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

以下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

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

# Logistic回归模型
def logistic_model(beta, X):
    return sigmoid(np.dot(X, beta))

# 对数似然函数
def log_likelihood(beta, X, y):
    p = logistic_model(beta, X)
    return np.sum(y*np.log(p) + (1-y)*np.log(1-p))

# 对数似然函数的梯度
def log_likelihood_gradient(beta, X, y):
    p = logistic_model(beta, X)
    return np.dot(X.T, y-p)

# 对数似然函数的海森矩阵
def log_likelihood_hessian(beta, X, y):
    p = logistic_model(beta, X)
    W = np.diag(p*(1-p))
    return np.dot(np.dot(X.T, W), X)

# 信赖域算法
def trust_region_method(beta_init, X, y):
    # 定义初始步长
    delta = 1.0
    # 定义信赖域半径
    rho = 0.5
    # 定义最大迭代次数
    max_iter = 100
    beta = beta_init
    for i in range(max_iter):
        # 计算梯度和海森矩阵
        grad = log_likelihood_gradient(beta, X, y)
        hess = log_likelihood_hessian(beta, X, y)
        # 求解步长
        p, success = minimize(lambda x: -log_likelihood(x, X, y), grad, method='Newton-CG', jac=log_likelihood_gradient, hess=log_likelihood_hessian, options={'xtol': 1e-8, 'disp': False})
        # 计算预测值
        p = logistic_model(beta+p, X)
        # 计算实际下降量
        actual_reduction = log_likelihood(beta+p, X, y) - log_likelihood(beta, X, y)
        # 计算模型预测下降量
        predicted_reduction = -np.dot(grad, p) - 0.5*np.dot(np.dot(p, hess), p)
        # 计算rho
        rho = actual_reduction / predicted_reduction
        # 根据rho更新beta
        if rho < 0.25:
            delta = 0.25 * delta
        else:
            if rho > 0.75 and np.abs(np.linalg.norm(p) - delta) < 1e-8:
                delta = min(2.0*delta, 1000)
        if rho > 0:
            beta = beta + p
    return beta

# 局部二次似方法
def local_quadratic_method(beta_init, X, y):
    # 定义最大迭代次数
    max_iter = 100
    beta = beta_init
    for i in range(max_iter):
        # 计算梯度和海森矩阵
        grad = log_likelihood_gradient(beta, X, y)
        hess = log_likelihood_hessian(beta, X, y)
        # 求解步长
        p = np.linalg.solve(hess, -grad)
        # 计算预测值
        p = logistic_model(beta+p, X)
        # 计算实际下降量
        actual_reduction = log_likelihood(beta+p, X, y) - log_likelihood(beta, X, y)
        # 计算模型预测下降量
        predicted_reduction = -np.dot(grad, p) - 0.5*np.dot(np.dot(p, hess), p)
        # 计算alpha
        alpha = actual_reduction / predicted_reduction
        # 根据alpha更新beta
        beta = beta + alpha*p
    return beta