1 Hidden Markov Model Observation : O1,O2,... States in time : q1, q2,... All states : s1, s2,... Si...
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1 Hidden Markov Model Hidden Markov Model Observation : O1,O2, . . . Observation : O1,O2, . . . States in time : q1, q2, . . . States in time : q1, q2, . . . All states : s1, s2, . . . All states : s1, s2, . . . t O O O O , , , , 3 2 1 t q q q q , , , , 3 2 1 Si S j ji a ij a
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Transcript of 1 Hidden Markov Model Observation : O1,O2,... States in time : q1, q2,... All states : s1, s2,... Si...
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Hidden Markov ModelHidden Markov Model
Observation : O1,O2, . . . Observation : O1,O2, . . .
States in time : q1, q2, . . .States in time : q1, q2, . . .
All states : s1, s2, . . .All states : s1, s2, . . .
tOOOO ,,,, 321
tqqqq ,,,, 321
Si Sjjiaija
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Hidden Markov Model (Cont’d)Hidden Markov Model (Cont’d)
Discrete Markov ModelDiscrete Markov Model
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1
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zktitjt
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sqsqsqsqP
Degree 1 Markov Model
33
Hidden Markov Model (Cont’d)Hidden Markov Model (Cont’d)
)|( 1, itjtji sqsqPa
ija : Transition Probability from Si to Sj ,
Nji ,1
44
Hidden Markov Model Hidden Markov Model ExampleExample
S1 : The weather is rainyS2 : The weather is cloudyS3 : The weather is sunny
8.01.01.0
2.06.02.0
3.03.04.0
}{ ijaA
rainy cloudy sunnyrainy
cloudy
sunny
55
Hidden Markov Model Example Hidden Markov Model Example (Cont’d)(Cont’d)
Question 1:How much is this probability:Sunny-Sunny-Sunny-Rainy-Rainy-Sunny-Cloudy-Cloudy
22311333 ssssssss
22321311313333 aaaaaaa
87654321 qqqqqqqq410536.1
66
Hidden Markov Model Example Hidden Markov Model Example (Cont’d)(Cont’d)
Question 2:The probability of staying in a state for d days if we are in state Si?
NisqP ii 1),( 1The probability of being in state i in time t=1
)()1()( 1 dPaassssP iiidiiijiii
d Days
77
HMM ComponentsHMM Components
N : Number Of StatesN : Number Of States
M : Number Of OutputsM : Number Of Outputs
A : State Transition Probability MatrixA : State Transition Probability Matrix
B : Output Occurrence Probability in B : Output Occurrence Probability in each stateeach state
: Primary Occurrence Probability: Primary Occurrence Probability),,( BA : Set of HMM Parameters
88
Three Basic HMM ProblemsThree Basic HMM Problems
Given an HMM Given an HMM and a sequence of and a sequence of observations observations O,O,what is the probability what is the probability ? ?
Given a model and a sequence of Given a model and a sequence of observations observations OO, what is the most likely , what is the most likely state sequence in the model that produced state sequence in the model that produced the observations?the observations?
Given a model Given a model and a sequence of and a sequence of observationsobservations O, O, how should we adjust how should we adjust model parameters in order to maximize model parameters in order to maximize ? ?
