Hidden Markov Models

Transcription

1 Hidden Markov Models Dr Philip Jackson Centre for Vision, Speech & Signal Processing University of Surrey, UK

2 Outline 1. Recognizing patterns 2. Markov and hidden-markov processes Likelihood calculation Decoding Training 3. Gaussian pdfs HMM-GMM 4. Practical issues 2

3 Recognizing patterns Pattern recognition is an important enabling technology in many machine intelligence applications, e.g., automatic speech recognition. A pattern is a structured sequence of observations. Various approaches have been employed: Rule-based heuristics Pattern matching dynamic time warping (deterministic) Hidden Markov models (stochastic) Classification artificial neural networks (discriminative) 3

4 Distance-from-template pattern matching Template the features of a typical example of the sequence to be recognized e.g., filterbank, linear prediction, PLP, cepstrum, MFCC Distance a measure of how well the features of a new test sequence match those of the reference template e.g., Euclidean distance, Mahalanobis distance, Itakura distance 4

5 Sequences of speech features Time-frequency representation of speech as observation sequences for a template and two test words: match, match, dummy 5

6 Dynamic time warping DTW is a method of pattern matching that allows for timescale variations in sequences of the same class. frames in template i d ( t, i) D( T, N) a b d c c t a a b c d frames in test utterance Two aligned instances of the same word (after Holmes & Holmes, 2001). Open circles mark permitted predecessors to the closed circle at (t, i). 6

7 Dynamic programming for time alignment Cumulative distance along the best path upto frame N in template and T th test frame is: D(T, N) = min t,i t 1..T i 1..N d(t, i) (1) where d(t, i) is distance between features from tth frame of test utterance and those from ith frame of template. Allowing transitions from current and previous frames only, we compute the cost recursively: D(t, i) = min [D(t, i 1), D(t 1, i 1), D(t 1, i)] + d(t, i) (2) 7

8 DTW algorithm 1. Initialise the cumulative distances for t = 1, D(1, i) = { d(1, i) for i = 1 D(1, i 1) + d(1, i) for i = 2,..., N (3) 2. Recur for t = 2,..., T, D(t, i) = D(t 1, i) + d(t, i) for i = 1 min [ D(t, i 1), D(t 1, i 1), D(t 1, i) ] + d(t, i) for i = 2,..., N (4) 3. Finalise, the cumulative distance up to the final point gives the total cost of the match: D(T, N) 8

9 Sequences of speech features Time-frequency representation of speech as observation sequences for a template and two test words: match, match, dummy 9

10 Inter-utterance distances Euclidean distances between template and test word features 10

11 Summary of Dynamic Time Warping The DTW approach allows efficient computation with limited flexibility in the alignment. It treats templates as deterministic with residual noise. Problems: 1. How much flexibility should we allow? 2. How should we penalise any warping? 3. How do we determine a fair distance metric? 4. How many templates should we register? 5. How do we select the best ones? Solution: Develop an inference framework to build templates based on the statistics of our data. 11

12 Characteristics of the desired model evolution of sequence should not be deterministic observations are coloured depending on class cannot directly observe class stochastic sequence + stochastic observations Applications: automatic speech recognition, optical character recognition, protein and DNA sequencing, speech synthesis, noise-robust data transmission, crytoanalsis, machine translation, image classification, etc. 12

13 Introduction to Markov models We can model stochastic sequences using a Markov chain, e.g., the state topology of an ergodic Markov model: For 1st-order Markov chains, probability of state occupation depends only on the previous step (Rabiner, 1989): P (x t = j x t 1 = i, x t 2 = h,...) P (x t = j x t 1 = i) (5) So, if we assume the RHS of eq. 5 is independent of time, we can write the state-transition probabilities with the properties a ij = P (x t = j x t 1 = i), 1 i, j N (6) a ij 0 and N j=1 a ij = 1 i, j 1..N 13

14 Weather prediction example Let us represent the state of the weather by a 1st-order, ergodic Markov model, M: state 1: state 2: state 3: raining cloudy sunny with state-transition probabilities expressed in matrix form: A = { a ij } = (7) 14

