一、HMM模型+维特比算法实例

1、问题描述

假设连续观察3天的海藻湿度为(Dry,Damp,Soggy),求这三天最可能的天气情况。

2、已知信息

①天气只有三类(Sunny,Cloudy,Rainy)，海藻湿度有四类{Dry,Dryish, Damp,Soggy }，而且海藻湿度和天气有一定的关系。

②隐藏的状态：Sunny, Cloudy, Rainy;

③观察状态序列：{Dry, Damp, Soggy}

④初始状态序列：

Sunny

Cloudy

Rainy

0.63

0.17

0.20

⑤状态转移矩阵：

Sunny

Cloudy

Rainy

Sunny

0.5

0.375

0.125

Cloudy

0.25

0.125

0.625

Rainy

0.25

0.375

0.375

⑥发射矩阵：

Dry

Dryish

Damp

Soggy

Sunny

0.6

0.2

0.15

0.05

Cloudy

0.25

0.25

0.25

0.25

Rainy

0.05

0.10

0.35

0.5

3、分析

由一阶HMM可知，Day2的天气仅取决于Day1；Day3的天气又只取决于Day2的天气。

4、计算过程

(1)Day1由于是初始状态，我们分别求

P(Day1-Sunny)=0.63*0.6;

P(Day1-Cloudy)=0.17*0.25;

P(Day1-Rain)=0.20*0.05;

Choose max{ P(Day1-Sunny) , P(Day1-Cloudy),P(Day1-Rainy)}, 得到P(Day1-Sunny)最大，得出第1天Sunny的概率最大。

(2)Day2的天气又取决于Day1的天气状况，同时也受Day2观察的海藻情况影响。

P(Day2-Sunny)= max{ P(Day1-Sunny)*0.5, P(Day1-Cloudy)*0.25,  P(Day1-Rainy)*0.25} *0.15;

P(Day2-Cloudy)= max{ P(Day1-Sunny)*0.375,  P(Day1-Cloudy)*0.125, P(Day1-Rainy)*0.625} *0.25;

P(Day2-Rainy)= max{ P(Day1-Sunny)*0.125,  P(Day1-Cloudy)*0.625 , P(Day1-Rainy)*0.375} *0.35;

Choosemax{ P(Day2-Sunny) , P(Day2-Cloudy), P(Day2-Rainy)},得到P(Day2-Cloudy)最大，得出第2天Cloudy的概率最大。

故{Sunny,Cloudy}是前两天最大可能的天气序列。

(3)Day3的天气又取决于Day2的天气状况，同时也受Day3观察的海藻情况影响。

P(Day3-Sunny)= max{ P(Day2-Sunny)*0.5, P(Day2-Cloudy)*0.25,  P(Day2-Rainy)*0.25} *0.05;

P(Day3-Cloudy)= max{ P(Day2-Sunny)*0.375,  P(Day2-Cloudy)*0.125, P(Day2-Rainy)*0.625} *0.25;

P(Day3-Rainy)= max{ P(Day2-Sunny)*0.125,  P(Day2-Cloudy)*0.625, P(Day2-Rainy)*0.375} *0. 05;

Choosemax{ P(Day3-Sunny) , P(Day3-Cloudy), P(Day3-Rainy)},得到P(Day3-Rainy)最大，得出第3天Rainy的概率最大。故{Sunny,Cloudy,Rainy}是这三天最可能的天气序列。

5.Python代码

流程图：

1 #-*- coding:utf-8 -*-

2 __author__ = 'Administrator'

3

4 init_vec={"sunny":0.63,"cloudy":0.17,"rainy":0.20}5 trans_mat={"sunny":{"sunny":0.5,"cloudy":0.375,"rainy":0.125},6 "cloudy":{"sunny":0.25,"cloudy":0.125,"rainy":0.625},7 "rainy":{"sunny":0.25,"cloudy":0.375,"rainy":0.375}}8 emit_mat={"sunny":{"dry":0.6,"dryish":0.2,"damp":0.15,"soggy":0.05},9 "cloudy":{"dry":0.25,"dryish":0.25,"damp":0.25,"soggy":0.25},10 "rainy":{"dry":0.05,"dryish":0.10,"damp":0.35,"soggy":0.50}}11 observes=["dry","damp","soggy"]12 states=["sunny","cloudy","rainy"]13

14 #列表中包含字典，就相当于二阶列表使用

15 tab=[{}] #只有一行

16 path=[{}]17 for t inrange(len(observes)):18 #print(t)

19 if t==0:20 temp=[]21 for state instates:22 prob=init_vec[state]*emit_mat[state].get(observes[t])23 tab[0][state]=prob24 temp.append((prob,state))25 best_prob,best_state=max(temp,key=lambdax:x[0])26 path[0][best_state]=best_prob27 else:28 tab.append({})29 path.append({})30 temp=[]31 for state1 instates:32 item=[]33 for state2 instates:34 prob=tab[t-1][state2]*trans_mat[state2].get(state1)*emit_mat[state1].get(observes[t])35 item.append((prob,state2))36 best_prob,best_state=max(item,key=lambdax:x[0])37 tab[t][state1]=best_prob38 temp.append((best_prob,state1))39 best_prob,best_state=max(temp,key=lambdax:x[0])40 path[t][best_state]=best_prob41 print(tab)42 print(path)

View Code

输出结果：

[{'rainy': 0.010000000000000002, 'cloudy': 0.0425, 'sunny': 0.378}, {'rainy': 0.0165375, 'cloudy': 0.0354375, 'sunny': 0.02835}, {'rainy': 0.01107421875, 'cloudy': 0.0026578125, 'sunny': 0.0007087500000000001}]

[{'sunny': 0.378}, {'cloudy': 0.0354375}, {'rainy': 0.01107421875}]