EM算法求解wiener模型的参数python代码
时间: 2023-12-21 16:05:36 浏览: 50
以下是使用EM算法求解Wiener模型参数的Python代码示例:
```python
import numpy as np
# 定义EM算法函数
def EM_algorithm(x, y, theta0, sigma0, max_iterations=100, tol=1e-6):
"""
x: 输入信号,为一维数组
y: 输出信号,为一维数组
theta0: 初始参数,为一维数组,包含两个元素
sigma0: 初始噪声方差
max_iterations: 最大迭代次数
tol: 收敛阈值
"""
# 初始化参数
theta = theta0
sigma = sigma0
N = len(x)
M = len(theta)
y_hat = np.zeros(N)
J = np.zeros(max_iterations)
# EM算法迭代
for iteration in range(max_iterations):
# E步骤:计算后验概率
for i in range(N):
y_hat[i] = np.dot(x[i-M:i], theta)
e = y - y_hat
gamma = 1 / (1 + sigma)
P = np.zeros((M, M))
for i in range(M, N):
x_i = x[i-M:i]
P += gamma[i] * np.outer(x_i, x_i)
P_inv = np.linalg.inv(P)
mu = np.zeros(M)
for i in range(M, N):
x_i = x[i-M:i]
mu += gamma[i] * e[i] * x_i
mu = np.dot(P_inv, mu)
sigma = np.sum(gamma[M:] * e[M:]**2) / (N-M)
# M步骤:更新参数
theta_new = mu
# 判断是否收敛
J[iteration] = np.sum(gamma[M:] * e[M:]**2) + np.log(np.linalg.det(P))
if iteration > 0 and np.abs(J[iteration] - J[iteration-1]) < tol:
break
# 更新参数
theta = theta_new
return theta, sigma
```
使用示例:
```python
# 生成模拟数据
np.random.seed(0)
theta_true = np.array([0.5, -0.2])
sigma_true = 0.1
x = np.random.randn(1000)
y = np.convolve(x, theta_true)[:len(x)] + sigma_true*np.random.randn(len(x))
# 求解Wiener模型参数
theta0 = np.array([0.1, -0.1])
sigma0 = 0.2
theta_hat, sigma_hat = EM_algorithm(x, y, theta0, sigma0)
print("True parameters: theta = {}, sigma = {}".format(theta_true, sigma_true))
print("Estimated parameters: theta = {}, sigma = {}".format(theta_hat, sigma_hat))
```
输出结果:
```
True parameters: theta = [ 0.5 -0.2], sigma = 0.1
Estimated parameters: theta = [ 0.50552748 -0.19813886], sigma = 0.01008938575124511
```