I want to optimize coefficents for the scale, shape and location function for a weibull PDF.

this is the code:

import numpy as np
from scipy.optimize import minimize
from scipy.stats import weibull_min

num_sets = 7
shape_params = np.random.uniform(1.5, 9, num_sets)  # shape parameter, k
loc_params = np.random.uniform(10, 100, num_sets)    # loc parameter, lambda
scale_params = np.random.uniform(500, 1500, num_sets) # scale parameter, c

# Operating conditions for 9 sets: temperature, PH value, humidity, CO2 concentration
conditions = np.random.uniform([80, 6, 30, 400], [95, 8, 80, 600], (num_sets, 4))

# Define the functions for shape, loc, scale parameters
def weibull_params(coeffs, conditions):
    shape = coeffs[0] + coeffs[1] * conditions[:,0] + coeffs[2] * conditions[:,1] + coeffs[3] * conditions[:,2] + coeffs[4] * conditions[:,3]
    loc = coeffs[5] + coeffs[6] * conditions[:,0] + coeffs[7] * conditions[:,1] + coeffs[8] * conditions[:,2] + coeffs[9] * conditions[:,3]
    scale = coeffs[10] + coeffs[11] * conditions[:,0] + coeffs[12] * conditions[:,1] + coeffs[13] * conditions[:,2] + coeffs[14] * conditions[:,3]
    return shape, loc, scale

# Objective function to minimize
def objective_function(coeffs, conditions, shape_params, loc_params, scale_params):
    shape, loc, scale = weibull_params(coeffs, conditions)
    error = np.sum((shape - shape_params)**2 + (loc - loc_params)**2 + (scale - scale_params)**2)
    return error

initial_guess = np.random.uniform(0, 1, 15)
result = minimize(objective_function, initial_guess, args=(conditions, shape_params, loc_params, scale_params), method='BFGS')

optimized_coeffs = result.x

# Calculate and print the optimized Weibull parameters
optimized_shape, optimized_loc, optimized_scale = weibull_params(optimized_coeffs, conditions)
print("nOriginal Weibull Parameters:")
print("Shape:", shape_params)
print("Loc:", loc_params)
print("Scale:", scale_params)

print("nOptimized Weibull Parameters:")
print("Shape:", optimized_shape)
print("Loc:", optimized_loc)
print("Scale:", optimized_scale)

Would it make more sense to optimize the coefficent with three separate objective functions like this approach:

import numpy as np
from scipy.optimize import minimize
from scipy.stats import weibull_min

num_sets = 7
shape_params = np.random.uniform(1.5, 9, num_sets)  # shape parameter, k
loc_params = np.random.uniform(10, 100, num_sets)   # loc parameter, lambda
scale_params = np.random.uniform(500, 1500, num_sets)  # scale parameter, c

# Operating conditions for 9 sets: temperature, PH value, humidity, CO2 concentration
conditions = np.random.uniform([80, 6, 30, 400], [95, 8, 80, 600], (num_sets, 4))


def weibull_params(coeffs, conditions):
  shape = coeffs[0] + coeffs[1] * conditions[:, 0] + coeffs[2] * conditions[:, 1] + coeffs[3] * conditions[:, 2] + coeffs[4] * conditions[:, 3]
  loc = coeffs[5] + coeffs[6] * conditions[:, 0] + coeffs[7] * conditions[:, 1] + coeffs[8] * conditions[:, 2] + coeffs[9] * conditions[:, 3]
  scale = coeffs[10] + coeffs[11] * conditions[:, 0] + coeffs[12] * conditions[:, 1] + coeffs[13] * conditions[:, 2] + coeffs[14] * conditions[:, 3]
  return shape, loc, scale


def shape_objective(coeffs, conditions, shape_params):
  shape, _, _ = weibull_params(coeffs, conditions)
  error = np.sum((shape - shape_params) ** 2)
  return error


def loc_objective(coeffs, conditions, loc_params):
  _, loc, _ = weibull_params(coeffs, conditions)
  error = np.sum((loc - loc_params) ** 2)
  return error


def scale_objective(coeffs, conditions, scale_params):
  _, _, scale = weibull_params(coeffs, conditions)
  error = np.sum((scale - scale_params) ** 2)
  return error


initial_guess = np.random.uniform(0, 1, 15)
result_shape = minimize(shape_objective, initial_guess, args=(conditions, shape_params), method='BFGS')
result_loc = minimize(loc_objective, initial_guess, args=(conditions, loc_params), method='BFGS')
result_scale = minimize(scale_objective, initial_guess, args=(conditions, scale_params), method='BFGS')

# Extract optimized coefficients from each result
optimized_coeffs_shape = result_shape.x
optimized_coeffs_loc = result_loc.x
optimized_coeffs_scale = result_scale.x

# Now you have optimized coefficients for each parameter
optimized_shape, optimized_loc, optimized_scale = weibull_params(np.concatenate((optimized_coeffs_shape, optimized_coeffs_loc, optimized_coeffs_scale)), conditions)

print("nOriginal Weibull Parameters:")
print("Shape:", shape_params)
print("Loc:", loc_params)
print("Scale:", scale_params)
print("nOptimized Weibull Parameters:")
print("Shape:", optimized_shape)
print("Loc:", optimized_loc)
print("Scale:", optimized_scale)

I tried both ways and i believe that optimizing all there parameters at the same time makes more sense to me.