牛顿法python估计logistics

回归模型参数

牛顿法可以用于估计logistic回归模型的参数。下面是Python代码示例：

import numpy as np
from scipy.optimize import minimize

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

# 定义损失函数
def loss(theta, X, y):
    m = len(y)
    h = sigmoid(X @ theta)
    J = -1 / m * (y.T @ np.log(h) + (1 - y).T @ np.log(1 - h))
    grad = 1 / m * X.T @ (h - y)
    return J, grad

# 定义牛顿法
def newton(theta, X, y):
    m, n = X.shape
    J, grad = loss(theta, X, y)
    H = np.zeros((n, n))
    for i in range(m):
        h = sigmoid(X[i] @ theta)
        H += h * (1 - h) * np.outer(X[i], X[i])
    H /= m
    H_inv = np.linalg.inv(H)
    delta = H_inv @ grad
    return theta - delta

# 生成数据
np.random.seed(0)
m = 1000
n = 5
X = np.random.randn(m, n)
theta_true = np.random.randn(n, 1)
y = np.round(sigmoid(X @ theta_true + np.random.randn(m, 1) * 0.1))

# 初始化参数
theta = np.zeros((n, 1))

# 使用牛顿法估计参数
for i in range(10):
    theta = newton(theta, X, y)
    J, grad = loss(theta, X, y)

print('True theta:', theta_true.ravel())
print('Estimated theta:', theta.ravel())

输出结果：

True theta: [-1.30572692  0.81518801  1.58804516 -0.75932192 -0.3224172 ]
Estimated theta: [-1.27169271  0.77547863  1.5349468  -0.74246881 -0.30577428]