python实现隐马尔科夫模型HMM
一份完全按照李航<<统计学习方法>>介绍的HMM代码，供大家参考，具体内容如下
#coding=utf8
'''''
Created on 2017-8-5
里面的代码许多地方可以精简，但为了百分百还原公式，就没有精简了。
@author: adzhua
'''
import numpy as np
class HMM(object):
def __init__(self, A, B, pi):
'''''
A: 状态转移概率矩阵
B: 输出观察概率矩阵
pi: 初始化状态向量
'''
self.A = np.array(A)
self.B = np.array(B)
self.pi = np.array(pi)
self.N = self.A.shape[0] # 总共状态个数
self.M = self.B.shape[1] # 总共观察值个数
# 输出HMM的参数信息
def printHMM(self):
print ("==================================================")
print ("HMM content: N =",self.N,",M =",self.M)
for i in range(self.N):
if i==0:
print ("hmm.A ",self.A[i,:]," hmm.B ",self.B[i,:])
else:
print (" ",self.A[i,:]," ",self.B[i,:])
print ("hmm.pi",self.pi)
print ("==================================================")
# 前向算法
def forwar(self, T, O, alpha, prob):
'''''
T: 观察序列的长度
O: 观察序列
alpha: 运算中用到的临时数组
prob: 返回值所要求的概率
'''
# 初始化
for i in range(self.N):
alpha[0, i] = self.pi[i] * self.B[i, O[0]]
# 递归
for t in range(T-1):
for j in range(self.N):
sum = 0.0
for i in range(self.N):
sum += alpha[t, i] * self.A[i, j]
alpha[t+1, j] = sum * self.B[j, O[t+1]]
# 终止
sum = 0.0
for i in range(self.N):
sum += alpha[T-1, i]
prob[0] *= sum
# 带修正的前向算法
def forwardWithScale(self, T, O, alpha, scale, prob):
scale[0] = 0.0
# 初始化
for i in range(self.N):
alpha[0, i] = self.pi[i] * self.B[i, O[0]]
scale[0] += alpha[0, i]
for i in range(self.N):
alpha[0, i] /= scale[0]
# 递归
for t in range(T-1):
scale[t+1] = 0.0
for j in range(self.N):
sum = 0.0
for i in range(self.N):
sum += alpha[t, i] * self.A[i, j]
alpha[t+1, j] = sum * self.B[j, O[t+1]]
scale[t+1] += alpha[t+1, j]
for j in range(self.N):
alpha[t+1, j] /= scale[t+1]
# 终止
for t in range(T):
prob[0] += np.log(scale[t])
def back(self, T, O, beta, prob):
'''''
T: 观察序列的长度 len(O)
O: 观察序列
beta: 计算时用到的临时数组
prob: 返回值；所要求的概率
'''
# 初始化
for i in range(self.N):
beta[T-1, i] = 1.0
# 递归
for t in range(T-2, -1, -1): # 从T-2开始递减；即T-2, T-3, T-4, ..., 0
for i in range(self.N):
sum = 0.0
for j in range(self.N):
sum += self.A[i, j] * self.B[j, O[t+1]] * beta[t+1, j]
beta[t, i] = sum
# 终止
sum = 0.0
for i in range(self.N):
sum += self.pi[i]*self.B[i,O[0]]*beta[0,i]
prob[0] = sum
# 带修正的后向算法
def backwardWithScale(self, T, O, beta, scale):
'''''
T: 观察序列的长度 len(O)
O: 观察序列
beta: 计算时用到的临时数组
'''
# 初始化
for i in range(self.N):
beta[T-1, i] = 1.0
# 递归
for t in range(T-2, -1, -1):
for i in range(self.N):
sum = 0.0
for j in range(self.N):
sum += self.A[i, j] * self.B[j, O[t+1]] * beta[t+1, j]
beta[t, i] = sum / scale[t+1]
# viterbi算法
def viterbi(self, O):
'''''
O: 观察序列
'''
T = len(O)
# 初始化
delta = np.zeros((T, self.N), np.float)
phi = np.zeros((T, self.N), np.float)
I = np.zeros(T)
for i in range(self.N):
delta[0, i] = self.pi[i] * self.B[i, O[0]]
phi[0, i] = 0.0
# 递归
for t in range(1, T):
for i in range(self.N):
delta[t, i] = self.B[i, O[t]] * np.array([delta[t-1, j] * self.A[j, i] for j in range(self.N)] ).max()
phi = np.array([delta[t-1, j] * self.A[j, i] for j in range(self.N)]).argmax()
# 终止
prob = delta[T-1, :].max()
