6.4 隐马尔可夫过程 (Hidden Markov)

6.4 隐马尔可夫过程(Hidden Markov) 隐马尔可夫模型作为信号处理的一种统计模型，今天正在信号处理的各个领域得到广泛应用。 HMM是一个输出符号序列的统计模型，具有N个状态S1,S2,...,SN,它按一定的周期从一个状态转移到另一个状态，每次转移时，输出一个符号。转移到什么状态，转移时输出什么符号，分别由状态转移概率和转移时的符号输出概率来确定。因为只能观测到输出符号序列，而不能观测到状态转移序列（即模型输出符号序列时，是通过了哪些状态路径，不能知道），所以称为隐马尔可夫模型。

p11=0.3 p22=0.4 p23=0.6 p12=0.5 S2 S3 S1 p13=0.2 有三个状态：初始态S1,中间态S2,终了态S3，HMM只输出两个符号a和b。

p11=0.3 p22=0.4 p23=0.6 p12=0.5 S2 S3 S1 p13=0.2 假定从S1出发到S3截止，输出的符号序列为aab，试求输出aab的概率。从S1到S3，并且输出aab，可能的路径有三条 S1->S1->S2->S3 S1->S2->S2->S3 S1->S1->S1->S3

p22=0.4 p11=0.3 p23=0.6 p12=0.5 S2 S3 S1 p13=0.2 S1->S1->S2->S3 0.3*0.8*0.5*1*0.6*0.5=0.036 S1->S2->S2->S3 0.5*1*0.4*0.3*0.6*0.5=0.018 S1->S1->S1->S3 0.3*0.8*0.3*0.8*0.2*1=0.01152 由于不知道输出路径，所以，输出aab有三种可能路径，输出aab的概率为 0.036+0.018+0.01152=0.06552 如果知道路径，那么输出aab的概率就是该路径的输出概率。

A concrete example Assume you have a friend who lives far away and who you call daily to talk about what each of you did that day. Your friend has only three things he's interested in: walking in the park, shopping, and cleaning his apartment. The choice of what to do is determined exclusively by the weather on a given day. You have no definite information about the weather where your friend lives, but you know general trends. Based on what he tells you he did each day, you try to guess what the weather must have been like.

You believe that the weather operates as a discrete Markov chain. There are two states, "Rainy" and "Sunny", but you cannot observe them directly, that is, they are hidden from you. On each day, there is a certain chance that your friend will perform one of the following activities, depending on the weather: "walk", "shop", or "clean". Since your friend tells you about his activities, those are the observations. The entire system is that of a hidden Markov model (HMM).

You know the general weather trends in the area and you know what your friend likes to do on average. In other words, the parameters of the HMM are known. In fact, you can write them down in the Python programming language: states = ('Rainy', 'Sunny') observations = ('walk', 'shop', 'clean') start_probability = {'Rainy': 0.6, 'Sunny': 0.4} transition_probability = { 'Rainy' : {'Rainy': 0.7, 'Sunny': 0.3}, 'Sunny' : {'Rainy': 0.4, 'Sunny': 0.6}, } emission_probability = { 'Rainy' : {'walk': 0.1, 'shop': 0.4, 'clean': 0.5}, 'Sunny' : {'walk': 0.6, 'shop': 0.3, 'clean': 0.1}, }

In this fragment, start_probability refers to your uncertainty about which state the HMM is in when your friend first calls you (all you know is that it tends to be rainy on average). The transition_probability refers to the change of the weather in the underlying Markov chain. In this example, there is only a 30% chance that tomorrow will be sunny if today is rainy. The emission_probability tells you how likely your friend is to perform a certain activity on each day. If it's rainy, there is a 50% chance that he is cleaning his apartment; if it's sunny, there is a 60% chance that he will go outside for a walk.

p22=0.6 p12=0.3 p11=0.7 雨天 s1 睛天 s2 雨天: Walk 0.1 Shop 0.4 clean 0.5 睛天: Walk 0.6 Shop 0.3 clean 0.1 p21=0.4

6.4 隐马尔可夫过程 (Hidden Markov)