【math】Hiden Markov Model 隐马尔可夫模型了解

news2024/10/3 0:50:58

文章目录

Introduction to Hidden Markov Model
- Introduction
- - Markov chain
  - Hidden Markov Model(HMM)
- Three Questions
- - Q1: evaluate problem -- Forward algorithm
  - Q2: decode problem -- Viterbi algorithm
  - Q3: learn problem -- Baum-Welch algorithm
- Application

Introduction to Hidden Markov Model

Introduction

Markov chains were first introduced in 1906 by Andrey Markov
HMM was developed by L. E. Baum and coworkers in the 1960s
HMM is simplest dynamic Bayesian network and a directed graphic model
Application: speech recognition, PageRank(Google), DNA analysis, …

Markov chain

A Markov chain is “a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event”.

在这里插入图片描述
Space or time can be either discrete(𝑋_𝑡:t=0, 1, 2,…) or continuous(𝑋_𝑡:t≥0). (we will focus on Markov chains in discrete space an time)

Example for Markov chain：

transition matrix 𝑄 :
5-step transition matrix is 𝑄^5 :

Hidden Markov Model(HMM)

HMM is a statistical Markov model in which the system being modeled is assumed to be a Markov process with unobserved (i.e. hidden) states.

State: x = (x1, x2, x3) ; 
Observation: y = (y1, y2, y3);
Transition matrix: A = (aij); 
Emission matrix: B = (bij)

在这里插入图片描述

Example:
在这里插入图片描述

Three Questions

Given the model 𝜆=[𝐴, 𝐵,𝜋], how to calculate the probability of producing the observation 𝒚={𝑦 $_1$ ,𝑦 $_2$ ,…,𝑦 $_𝑛$ | 𝑦 $_𝑖$ ∈𝑂}? In other words, how to evaluate the matching degree between the model and the observation ?
Given the model 𝜆=[𝐴, 𝐵,𝜋] and the observation 𝒚={𝑦 $_1$ ,𝑦 $_2$ ,…,𝑦 $_𝑛$ | 𝑦 $_𝑖$ ∈𝑂}, how to find most probable state 𝒙={𝑥 $_1$ ,𝑥 $_2$ ,…,𝑥 $_𝑛$ |𝑥 $_𝑖$ ∈𝑆} ? In other words, how to infer the hidden state from the observation ?
Given the observation 𝒚={𝑦 $_1$ ,𝑦 $_2$ ,…,𝑦 $_𝑛$ | 𝑦 $_𝑖$ ∈𝑂}, how to adjust model parameters 𝜆=[𝐴, 𝐵,𝜋] to maximize to the probability 𝑃(𝒚│𝜆) ? In other words, how to train the model to describe the observation more accurately ?

Q1: evaluate problem – Forward algorithm

Q1: how to evaluate the matching degree between the model and the observation ? ( forward algorithm)

the probability of observing event 𝑦 $_1$ : 𝑃 $_{𝑖0}$ = 𝑃 $_𝑖$ (𝑂=𝑦 ${_1}$ ) = 𝜋 $_{𝑖}𝑏_{1𝑖}$
the probability of observing event 𝑦 $_{𝑗+1}$ (𝑗≥1): 𝑃 $_𝑖𝑗$ =𝑏 $_{𝑗,𝑖+1}$ ∑ $_𝑘$ 𝑃 $_{𝑖,𝑗−1}$ 𝑎 $_{𝑖𝑗}$
𝑃(𝒚)=∑ $_{𝑘}$ 𝑃 $_{𝑖𝑗}$ ∗𝑎 $_{𝑗0}$

Q2: decode problem – Viterbi algorithm

Q2: how to infer the hidden state from the observation ? (Viterbi algorithm)

Observation 𝒚=(𝑦 $_1$ , 𝑦 $_2$ ,…, 𝑦 $_𝑇$ ), initial prob. 𝝅=(𝜋 $_1$ ,𝜋 $_2$ ,…, 𝜋 $_𝐾$ ), transition matrix 𝐴, emission matrix 𝐵.
在这里插入图片描述
Viterbi algorithm(optimal solution) backtracking method; It retains the optimal solution of each choice in the previous step and finds the optimal selection path through the backtracking method.

Example:

The observation 𝒚={“′Normal′, ′Cold′, ′Dizzy′” } , 𝜆=[𝐴, 𝐵,𝜋], the hidden state 𝒙={𝑥 $_1$ ,𝑥 $_2$ ,𝑥 $_3$ }= ?

Q3: learn problem – Baum-Welch algorithm

Q3: how to train the model to describe the observation more accurately ? (Baum-Welch algorithm)

The Baum–Welch algorithm uses the well known EM algorithm to find the maximum likelihood estimate of the parameters of a hidden Markov model given a set of observed feature vectors.
在这里插入图片描述

Baum–Welch algorithm(forward-backward alg.) It approximates the optimal parameters through iteration.

在这里插入图片描述

(1) Likelihood function :
𝑙𝑜𝑔𝑃(Y,𝐼|𝜆)
(2) Expectation of EM algorithm:
𝑄(𝜆,𝜆 ̂ )= ∑ $_𝐼$ 𝑙𝑜𝑔𝑃(𝑌,𝐼|𝜆)𝑃(𝑌,𝐼|𝜆 ̂ )
(3) *Maximization of EM algorithm:
max 𝑄(𝜆,𝜆 ̂ )

use Lagrangian multiplier method and take the partial derivative of Lagrangian funcition。

Application

CpG island:

In the human genome wherever the dinucleotide CG occurs, the C nucleotide is typically chemically modified by methylation.
around the promoters or ‘start’ regions
CpG is typically a few hundred to a few thousand bases long.

three questions: of CpG island:
Given the model of distinguished the CpG island, how to calculate the probability of the observation sequence ?
Given a short stretch of genomic sequence, how would we decide if it comes from a CpG island or not ?
Given a long piece of sequence, how would we find the CpG islands in it?

reference：

Markov chain Defination
A Revealing Introduction to Hidden Markov Models
Top 10 Algorithms in Data Mining
An Introduction to Hidden Markov Models for Biological Sequences
python-hmmlearn-example
Viterbi algorithm
Baum-Welch blog
Biological Sequence Analysis. Probabilistic Models of Proteins and Nucleic Acids. R. Durbin, S. Eddy, A. Krogh and G. Mitchison

未完待续…

本文来自互联网用户投稿，该文观点仅代表作者本人，不代表本站立场。本站仅提供信息存储空间服务，不拥有所有权，不承担相关法律责任。如若转载，请注明出处：http://www.coloradmin.cn/o/7211.html

如若内容造成侵权/违法违规/事实不符，请联系多彩编程网进行投诉反馈，一经查实，立即删除！