李航《统计学习方法》第2版第10章HMM课后习题答案

HMM这一章比较简单，把原理弄熟，看着公式，用numpy实现很简单。

习题10.1

在这里插入图片描述
懒得手算，直接代码：

import numpy as np
import pandas as pd

class MyHMM:
    def __init__(self,p,A,B,Ob,method="forward"):
        self.p=p  # 初始状态概率
        self.A=A  # 状态转移概率分布
        self.B=B  # 观测概率分布
        self.Ob=Ob # 观测序列
        self.method=method # 采用算法

    def forward(self):
        """前向算法"""
        alpha=self.p*self.B[self.Ob[0]]
        for i in range(1,self.Ob.size):
            alpha=alpha.dot(self.A)*self.B[self.Ob[i]]
        return alpha.sum()

    def back(self):
        """后向算法"""
        beta=np.ones(self.p.size)
        for i in range(1,self.Ob.size):
            beta=self.A.dot(self.B[self.Ob[self.Ob.size-i]]*beta)
        res=self.p.dot(self.B[self.Ob[0]]*beta)
        return res

    def fit(self):
        if "forward" in self.method:
            print("前向算法")
            res=self.forward()
        elif "back" in self.method:
            print("后向算法")
            res=self.back()
        else:
            raise ValueError
        return res

    def best_way(self):
        """最优路径"""
        index=[]
        sigema=self.p*self.B[self.Ob[0]]

        for i in range(1,self.Ob.size):
            _sigema=sigema.values*(self.A.T)
            col_index=np.argmax(_sigema,axis=1)
            row_index=np.arange(self.p.size)
            sigema=_sigema[row_index,col_index]*self.B[self.Ob[i]]
            index.append(dict(enumerate(col_index)))
        best_index=[]
        k=np.argmax(sigema)
        best_index.append(k)
        for i in range(len(index)):
            k=index[len(index)-1-i][k]
            best_index.append(k)
        best_index=np.array(best_index[::-1])+1  #因为索引从0开始的，所以加1
        return best_index

def main():
    A=np.array([[0.5,0.2,0.3],
                [0.3,0.5,0.2],
                [0.2,0.3,0.5]
                ])
    B=pd.DataFrame(np.array([[0.5,0.5],
                             [0.4,0.6],
                             [0.7,0.3]]),columns=["红","白"])
    p=np.array([0.2,0.4,0.4])
    Ob=np.array(["红","白","红","白"])
    clf=MyHMM(p,A,B,Ob,method="back")
    res=clf.fit()
    print('P(O|入)：',round(res,2))
    best_way=clf.best_way()
    print('维特比求得的最优路径为：',best_way)

if __name__=="__main__":
    main()

习题10.2

在这里插入图片描述

import numpy as np

def fore_algorithm(A, B, p_i, o, T, N):
    # 设初值，alpha_1(i) = pt_(i)b_i(o(i))
    alpha = np.zeros((T, N))
    for i in range(N):
        h = o[0]
        alpha[0][i] = p_i[i] * B[i][h]
    #递推
    for t in range(T-1):
        h = o[t+1]
        for i in range(N):
            a = 0
            for j in range(N):
                a += (alpha[t][j] * A[j][i])
            alpha[t+1][i] = a * B[i][h]
    #终止
    P = 0
    for i in range(N):
        P += alpha[T-1][i]
    return P, alpha

def back_algorithm(A, B, p_i, o, T, N):
    #设置初值，beta_t(i)=1
    beta = np.ones((T, N))
    #递推
    for t in range(T-1):
        t = T - t - 2
        h = o[t + 1]
        h = int(h)

        for i in range(N):
            beta[t][i] = 0
            for j in range(N):
                beta[t][i] += A[i][j] * B[j][h] * beta[t+1][j]
    #终止
    P = 0
    for i in range(N):
        h = o[0]
        h = int(h)
        P += p_i[i] * B[i][h] * beta[0][i]
    return P, beta

if __name__ == "__main__":
    T = 8
    N = 3
    A = [[0.5, 0.1, 0.4], [0.3, 0.5, 0.2], [0.2, 0.2, 0.6]]
    B = [[0.5, 0.5], [0.4, 0.6], [0.7, 0.3]]
    pi = [0.2, 0.3, 0.5]
    O = ['红', '白', '红', '红', '白', '红', '白', '白']
    o = np.zeros(T, np.int)
    for i in range(T):
        if O[i] == '白':
            o[i] = 1
        else:
            o[i] = 0
    PF, alpha = fore_algorithm(A, B, pi, o, T, N)
    PB, beta = back_algorithm(A, B, pi, o, T, N)
    print("PF:", PF, "PB:", PB)
    #P(i_4=q_3|O,\lambda) = alpah_4(3)* beta_4(3)
    P = alpha[4-1][3-1] * beta[4-1][3-1]
    print("前向后向概率计算可得 P(i4=q3|O,lambda)=", P / PF)