机器学习算法隐马尔可夫模型算法理论知识及算法实现
代码实现最全整理汇总
HMM算法有三个核心问题：
1.评估问题：给定观测序列和模型参数，计算观测序列出现的概率。
2.解码问题：给定观测序列和模型参数，推测最可能的隐藏状态序列。
3.学习问题：给定观测序列，估计模型参数的最大似然估计。
下面是隐马尔可夫模型算法实现的代码示例（基于Python）：
1.基本定义
```python
import numpy as np
#定义隐马尔可夫模型
class HMM(object):
def __init__(self, A, B, pi):
self.A = A # 状态转移概率矩阵
self.B = B # 观测概率矩阵
self.pi = pi # 初始状态概率向量
self.N = self.A.shape[0] # 状态数
self.M = self.B.shape[1] # 观测数
```
2.评估问题
```python
#前向算法
def forward(self, O):
T = len(O)
alpha = np.zeros((T, self.N))
alpha[0] = self.pi * self.B[:, O[0]]
for t in range(1, T):
for i in range(self.N):
alpha[t, i] = np.sum(alpha[t-1] * self.A[:, i]) * self.B[i, O[t]]
return alpha
#后向算法
def backward(self, O):
T = len(O)
beta = np.zeros((T, self.N))
beta[T-1] = 1
for t in range(T-2, -1, -1):
for i in range(self.N):
beta[t, i] = np.sum(beta[t+1] * self.A[i] * self.B[:, O[t+1]])
return beta
#前向后向算法，计算观测序列的概率
def evaluate(self, O):
alpha = self.forward(O)
return np.sum(alpha[-1])
```
3.解码问题
```python
#维特比算法
def viterbi(self, O):
T = len(O)
delta = np.zeros((T, self.N))
psi = np.zeros((T, self.N), dtype=int)
delta[0] = self.pi * self.B[:, O[0]]
for t in range(1, T):
for i in range(self.N):
delta[t, i] = np.max(delta[t-1] * self.A[:, i]) * self.B[i, O[t]]
psi[t, i] = np.argmax(delta[t-1] * self.A[:, i])
z = np.zeros(T, dtype=int)
z[-1] = np.argmax(delta[-1])
for t in range(T-2, -1, -1):
z[t] = psi[t+1, z[t+1]]
return z
```
4. 学习问题（Baum-Welch算法）
```python
def learn(self, O_list, max_iter=100):
P_O_list = [self.evaluate(O) for O in O_list]
for n in range(max_iter):
gamma_sum = np.zeros((self.N, self.N))
epsilon_sum = np.zeros((self.N, self.M))
pi_new = self.pi * 0
for O, P_O in zip(O_list, P_O_list):
alpha = self.forward(O)
beta = self.backward(O)
gamma = alpha * beta / P_O
T = len(O)
epsilon = np.zeros((self.N, self.N, T))
for t in range(T-1):
for i in range(self.N):
for j in range(self.N):
epsilon[i, j, t] = alpha[t, i] * self.A[i, j] * self.B[j, O[t+1]] * beta[t+1, j] / P_O
gamma_sum += np.sum(gamma[:T-1], axis=0)
epsilon_sum += np.sum(epsilon[:, :, :T-1], axis=2)
pi_new += gamma[0]
self.A = epsilon_sum / gamma_sum[:, np.newaxis]
self.B = np.zeros((self.N, self.M))
for k in range(self.M):
mask = np.array([o == k for O in O_list for o in O])
self.B[:, k] = np.sum(np.stack([gamma[mask, i] for i in range(self.N)]), axis=1) / np.sum(gamma, axis=0)
self.pi = pi_new / len(O_list)
P_O_list_new = [self.evaluate(O) for O in O_list]
if np.linalg.norm(np.array(P_O_list) -
np.array(P_O_list_new)) < 1e-6:
break
P_O_list = P_O_list_new
```
这些代码实现了隐马尔可夫模型的基本功能，包括评估问题、解码问题和学习问题。
根据具体需求，可以进一步进行优化和扩展。