Upload
kevin-casey
View
213
Download
1
Embed Size (px)
Citation preview
HIDDEN MARKOV CHAINSHIDDEN MARKOV CHAINS
Prof. Alagar RanganProf. Alagar RanganDept of Industrial Engineering Dept of Industrial Engineering Eastern Mediterranean UniversityEastern Mediterranean UniversityNorth CyprusNorth Cyprus
Source: Probability ModelsSource: Probability ModelsSheldon M.RossSheldon M.Ross
MARKOV CHAINSMARKOV CHAINS
Toss a coin repeatedly .Toss a coin repeatedly .
Denote Head=1Denote Head=1
Tail =0Tail =0
Let Yn=outcome of nLet Yn=outcome of nthth Toss Toss
P( Yn=1) = pP( Yn=1) = p
P( Yn=0) = qP( Yn=0) = q
Y1,Y2,… are iid random variables.Y1,Y2,… are iid random variables.
Sn = Y1 + Y2 + …+ YnSn = Y1 + Y2 + …+ Yn
Sn is the accumulated number of Heads in Sn is the accumulated number of Heads in the first n trials. the first n trials.
Sn ~ Markov chainSn ~ Markov chain ; ;
Time n=0,1,2,…Time n=0,1,2,…
States j=0,1,2,…States j=0,1,2,…
qjjP SS nn
1
YSS nnn 11
pjjP SS nn
1
1
)(
,...,,
1
1211
sayijP
ijP
pXXXXXXX
ijnn
nnnn
Xn ~ Markov ChainXn ~ Markov Chain
One step Transition One step Transition
Probability MatrixProbability Matrix
...
...
...
...
...
...
222120
121110
020100
ppppppppp
P
0
1
2
.
.
.
0 1 2 . . .
pXXn
ijmnmijP
)(
n - step Transition Probabilitiesn - step Transition Probabilities
The corresponding Matrix The corresponding Matrix
Simple Results:Simple Results: (a) (a) (b) Expected sojourn time in a state (b) Expected sojourn time in a state (c) Steady state Probability (c) Steady state Probability
......
...)(11
)(10
...)(01
)(00
)(
pp
pp
P nn
nn
n
PPnn )(
jj
....,.10
, P
)(
,...,,
1
1211
sayijP
ijP
pXXXXXXX
ijnn
nnnn
Xn ~ Markov ChainXn ~ Markov Chain
0 1 2 . . .0 1 2 . . .
One step Transition One step Transition
Probability MatrixProbability Matrix
...
...
...
...
...
...
222120
121110
020100
ppppppppp
P
Examples:Examples:Weather ForecastingWeather ForecastingStates : Dry day, Wet day, state of the nth dayStates : Dry day, Wet day, state of the nth day
Communication SystemCommunication SystemStates : signals 0 , 1States : signals 0 , 1 signals leaving the nth stage of the system.signals leaving the nth stage of the system.
Moods of a ProfessorMoods of a ProfessorStates: cheerful , ok , unhappy.States: cheerful , ok , unhappy. (C) (O) (U)(C) (O) (U) Mood of the Professor on the nth day.Mood of the Professor on the nth day.
X n
1 2 3
X n
pq
qp
P
1
0
1
X n
X n 1
0
X n
6.4.
2.8.
P
Dry Wet
Dry
Wet
X n
X n 1
5.3.2.
3.4.3.
1.4.5.
P
UX n
X n 1
C O
U
C
O
Hidden Markov Chain Hidden Markov Chain ModelsModels Let Xn be a Markov chain with one step Transition Let Xn be a Markov chain with one step Transition
Probability Probability MatrixMatrix
Let S be a set of signals.Let S be a set of signals. A signal from S is emitted each time the Markov chain A signal from S is emitted each time the Markov chain
enters a state.enters a state.
