (x - x0))) 

** 2)) / m 

        dx0 = np.sum(error * L * k * np.exp(-k * (x - x0)) / ((1 + np.exp(-k * (x - x0))) ** 2)) 

/ m 

 

        # Update parameters 

        L -= alpha * dL 

        k -= alpha * dk 

        x0 -= alpha * dx0 

        # Convergence condition 

        if i > 1 and abs(cost_history[-1] - cost_history[-2]) < 1e-6: 

            break 

    return L, k, x0, cost_history 

 

# Initialize parameters 

L = 1 

k = 1 

x0 = 0.5 

alpha = 0.1 

iterations = 10000 

L_final, k_final, x0_final, cost_history = gradient_descent(x, y, L, k, x0, alpha, iterations) 

print(f"Optimal parameters: L={L_final}, k={k_final}, x0={x0_final}") 

 

 

 

# Predict using the trained model 

x_range = np.linspace(min(x), max(x), 100) 

y_pred = sigmoid(x_range, L_final, k_final, x0_final) 

 

# Convert back to original scale 

plt.scatter(x_data, y_data, label='Original Data') 

plt.plot(x_range * max(x_data), y_pred * max(y_data), color='red', label='Fitted Logistic Curve') 

plt.xlabel("Year") 

plt.ylabel("GDP") 

plt.legend() 

plt.title("China GDP Non-linear Regression") 

plt.grid(True) 

plt.show() 

 

from sklearn.metrics import mean_squared_error, r2_score 

y_fit = sigmoid(x, L_final, k_final, x0_final) 

mse = mean_squared_error(y, y_fit) 

r2 = r2_score(y, y_fit) 

print(f"MSE: {mse:.6f}") 

print(f"R² Score: {r2:.6f}")