前向算法Python实现

循环方式

import numpy as np
def hmm_forward(Q, V, A, B, pi, T, O, p):
    """
    :param Q: 状态集合
    :param V: 观测集合
    :param A: 状态转移概率矩阵
    :param B: 观测概率矩阵
    :param pi: 初始概率分布
    :param T: 观测序列和状态序列的长度
    :param O: 观测序列
    :param p: 存储各个状态的前向概率的列表，初始为空
    """
    for t in range(T):
        # 计算初值
        if t == 0:
            for i in range(len(Q)):
                p.append(pi[i] * B[i, V[O[0]]])
        # 初值计算完毕后，进行下一时刻的递推运算
        else:
            alpha_t_ = 0
            alpha_t_t = []
            for i in range(len(Q)):
                for j in range(len(Q)):
                    alpha_t_ += p[j] * A[j, i]
                alpha_t_t.append(alpha_t_ * B[i, V[O[t]]])
                alpha_t_ = 0
            p = alpha_t_t
    return sum(p)
# 《统计学习方法》书上例10.2
Q = [1, 2, 3]
V = {'红':0, '白':1}
A = np.array([[0.5, 0.2, 0.3], [0.3, 0.5, 0.2], [0.2, 0.3, 0.5]])
B = np.array([[0.5, 0.5], [0.4, 0.6], [0.7, 0.3]])
pi = [0.2, 0.4, 0.4]
T = 3
O = ['红', '白', '红']
p = []
print(hmm_forward(Q, V, A, B, pi, T, O, p)) # 0.130218

递归方式

import numpy as np
def hmm_forward_(Q, V, A, B, pi, T, O, p, T_final):
    """
    :param T_final:递归的终止条件
    """
    if T == 0:
        for i in range(len(Q)):
            p.append(pi[i] * B[i, V[O[0]]])
    else:
        alpha_t_ = 0
        alpha_t_t = []
        for i in range(len(Q)):
            for j in range(len(Q)):
                alpha_t_ += p[j] * A[j, i]
            alpha_t_t.append(alpha_t_ * B[i, V[O[T]]])
            alpha_t_ = 0
        p = alpha_t_t
    if T >= T_final:
        return sum(p)
    return hmm_forward_(Q, V, A, B, pi, T+1, O, p, T_final)

Q = [1, 2, 3]
V = {'红':0, '白':1}
A = np.array([[0.5, 0.2, 0.3], [0.3, 0.5, 0.2], [0.2, 0.3, 0.5]])
B = np.array([[0.5, 0.5], [0.4, 0.6], [0.7, 0.3]])
pi = [0.2, 0.4, 0.4]
T = 0
O = ['红', '白', '红']
p = []
T_final = 2  # T的长度是3，T的取值是(0时刻, 1时刻, 2时刻)
print(hmm_forward_(Q, V, A, B, pi, T, O, p, T_final))

后向算法Python实现

循环方式

import numpy as np
def hmm_backward(Q, V, A, B, pi, T, O, beta_t, T_final):
    for t in range(T, -1, -1):
        if t == T_final:
            beta_t = beta_t
        else:
            beta_t_ = 0
            beta_t_t = []
            for i in range(len(Q)):
                for j in range(len(Q)):
                    beta_t_ += A[i, j] * B[j, V[O[t + 1]]] * beta_t[j]
                beta_t_t.append(beta_t_)
                beta_t_ = 0
            beta_t = beta_t_t
        if t == 0:
            p=[]
            for i in range(len(Q)):
                p.append(pi[i] * B[i, V[O[0]]] * beta_t[i])
            beta_t = p
    return sum(beta_t)
# 《统计学习方法》课后题10.1
Q = [1, 2, 3]
V = {'红':0, '白':1}
A = np.array([[0.5, 0.2, 0.3], [0.3, 0.5, 0.2], [0.2, 0.3, 0.5]])
B = np.array([[0.5, 0.5], [0.4, 0.6], [0.7, 0.3]])
pi = [0.2, 0.4, 0.4]
T = 3
O = ['红', '白', '红', '白']
beta_t = [1, 1, 1]
T_final = 3
print(hmm_backward_(Q, V, A, B, pi, T, O, beta_t, T_final))  # 0.06009

递归方式

import numpy as np
def hmm_backward(Q, V, A, B, pi, T, O, beta_t, T_final):
    if T == T_final:
        beta_t = beta_t
    else:
        beta_t_ = 0
        beta_t_t = []
        for i in range(len(Q)):
            for j in range(len(Q)):
                beta_t_ += A[i, j] * B[j, V[O[T+1]]] * beta_t[j]
            beta_t_t.append(beta_t_)
            beta_t_ = 0
        beta_t = beta_t_t
    if T == 0:
        p=[]
        for i in range(len(Q)):
            p.append(pi[i] * B[i, V[O[0]]] * beta_t[i])
        beta_t = p
        return sum(beta_t)
    return hmm_backward(Q, V, A, B, pi, T-1, O, beta_t, T_final)

Q = [1, 2, 3]
V = {'红':0, '白':1}
A = np.array([[0.5, 0.2, 0.3], [0.3, 0.5, 0.2], [0.2, 0.3, 0.5]])
B = np.array([[0.5, 0.5], [0.4, 0.6], [0.7, 0.3]])
pi = [0.2, 0.4, 0.4]
T = 3
O = ['红', '白', '红', '白']
beta_t = [1, 1, 1]
T_final = 3
print(hmm_backward_(Q, V, A, B, pi, T, O, beta_t, T_final))  # 0.06009