【ML从入门到入土系列09】HMM
文章目录
- 1 理论
- 2 代码
- 3 参考
1 理论
HMM解决的三大问题如下，即概率计算，学习以及预测问题。分别采用前向/后向算法，Viterbi算法，Baum-Welch算法进行求解。
2 代码
class HiddenMarkov:
# 前向算法
def forward(self, Q, V, A, B, O, PI):
N = len(Q) #可能存在的状态数量
M = len(O) # 观测序列的大小
alphas = np.zeros((N, M)) # alpha值
T = M
for t in range(T): # 遍历每一时刻，算出alpha值
indexOfO = V.index(O[t]) # 找出序列对应的索引
for i in range(N):
if t == 0: # 计算初值
alphas[i][t] = PI[t][i] * B[i][indexOfO]
print(
'alpha1(%d)=p%db%db(o1)=%f' % (i, i, i, alphas[i][t]))
else:
alphas[i][t] = np.dot(
[alpha[t - 1] for alpha in alphas],
[a[i] for a in A]) * B[i][indexOfO]
print('alpha%d(%d)=[sigma alpha%d(i)ai%d]b%d(o%d)=%f' %
(t, i, t - 1, i, i, t, alphas[i][t]))
P = np.sum([alpha[M - 1] for alpha in alphas])
# 后向算法
def backward(self, Q, V, A, B, O, PI):
N = len(Q) # 可能存在的状态数量
M = len(O) # 观测序列的大小
betas = np.ones((N, M)) # beta
for i in range(N):
print('beta%d(%d)=1' % (M, i))
for t in range(M - 2, -1, -1):
indexOfO = V.index(O[t + 1]) # 找出序列对应的索引
for i in range(N):
betas[i][t] = np.dot(
np.multiply(A[i], [b[indexOfO] for b in B]),
[beta[t + 1] for beta in betas])
realT = t + 1
realI = i + 1
print(
'beta%d(%d)=[sigma a%djbj(o%d)]beta%d(j)=(' %
(realT, realI, realI, realT + 1, realT + 1),
end='')
for j in range(N):
print(
"%.2f*%.2f*%.2f+" % (A[i][j], B[j][indexOfO],
betas[j][t + 1]),
end='')
print("0)=%.3f" % betas[i][t])
indexOfO = V.index(O[0])
P = np.dot(
np.multiply(PI, [b[indexOfO] for b in B]),
[beta[0] for beta in betas])
print("P(O|lambda)=", end="")
for i in range(N):
print(
"%.1f*%.1f*%.5f+" % (PI[0][i], B[i][indexOfO], betas[i][0]),
end="")
print("0=%f" % P)
# viterbi算法
def viterbi(self, Q, V, A, B, O, PI):
N = len(Q) #可能存在的状态数量
M = len(O) # 观测序列的大小
deltas = np.zeros((N, M))
psis = np.zeros((N, M))
I = np.zeros((1, M))
for t in range(M):
realT = t + 1
indexOfO = V.index(O[t]) # 找出序列对应的索引
for i in range(N):
realI = i + 1
if t == 0:
deltas[i][t] = PI[0][i] * B[i][indexOfO]
psis[i][t] = 0
print('delta1(%d)=pi%d * b%d(o1)=%.2f * %.2f=%.2f' %
(realI, realI, realI, PI[0][i], B[i][indexOfO],
deltas[i][t]))
print('psis1(%d)=0' % (realI))
else:
deltas[i][t] = np.max(
np.multiply([delta[t - 1] for delta in deltas],
[a[i] for a in A])) * B[i][indexOfO]
print(
'delta%d(%d)=max[delta%d(j)aj%d]b%d(o%d)=%.2f*%.2f=%.5f'
% (realT, realI, realT - 1, realI, realI, realT,
np.max(
np.multiply([delta[t - 1] for delta in deltas],
[a[i] for a in A])), B[i][indexOfO],
deltas[i][t]))
psis[i][t] = np.argmax(
np.multiply(
[delta[t - 1] for delta in deltas],
[a[i]
for a in A])) + 1 #由于其返回的是索引，因此应+1才能和正常的下标值相符合
print('psis%d(%d)=argmax[delta%d(j)aj%d]=%d' %
(realT, realI, realT - 1, realI, psis[i][t]))
print(deltas)
print(psis)
I[0][M - 1] = np.argmax([delta[M - 1] for delta in deltas
]) + 1
print('i%d=argmax[deltaT(i)]=%d' % (M, I[0][M - 1]))
for t in range(M - 2, -1, -1):
I[0][t] = psis[int(I[0][t + 1]) - 1][t + 1]
print('i%d=psis%d(i%d)=%d' % (t + 1, t + 2, t + 2, I[0][t]))
print("状态序列I：", I)
3 参考
理论：周志华《机器学习》，李航《统计学习方法》
代码：https://github.com/fengdu78/lihang-code