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99
First Problem SolutionFirst Problem Solution
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ttt
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And
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First Problem Solution (Cont’d)First Problem Solution (Cont’d)
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Account Order : )2( TTNO
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Forward Backward ApproachForward Backward Approach
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1) Initialization
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Forward Backward Approach Forward Backward Approach (Cont’d)(Cont’d)
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1 2) Induction :
3) Termination :
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Account Order : )( 2TNO
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Backward Variable ApproachBackward Variable Approach
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NiiT 1,1)(1) Initialization
2)Induction
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Second Problem SolutionSecond Problem Solution
Finding the most likely state sequenceFinding the most likely state sequence
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t
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iqoP
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Individually most likely state :
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Viterbi AlgorithmViterbi Algorithm
Define : Define :
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P is the most likely state sequence with this conditions : state i , time t and observation o
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Viterbi Algorithm (Cont’d)Viterbi Algorithm (Cont’d)
)(].)(max[)( 11 tjijti
t obaij
1) Initialization
0)(
1),()(
1
11
i
Niobi ii
)(it Is the most likely state before state i at time t-1
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Viterbi Algorithm (Cont’d)Viterbi Algorithm (Cont’d)
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aij
obaij
ijtNi
t
tjijtNi
t
1,2
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)(])([max)(
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2) Recursion
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Viterbi Algorithm (Cont’d)Viterbi Algorithm (Cont’d)
)]([maxarg
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ip
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3) Termination:
4)Backtracking:
1,,2,1),( *11
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Third Problem SolutionThird Problem Solution
Parameters Estimation using Baum-Parameters Estimation using Baum-Welch Or Expectation Maximization Welch Or Expectation Maximization (EM) Approach(EM) Approach
Define:
N
i
N
jttjijt
ttjijt
tt
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jobai
jobai
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jqiqoP
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Third Problem Solution Third Problem Solution (Cont’d)(Cont’d)
N
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tt i
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: Expectation value of the number of jumps from state i
: Expectation value of the number of jumps from state i to state j
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Third Problem Solution Third Problem Solution (Cont’d)(Cont’d)
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Baum Auxiliary FunctionBaum Auxiliary Function
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By this approach we will reach to a local optimum
2323
Restrictions Of Restrictions Of Reestimation FormulasReestimation Formulas
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N
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jij
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kj
1,1)(1
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Continuous Observation Continuous Observation DensityDensity
We have amounts of a PDF instead of We have amounts of a PDF instead of
We haveWe have
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dooboCob j
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Mixture Coefficients
Average Variance
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Continuous Observation Continuous Observation DensityDensity
Mixture in HMMMixture in HMM
),,()( jkjkjkk
j oCMaxob
M2|1M1|1
M4|1M3|1
M2|3M1|3
M4|3M3|3
M2|2M1|2
M4|2M3|2
S1 S2 S3Dominant Mixture:
2626
Continuous Observation Continuous Observation Density (Cont’d)Density (Cont’d)
Model Parameters:Model Parameters:
),,,,( CA
N×N N×M×K×KN×M×KN×M1×N
N : Number Of StatesM : Number Of Mixtures In Each StateK : Dimension Of Observation Vector
2727
Continuous Observation Continuous Observation Density (Cont’d)Density (Cont’d)
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t
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Continuous Observation Continuous Observation Density (Cont’d)Density (Cont’d)
T
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),(
)()(),(
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2929
State Duration ModelingState Duration Modeling
Si Sj
Probability of staying d times in state i :
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jia
ija
3030
State Duration Modeling State Duration Modeling (Cont’d)(Cont’d)
Si Sjjia
……. …….
HMM With clear duration
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3131
State Duration Modeling State Duration Modeling (Cont’d)(Cont’d)
HMM consideration with State Duration :HMM consideration with State Duration :– Selecting using ‘sSelecting using ‘s– Selecting usingSelecting using– Selecting Observation Sequence Selecting Observation Sequence
using using in practice we assume the following in practice we assume the following
independence:independence:
– Selecting next state using transition probabilities Selecting next state using transition probabilities . We also have an additional constraint: . We also have an additional constraint:
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Training In HMMTraining In HMM
Maximum Likelihood (ML)Maximum Likelihood (ML)
Maximum Mutual Information (MMI)Maximum Mutual Information (MMI)
Minimum Discrimination Information (MDI)Minimum Discrimination Information (MDI)
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Training In HMMTraining In HMM
Maximum Likelihood (ML)Maximum Likelihood (ML)
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Training In HMM (Cont’d)Training In HMM (Cont’d)
Maximum Mutual Information (MMI)Maximum Mutual Information (MMI)
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Mutual Information
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Training In HMM (Cont’d)Training In HMM (Cont’d)
Minimum Discrimination Information Minimum Discrimination Information (MDI)(MDI)
dooP
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),,,( 21 TOOOO
),,,( 21 tRRRR Observation :
Auto correlation :
):(inf),( PQIPR )(RQ
