网上关于HMM的学习资料、博客有很多，基本都是左边摘抄一点，右边摘抄一点，这里一个图，那里一个图，公式中有的变量说不清道不明，学起来很费劲。
经过浏览几篇博文（其实有的地方写的也比较乱），在7张4开的草稿纸上写公式、单步跟踪程序，终于还是搞清楚了HMM的原理。
HMM学习过程：
1、搜索相关博客：
隐马尔可夫模型[博客]（图示比较详细，前部分还可以，后部分公式有点乱）：http://www.leexiang.com/hidden-markov-model
HMM简介、forward算法和viterbi算法[博客]（含源码，算法描述不是很清晰，但是有源码可看）http://www.cnblogs.com/zhangchaoyang/articles/2219571.html
forward-backward算法[博客]（含源码，算法描述不是很清晰，但是有源码可看）：http://www.cnblogs.com/zhangchaoyang/articles/2220398.html
隐马尔科夫模型PPT—刘秉权[百度文库]（算法流程、公式、参数都比较详细，有理论基础之后是很好的总结资源，但是没有具体例子，无基础的同学学习起来不是很形象。）：http://wenku.baidu.com/view/2f0d944769eae009581bec04.html
----其他代码资源（没有理论基础，只看代码很难看懂HMM的原理）---
UMDHMM的C语言实现：http://www.kanungo.com/software/umdhmm-v1.02.zip
GitHub上一个UMDHMM的Python实现：https://github.com/dkyang/UMDHMM-python
2、根据隐马尔科夫模型PPT—刘秉权[百度文库]，在5张4开草稿纸上把HMM流程顺一遍，下边是整理的笔记：
HMM三个算法的作用：
forward算法：（评估）给定一HMM模型，计算一观察序列O1O2...OLEN出现的概率。
viterbi算法：（解码）给定一HMM模型，计算一观察序列O1O2...OLEN对应的最可能的隐藏序列H1H2...HLEN及该隐藏序列出现的概率。
forward-backward算法：（学习）给定一观察序列O1O2...OLEN，求解能够拟合这个序列的HMM模型。
HMM三个算法之间的关系：
forward算法中的forward变量就是forward-backforward算法中的forward变量，而backward变量与forward变量是类似的；
forward-backward算法是为了通过类似最大似然估计的方法找到局部最优的模型参数，在迭代过程中forward变量和backward变量起了很大作用；
viterbi算法和forward算法很相似，只是forward算法迭代过程需要的是sum，viterbi算法迭代过程需要的是max，而且viterbi算法除了输出概率，还要用逆推过程求解路径；
当用forwar-backward算法求解出模型参数之后，用户给出一个观察序列，用viterbi算法就能求出最可能的隐藏序列以及概率了。
首先是forward算法的Python实现：
#-*-coding:utf-8-*-
__author__ = 'ZhangHe'
def forward(N,M,A,B,P,observed):
p = 0.0
#观察到的状态数目
LEN = len(observed)
#中间概率LEN*M
Q = [([0]*N) for i in range(LEN)]
#第一个观察到的状态,状态的初始概率乘上隐藏状态到观察状态的条件概率。
for j in range(N):
Q[0][j] = P[j]*B[j][observation.index(observed[0])]
#第一个之后的状态，首先从前一天的每个状态，转移到当前状态的概率求和，然后乘上隐藏状态到观察状态的条件概率。
for i in range(1,LEN):
for j in range(N):
sum = 0.0
for k in range(N):
sum += Q[i-1][k]*A[k][j]
Q[i][j] = sum * B[j][observation.index(observed[i])]
for i in range(N):
p += Q[LEN-1][i]
return p
# 3 种隐藏层状态:sun cloud rain
hidden = []
hidden.append('sun')
hidden.append('cloud')
hidden.append('rain')
N = len(hidden)
# 4 种观察层状态:dry dryish damp soggy
observation = []
observation.append('dry')
observation.append('damp')
observation.append('soggy')
M = len(observation)
# 初始状态矩阵（1*N第一天是sun，cloud，rain的概率）
P = (0.3,0.3,0.4)
# 状态转移矩阵A（N*N 隐藏层状态之间互相转变的概率）
A=((0.2,0.3,0.5),(0.1,0.5,0.4),(0.6,0.1,0.3))
# 混淆矩阵B（N*M 隐藏层状态对应的观察层状态的概率）
B=((0.1,0.5,0.4),(0.2,0.4,0.4),(0.3,0.6,0.1))
#假设观察到一组序列为observed，输出HMM模型（N，M，A，B，P）产生观察序列observed的概率
observed = ['dry']
print forward(N,M,A,B,P,observed)
observed = ['damp']
print forward(N,M,A,B,P,observed)
observed = ['dry','damp']
print forward(N,M,A,B,P,observed)
observed = ['dry','damp','soggy']
print forward(N,M,A,B,P,observed)
输出结果：
0.21
0.51
0.1074
0.030162
其中前两个结果和手工计算的一样；
后两个结果没有手工计算，但是在调试程序过程中单步跟踪运行代码，运行过程与手工计算过程相同。
然后是Viterbi算法的Python实现：
def viterbi(N,M,A,B,P,hidden,observed):
sta = []
LEN = len(observed)
Q = [([0]*N) for i in range(LEN)]
path = [([0]*N) for i in range(LEN)]
#第一天计算，状态的初始概率*隐藏状态到观察状态的条件概率
for j in range(N):
Q[0][j]=P[j]*B[j][observation.index(observed[0])]
path[0][j] = -1
# 第一天以后的计算
# 前一天的每个状态转移到当前状态的概率最大值
# *
# 隐藏状态到观察状态的条件概率
for i in range(1,LEN):
for j in range(N):
max = 0.0
index = 0
