Hidden markov model

Hidden Markov Model

Prepared by : Haitham Abdel-atty AbdullahSupervised by : Prof. Taymor T. Nazmy

Agenda

Introduction Markov Model Hidden Markov Model Problems in HMM Applications HMM in speech recognition References

Introduction

Stochastic process (random process) :

System that changes over time in an uncertain manner. Is a collection of random variables, representing the

evolution of some system of random values over time. This is the probabilistic counterpart to a deterministic process.

Instead of describing a process which can only evolve in one way, in a stochastic or random process there is some indeterminacy: even if the initial condition (or starting point) is known, there are several (often infinitely many) directions in which the process may evolve.

Introduction (Cont.)

Deterministic process example :


Stochastic process example :

Techniques to model the Stochastic process

Branching processGaussian process Hidden Markov model Markov process



In 1906, Andrey Markov introduced the Markov chains.

He produced the first theoretical results for stochastic processes by using the term “chain” for the first time.

It is required to possess a property that is usually characterized as "memoryless" : the probability distribution of the next state depends only on the current state and not on the sequence of events that preceded it. (also called Markov Property)

What’s HMM?

Hidden Markov Model

Markov ModelHidden

What is ‘hidden’? What is ‘Markov model’?

Markov Model

Markov Model

Is a stochastic model used to model randomly changing systems where it is assumed that future states depend only on the present state and not on the sequence of events that preceded it

Markov Model (Cont.)

Example 1 : Let’s talk about the weather, Here in Cairo we assume that we have three

types of weather sunny, rainy, and cloudy. Let’s assume for the moment the weather lasts all day, i.e. it doesn’t change from rainy to sunny in that the middle of the day.

By carefully examining the weather for a long time, we found following weather change pattern.


Question :

What is the probability that the weather for the next 6 days will be “cloudy-rainy-rainy-sunny-cloudy-sunny” when today is sunny given our weather Markov model ?


Definitions : Observable states :

Observed sequence :

State transition matrix :


},,,{ 21 Tqqq

},,2,1{ N

Definitions : Initial state probability :

Markov assumption ( Markov Property ) :


Definitions : Sequence probability of Markov model :


Remember Markov assumption

???)sunny-cloudy-sunny-rainy-rainy-cloudy -sunny ( P

The answer :O = {“cloudy-rainy-rainy-sunny-cloudy-sunny”}.

when today is sunny

Assume that S1 : rainy ,

S2 : cloudy,

S3 : sunny.

P(O | model ) = P(sunny-cloudy-rainy-rainy-sunny-cloudy-sunny |model)

= P(S3, S2, S1, S1, S3, S2, S3 | model )

= P(S3) . P(S2|S3) . P(S1|S2) . P(S1|S1)

P(S3|S1) . P(S2|S3) . P(S3|S2).

= 1 . (0.1) . (0.3) . (0.4) . (0.3) . (0.1) . (0.2)

= 0.00007


What’s HMM?

Hidden Markov Model

Markov ModelHidden

What is ‘hidden’? What is ‘Markov model’?

So far we have considered Markov models in which each state corresponded to an observable (physical) event. This model is too restrictive to be applicable to many problems of interest, so we extend the concept of Markov models to include the case where the observation is a probabilistic function of the state.

Hidden Markov Model

The adjective 'hidden' refers to the state sequence through which the model passes, not to the parameters of the model.

Notation : (1) N: Number of states.

(2) M: Number of symbols observable in states.

(3) A: State transition probability distribution

(4) B: Observation symbol probability distribution

(5) Initial state distribution

Hidden Markov Model

),,( BA

HMM Core Problems

Problem 1 : Finding the probability of an observed sequence. What is the ???

Solution : Sum over all possible paths of the state sequence

that generate the given observation sequence, using forward algorithm .

HMM Core Problems (cont.)

) | P(O

Forward algorithm


Example : What is the probability of the

sequence of observation :

O = {shopping, cleaning, walking, cleaning}

given that HMM model ?


Solution :


R

S

R

S

R

S

R

S

shopping cleaning walking cleaning

Day 1 Day 2 Day 3 Day 4

24.0)1(1

12.0)2(1

Step 1 Step 2 (repeat step 2 to the end)

108.0)1(2

0114.0)2(2

008.0)1(3

386.0)2(3

08.0)1(4

023.0)2(4

12.03.0*4.0)2(1

24.04.0*6.0)1(1

Solution :


R

S

R

S

R

S

R

S

shopping cleaning walking cleaning


24.0)1(1

12.0)2(1

108.0)1(2

0114.0)2(2

008.0)1(3

386.0)2(3

08.0)1(4

023.0)2(4

Step 3

Problem 2 : Given observation, what is the most probable

transition sequence ?

Solution : We can find the most probable transition sequence

using Viterbi Algorithm.


Example : Given sequence of observation :

O = {shopping, cleaning, walking, cleaning}

what is the most probable transition sequence of hidden states ?


Solution :


R

S

R

S

R

S

shopping

walking cleaning

Day 1 Day 3 Day 4

24.0)1(1

12.0)2(1

Step 112.03.0*4.0)2(1

24.04.0*6.0)1(1

R

S

cleaning

Day 2

Solution :


R

S

R

S

R

S

R

S

shopping

cleaning walking cleaning


24.0)1(1

12.0)2(1

Step 2

084.0)2(2

072.0)2(2

024.0)2(1*12*)2(1)(

084.0)2(1*11*)1(1)(

ObaRP

ObaRP

0.084

0.024

0.0072

0.072

Solution :


R

S

R

S

walking cleaning


Step 2

018.0)1(3

0194.0)2(3

023.0)1(4

077.0)2(4

0.001

0.018

0.0041

0.0194

0.002

0.077

0.00020.023

R

S

R

S

shopping

cleaning

Day 1 Day 2

24.0)1(1

12.0)2(1

084.0)2(2

072.0)2(2

0.084

0.024

0.0072

0.072

Solution :


R

S

R

S

R

S

R

S

shopping

cleaning walking cleaning


24.0)1(1

12.0)2(1

084.0)2(2 018.0)1(3

0194.0)2(3

023.0)1(4

077.0)2(4

0.084

0.0072

0.072

0.024 072.0)2(2

0.001

0.018

0.0041

0.0194

0.002

0.077

0.00020.023

Applications

Speech recognition• Recognizing spoken words and phrases

Text processing• Parsing raw records into structured records

Bioinformatics• Protein sequence prediction

Financial• Stock market forecasts (price pattern prediction)• Comparison shopping services

HMM in speech recognition

The basic idea is to find the most likely string of words given some acoustic (voiced) input.

HMM in speech recognition (Cont.)

The units (levels) of speech recognition systems

References

Questions

Thank You

Technology

Hidden markov model