语音识别入门第四节：隐马尔可夫模型（HMM）（实战篇）

本节理论笔记见：语音识别入门第四节：隐马尔可夫模型（HMM）

前向算法核心代码如下：

    a = np.zeros((T, N))
    a[0, :] = [pi[k] * B[k][O[0]] for k in range(N)]
    for t in range(1, T, 1):
        a[t, :] = [sum([a[t-1][j] * A[j][i] for j in range(N)]) * B[i][O[t]]for i in range(N)]
    prob = sum(a[T-1])

后向算法核心代码如下：

    beta = np.ones((T, N))
    for t in range(T - 2, -1, -1):
        beta[t, :] = [sum([A[i][j] * B[j][O[t+1]] * beta[t+1][j] for j in range(N)]) for i in range(N)]
    prob = sum([pi[i] * B[i][O[0]] * beta[0][i] for i in range(N)])

Viterbi算法核心代码如下：

    delta = np.zeros((T, N))
    psi = np.zeros((T, N))
    delta[0, :] = [pi[i] * B[i][O[0]] for i in range(N)]

    for t in range(1, T, 1):
        delta[t, :] = [max([delta[t - 1][j] * A[j][i] * B[i][O[t]] for j in range(N)]) for i in range(N)]
        psi[t, :] = [np.argmax([delta[t - 1][j] * A[j][i] for j in range(N)]) for i in range(N)]

    best_prob = max(delta[T - 1])
    best_path[T-1] = np.argmax(delta[T - 1])
    for t in range(T - 2, -1, -1):
        best_path[t] = int(psi[t + 1][best_path[t + 1]])

完整代码如下：

# Author: Kaituo Xu, Fan Yu
import numpy as np


def forward_algorithm(O, HMM_model):
    """HMM Forward Algorithm.
    Args:
        O: (o1, o2, ..., oT), observations
        HMM_model: (pi, A, B), (init state prob, transition prob, emitting prob)
    Return:
        prob: the probability of HMM_model generating O.
    """
    pi, A, B = HMM_model
    T = len(O)
    N = len(pi)
    prob = 0.0
    # Begin Assignment
    a = np.zeros((T, N))
    a[0, :] = [pi[k] * B[k][O[0]] for k in range(N)]
    for t in range(1, T, 1):
        a[t, :] = [sum([a[t-1][j] * A[j][i] for j in range(N)]) * B[i][O[t]]for i in range(N)]
    prob = sum(a[T-1])
    # Put Your Code Here

    # End Assignment
    return prob

def backward_algorithm(O, HMM_model):
    """HMM Backward Algorithm.
    Args:
        O: (o1, o2, ..., oT), observations
        HMM_model: (pi, A, B), (init state prob, transition prob, emitting prob)
    Return:
        prob: the probability of HMM_model generating O.
    """
    pi, A, B = HMM_model
    T = len(O)
    N = len(pi)
    prob = 0.0
    # Begin Assignment
    beta = np.ones((T, N))
    for t in range(T - 2, -1, -1):
        beta[t, :] = [sum([A[i][j] * B[j][O[t+1]] * beta[t+1][j] for j in range(N)]) for i in range(N)]
    prob = sum([pi[i] * B[i][O[0]] * beta[0][i] for i in range(N)])
    # Put Your Code Here

    # End Assignment
    return prob
 

def Viterbi_algorithm(O, HMM_model):
    """Viterbi decoding.
    Args:
        O: (o1, o2, ..., oT), observations
        HMM_model: (pi, A, B), (init state prob, transition prob, emitting prob)
    Returns:
        best_prob: the probability of the best state sequence
        best_path: the best state sequence
    """
    pi, A, B = HMM_model
    T = len(O)
    N = len(pi)
    best_prob, best_path = 0.0, {}
    # Begin Assignment
    delta = np.zeros((T, N))
    psi = np.zeros((T, N))
    delta[0, :] = [pi[i] * B[i][O[0]] for i in range(N)]

    for t in range(1, T, 1):
        delta[t, :] = [max([delta[t - 1][j] * A[j][i] * B[i][O[t]] for j in range(N)]) for i in range(N)]
        psi[t, :] = [np.argmax([delta[t - 1][j] * A[j][i] for j in range(N)]) for i in range(N)]

    best_prob = max(delta[T - 1])
    best_path[T-1] = np.argmax(delta[T - 1])
    for t in range(T - 2, -1, -1):
        best_path[t] = int(psi[t + 1][best_path[t + 1]])

    # End Assignment
    return best_prob, best_path


if __name__ == "__main__":
    color2id = { "RED": 0, "WHITE": 1 }
    # model parameters
    pi = [0.2, 0.4, 0.4]
    A = [[0.5, 0.2, 0.3],
         [0.3, 0.5, 0.2],
         [0.2, 0.3, 0.5]]
    B = [[0.5, 0.5],
         [0.4, 0.6],
         [0.7, 0.3]]
    # input
    observations = (0, 1, 0)
    HMM_model = (pi, A, B)
    # process
    observ_prob_forward = forward_algorithm(observations, HMM_model)
    print(observ_prob_forward)

    observ_prob_backward = backward_algorithm(observations, HMM_model)
    print(observ_prob_backward)

    best_prob, best_path = Viterbi_algorithm(observations, HMM_model) 
    print(best_prob, best_path)