Hidden Markov Models. Hongxin Zhang State Key Lab of CAD&CG, ZJU

Transcription

1 Hdden Markov Models Hongxn Zhang State Key Lab of CAD&CG, ZJU

2 utlne Background Markov Chans Hdden Markov Models

3 Example: Vdeo extures Problem statement vdeo clp vdeo texture SIGGRAPH 000. Schoedl et. al.

4 he approach How do we fnd good transtons?

5 Fndng good transtons Compute L dstance D, j between all frames frame frame j Smlar frames make good transtons

6 Demo: Fsh ank

7 Mathematc model of Vdeo exture A sequence of random varables {ADEABEDADBCAD} A sequence of random varables {BDACBDCACDBCADCBADCA} Mathematc Markov Model he future s ndependent of the past and gven by the present.

8 Markov Property Formal defnton Let X={X n } n=0n be a sequence of random varables takng values s k N f and only f P(X m =s m X 0 =s 0,,X m- =s m- = P(X m =s m X m- =s m- then the X fulflls Markov property Informal defnton he future s ndependent of the past gven the present.

9 Hstory of MC Markov chan theory developed around 900. Hdden Markov Models developed n late 960 s. Used extensvely n speech recognton n Introduced to computer scence n 989. Applcatons Bonformatcs. Sgnal Processng Data analyss and Pattern recognton

10 Markov Chan A Markov chan s specfed by A state space S = { s, s..., s n } An ntal dstrbuton a 0 A transton matrx A Where A(n j = a j = P(q t =s j q t- =s Graphcal Representaton as a drected graph where Vertces represent states Edges represent transtons wth postve probablty

11 Probablty Axoms Margnal Probablty sum the jont probablty P( x a P( x a, y ya Condtonal Probablty Y P( x a, y bj P( x a y bj f P( y bj 0. P( y b j

12 Calculatng wth Markov chans Probablty of an observaton sequence: Let X={x t } L t=0 be an observaton sequence from the Markov chan {S, a 0, A}

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15 Motvaton of Hdden Markov Models Hdden states he state of the entty we want to model s often not observable: he state s then sad to be hdden. bservables Sometmes we can nstead observe the state of enttes nfluenced by the hdden state. A system can be modeled by an HMM f: he sequence of hdden states s Markov he sequence of observatons are ndependent (or Markov gven the hdden

16 Hdden Markov Model Defnton M={S,V,A,B, } Set of states S = { s, s,, s N } bservaton symbols V = { v, v,, v M } ranston probabltes A between any two states a j = P(q t =s j q t- =s Emsson probabltes B wthn each state b j ( t = P( t =v j q t = s j Start probabltes = {a 0 } Use = (A, B, to ndcate the parameter set of the model. 3 4 n q q q 3 q 4 q n

17 Generatng a sequence by the model Gven a HMM, we can generate a sequence of length n as follows:. Start at state q accordng to prob a 0t. Emt letter o accordng to prob e t (o 3. Go to state q accordng to prob a tt 4. untl emttng o n 0 a 0 N N N N b (o o o o 3 o n

18 Example

19 Calculatng wth Hdden Markov Model Consder one such fxed state sequence Q qq q he observaton sequence for the Q s P( Q, P( t t b ( b q t, ( b ( q q q 3 4 n q q q 3 q 4 q n

20 Calculatng wth Hdden Markov Model (cont. he probablty of such a state sequence Q P( Q a a a a 0q q q q q q q 3 he probablty that and Q occur smultaneously, s smply the product of the above two terms,.e., P(, Q P( Q, P( Q P(, Q a b ( a b ( a a b ( 0q q q q q q q q q q 3

21 Example

22 he three man questons on HMMs. Evaluaton GIVEN a HMM M=(S, V, A, B,, and a sequence, FIND P[ M]. Decodng GIVEN a HMM M=(S, V, A, B,, and a sequence, FIND the sequence Q of states that maxmzes P(, Q 3. Learnng GIVEN FIND a HMM M=(S, V, A, B,, wth unspecfed transton/emsson probabltes and a sequence Q, parameters = (e (., a j that maxmze P[x ]

