Chapter 6 Hidden Markov Models. Chaochun Wei Spring 2018

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1 Chapter 6 Hdden Markov Modes Chaochun We Sprng 208

2 Contents Readng materas Introducton to Hdden Markov Mode Markov chans Hdden Markov Modes Parameter estmaton for HMMs 2

3 Readng Rabner, L.(989) A Tutora on Hdden Markov Modes and Seected Appcatons n Speech Recognton. Proceedngs of the IEEE, 77 (2) Rabner, L., and Juang, Bng-Hwang, (993), Fundamentas of Speech Recognton, Prentce Ha. 3

4 Markov chan: a process that the current state depends on at most a mted number of prevous states Weather Sunny, Ran, Ran, Sunny, Coudy, Coudy,. Stock market ndex Up, up, down, down, down, up, up, up,. Gr frend s mood Hgh, ow, ow, hgh, hgh, hgh, Genome sequence ATGTTAGATATAACAGATAA Fp cons HTTTHHHHHH 4

5 5 Hdden Markov Mode HMM for two based cons fppng End e ( H ) 0.8, e ( T ) 0.2, e2( H ) 0.3, e2 ( T ) 0.7 TTHHTTHTTTTTHTHHHHHTHTH Observed sequence x Hdden state sequence arg max P( x, )

6 Hdden Markov Mode Eements of an HMM (N, M, A, B, Int). N: number of states n the mode S={S, S 2,, S N }, and the state at tme t s q t. 2. M: aphabet sze (the number of observaton symbos) V={v, v 2,, v M } 3. A: state transton probabty dstrbuton A={a j } where a j =P[q t+ =S j q t =S ],,j N 4. E: emsson probabty E={e j (k)} (observaton symbos probabty dstrbuton n state j), where e j (k)=p[v k at t q t = S j }, j N, k M 5. Int: nta state probabty Int={I }, where I =P[q =S ], N.

7 HMM s a generatve mode HMM can be used as a generator to produce an observaton sequence O=O O 2 O T, where each O t s one of the symbos from V, and T s the number of observatons n the sequence.. Choose an nta state q =S accordng to Int; 2. Set t=; 3. Choose O t =v k accordng to e (k) (the symbo probabty dstrbuton n state S ); 4. Transt to a new state q t+ =S j accordng to a j ; 5. Set t=t+; return to step 3 f t<t; otherwse termnate the procedure.

8 HMM s a generatve mode HMM for two based cons fppng End e ( H ) 0.8, e ( T ) 0.2, e2( H ) 0.3, e2 ( T ) 0.7 TTHHTTHTTTTTHTHHHHHTHTH Observed sequence x Hdden state sequence P ( x, ) Int * e ( x(0))* ( a e T ( x( ))

9 HMM s a generatve mode HMM for two based cons fppng Begn End e ( H ) 0.8, e ( T ) 0.2, e2( H ) 0.3, e2 ( T ) 0.7 TTHHT Observed sequence x 22 Hdden state sequence P( x, )?

10 HMM s a generatve mode HMM for two based cons fppng Begn End e ( H ) 0.8, e ( T ) 0.2, e2( H ) 0.3, e2 ( T ) 0.7 P( x, ) Int * e TTHHT ( T)*( a Observed sequence x 22 Hdden state sequence e 0 * e 0 ( T))*( a ( x(0))* 2 e 2 (H))*( a ( x( )) (H))*( a ( T)) *0.2* ( 0.9*0.2)( * 0.*0.3)( * 0.2*0.3)( * 0.7* 0.2) ( a 0T 22 e 2 e 2 e

11 Hdden Markov Mode HMM:λ={A, B,Int} Three basc probems for HMMs. Gven the observaton sequence O=O O 2 O T, and a mode λ={a, B, Int}, how to compute P(O λ)? 2. Gven the observaton sequence O=O O 2 O T, and a mode λ={a, B, Int}, how to choose a correspondng state sequence Q=q q 2 q T, whch s optma n some meanngfu sense.. 3. How to estmate mode parameters λ={a, B, Int} to maxmze P(O λ).

12 Hdden Markov Mode HMM:λ={A, B, Int} Three basc probems for HMMs. From the observaton sequence and the mode to a jont probabty; 2. Fnd the best hdden state sequence; 3. Optmze the mode parameters;

13 3 states Most Probabe Path and Vterb Agorthm L- L max Let f ( ) max (Pr( x0,..., x, x, 0,...,, { 0,..., } )) Recurson (= L) f ( ) e ptr ( ) ( x ) max( f k arg max( f k k k ( ) a ( ) a Tme compexty O ( N 2 L) space compexty O(NL) Souton to probem 2: prob of best state sequence k k ); ).

