Hidden Markov Model
-
Upload
zeus-sherman -
Category
Documents
-
view
71 -
download
1
description
Transcript of Hidden Markov Model
11
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
22
Hidden Markov Model (Cont’d)Hidden Markov Model (Cont’d)
Discrete Markov ModelDiscrete Markov Model
)|(
),,,|(
1
121
itjt
zktitjt
sqsqP
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 ? ?
)|( OP
)|( OP
99
First Problem SolutionFirst Problem Solution
)(),|(),|(11
tq
T
ttt
T
tobqoPqoP
t
TT qqqqqqq aaaqP132211
)|(
)()|(),( yPyxPyxP
)|(),|()|,( zyPzyxPzyxP We Know That:
And
1010
First Problem Solution (Cont’d)First Problem Solution (Cont’d)
)|(),|()|,( qPqoPqoP
)()()(
)|,(
122111 21 Tqqqqqqqq obaobaob
qoP
TTT
T
TTTqqq
Tqqqqqqqq
q
obaobaob
qoPoP
21
122111)()()(
)|,()|(
21
Account Order : )2( TTNO
1111
Forward Backward ApproachForward Backward Approach
)|,,,,()( 21 iqoooPi ttt
Niobi ii 1),()( 11
Computing )(it
1) Initialization
1212
Forward Backward Approach Forward Backward Approach (Cont’d)(Cont’d)
NjTt
obaij tjij
N
itt
1,11
)(])([)( 11
1 2) Induction :
3) Termination :
N
iT ioP
1
)()|(
Account Order : )( 2TNO
1313
Backward Variable ApproachBackward Variable Approach
),|,,,()( 21 iqoooPi tTttt
NiiT 1,1)(1) Initialization
2)Induction
NjAndTTt
jobaiN
jttjijt
11,,2,1
)()()(1
11
1414
Second Problem SolutionSecond Problem Solution
Finding the most likely state sequenceFinding the most likely state sequence
N
itt
ttN
it
t
ttt
ii
ii
iqoP
iqoP
oP
iqoPoiqPi
11
)()(
)()(
)|,(
)|,(
)|(
)|,(),|()(
Individually most likely state :
NntTtiq tt 1,1)],(max[arg*
1515
Viterbi AlgorithmViterbi Algorithm
Define : Define :
Ni
qqq
oooiqqqqP
i
t
ttt
t
1
,,,
]|,,,,,,,,[max
)(
121
21121
P is the most likely state sequence with this conditions : state i , time t and observation o
1616
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
1717
Viterbi Algorithm (Cont’d)Viterbi Algorithm (Cont’d)
NjTt
aij
obaij
ijtNi
t
tjijtNi
t
1,2
])([maxarg)(
)(])([max)(
11
11
2) Recursion
1818
Viterbi Algorithm (Cont’d)Viterbi Algorithm (Cont’d)
)]([maxarg
)]([max
1
*
1
*
iq
ip
TNi
T
TNi
3) Termination:
4)Backtracking:
1,,2,1),( *11
* TTtqq ttt
1919
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
ttt
jobai
jobai
oP
jqiqoP
ojqiqPji
1 111
11
1
1
)()()(
)()()(
)|(
)|,,(
),|,(),(
2020
Third Problem Solution Third Problem Solution (Cont’d)(Cont’d)
N
jtt jii
1
),()(
1
1
)(T
tt i
T
tt ji
1
),(
: Expectation value of the number of jumps from state i
: Expectation value of the number of jumps from state i to state j
2121
Third Problem Solution Third Problem Solution (Cont’d)(Cont’d)
)(1 ii
T
tt
T
tt
ij
i
jia
1
1
)(
),(
T
tt
Vo
T
tt
j
j
j
kb kt
1
1
)(
)(
)(
2222
Baum Auxiliary FunctionBaum Auxiliary Function
q
qoPqoPQ )|,(log)'|,()|( '
)|()|(
)',(),(: ''
oPoP
QQif
By this approach we will reach to a local optimum
2323
Restrictions Of Restrictions Of Reestimation FormulasReestimation Formulas
11
N
ii
NiaN
jij
1,11
NjkbM
kj
1,1)(1
2424
Continuous Observation Continuous Observation DensityDensity
We have amounts of a PDF instead of We have amounts of a PDF instead of
We haveWe have
)|()( jqVoPkb tktj
1)(,),,()(1
dooboCob j
M
kjkjkjkj
Mixture Coefficients
Average Variance
2525
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)
T
t
M
kt
T
tt
jk
kj
kjC
1 1
1
),(
),(
T
tt
t
T
tt
jk
kj
okj
1
1
),(
),(
2828
Continuous Observation Continuous Observation Density (Cont’d)Density (Cont’d)
T
tt
jktjkt
T
tt
jk
kj
ookj
1
1
),(
)()(),(
),( kjt Probability of event j’th state and k’th mixture at time t
2929
State Duration ModelingState Duration Modeling
Si Sj
Probability of staying d times in state i :
)1()( 1ii
diii aadP
jia
ija
3030
State Duration Modeling State Duration Modeling (Cont’d)(Cont’d)
Si Sjjia
……. …….
HMM With clear duration
ija )(dPj)(dPi
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:
),(),,,(1
1
11 121 tq
d
tdq OtbOOOb
iiq 1
dOOO ,,, 21 )(
1dPq1d
21qqa
),,,(11 21 dq OOOb
jq 2
011qqa
3232
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)
3333
Training In HMMTraining In HMM
Maximum Likelihood (ML)Maximum Likelihood (ML)
)|( 1oP
)|( 2oP)|( 3oP
)|( noP
.
.
.
)]|([*V
rOPMaximumP
ObservationSequence
3434
Training In HMM (Cont’d)Training In HMM (Cont’d)
Maximum Mutual Information (MMI)Maximum Mutual Information (MMI)
)()(
)|,(log),(
POP
OPOI
v
ww
v
wPwOP
OPOI
1
)(),|(log
)|(log),(
Mutual Information
}{, v