HMM 隐马尔可夫模型初学（二）

1、HMM，Hidden Markov model 隐马尔科夫模型

（1）天气举例

• 假设不能直接观察天气阴晴雨情况，只能看到地面的潮湿情况（假如分为非常潮湿，一般潮湿，不潮湿三种对应A，B，C三种评级）。现在我一连观察了一周的地面潮湿情况（AABBCBA），是否能够判断这一周的天气？
• 如上所述，有两类状态：一类是地面潮湿状态 observation stata（A、B、C）；一类是天气情况 latent stata（阴晴雨）；
• 之前已经知道了天气的状态转移概率矩阵，现在主要就需要知道天气情况与地面潮湿情况的对应关系，从而预测天气情况。

（2）Hidden Markov model

HMM理解

• HMM随机生成的状态随机序列被称为状态序列；每个状态生成一个观测，由此产生的观测随机序列，被称为观测序列。
• HMM是一个双重随机过程---具有一定状态的隐马尔可夫链（天气情况）和随机的观测序列（地面潮湿情况）。

• The HMM is a discrete time model: for each point in time t, we have one hidden state that generates one observed event for that time point t.

  
  
  HMM

HMM应用条件

• The hidden states follow a Markov process, i.e., the states over time are not independent of one another, but the current state depends on the previous state only (and not on earlier states)
  即MM马尔科夫的基本假设，当前状态只与上一状态有关

  
  
  assumption1

• The distribution generating the observation depends on the state of an underlying, hidden state,。
  某时刻的观测结果只与当前时刻对应的隐状态有关

  
  
  assumption2

发射矩阵 emission distribution matrix

• 即每种隐状态对应的，可能出现的观测结果的概率情况。

• 例如晴天天气里，地面出现A、B、C三种情况的概率分别为0.1，0.3，0.6；阴天分别为0.2，0.5，0.3；雨天分别为0.5，0.3，0.2

  
  
  EM

HMM三要素

• 综上，以及MM知识点，描述HMM，需要三大参数

  
  
  three sets of parameters
• 此外还有隐状态集合，预测值集合
  关于预测值，一般以分类类型结果为主；
  关于隐状态，我认为在实际探索的过程中往往并不清楚状态的含义，而是根据预测值及专业知识分析状态意义。例如我把天气视为观测结果，那么隐状态可以是梅雨期？台风期？甚至可以说是太阳公公心情的好坏状态。

HMM模型可以解决的三类问题！！！划重点

（1）概率计算问题

• 给定模型λ=(A,B,π)和观测序列Q={q1,q2,...,qT}，计算模型λ下观测到序列Q出现的概率P(Q|λ)；
• 例如给定天气与地面潮湿情况的HMM三要素，及一组地面潮湿情况数据。计算这组数据的出现概率是多大
• 方法：前向-后向算法

（2）学习问题

• 已知观测序列Q={q1,q2,...,qT}，估计模型λ=(A,B,π)的参数，使得在该模型下观测序列P(Q|λ)最大；
• 例如已知一组地面潮湿情况数据，估计天气与地面潮湿情况的HMM三要素，从而使出现这组数据的可能性最大；
• 方法：Maximum likelihood, Expectation Maximization or Baum-Welch algorithm, and Bayesian estimation

（3）预测问题

• 给定模型λ=(A,B,π)和观测序列Q={q1,q2,...,qT}，求给定观测序列条件概率P(I|Q，λ)最大的状态序列I。
• 例如已知一周的地面潮湿情况数据，并且已知天气与地面潮湿情况的HMM三要素，从而估计这一周最有可能的天气情况
• 方法：Viterbi algorithm

个人认为在实验探索中，观察预测值比较容易获得，由此学习建模，估计参数。然后根据模型结果，进行其它观察结果的隐状态序列的预测。

2、R代码实操

• 已知某特殊DNA序列存在两个区（hidden stata），分别是AT-rich与CG-rich。

• 前者富含AT碱基，后者CG碱基，具体碱基概率分布(emission matrix)如下图。

  
  
