import numpy as np
from scipy.optimize import minimize

# load data
xm = np.array([18.3447,79.86538,85.09788,10.5211,44.4556, \
               69.567,8.960,86.197,66.857,16.875, \
               52.2697,93.917,24.35,5.118,25.126, \
               34.037,61.4445,42.704,39.531,29.988])

ym = np.array([5.072,7.1588,7.263,4.255,6.282, \
               6.9118,4.044,7.2595,6.898,4.8744, \
               6.5179,7.3434,5.4316,3.38,5.464, \
               5.90,6.80,6.193,6.070,5.737])

# calculate y
def calc_y(x):
    a,b,c = x
    y = a + b/xm + c*np.log(xm)
    return y

# define objective
def objective(x):
    return np.sum(((calc_y(x)-ym)/ym)**2)

# initial guesses
x0 = np.zeros(3)

# show initial objective
print('Initial SSE Objective: ' + str(objective(x0)))

# optimize
# bounds on variables
bnds100 = (-100.0, 100.0)
no_bnds = (-1.0e10, 1.0e10)
bnds = (no_bnds, no_bnds, bnds100)
solution = minimize(objective,x0,method='SLSQP',bounds=bnds)
x = solution.x
y = calc_y(x)

# show final objective
print('Final SSE Objective: ' + str(objective(x)))

# print solution
print('Solution')
print('a = ' + str(x[0]))
print('b = ' + str(x[1]))
print('c = ' + str(x[2]))

# plot solution
import matplotlib.pyplot as plt
plt.figure(1)
plt.plot(xm,ym,'ro')
plt.plot(xm,y,'bx');
plt.xlabel('x')
plt.ylabel('y')
plt.legend(['Measured','Predicted'],loc='best')
plt.savefig('results.png')
plt.show()