隐马尔科夫模型——前向后向算法python实现

描述：隐马尔科夫模型的三个基本问题之一：概率计算问题。给定模型λ=（A，B，π）和观测序列O=(o1，o2，...，oT)，计算在模型λ下观测序列O出现的概率P(O|λ)

概率计算问题有三种求解方法：

　　直接计算法（时间复杂度为O(TN^T)，计算量非常大，不易实现）

　　前向算法：A：状态转移概率矩阵；B：观测概率矩阵；Pi：初始状态概率向量；O：观测序列

def forward(A, B, Pi, O):
    """计算前向概率"""
    m = np.shape(A)[0]
    n = np.shape(O)[0]
    alpha = np.zeros((n, m))

    for i in range(m):
        alpha[0][i] = Pi[0][i]*B[i][0]

    for t in range(1, n):
        for i in range(m):
            sum = 0
            for j in range(m):
                sum += alpha[t-1][j]*A[j][i]
            alpha[t][i] = sum * B[i][O[t]-1]

    return alpha

def calc_forward(alpha):
    """前向算法计算观测序列O出现的概率"""
    p = 0
    m = np.shape(alpha)[0]
    for i in range(m):
        p += alpha[-1][i]

    return p

　　后向算法：A：状态转移概率矩阵；B：观测概率矩阵；Pi：初始状态概率向量；O：观测序列

# 方法一
def backward1(A, B, Pi, O):
    """计算后向概率"""
    m = np.shape(A)[0]
    n = np.shape(O)[0]
    beta = np.zeros((n, m))
    sum_beta = np.zeros((m, m))

    for i in range(m):
        beta[n-1][i] = 1

    for t in range(n-1, 0, -1):
        for i in range(m):
            for j in range(m):
                sum_beta[i, j] = A[i, j]*B[j, O[t]-1]*beta[t, j]
        beta[t-1][:] = sum_beta.sum(1)

    return beta

# 方法二
def backward(A, B, Pi, O):
    """计算后向概率"""
    m = np.shape(A)[0]
    n = np.shape(O)[0]
    beta = np.zeros((n, m))
    sum_beta = np.zeros((m, m))

    for i in range(m):
        beta[n-1][i] = 1

    for t in range(n-1, 0, -1):
        for i in range(m):
            sum_beta[i, :] = A[i, :]*B[:, O[t]-1]*beta[t, :]
        beta[t-1][:] = sum_beta.sum(1)

    return beta

def calc_backward(beta, Pi, B):
    """后向算法计算观测概率O出现的概率"""
    r = Pi[0]*B[:, 0]*beta[0, :]
    p = r.sum()

    return p