If the Markov chain enters state j, then the signal s is If the Markov chain enters state j, then the signal s is emitted with probability , withemitted with probability , with
pij
P
jsP 1 SsjsP
jspjsP XS 11
jspjsP XSSXSXS nnn
,,...,,,,12211
The above model in which the sequence of signals The above model in which the sequence of signals S1,S2,…S1,S2,…
is observed while the sequence of the underlying Markov is observed while the sequence of the underlying Markov chain states X1,X2,… is unobserved is called a chain states X1,X2,… is unobserved is called a hidden hidden Markov chain Markov chain Model.Model.
signalsignal
timetime
state of the state of the chainchain
X nX n 1
jsp itp
i j
pij
ExamplesExamples : : Production ProcessProduction Process StateState SignalSignal
Good state(1)Good state(1)
Poor state(2)Poor state(2)
acceptable quality .99acceptable quality .99
ProductioProduction n ProcessProcess
acceptable .04acceptable .04
unacceptable .01unacceptable .01
unacceptable .unacceptable .9696
10
1.90.P
1 2
1
2
Moods of the ProfessorMoods of the Professor
Professor Professor
Condition of a Patient subject to Therapy.Condition of a Patient subject to Therapy.
Signal ProcessingSignal Processing
C
O
U
Grades High averageGrades average
PatientImproving
Deteriorating
Red Cell count high
Red Cell count low
Signals sent
0
1
Signals received as 0
Signals received as 1
Let be the random vector of the first n Let be the random vector of the first n signalssignals..
For a fixed sequence of signals, let and For a fixed sequence of signals, let and
It can be shown thatIt can be shown that
Now starting with Now starting with
SSSS n
n,...,,21
ssss nn,...,,21
sSXF n
n
nnjPj ,
in
n
n
nn
n
nn
n
n
i
j
P
jPjP
FF
sSsSXsSX
,
.....3,2,1.1 njii
nnnpjPj FsF
ippiPj ssSXF i 11
1
11,
1
2
We can recursively determineWe can recursively determine
using , which will determine . using , which will determine .
Note Note
We can also compute the above using We can also compute the above using backward recursion usingbackward recursion using
iuptoii FFF n,...,,
32
2 1
j
nn
njP FsS
iPi XsSsSB knnkkk
,...,
11
Example:Example:
LetLet
Let the first 3 items produced be a,u,aLet the first 3 items produced be a,u,a
96.204.2
99.11.1
01.
19.
2112
2211
apup
apup
PPPP
8.11
XP
192.)96)(.2(.22
792.)99)(.8(.11
,,
121
111
3
spFspF
s
p
p
aua
iandi FF 32Similarly calculating usingSimilarly calculating using
Predicting the states.Predicting the states.
Suppose the first observed n signals are Suppose the first observed n signals are
We wish to predict the first n states of the Markov We wish to predict the first n states of the Markov chain using this data. chain using this data.
364.)2()1(
)1(),,(1
33
333
FF
FsX auaP
2
sss nn,...,1
Case 1Case 1
We wish to maximize the expected number of states We wish to maximize the expected number of states that are correctly predicted.that are correctly predicted.
For each k=1,2,…,n , we calculateFor each k=1,2,…,n , we calculate
choose that j which maximizes choose that j which maximizes
the above as the predictor of . the above as the predictor of .
sSX n
n
kjP
X k
Case 2Case 2 A different problem arises if we regard the sequence A different problem arises if we regard the sequence
of states as a single entity.of states as a single entity.
For instance in signal processing while For instance in signal processing while
may be actual message sent , would be may be actual message sent , would be what is received. Thus the objective is to predict the what is received. Thus the objective is to predict the actual message in its entirety.actual message in its entirety.
XXX n,...,,21
SSS n,...,,21
Let our problem is to find the Let our problem is to find the
sequence of states that maximizes sequence of states that maximizes
To solve the above we letTo solve the above we let
XXXX kk,...,,21
iii k,...,,21
sS
sSiiiX
sSiiiX
n
n
n
n
nn
n
n
nn
P
P
P
,...,,
,...,,
21
21
sSXiiiXPV k
k
kkkkj
iiij
k
,,,...,,1211
,...,,max
121
We can show using probabilistic argumentsWe can show using probabilistic arguments
Starting withStarting with
We can recursively determine for each .We can recursively determine for each .
This procedure is known as This procedure is known as Viterbi Algorithm.Viterbi Algorithm.
jpji SpsSXPV j 11111,
ijj VpsPV kiji
kk 1.max
jV nj