python实现隐马尔科夫模型HMM

#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()