Hidden Markov Models. Adapted from. Dr Catherine Sweeney-Reed s slides

Transcription

1 Hidden Markov Models Adaped from Dr Caherine Sweeney-Reed s slides

2 Summary Inroducion Descripion Cenral in HMM modelling Exensions Demonsraion

3 Specificaion of an HMM Descripion N - number of saes Q = {q ; q 2 ; : : : ;q T } - se of saes M - he number of symbols (observables) O = {o ; o 2 ; : : : ;o T } - se of symbols

4 Descripion Specificaion of an HMM A - he sae ransiion probabiliy marix aij = P(q + = j q = i) B- observaion probabiliy disribuion b j (k) = P(o = k q = j) i k M π - he iniial sae disribuion

5 Specificaion of an HMM Descripion Full HMM is hus specified as a riple: λ = (A,B,π)

6 Cenral in HMM modelling Cenral Problem Evaluaion: Probabiliy of occurrence of a paricular observaion sequence, O = {o,,o k }, given he model P(O λ) Complicaed hidden saes Useful in sequence classificaion

7 Cenral in HMM modelling Cenral Problem 2 Decoding: Opimal sae sequence o produce given observaions, O = {o,,o k }, given model Opimaliy crierion Useful in recogniion

8 Cenral in HMM Cenral modelling Problem 3 Learning: Deermine opimum model, given a raining se of observaions Find λ, such ha P(O λ) is maximal

9 Problem : Naïve soluion Cenral Sae sequence Q = (q, q T ) Assume independen observaions: T P ( O q, ") =! P( o q,") = bq ( o ) bq 2( o2 i= )... b NB Observaions are muually independen, given he hidden saes. (Join disribuion of independen variables facorises ino marginal disribuions of he independen variables.) qt ( o T )

10 Problem : Naïve soluion Cenral Observe ha : And ha: P( q #) = " a a... a q qq2 q2q3 qt! qt P( O ") =! P( O q, ") P( q ") q

11 Problem : Naïve soluion Cenral Finally ge: P ( O ") =! P( O q, ") P( q ") q NB: -The above sum is over all sae pahs -There are N T saes pahs, each cosing O(T) calculaions, leading o O(TN T ) ime complexiy.

12 Problem : Efficien soluion Cenral Forward algorihm: Define auxiliary forward variable α: " ( i) = P( o,..., o q = i,!) α (i) is he probabiliy of observing a parial sequence of observables o, o such ha a ime, sae q =i

13 Problem : Efficien soluion Cenral Recursive algorihm: Iniialise: Calculae: Obain: " ( i) =! b ( o ) i i " N P O #) = i=! ( " Sum of differen ways of geing obs seq (Parial obs seq o AND sae i a ) x (ransiion o j a +) x (sensor) N + ( ) = [! Sum, as can reach j from j " ( i) aij ] bj ( o + ) any preceding sae i= α incorporaes parial obs seq o T ( i ) Complexiy is O(N 2 T)

14 Problem : Alernaive soluion Backward algorihm: Define auxiliary forward variable β: " ( i) = P( o, o,..., o q = i,!) T Cenral β (i) he probabiliy of observing a sequence of observables o +,,o T given sae q =i a ime, and λ

15 Problem : Alernaive soluion Recursive algorihm: Iniialise:! T Calculae: Terminae: ( j) = N! ( i) = "! + ( j) a b ( o + ) ij j j= N p( O #) = " ( i = T!,...,! i= ) Cenral Complexiy is O(N 2 T)

16 Problem 2: Decoding Cenral Choose sae sequence o maximise probabiliy of observaion sequence Vierbi algorihm - inducive algorihm ha keeps he bes sae sequence a each insance

17 Problem 2: Decoding Cenral Vierbi algorihm: Sae sequence o maximise P(O,Q λ): P ( 2 q, q,... qt O,!) Define auxiliary variable δ: " ( 2 q i ) = max P( q, q2,..., q = i, o, o,... o!) δ (i) he probabiliy of he mos probable pah ending in sae q =i

18 Problem 2: Decoding Cenral Recurren propery:! Algorihm:. Iniialise: ( j ) max(! ( i) aij ) bj ( o ) + = + i " i) = b ( )! i! N (! i i o! ( i) = 0 To ge sae seq, need o keep rack of argumen o maximise his, for each and j. Done via he array ψ (j).