for k in range(N):
if(Q[i-1][k]*A[k][j] > max):
max = Q[i-1][k]*A[k][j]
index = k
Q[i][j] = max * B[j][observation.index(observed[i])]
path[i][j] = index
#找到最后一天天气呈现哪种观察状态的概率最大
max = 0.0
idx = 0
for i in range(N):
if(Q[LEN-1][i]>max):
max = Q[LEN-1][i]
idx = i
print "最可能隐藏序列的概率："+str(max)
sta.append(hidden[idx])
#逆推回去找到每天出现哪个隐藏状态的概率最大
for i in range(LEN-1,0,-1):
idx = path[i][idx]
sta.append(hidden[idx])
sta.reverse()
return sta;
# 3 种隐藏层状态:sun cloud rain
hidden = []
hidden.append('sun')
hidden.append('cloud')
hidden.append('rain')
N = len(hidden)
# 4 种观察层状态:dry dryish damp soggy
observation = []
observation.append('dry')
observation.append('damp')
observation.append('soggy')
M = len(observation)
# 初始状态矩阵（1*N第一天是sun，cloud，rain的概率）
P = (0.3,0.3,0.4)
# 状态转移矩阵A（N*N 隐藏层状态之间互相转变的概率）
A=((0.2,0.3,0.5),(0.1,0.5,0.4),(0.6,0.1,0.3))
# 混淆矩阵B（N*M 隐藏层状态对应的观察层状态的概率）
B=((0.1,0.5,0.4),(0.2,0.4,0.4),(0.3,0.6,0.1))
#假设观察到一组序列为observed，输出HMM模型（N，M，A，B，P）产生观察序列observed的概率
observed = ['dry','damp','soggy']
print viterbi(N,M,A,B,P,hidden,observed)
输出：
最可能隐藏序列的概率：0.005184
['rain', 'rain', 'sun']
GITHUB上一个Python实现的完整HMM：
import numpy as np
DELTA = 0.001
class HMM:
def __init__(self, pi, A, B):
self.pi = pi
self.A = A
self.B = B
self.M = B.shape[1]
self.N = A.shape[0]
def forward(self,obs):
T = len(obs)
N = self.N
alpha = np.zeros([N,T])
alpha[:,0] = self.pi[:] * self.B[:,obs[0]-1]
for t in xrange(1,T):
for n in xrange(0,N):
alpha[n,t] = np.sum(alpha[:,t-1] * self.A[:,n]) * self.B[n,obs[t]-1]
prob = np.sum(alpha[:,T-1])
return prob, alpha
def forward_with_scale(self, obs):
"""see scaling chapter in "A tutorial on hidden Markov models and
selected applications in speech recognition."
"""
T = len(obs)
N = self.N
alpha = np.zeros([N,T])
scale = np.zeros(T)
alpha[:,0] = self.pi[:] * self.B[:,obs[0]-1]
scale[0] = np.sum(alpha[:,0])
alpha[:,0] /= scale[0]
for t in xrange(1,T):
for n in xrange(0,N):
alpha[n,t] = np.sum(alpha[:,t-1] * self.A[:,n]) * self.B[n,obs[t]-1]
scale[t] = np.sum(alpha[:,t])
alpha[:,t] /= scale[t]
logprob = np.sum(np.log(scale[:]))
return logprob, alpha, scale
def backward(self, obs):
T = len(obs)
N = self.N
beta = np.zeros([N,T])
beta[:,T-1] = 1
for t in reversed(xrange(0,T-1)):
for n in xrange(0,N):
beta[n,t] = np.sum(self.B[:,obs[t+1]-1] * self.A[n,:] * beta[:,t+1])
prob = np.sum(beta[:,0])
return prob, beta
def backward_with_scale(self, obs, scale):
T = len(obs)
N = self.N
beta = np.zeros([N,T])
beta[:,T-1] = 1 / scale[T-1]
for t in reversed(xrange(0,T-1)):
for n in xrange(0,N):
beta[n,t] = np.sum(self.B[:,obs[t+1]-1] * self.A[n,:] * beta[:,t+1])
beta[n,t] /= scale[t]
return beta
def viterbi(self, obs):
T = len(obs)
N = self.N
psi = np.zeros([N,T]) # reverse pointer
delta = np.zeros([N,T])
q = np.zeros(T)
temp = np.zeros(N)
delta[:,0] = self.pi[:] * self.B[:,obs[0]-1]
for t in xrange(1,T):
for n in xrange(0,N):
temp = delta[:,t-1] * self.A[:,n]
max_ind = argmax(temp)
psi[n,t] = max_ind
delta[n,t] = self.B[n,obs[t]-1] * temp[max_ind]
max_ind = argmax(delta[:,T-1])
q[T-1] = max_ind
prob = delta[:,T-1][max_ind]
for t in reversed(xrange(1,T-1)):
q[t] = psi[q[t+1],t+1]
return prob, q, delta
