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1 Hidden Markov Model Prof. Yan Wang Woodruff School of Mechanical Engineering Georgia Institute of echnology Atlanta, GA 30332, U.S.A.
2 Learning Objectives o familiarize the hidden Markov model as a generalization of Markov chain o understand the three basic problems (evaluation, decoding, and learning) in HMM model construction and applications
3 Hidden Markov Model (HMM) HMM is an extension of regular Markov chain State variables q s are not directly observable All statistical inference about the Markov chain itself has to be done in terms of observable o s observable o o t o t o t+ o hidden q t q q t q t+ q
4 HMM o s are conditionally independent given {q t }. However, {o t } is not an independent sequence, nor a Markov chain itself. observable o o t o t o t+ o hidden q t q q t q t+ q
5 HMM Components State sequence: Q={q,q 2,,q } with N possible values Observation sequence: O={o,o 2,,o } with M possible symbols {v,v 2, v M } State transition matrix: A=(a ij ) N N where a ij =P(q t+ =j q t =i) Observation matrix: B=(b ik ) N M where b ik =b i (v k )=P(o t =v k q t =i) Initial state distribution: Δ=(δ i ) N where δ i =P(q =i) Model s parameters λ={a, B, Δ}
6 hree Basic Problems in HMM Evaluation: Given a model with parameters λ and a sequence of observations O={o,o 2,,o }, what is the probability that the model generates those observations P(O λ)? Decoding: Given a model with parameters λ and a sequence of observations O={o,o 2,,o }, what is the most likely state sequence Q={q,q 2,,q } in the model that produces the observations? Learning: Given a set of observations O={o,o 2,,o }, how can we find a model with the parameters λ with the maximum likelihood P(O λ)?
7 Evaluation Problem Given an O={o,o 2,,o }, P(O λ)=? Naïve algorithm because of lemma 3. ( d) ( d) ( d) λ λ λ λ PO ( ) = POQ ( d) (, ) = PO ( d) ( Q, ) PQ ( ) Q where Q (d) is one of all possible combinations of state sequences assume conditional independence between observations ( d ) (, λ) = (, λ) = ( ) t= t t t= q t t ( d ) λ = δ λ = δ q t= t+ t q t= q q PO Q Po q b o PQ ( ) Pq ( q, ) a t t+ However, the number of possible combinations of state sequences is huge! Q
8 Evaluation Problem Define a forward variable Forward Algorithm which can be recursively calculated by Multiscale Systems Engineering Research Group α () i = P( o, o,, o, q = i λ) t 2 t t α () i = P( o, q = i λ) = P( q = i λ) P( o q = i, λ) = δb( o ) i i α () i = P( o,, o, q = i λ) t+ t+ t+ = Pq ( = i λ) Po (,, o q = i, λ) t+ t+ t+ = Pq ( = i λ) Po ( q = i, λ) Po (,, o q = i, λ) t+ t+ t+ t t+ = Po ( q = i, λ) Po (,, o, q = i λ) t+ t+ t t+ = Po ( q = i, λ) Po (,, o, q, q = i λ) t+ t+ q t t t+ = Po ( q = t+ t+ i t+ q q, i t t t t = b( o ) a α ( q ) t i, λ) P( q = i q, λ) P( o,, o, q λ) q t t+ t t t N N i= i= PO ( λ) = POq (, = i λ) = α ( i )
9 Evaluation Problem Backward Algorithm Define a backward variable which can be recursively calculated by β () i = β () i = P( o,, o q = i, λ) t t+ t = Po ( q = i, λ) Po (,, o q = i, λ) t+ t t+ 2 t t + β () i = P ( o, o,, o q = i, λ) t t+ t+ 2 t = Po ( q = i, λ) Po (,, o, q = i λ)/ Pq ( = i λ) t+ t t+ 2 t t = Po ( q = i, λ) Po (,, o, q = iq, λ)/ Pq ( = i λ) t+ t q t+ 2 t t+ t = Po ( q = i, λ) t+ t t + t + t + t + Po (,, o q = iq,, λ) Pq ( t+ 2 t t+ t t+ t q t+ 2 t+ t+ t i t+ q i, q t+ t+ q Pq ( = i λ) = Po ( q = i, λ) Po (,, o q, λ) Pq ( q = i, λ) = b( o ) a β ( q ) t = iq, λ) t +
10 Decoding Problem Given an O={o,o 2,,o }, find a Q * ={q *,q 2*,,q * } with the maximum of P(O Q) Viterbi Algorithm is a dynamic programming method to solve the decoding problem
11 Dynamic Programming Breaking down complex problems into subproblems in a recursive manner. e.g. search the shortest path in the raveling Salesman Problem
12 Define an auxiliary probability which is the highest probability that a single path leads to q t =i at time t Recursively Decoding Problem Viterbi Algorithm ρ (): i = max P ( q,, q, q = i, o,, o λ ) t q,, q t t t t { } { ρ ipq j q iλ } Po q t t+ t t+ t+ j { ρ ia } b o ρ () j = max ρ () ipq ( = j q = i, λ) Po ( q = j) t+ i t t+ t t+ t+ = max ( ) ( = =, ) ( = ) = i max ( ) ( ) i t ij j t+
