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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 :
Introduction (Cont.)
Stochastic process example :
Techniques to model the Stochastic process
Branching processGaussian process Hidden Markov model Markov process
Introduction (Cont.)
Introduction (Cont.)
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.
Markov Model (Cont.)
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 ?
Markov Model (Cont.)
Definitions : Observable states :
Observed sequence :
State transition matrix :
Markov Model (Cont.)
},,,{ 21 Tqqq
},,2,1{ N
Definitions : Initial state probability :
Markov assumption ( Markov Property ) :
Markov Model (Cont.)
Definitions : Sequence probability of Markov model :
Markov Model (Cont.)
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
Markov Model (Cont.)
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
HMM Core Problems (cont.)
Example : What is the probability of the
sequence of observation :
O = {shopping, cleaning, walking, cleaning}
given that HMM model ?
HMM Core Problems (cont.)
Solution :
HMM Core Problems (cont.)
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 :
HMM Core Problems (cont.)
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
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.
HMM Core Problems (cont.)
Example : Given sequence of observation :
O = {shopping, cleaning, walking, cleaning}
what is the most probable transition sequence of hidden states ?
HMM Core Problems (cont.)
Solution :
HMM Core Problems (cont.)
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 :
HMM Core Problems (cont.)
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 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 :
HMM Core Problems (cont.)
R
S
R
S
walking cleaning
Day 1 Day 2 Day 3 Day 4
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 :
HMM Core Problems (cont.)
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
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
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