Hidden markov models
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Hidden Markov Models. 戴玉書. L.R Rabiner, B. H. Juang, An Introduction to Hidden Markov Models Ara V. Nefian and Monson H. Hayeslll, Face detection and recognition using Hidden Markov Models. Outline. Markov Chain & Markov Models

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Hidden markov models

Hidden Markov Models

戴玉書

L.R Rabiner, B. H. Juang, An Introduction to Hidden Markov Models

Ara V. Nefian and Monson H. Hayeslll, Face detection and recognition using Hidden Markov Models


Outline

Outline

  • Markov Chain & Markov Models

  • Hidden Markov Models

  • HMM Problem

    -Evaluation

    -Decoding

    -Learning

  • Application


Outline1

Outline

  • Markov Chain & Markov Models

  • Hidden Markov Models

  • HMM Problem

    -Evaluation

    -Decoding

    -Learning

  • Application


Markov chain property

Markov chain property:

  • Probability of each subsequent state depends only on what was the previous state


Markov models

Markov Models

State

State

State


Outline2

Outline

  • Markov Chain & Markov Models

  • Hidden Markov Models

  • HMM Problem

    -Evaluation

    -Decoding

    -Learning

  • Application


Hidden markov models1

Hidden Markov Models

  • If you don’t have complete state information, but some

    observations at each state

N - number of states :

M - the number of observables:

……

q1

q2

q3

q4


Hidden markov models2

Hidden Markov Models

State:{ , , }

Observable:{ , }

0.1

0.3

0.9

0.7

0.8

0.2


Hidden markov models3

Hidden Markov Models

  • M=(A, B, )

 = initial probabilities : =(i) , i= P(si)


Outline3

Outline

  • Markov Chain & Markov Models

  • Hidden Markov Models

  • HMM Problem

    -Evaluation

    -Decoding

    -Learning

  • Application


Evaluation

Evaluation

  • Determine the probability that a particular sequence of symbols O was generated by that model


Forward recursion

Forward recursion

  • Initialization:

  • Forward recursion:

  • Termination:


Backward recursion

Backward recursion

  • Initialization:

  • Backward recursion:

  • Termination:


Outline4

Outline

  • Markov Chain & Markov Models

  • Hidden Markov Models

  • HMM Problem

    -Evaluation

    -Decoding

    -Learning

  • Application


Decoding

Decoding

  • Given a set of symbols O determine the most likely

    sequence of hidden states Q that led to the

    observations

  • We want to find the state sequence Q which

  • maximizes P(Q|o1,o2,...,oT)


Viterbi algorithm

s1

si

sN

sj

qt-1 qt

a1j

aij

aNj

Viterbi algorithm

General idea:

if best path ending in qt= sj goes through qt-1= si then it should coincide with best path ending in qt-1= si


Viterbi algorithm1

Viterbi algorithm

  • Initialization:

  • Forward recursion:

  • Termination:


Viterbi algorithm2

Viterbi algorithm


Outline5

Outline

  • Markov Chain & Markov Models

  • Hidden Markov Models

  • HMM Problem

    -Evaluation

    -Decoding

    -Learning

  • Application


Learning problem

Learning problem

  • Given a coarse structure of the model, determine HMM parameters M=(A, B, ) that best fit training

    data

    determine these parameters


Baum welch algorithm

Baum-Welch algorithm

  • Define variable t(i,j) as the probability of being in state si at time t and in state sj at time t+1, given the observation sequence o1, o2, ... ,oT


Baum welch algorithm1

Baum-Welch algorithm

  • Define variable k(i) as the probability of being in state si at time t, given the observation sequence

    o1,o2 ,...,oT


Outline6

Outline

  • Markov Chain & Markov Models

  • Hidden Markov Models

  • HMM Problem

    -Evaluation problem

    -Decoding problem

    -Learning problem

  • Application


Example 1 character recognition

s1

s2

s3

Example 1 -character recognition

  • The structure of hidden states:

  • Observation = number of islands in the vertical slice


Example 1 character recognition1

Example 1 -character recognition

{1,3,2,1}

  • After character image segmentation the following sequence

    of island numbers in 4 slices was observed :


Example 2 face detection recognition

Example 2- face detection & recognition

  • The structure of hidden states:


Example 2 face detection

Example 2- face detection

  • A set of face images is used in the training of one HMM model

    N =6 states

Image:48, Training:9, Correct detection:90%,Pixels:60X90


Example 2 face recognition

Example 2- face recognition

  • Each individual in the database is represent by an HMM face model

  • A set of images representing different instances of same face are used to train each HMM

N =6 states


Example 2 face recognition1

Example 2- face recognition

Image:400, Training :Half, Individual:40, Pixels:92X112


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