gamma[t][i] = alpha[t][i] * beta[t][i] / sum_alpha_beta_of_t
if __name__ == '__main__':
# 测试
pi = np.array([0.2, 0.5, 0.3])
A = np.array([[0.5, 0.4, 0.1], [0.2, 0.2, 0.6], [0.2, 0.5, 0.3]])
B = np.array([[0.4, 0.6], [0.8, 0.2], [0.5, 0.5]])
Q = '白黑白白黑'
beta = np.zeros((len(Q), len(A)))
alpha = np.zeros((len(Q), len(A)))
gamma = np.zeros((len(Q), len(A)))
# 开始计算
# 1. 计算beta
backward.calc_beta(pi, A, B, Q, beta, common.convert_obs_seq_2_index)
# 输出最终结果
print("beta矩阵:")
print(beta)
# 2. 计算alpha
forward.calc_alpha(pi, A, B, Q, alpha, common.convert_obs_seq_2_index)
# 输出最终结果
print("alpha矩阵:")
print(alpha)
# 3. 计算gamm矩阵
calc_gamma(alpha, beta, gamma)
# 输出最终结果
print("gamma矩阵:")
print(gamma)
# 选择每个时刻最大的概率作为预测概率
def baum_welch(pi, A, B, Q, max_iter=3, fetch_index_by_obs_seq=None):
"""
根据传入的初始概率矩阵(pi、A、B)以及观测序列Q，使用baum_welch算法进行迭代求解最终的pi、A、B的值；最大迭代次数默认为3；最终更新结果保存在传入的参数矩阵中(pi\A\B)
"""
# 0. 初始化相关变量
# 初始化序列转换为索引的方法
fetch_index_by_obs_seq_f = fetch_index_by_obs_seq
if fetch_index_by_obs_seq_f is None:
fetch_index_by_obs_seq_f = lambda obs, obs_index: ord(obs[obs_index])
# 初始化相关的参数值: n、m、T
T = len(Q)
n = len(A)
m = len(B[0])
alpha = np.zeros((T, n))
beta = np.zeros((T, n))
gamma = np.zeros((T, n))
ksi = np.zeros((T - 1, n, n))
n_range = range(n)
m_range = range(m)
t_range = range(T)
t_1_range = range(T - 1)
# 1. 迭代更新(EM算法思想类型)
for time in range(max_iter):
# a. 在当前的pi，A，B的情况下对观测序列Q分别计算alpha、beta、gamma和ksi
forward.calc_alpha(pi, A, B, Q, alpha, fetch_index_by_obs_seq_f)
backward.calc_beta(pi, A, B, Q, beta, fetch_index_by_obs_seq_f)
single.calc_gamma(alpha, beta, gamma)
continuous.calc_ksi(alpha, beta, A, B, Q, ksi,
fetch_index_by_obs_seq_f)
# b. 更新pi、A、B的值
# b.1. 更新pi值
for i in n_range:
pi[i] = gamma[0][i]
# b.2. 更新状态转移矩阵A的值
tmp1 = np.zeros(T - 1)
tmp2 = np.zeros(T - 1)
for i in n_range:
for j in n_range:
# 获取所有时刻从状态i转移到状态j的值
for t in t_1_range:
tmp1[t] = ksi[t][i][j]
tmp2[t] = gamma[t][i]
# 更新状态i到状态j的转移概率
A[i][j] = np.sum(tmp1) / np.sum(tmp2)
# b.3. 更新状态和观测值之间的转移矩阵
for i in n_range:
for k in m_range:
tmp1 = np.zeros(T)
tmp2 = np.zeros(T)
# 获取所有时刻从状态i转移到观测值k的概率和
number = 0
for t in t_range:
if k == fetch_index_by_obs_seq_f(Q, t):
# 如果序列Q中时刻t对应的观测值就是k，那么进行统计这个时刻t为状态i的概率值
tmp1[t] = gamma[t][i]
number += 1
tmp2[t] = gamma[t][i]
# 更新状态i到观测值k之间的转移概率
if number == 0:
# 没有转移，所以为0
B[i][k] = 0
else:
# 有具体值，那么进行更新操作
B[i][k] = np.sum(tmp1) / np.sum(tmp2)