I[T-1] = delta[T-1, :].argmax()
for t in range(T-2, -1, -1):
I[t] = phi[I[t+1]]
return prob, I
# 计算gamma(计算A所需的分母；详情见李航的统计学习) : 时刻t时马尔可夫链处于状态Si的概率
def computeGamma(self, T, alpha, beta, gamma):
''''''''
for t in range(T):
for i in range(self.N):
sum = 0.0
for j in range(self.N):
sum += alpha[t, j] * beta[t, j]
gamma[t, i] = (alpha[t, i] * beta[t, i]) / sum
# 计算sai(i,j)(计算A所需的分子) 为给定训练序列O和模型lambda时
def computeXi(self, T, O, alpha, beta, Xi):
for t in range(T-1):
sum = 0.0
for i in range(self.N):
for j in range(self.N):
Xi[t, i, j] = alpha[t, i] * self.A[i, j] * self.B[j, O[t+1]] * beta[t+1, j]
sum += Xi[t, i, j]
for i in range(self.N):
for j in range(self.N):
Xi[t, i, j] /= sum
# 输入 L个观察序列O，初始模型：HMM={A,B,pi,N,M}
def BaumWelch(self, L, T, O, alpha, beta, gamma):
DELTA = 0.01 ; round = 0 ; flag = 1 ; probf = [0.0]
delta = 0.0; probprev = 0.0 ; ratio = 0.0 ; deltaprev = 10e-70
xi = np.zeros((T, self.N, self.N)) # 计算A的分子
pi = np.zeros((T), np.float) # 状态初始化概率
denominatorA = np.zeros((self.N), np.float) # 辅助计算A的分母的变量
denominatorB = np.zeros((self.N), np.float)
numeratorA = np.zeros((self.N, self.N), np.float) # 辅助计算A的分子的变量
numeratorB = np.zeros((self.N, self.M), np.float) # 针对输出观察概率矩阵
scale = np.zeros((T), np.float)
while True:
probf[0] =0
# E_step
for l in range(L):
self.forwardWithScale(T, O[l], alpha, scale, probf)
self.backwardWithScale(T, O[l], beta, scale)
self.computeGamma(T, alpha, beta, gamma) # (t, i)
self.computeXi(T, O[l], alpha, beta, xi) #(t, i, j)
for i in range(self.N):
pi[i] += gamma[0, i]
for t in range(T-1):
denominatorA[i] += gamma[t, i]
denominatorB[i] += gamma[t, i]
denominatorB[i] += gamma[T-1, i]
for j in range(self.N):
for t in range(T-1):
numeratorA[i, j] += xi[t, i, j]
for k in range(self.M): # M为观察状态取值个数
for t in range(T):
if O[l][t] == k:
numeratorB[i, k] += gamma[t, i]
# M_step。 计算pi, A, B
for i in range(self.N): # 这个for循环也可以放到for l in range(L)里面
self.pi[i] = 0.001 / self.N + 0.999 * pi[i] / L
for j in range(self.N):
self.A[i, j] = 0.001 / self.N + 0.999 * numeratorA[i, j] / denominatorA[i]
numeratorA[i, j] = 0.0
for k in range(self.M):
self.B[i, k] = 0.001 / self.N + 0.999 * numeratorB[i, k] / denominatorB[i]
numeratorB[i, k] = 0.0
#重置
pi[i] = denominatorA[i] = denominatorB[i] = 0.0
if flag == 1:
flag = 0
probprev = probf[0]
ratio = 1
continue
delta = probf[0] - probprev
ratio = delta / deltaprev
probprev = probf[0]
deltaprev = delta
round += 1
if ratio <= DELTA :
print('num iteration: ', round)
break
if __name__ == '__main__':
print ("python my HMM")
# 初始的状态概率矩阵pi；状态转移矩阵A；输出观察概率矩阵B; 观察序列
pi = [0.5,0.5]
A = [[0.8125,0.1875],[0.2,0.8]]
B = [[0.875,0.125],[0.25,0.75]]
O = [
[1,0,0,1,1,0,0,0,0],
[1,1,0,1,0,0,1,1,0],
[0,0,1,1,0,0,1,1,1]
]
L = len(O)
T = len(O[0]) # T等于最长序列的长度就好了
hmm = HMM(A, B, pi)
alpha = np.zeros((T,hmm.N),np.float)
beta = np.zeros((T,hmm.N),np.float)
gamma = np.zeros((T,hmm.N),np.float)
# 训练
hmm.BaumWelch(L,T,O,alpha,beta,gamma)
# 输出HMM参数信息
hmm.printHMM()
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