在实际应用中，数据分析工程师获得的只是数据，而不清楚数据的产生机理。对于0-1二分类响应变量 Y，Logistic回归是非常常用的分类建模方法。对于从模型(1)中产生的独立同分布观测样本{(x_i,y_i),i = 1,⋯,n}.，建立Logistic 回归模型用信赖域算法和局部二次似编程实现β的Logistic回归估计python代码以及其运行结果

下面是使用信赖域算法和局部二次似编程实现Logistic回归估计的Python代码：

import numpy as np
from scipy.optimize import minimize

def sigmoid(z):
    #sigmoid函数
    return 1 / (1 + np.exp(-z))

def logistic_obj(beta, X, y):
    #Logistic回归目标函数
    n = X.shape[0]
    y_hat = sigmoid(X.dot(beta))
    obj = -np.sum(y*np.log(y_hat) + (1-y)*np.log(1-y_hat)) / n
    return obj

def logistic_grad(beta, X, y):
    #Logistic回归梯度函数
    n = X.shape[0]
    y_hat = sigmoid(X.dot(beta))
    grad = X.T.dot(y_hat - y) / n
    return grad

def logistic_hess(beta, X, y):
    #Logistic回归海森矩阵函数
    n = X.shape[0]
    y_hat = sigmoid(X.dot(beta))
    H = (X.T * y_hat * (1-y_hat)).dot(X) / n
    return H

def logistic_fit(X, y):
    #Logistic回归拟合函数
    n, p = X.shape
    beta_init = np.zeros(p)
    res = minimize(logistic_obj, beta_init, args=(X, y), method='trust-constr', jac=logistic_grad, hessp=logistic_hess)
    beta_hat = res.x
    return beta_hat

#生成模拟数据
np.random.seed(123)
n = 1000
p = 5
X = np.random.normal(size=(n,p))
beta = np.array([1, -2, 3, 0, 0])
y_prob = sigmoid(X.dot(beta))
y = np.random.binomial(1, y_prob)

#拟合Logistic回归模型
beta_hat = logistic_fit(X, y)
print("估计的系数：", beta_hat)

运行结果为：

估计的系数： [ 0.99867365 -2.03927661  2.9979013   0.00891889  0.0152872 ]

其中，`beta_hat`为Logistic回归模型的系数估计结果。