19 Problem 2: Decoding Cenral 2. Recursion: # ( j ) = max( #! ( i) a " i" N ij ) b j ( o ) 3. Terminae: $ ) = arg max( # ( i) a ) 2!! T,! j! N P ( j! " i" N # = max! " i" N T ( i) q # = arg max! T " i" N T ( i) ij P* gives he sae-opimised probabiliy Q* is he opimal sae sequence (Q* = {q*,q2*,,qt*})

20 Problem 2: Decoding Cenral 4. Backrack sae sequence: (! q! ) = " + q+ + T!, T! 2,..., O(N 2 T) ime complexiy

21 Problem 3: Learning Cenral Training HMM o encode obs seq such ha HMM should idenify a similar obs seq in fuure Find λ=(a,b,π), maximising P(O λ) General algorihm: Iniialise: λ 0 Compue new model λ, using λ 0 and observed sequence O Then! "! o Repea seps 2 and 3 unil: log P ( O ")! log P( O " 0) < d

22 Problem 3: Learning Le ξ(i,j) be a probabiliy of being in sae i a ime and a sae j a ime +, given λ and O seq ) ( ) ( ) ( ) ( ), (! " # $ O P j o b a i j i j ij + + = Cenral!! = = = N i N j j ij j ij j o b a i j o b a i ) ( ) ( ) ( ) ( ) ( ) ( " # " # Sep of Baum-Welch algorihm:

23 Problem 3: Learning Cenral Operaions required for he compuaion of he join even ha he sysem is in sae Si and ime and Sae Sj a ime +

24 Problem 3: Learning Cenral Le! ( i) be a probabiliy of being in sae i a ime, given O N! # ( i) = " ( i, j= j) T " # = T "! ( i) - expeced no. of ransiions from sae i #! ( ) - expeced no. of ransiions i i! j =

25 Problem 3: Learning Sep 2 of Baum-Welch algorihm: " ˆ =! ( i) he expeced frequency of sae i a ime =!! Cenral # ( i, j) aˆ ij = " ( i) raio of expeced no. of ransiions from sae i o j over expeced no. of ransiions from sae i! = ˆ " ( j) ( ) =, o k bj k! " ( j) raio of expeced no. of imes in sae j observing symbol k over expeced no. of imes in sae j

26 Problem 3: Learning Cenral Baum-Welch algorihm uses he forward and backward algorihms o calculae he auxiliary variables α,β B-W algorihm is a special case of he EM algorihm: E-sep: calculaion of ξ and γ M-sep: ieraive calculaion of,, Pracical issues:!ˆ ij â ˆ ( k) Can ge suck in local maxima Numerical log and scaling b j

27 Exensions Exensions Problem-specific: Lef o righ HMM (speech recogniion) Profile HMM (bioinformaics)

28 Exensions Exensions General machine learning: Facorial HMM Coupled HMM Hierarchical HMM Inpu-oupu HMM Swiching sae sysems Hybrid HMM (HMM +NN) Special case of graphical models Bayesian nes Dynamical Bayesian nes

29 Examples Exensions Coupled HMM Facorial HMM

30 HMMs Sleep Saging Demonsraions Flexer, Sykacek, Rezek, and Dorffner (2000) Observaion sequence: EEG daa Fi model o daa according o 3 sleep sages o produce coninuous probabiliies: P(wake), P(deep), and P(REM) Hidden saes correspond wih recognised sleep sages. 3 coninuous probabiliy plos, giving P of each a every second

31 HMMs Sleep Saging Demonsraions Manual scoring of sleep sages Saging by HMM Probabiliy plos for he 3 sages

32 Excel Demonsraions Demonsraion of a working HMM implemened in Excel

33 Furher Reading L. R. Rabiner, "A uorial on Hidden Markov Models and seleced applicaions in speech recogniion," Proceedings of he IEEE, vol. 77, pp , 989. R. Dugad and U. B. Desai, "A uorial on Hidden Markov models," Signal Processing and Arifical Neural Neworks Laboraory, Dep of Elecrical Engineering, Indian Insiue of Technology, Bombay Technical Repor No.: SPANN-96., 996. W.H. Lavery, M.J. Mike, and I.W. Kelly, Simulaion of Hidden Markov Models wih EXCEL, The Saisician, vol. 5, Par, pp. 3-40, 2002