23 Evaluaton Fnd the lkelhood a sequence s generated by the model A straghtforward way ( 穷举法 he probablty of s obtaned by summng all possble state sequences q gvng q q q q q q q q q q q q q Q all b a a b a b Q P Q P P,, 3 ( ( ( (, ( ( Complexty s (N Calculatons s unfeasble N N N N o o o 3 o n N 0 b (o a 0

24 he Forward Algorthm A more elaborate algorthm he Forward Algorthm N N K N o o o 3 o n N 0 a 0 a 0 a 0n a a N P ( ( ( ] ( [ ( b a N N a n N P ( (

25 he Forward Algorthm he Forward varable t ( P( t, qt S We can compute α( for all N,, Intalzaton: α ( = a 0 b 0 ( Iteraton: ermnaton: N = N t ( [ t ( aj ] bj ( t t N P ( ( 0 a 0 a 0 a 0n N a a a n N KN N o o o 3 o n

26 he Backward Algorthm he backward varable Smlar, we can compute backward varable for all N,, Intalzaton: Iteraton: ermnaton: N t j b a t t j N j j t,,,, ( ( ( N j j j b a P 0 ( ( ( N N K N o o o 3 o n N 0 a 0 a 0 a 0n a a N a n, ( ( t t t t S q P (,,..., N

27 Consder P S q P S q P (, ( ( hus ( (, ( Also P S q P S q P t t t t, ( S q P (,, ( P S q P t t t t t (, P(, ( P S q S q P t t t t t t ( P(, ( P S q S q P t t t t t Forward, α k ( Backward, β k (

28 Decodng GIVEN a HMM, and a sequence. Suppose that we know the parameters of the Hdden Markov Model and the observed sequence of observatons,,...,. FIND the sequence Q of states that maxmzes P(Q, Determnng the sequence of States q, q,..., q, whch s optmal n some meanngful sense. (.e. best explan the observatons

29 Decodng Consder o maxmze the above probablty s equvalent to maxmzng ( (,, ( P Q P Q P, ( Q P o o o o b a b a b a b a ln ln ln max( (, max ln( (, max o o o b a b a b a Q P Q P N N K N o o o 3 o n N 0 a 0 A best path fndng problem

30 Vterb Algorthm [Dynamc programmng] Intalzaton: δ ( = a 0 b (, ψ ( = 0. Recurson: = N δ t (j = max [δ t- ( a j ]b j ( t t= j=n ψ (j = argmax [δ t- ( a j ] t= j=n ermnaton: P* = max δ ( q * = argmax [δ ( ] raceback: q t * = ψ (q* t+ 0 a 0 t=-,-,,. N N K N N o o o 3 o n

31 he Vterb Algorthm State x x x 3..x N V j ( K Smlar to algnng a set of states to a sequence me: (K N Space: (KN

32 Learnng Estmaton of Parameters of a Hdden Markov Model. Both the sequence of observatons and the sequence of states Q s observed learnng = (A, B,. nly the sequence of observatons are observed learnng Q and = (A, B,

33 Maxmal Lkelhood Estmaton Gven and Q, the Lkelhood s gven by: o o o o b a b a b a b a B A L 3 3 3,,

34 Maxmal Lkelhood Estmaton the log-lkelhood s: l A, B, ln LA, B, lna ln ln b o a a lnb lna lnb M where f f 0 ln M M M o a fj ln aj bo 0 ln ln j o the number of tmes state occurs n the frst state f the number of tmes state changes to state j. j (or f y p y o y the sum of all observatons n the dscrete case o where q S t t o

35 Maxmal Lkelhood Estmaton In such case these parameters computed by Maxmum Lkelhood Estmaton are: f 0 ˆ a bˆ aˆ j f j M j f j, and = the MLE of b computed from the observatons o t where q t = S.

