隱馬爾科夫模型——前向后向算法python實現

描述：隱馬爾科夫模型的三個基本問題之一：概率計算問題。給定模型λ=（A，B，π）和觀測序列O=(o1，o2，...，oT)，計算在模型λ下觀測序列O出現的概率P(O|λ)

概率計算問題有三種求解方法：

　　直接計算法（時間復雜度為O(TN^T)，計算量非常大，不易實現）

　　前向算法：A：狀態轉移概率矩陣；B：觀測概率矩陣；Pi：初始狀態概率向量；O：觀測序列

def forward(A, B, Pi, O):
    """計算前向概率"""
    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):
    """前向算法計算觀測序列O出現的概率"""
    p = 0
    m = np.shape(alpha)[0]
    for i in range(m):
        p += alpha[-1][i]

    return p

 

　　后向算法：A：狀態轉移概率矩陣；B：觀測概率矩陣；Pi：初始狀態概率向量；O：觀測序列

# 方法一
def backward1(A, B, Pi, O):
    """計算后向概率"""
    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):
    """計算后向概率"""
    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):
    """后向算法計算觀測概率O出現的概率"""
    r = Pi[0]*B[:, 0]*beta[0, :]
    p = r.sum()

    return p