14 Vterb for the HMM for two based cons fppng Begn End e ( H ) 0.8, e ( T ) 0.2, e2( H ) 0.3, e2 ( T ) 0.7 TTHHT Observed sequence x 22 Hdden state sequence T T H H T

Vterb for the HMM for two based cons fppng 0.9 0.2 Begn 0. 2 0.7 0. End e ( H ) 0.8, e ( T ) 0.2, e2( H ) 0.3, e2 ( T ) 0.7 TTHHT Observed sequence x 22 Hdden state sequence T T H H T 2 0 2 3 4 0.

15 Vterb for the HMM for two based cons fppng Begn End e ( H ) 0.8, e ( T ) 0.2, e2( H ) 0.3, e2 ( T ) 0.7 TTHHT Observed sequence x 22 Hdden state sequence T T H H T Max0.2*(0. 9 *0.2, 0) = Max0.7*( 0. 2 * 0.,0) = 0.04 max0.8*(0.036* 0.9, 0.04*0.7 ) = Max0.3*(0.036* 0., 0.04*0.2 ) = Max 0.8 * ( *0.9, 0.008*0.7 ) = Max0.3(0.0259* 0., 0.008*0.2) = =max0.2*( * 0.9, *0.7) = =max0.3( *0., *0.7) =

16 states 6 Probabty of A the Possbe Paths and Forward Agorthm 2 Let L- L f Intazaton (= L) ( ) Pr( x0,..., x, ) f + ( ) e ( 0) 0 x Recurson (= L) Probabty of a the probabe paths Souton to probem P ( x) f k ( L) P( x, ) f ( ) e ( x ) ( f ( ) a ) k k k k

17 7 states Posteror Probabty and Forward and Backward Agorthm L- L + Posteror Probabty P( k x) P( k, P( x) x)

18 states 8 Backward Agorthm L- L + Let bk( ) Pr( x, x2..., x L, k) Intazaton (= L) b L ( ), N Recurson (= L) Probabty of a the probabe paths b ( ) ( akek ( x )) b ( ), L, L 2,...,0; N P ( x) (0) P( x, ) k k b k

19 Optmze the mode parameters HMM:λ={A, B, Int} Wth annotatons Maxmum key-hood Wthout annotatons EM agorthm

20 Optmze the mode parameters HMM:λ={A, B, Int}, Wthout annotatons Baum-Wech method (EM method) Let ),, ( ), ( x j P j then )) ( ) ( ) ( ( ) ( ) ( ) ( ), ( j b x e a f j b x e a f j j j N j N j j Let ), ( ) ( j N j then ) ( 0 N = expected number of transtons from S ), ( 0 j N = expected number of transton S to S j

21 Optmze the mode parameters(2) HMM:λ={A, B, Int}, Wthout annotatons Baum-Wech method (EM method) Then, Int expected frequency n S at tme 0 = 0 ( ) a, j exp ected exp ected number of transton s from S number of transton s from S to S j L 0 L 0 (, j ) ( ) e( k) exp ected number of tmes exp ected number n state of tmes and obervng n state symbo v k 0 s. t. x v L L t 0 ( ) k ( )

22 22 One more exampe: Fppng two cons O= HHTHHTTTHT, P(O λ)=?

Forward Agorthms H H T H H T T T H T End A 0.

23 Forward Agorthms H H T H H T T T H T End A B e-5 23

24 O= HHTHHTTTHT argmax(p(o, Q, λ)) 24

25 Vterb Agorthms A -> A -> A -> A -> A -> B -> B -> B -> B -> B H H T H H T T T H T End A B e-6 25

26 See Probem 3. Mode parameter estmaton Rabner, L.(989) A Tutora on Hdden Markov Modes and Seected Appcatons n Speech Recognton. Proceedngs of the IEEE, 77 (2) Rabner, L., and Juang, Bng-Hwang, (993), Fundamentas of Speech Recognton, Prentce Ha. 26

27 Gene Structure predcton wth HMM Inta Exon Intron Interna Exon Intron Intron Termna Exon 5 UTR ATG GT AG TAA TAG TGA 3 UTR A gene s a hghy structured regon of DNA, t s a functona unt of nhertance. 27

28 A Typca Human Gene Structure 28

29 Gene Predcton Mode HMM (27 states) Each state For a gene structure State-specfc modes (Generazed HMM) Length dstrbuton Sequence content 29

30 Another exampe: Par HMM for oca agnment X q x RX 2 RX q x Begn q x RY M p x y j RY 2 End q y j Y q y j q y j 30

Introduction ( Week 1-2) Course introduction A brief introduction to molecular biology A brief introduction to sequence comparison Part I: Algorithms

Introduction ( Week 1-2) Course introduction A brief introduction to molecular biology A brief introduction to sequence comparison Part I: Algorithms Course organzaon Inroducon Wee -2) Course nroducon A bref nroducon o molecular bology A bref nroducon o sequence comparson Par I: Algorhms for Sequence Analyss Wee 3-8) Chaper -3, Models and heores» Probably