13 Decoding Problem Viterbi Algorithm cont d Define an auxiliary variable to store the optimal state at time t to reach state j at time t+. he algorithm is. initialize ρ () j = δ b ( o ) j, j =,, N j j ψ () j = 0 2. recursion ρ { } { } ψ ( j) = arg max ρ ( iab ) ( o ) = arg max ρ ( ia ) t+ t ij j t+ t ij i i ψ ( ) { ρ } { ρ ia } () j = max () ia b( o ) t+ i t ij j t+ ( j) = arg max ( ) t+ t ij i Multiscale Systems Engineering Research Group
14 Decoding Problem Viterbi Algorithm cont d 3. terminate he optimal probability P * = { ρ j } max ( ) j he optimal final state q * = { ρ j } arg max ( ) j 4. backtrack state sequence q = ψ + ( q + ) * * t t t
15 Learning Problem Given an O={o,o 2,,o }, find a λ * ={A *, B *, Δ * } with the maximum likelihood P(O λ) { PO } { PO } * λ = arg max ( λ) = arg max log ( λ) λ λ Global optimum needs to search all possible state and observation sequences λ = { d d PO Q λ } arg max log (, ) * ( ) ( ) λ O ( d) ( d) Q Instead, Baum-Welch Algorithm is usually used to search heuristically
16 Learning Problem Baum-Welch Algorithm Algorithm.Guess some parameters λ 2. Determine some probable paths {Q (),,Q (d) } 3. Estimate the number of transitions aˆij, from state i to state j, given the current estimate of λ. 4. Estimate the number of the observation v k emitted from state i as b ˆ( v i k ) 5. Estimate the initial distribution ˆ δ i 6. Re-estimate λ from A ij s and B i (v k ) s 7. If λ and λ is close enough, stop; otherwise, assign λ = λ and go back to step 2.
17 Learning Problem Baum-Welch Algorithm cont d Define an auxiliary likelihood ξ (, ij) = Pq ( = iq, = j o,, o, λ) t t t+ which is the probability that a transition q t =i and q t+ =j occurs at time t given the complete observations {o,o 2,,o } and model parameters λ
18 Learning Problem Baum-Welch Algorithm cont d ξ (, ij) = Pq ( = iq, = j o,, o, λ) t t t+ = = = = = Pq ( = iq, = jo,,, o λ) t t+ Po (,, o λ) Po (,, o q = iq, = j, λ) Pq ( = iq, = j λ) t t+ t t+ Po (,, o λ) Po (,, o q = i λ) Po ( q = j, λ) t t t+ t+ Po (,, o q = j λ) Pq ( = j q = i, λ) Pq ( = i λ) t+ 2 t+ t+ t t Po (,, o λ) Po (,, o, q = i λ) Pq ( = j q = i, λ) t t t+ t Po ( q = j, λ) Po (,, o q = j λ) t+ t+ t+ 2 t+ Po (,, o λ) α () iab( o ) β ( j) t ij i t+ t+ N α () i i = Multiscale Systems Engineering Research Group
19 aˆ Learning Problem Baum-Welch Algorithm cont d Estimate parameters Pq ( = iq, = j o,, o, λ) ξ (, i j) = = Pq ( = i o,, o, λ) ξ (, ik) t= t t+ t= t ij N t= t t= k= t Po = v q = iq = j o o ˆ( ) t = t k t t+ = i k Pq ( = i o,, o, λ) t = t b v = t = t= k= (,,,,, λ) ξ (, ij) o = v t t N k ξ (, ik) t N N ˆ δ = Pq ( = iq, = k o,, o, λ) = ξ (, ik ) i k= 2 k=
20 Learning Problem Baum-Welch Algorithm cont d How to measure two models, λ and λ, are close enough? p = ( d ) P O λ O ( d ) ( ) Cross Entropy = p ( x)log p ( x) p ( x) 2
21 HMM Applications HMM has been applied in many fields that are based on the analysis of discrete-valued time series, such as speech recognition (Rabiner, 989) Image recognition genetic profile and classification (Eddy, 998)
22 Kalman Filter Can be regarded as a special case of HMM Also known as the Gaussian linear statespace model State series are linearly dependent on history, subject to process white noises. x = Cx + w t t t Observations are also linearly dependent on states, subject to measurement noises. y = Dx + v t t t
23 Summary Hidden Markov model is a generalization Markov chain
24 Further Readings Rabiner L.R. (989) A tutorial on hidden Markov models and selected applications in speech recognition. Proceedings of the IEEE, 77(2): Eddy S.R. (998) Profile hidden Markov models. Bioinformatics Review, 4(9): Dempster A.P., Laird N.M., and Rubin D.B. (977) Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society, Series B. 39(): -38 Wu C.F.J. (983) On the convergence properties of the EM algorithm. he Annals of Statistics, (): HMM Software Packages
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