python实现HMM

一份完全按照李航<<统计学习方法>>介绍的HMM代码。

#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 npclass 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()