#!/usr/bin/env python3
# -*- coding: utf-8 -*-
"""
Created on Tue Apr 22 08:06:00 2025

@author: amriksen
"""

import numpy as np
import matplotlib.pyplot as plt

# Synthetic data: hours studied per week (x) and marks obtained (y)
np.random.seed(42)
x = np.linspace(0, 10, 30)  # hours per week from 0 to 10
# assume marks = base + b*x + c*x^2 + noise
a_true, b_true, c_true = 50, 10, -0.5  # underlying parameters
y = a_true + b_true * x + c_true * x**2 + np.random.randn(len(x)) * 5.0


# Construct design matrix X with a column of ones (intercept) and x
X = np.vstack((np.ones_like(x), x)).T

# Compute OLS coefficients: beta = (X^T X)^{-1} X^T y
beta = np.linalg.inv(X.T @ X) @ X.T @ y
intercept, slope = beta

print(f"Estimated intercept: {intercept:.3f}")
print(f"Estimated slope: {slope:.3f}")

# Plot the data and the fitted regression line
plt.scatter(x, y, label='Data points')
plt.plot(x, X @ beta, label='OLS fit', linewidth=2)
plt.xlabel('x')
plt.ylabel('y')
plt.title('Ordinary Least Squares Regression (Multivariate Form)')
plt.legend()
plt.show()