Python中Weibull分布三参数系数：同时优化还是分开优化？

我需要优化Weibull PDF的scale、shape、location函数的系数，为此编写了两种Python实现方案，想请教哪种选择更合理？

方案一：同时优化所有参数系数 #

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)

方案二：使用三个独立目标函数分别优化 #

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)

你的判断是对的，同时优化所有参数系数的方案（方案一）更合理，原因如下：

• 参数关联性：Weibull分布的shape、loc、scale三个参数并非完全独立，它们共同决定了分布的整体形态。分开优化时，单个参数的最优解可能会破坏三者的内在联系，导致整体分布不符合实际逻辑；而同时优化能兼顾参数间的关联，得到全局更优的系数组合。
• 优化效率：方案一只需执行一次优化流程，计算量更小；方案二需要三次独立优化，不仅耗时更长，还可能因多次初始化导致结果不稳定。
• 结果一致性：方案二中存在明显的代码错误——np.concatenate会把原本15维的系数变成45维，而weibull_params函数只接受15维输入，直接运行会报错。退一步说，即使修正拼接方式，三个独立优化的结果也可能在部分操作条件下出现参数组合冲突，而方案一的结果是统一优化得到的，能保证所有条件下参数组合的一致性。

内容的提问来源于stack exchange，提问作者newtopy