def viterbi_log(self, obs):
T = len(obs)
N = self.N
psi = np.zeros([N,T])
delta = np.zeros([N,T])
pi = np.zeros(self.pi.shape)
A = np.zeros(self.A.shape)
biot = np.zeros([N,T])
pi = np.log(self.pi)
A = np.log(self.A)
biot = np.log(self.B[:,obs[:]-1])
delta[:,0] = pi[:] + biot[:,0]
for t in xrange(1,T):
for n in xrange(0,N):
temp = delta[:,t-1] + self.A[:,n]
max_ind = argmax(temp)
psi[n,t] = max_ind
delta[n,t] = temp[max_ind] + biot[n,t]
max_ind = argmax(delta[:,T-1])
q[T-1] = max_ind
logprob = delta[max_ind,T-1]
for t in reversed(xrange(1,T-1)):
q[t] = psi[q[t+1],t+1]
return logprob, q, delta
def baum_welch(self, obs):
T = len(obs)
M = self.M
N = self.N
alpha = np.zeros([N,T])
beta = np.zeros([N,T])
scale = np.zeros(T)
gamma = np.zeros([N,T])
xi = np.zeros([N,N,T-1])
# caculate initial parameters
logprobprev, alpha, scale = self.forward_with_scale(obs)
beta = self.backward_with_scale(obs, scale)
gamma = self.compute_gamma(alpha, beta)
xi = self.compute_xi(obs, alpha, beta)
logprobinit = logprobprev
# start interative
while True:
# E-step
self.pi = 0.001 + 0.999*gamma[:,0]
for i in xrange(N):
denominator = np.sum(gamma[i,0:T-1])
for j in xrange(N):
numerator = np.sum(xi[i,j,0:T-1])
self.A[i,j] = numerator / denominator
self.A = 0.001 + 0.999*self.A
for j in xrange(0,N):
denominator = np.sum(gamma[j,:])
for k in xrange(0,M):
numerator = 0.0
for t in xrange(0,T):
if obs[t]-1 == k:
numerator += gamma[j,t]
self.B[j,k] = numerator / denominator
self.B = 0.001 + 0.999*self.B
# M-step
logprobcur, alpha, scale = self.forward_with_scale(obs)
beta = self.backward_with_scale(obs, scale)
gamma = self.compute_gamma(alpha, beta)
xi = self.compute_xi(obs, alpha, beta)
delta = logprobcur - logprobprev
logprobprev = logprobcur
# print "delta is ", delta
if delta <= DELTA:
break
logprobfinal = logprobcur
return logprobinit, logprobfinal
def compute_gamma(self, alpha, beta):
gamma = np.zeros(alpha.shape)
gamma = alpha[:,:] * beta[:,:]
gamma = gamma / np.sum(gamma,0)
return gamma
def compute_xi(self, obs, alpha, beta):
T = len(obs)
N = self.N
xi = np.zeros((N, N, T-1))
for t in xrange(0,T-1):
for i in xrange(0,N):
for j in xrange(0,N):
xi[i,j,t] = alpha[i,t] * self.A[i,j] * \
self.B[j,obs[t+1]-1] * beta[j,t+1]
xi[:,:,t] /= np.sum(np.sum(xi[:,:,t],1),0)
return xi
def read_hmm(hmmfile):
fhmm = open(hmmfile,'r')
M = int(fhmm.readline().split(' ')[1])
N = int(fhmm.readline().split(' ')[1])
A = np.array([])
fhmm.readline()
for i in xrange(N):
line = fhmm.readline()
if i == 0:
A = np.array(map(float,line.split(',')))
else:
A = np.vstack((A,map(float,line.split(','))))
B = np.array([])
fhmm.readline()
for i in xrange(N):
line = fhmm.readline()
if i == 0:
B = np.array(map(float,line.split(',')))
else:
B = np.vstack((B,map(float,line.split(','))))
fhmm.readline()
line = fhmm.readline()
pi = np.array(map(float,line.split(',')))
fhmm.close()
return M, N, pi, A, B
def read_sequence(seqfile):
fseq = open(seqfile,'r')
T = int(fseq.readline().split(' ')[1])
line = fseq.readline()
obs = np.array(map(int,line.split(',')))
fseq.close()
return T, obs
本文转自ZH奶酪博客园博客，原文链接：http://www.cnblogs.com/CheeseZH/p/4229910.html，如需转载请自行联系原作者
