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## PowerPoint Slideshow about ' Hidden Markov Models' - hector-skinner

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### Hidden Markov Models

Outline

Outline

Outline

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
- Hidden Markov Models
- HMM Problem
-Evaluation

-Decoding

-Learning

- Application

Outline

- Markov Chain & Markov Models
- Hidden Markov Models
- HMM Problem
-Evaluation

-Decoding

-Learning

- Application

Markov chain property:

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

Outline

- Markov Chain & Markov Models
- Hidden Markov Models
- HMM Problem
-Evaluation

-Decoding

-Learning

- Application

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

- Markov Chain & Markov Models
- Hidden Markov Models
- HMM Problem
-Evaluation

-Decoding

-Learning

- Application

Evaluation

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

Forward recursion

- Initialization:
- Forward recursion:
- Termination:

Backward recursion

- Initialization:
- Backward recursion:
- Termination:

- Markov Chain & Markov Models
- Hidden Markov Models
- HMM Problem
-Evaluation

-Decoding

-Learning

- Application

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)

s1

si

sN

sj

qt-1 qt

a1j

aij

aNj

Viterbi algorithmGeneral 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 algorithm

- Initialization:
- Forward recursion:
- Termination:

- Markov Chain & Markov Models
- Hidden Markov Models
- HMM Problem
-Evaluation

-Decoding

-Learning

- Application

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

- 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 algorithm

- Define variable k(i) as the probability of being in state si at time t, given the observation sequence
o1,o2 ,...,oT

Outline

- Markov Chain & Markov Models
- Hidden Markov Models
- HMM Problem
-Evaluation problem

-Decoding problem

-Learning problem

- Application

s1

s2

s3

Example 1 -character recognition- The structure of hidden states:
- Observation = number of islands in the vertical slice

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

- The structure of hidden states:

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

- 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 recognition

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

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