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import numpy as np
from scipy.optimize import curve_fit
from sklearn.metrics import mean_squared_error, r2_score
import matplotlib.pyplot as plt

# Exponential function
def exponential_func(x, a, b):
    return a * np.exp(b * x)

# Generate sparse sample data
np.random.seed(42)
x_data_sparse = np.linspace(0, 5, 10)  # Reduce the number of data points
y_true_sparse = 2 * np.exp(0.5 * x_data_sparse) + np.random.normal(0, 0.5, len(x_data_sparse))  # Example exponential data with noise

# Fit the exponential function to the sparse data
initial_guess = [1, 1]  # Initial guess for the parameters (a, b)
params_sparse, covariance_sparse = curve_fit(exponential_func, x_data_sparse, y_true_sparse, p0=initial_guess)

# Extract fitted parameters
a_fit_sparse, b_fit_sparse = params_sparse

# Generate fitted data using the parameters
y_fit_sparse = exponential_func(x_data_sparse, a_fit_sparse, b_fit_sparse)

# Calculate accuracy metrics for sparse data
mse_sparse = mean_squared_error(y_true_sparse, y_fit_sparse)
r2_sparse = r2_score(y_true_sparse, y_fit_sparse)

print(f'Mean Squared Error (MSE) for Sparse Data: {mse_sparse:.4f}')
print(f'R-squared (R2) for Sparse Data: {r2_sparse:.4f}')

# Plot the sparse original data and the fitted curve
plt.scatter(x_data_sparse, y_true_sparse, label='Sparse Original Data')
plt.plot(x_data_sparse, y_fit_sparse, color='red', label=f'Sparse Exponential Fit: y = {a_fit_sparse:.2f} * e^({b_fit_sparse:.2f} * x)')
plt.legend()
plt.xlabel('X')
plt.ylabel('Y')
plt.title('Sparse Exponential Regression Example')
plt.show()