  HMM

• 此外也知道了初始状态分布概率以及状态概率转移矩阵，据此随机生成一段符合要求的碱基序列。

• 第一步：准备TM、EM

states              <- c("AT-rich", "GC-rich") # Define the names of the states
ATrichprobs         <- c(0.7, 0.3)             # Set the probabilities of switching states, where the previous state was "AT-rich"
GCrichprobs         <- c(0.1, 0.9)             # Set the probabilities of switching states, where the previous state was "GC-rich"
thetransitionmatrix <- matrix(c(ATrichprobs, GCrichprobs), 2, 2, byrow = TRUE) # Create a 2 x 2 matrix
rownames(thetransitionmatrix) <- states
colnames(thetransitionmatrix) <- states
thetransitionmatrix     

nucleotides         <- c("A", "C", "G", "T")   # Define the alphabet of nucleotides
ATrichstateprobs    <- c(0.39, 0.1, 0.1, 0.41) # Set the values of the probabilities, for the AT-rich state
GCrichstateprobs    <- c(0.1, 0.41, 0.39, 0.1) # Set the values of the probabilities, for the GC-rich state
theemissionmatrix <- matrix(c(ATrichstateprobs, GCrichstateprobs), 2, 4, byrow = TRUE) # Create a 2 x 4 matrix
rownames(theemissionmatrix) <- states
colnames(theemissionmatrix) <- nucleotides
theemissionmatrix   

• 第二步：编写函数

# Function to generate a DNA sequence, given a HMM and the length of the sequence to be generated.
generatehmmseq <- function(transitionmatrix, emissionmatrix, initialprobs, seqlength)
{
  nucleotides     <- c("A", "C", "G", "T")   # Define the alphabet of nucleotides
  states          <- c("AT-rich", "GC-rich") # Define the names of the states
  mysequence      <- character()             # Create a vector for storing the new sequence
  mystates        <- character()             # Create a vector for storing the state that each position in the new sequence
  # was generated by
  # Choose the state for the first position in the sequence:
  firststate      <- sample(states, 1, rep=TRUE, prob=initialprobs)
  # Get the probabilities of the current nucleotide, given that we are in the state "firststate":
  probabilities   <- emissionmatrix[firststate,]
  # Choose the nucleotide for the first position in the sequence:
  firstnucleotide <- sample(nucleotides, 1, rep=TRUE, prob=probabilities)
  mysequence[1]   <- firstnucleotide         # Store the nucleotide for the first position of the sequence
  mystates[1]     <- firststate              # Store the state that the first position in the sequence was generated by
  
  for (i in 2:seqlength)
  {
    prevstate    <- mystates[i-1]           # Get the state that the previous nucleotide in the sequence was generated by
    # Get the probabilities of the current state, given that the previous nucleotide was generated by state "prevstate"
    stateprobs   <- transitionmatrix[prevstate,]
    # Choose the state for the ith position in the sequence:
    state        <- sample(states, 1, rep=TRUE, prob=stateprobs)
    # Get the probabilities of the current nucleotide, given that we are in the state "state":
    probabilities <- emissionmatrix[state,]
    # Choose the nucleotide for the ith position in the sequence:
    nucleotide   <- sample(nucleotides, 1, rep=TRUE, prob=probabilities)
    mysequence[i] <- nucleotide             # Store the nucleotide for the current position of the sequence
    mystates[i]  <- state                   # Store the state that the current position in the sequence was generated by
  }
  
  for (i in 1:length(mysequence))
  {
    nucleotide   <- mysequence[i]
    state        <- mystates[i]
    print(paste("Position", i, ", State", state, ", Nucleotide = ", nucleotide))
  }
}

• 第三步：给定初始状态概率进行预测

theinitialprobs <- c(0.5, 0.5)
generatehmmseq(thetransitionmatrix, theemissionmatrix, theinitialprobs, 30)

result

R实操代码本身意义可能不大，但对于我们具体了解HMM模型很有帮助。在具体应用HMM模型时，更多的是采用相应的R包进行分析。这类R包有不少，会挑选几个进行示例学习。

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参考文章
1、Hidden Markov Models — Bioinformatics 0.1 documentation
2、01 隐马尔可夫模型 - 马尔可夫链、HMM参数和性质 -
3、Multilevel HMM tutorial
4、马尔可夫链_百度百科