36 Maxmal Lkelhood Estmaton nly the sequence of observatons are observed It s dffcult to fnd the Maxmum Lkelhood Estmates drectly from the Lkelhood functon. he echnques that are used are. he Segmental K-means Algorth. he Baum-Welch (E-M Algorthm o o o o b a b a b a b a B A L, ,,

37 he Baum-Welch Algorthm he E-M algorthm was desgned orgnally to handle Mssng observatons. In ths case the mssng observatons are the states {q, q,..., q }. Assumng a model, the states are estmated by fndng ther expected values under ths model. (he E part of the E-M algorthm.

38 he Baum-Welch Algorthm Wth these values the model s estmated by Maxmum Lkelhood Estmaton (he M part of the E-M algorthm. he process s repeated untl the estmated model converges.

39 he Baum-Welch Algorthm Intalzaton: Pck the best-guess for model parameters (or arbtrary Iteraton: Forward Backward Calculate A kl, E k (b Calculate new model parameters a kl, e k (b Calculate new log-lkelhood P(x GUARANEED BE HIGHER BY EXPECAIN-MAXIMIZAIN Untl P(x does not change much

40 he Baum-Welch Algorthm Let f, Q L, Q, denote the jont dstrbuton of Q,. Consder the functon: Q E ln L, Q,, Q, Startng wth an ntal estmate of. A sequence of estmates (m are formed ( by fndng m to maxmze ( m Q, wth respect to. X (

41 he Baum-Welch Algorthm he sequence of estmates (m converge to a local maxmum of the lkelhood. L Q, f Q

42 Speech Recognton n-lne documents of Java Speech API n-lne documents of Free S n-lne documents of Sphnx-II

43 Bref Hstory of CMU Sphnx Sphnx-I (987 he frst user ndependent, hgh performance ASR of the world. Wrtten n C by Ka-Fu Lee ( 李開復博士, 現任 Google 副總裁. Sphnx-II (99 Wrtten by Xuedong Huang n C. ( 黃學東博士, 現為 Mcrosoft Speech.NE 團隊領導人 5-state HMM / N-gram LM. Sphnx-III (996 Bult by Erc hayer and Mosur Ravshankar. Slower than Sphnx-II but the desgn s more flexble. Sphnx-4 (rgnally Sphnx 3j Refactored from Sphnx 3. Fully mplemented n Java. (Not fnshed yet

44 Components of CMU Sphnx

45 Knowledge Base he data that drves the decoder. hree sets of data Acoustc Model. Language Model. Lexcon (Dctonary.

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49 Acoustc Model /model/hmm/6k Database of statstcal model. Each statstcal model represents a phoneme. Acoustc Models are traned by analyzng large amount of speech data.

50 HMM n Acoustc Model HMM represent each unt of speech n the Acoustc Model. ypcal HMM use 3-5 states to model a phoneme. Each state of HMM s represented by a set of Gaussan mxture densty functons. Sphnx default phone set.

51 Mxture of Gaussans Represent each state n HMM. Each set of Gaussan Mxtures are called senones. HMM can share senones.

52 Mxture of Gaussans

53 Language Model Descrbes what s lkely to be spoken n a partcular context Word transtons are defned n terms of transton probabltes Helps to constran the search space

54 N-gram Language Model Probablty of word N dependent on word N-, N-,... Bgrams and trgrams most commonly used Used for large vocabulary applcatons such as dctaton ypcally traned by very large (mllons of words corpus

55 Markov Random feld See webpage /MRF_Book.html

56 Belef Network (Propagaton Y. Wess and W.. Freeman Correctness of Belef Propagaton n Gaussan Graphcal Models of Arbtrary opology. n: Advances n Neural Informaton Processng Systems, edted by S. A. Solla,. K. Leen, and K-R Muller, 000. MERL-R99-38.

57 Homework Read the moton texture sggraph paper.