#%% Import of packages

import numpy as np

#%% Data

t_array = np.arange(2000, 2009+1)
kpi_obs_array = np.array([75.5, 77.7, 78.7, 80.7, 81.0,
                        82.3, 84.2, 84.8, 88.0, 89.9])
    
#%% Regression model

phi_0 = np.array(t_array)
phi_1 = np.array(np.ones(len(kpi_obs_array)))
Phi = np.array([phi_0, phi_1]).T
Y = np.array([kpi_obs_array]).T

#%% Calculating LS estimate

theta = np.linalg.inv(Phi.T @ Phi) @ Phi.T @ Y
a_opt = theta[0][0]
b_opt = theta[1][0]

#%% Calculating value of objective function

kpi_pred_array = a_opt*t_array + b_opt
pe = kpi_obs_array - kpi_pred_array # prediction error
sspe_min = sum(pe*pe)  # sum of elementwise products of pe

#%% Presentation of result

print(f'a_opt = {a_opt:.3e}')
print(f'b_opt = {b_opt:.3e}')
print(f'sspe_min = {sspe_min:.3e}')