15 Weather predictor probability calculation Given today s weather, what is the probability of directly observing the sequence of weather states rain-sun-sun with model M? A = rain cloud sun rain cloud sun P (X M) = P (X = {1, 3, 3} M) = P (x 1 = rain today) P (x 2 = sun x 1 = rain) P (x 3 = sun x 2 = sun) = a 21 a 13 a 33 = =

16 Start and end of a stochastic state sequence Null states deal with the start and end of sequences, as in the state topology of this left-right Markov model: a 11 a 22 a 33 π 1 a 12 a 23 η Entry probabilities at t=1 for each state i are defined π i = P (x 1 = i) 1 i N (8) with the properties π i 0, and N i=1 π i = 1 for i 1..N Exit probabilities at t=t are similarly defined η i = P (x T = i) 1 i N (9) with properties η i 0, and η i + N j=1 a ij = 1 for i 1..N 16

17 Example: probability of MM state sequence Consider the state topology state transition probabilities A = The probability of state sequence X = {1, 2, 2} is P (X M) = π 1 a 12 a 22 η 2 = =

18 Summary of Markov models State topology diagram: Entry probabilities π = {π i } = [ ] and exit probabilities η = {η i } = [ ] T are combined with state transition probabilities in complete A matrix: A = Probability of a given state sequence X: P (X M) = π x1 T a xt 1 x t t=2 η xt (10) 18

19 Introduction to hidden Markov models Hidden Markov Models (HMMs) use a Markov chain to model stochastic state sequences which emit stochastic observations, e.g., the state topology of an ergodic HMM: b b 3 2 b2 Probability of state i generating a discrete observation o t, which has one of a finite set of values, k 1..K, is b i (o t ) = P (o t = k x t = i). (11) Probability distribution of a continuous observation o t, which can have one of an infinite set of values, is b i (o t ) = p(o t x t = i). (12) We begin by considering only discrete observations. 19

20 Observations in discretised feature space c 2 k =1 k =2 k =3 c 1 Discrete output probability histogram P(o) K k 20

21 Parameters of a discrete HMM, λ State transition probabilities, A = {π j, a ij, η i } = {P (x t = j x t 1 = i)} where N is the number of states for 1 i, j N Discrete output probabilities, B = {b i (k)} = {P (o t = k x t = i)} where K is the number of observation types for 1 i N 1 k K a 11 a 22 a 33 a 44 π 1 a 12 a 23 a b 1(o 1) b 1(o 2) b 2(o 3) b 3(o 4) b 3(o 5) η 4 b 4(o 6) generating a state sequence X = { 1, 1, 2, 3, 3, 4 } and observations O = {o 1, o 2,..., o 6 } o 1 o 2 o 3 o 4 o 5 o 6 21

22 HMM probability calculation The joint likelihood of state and observation sequences is P (O, X λ) = P (X λ) P (O X, λ) (13) a 11 a 22 a 33 a 44 π 1 a 12 a 23 a b 1(o 1) b 1(o 2) b 2(o 3) b 3(o 4) b 3(o 5) η 4 b 4(o 6) o 1 o 2 o 3 o 4 o 5 o 6 The state sequence X = {1, 1, 2, 3, 3, 4} produces the set of observations O = {o 1, o 2,..., o 6 }: P (X λ) = π 1 a 11 a 12 a 23 a 33 a 34 η 4 P (O X, λ) = b 1 (o 1 ) b 1 (o 2 ) b 2 (o 3 ) b 3 (o 4 ) b 3 (o 5 ) b 4 (o 6 ) P (O, X λ) = π x1 b x1 (o 1 ) T t=2 a xt 1 x t b xt (o t ) η xt (14) 22

23 Example: probability of HMM state sequence Consider state topology and state transition matrix: A = o1 o 2 o 3 Output probabilities: B = [ b1 b 2 ] = R G B [ ] Probability of observations with state sequence X = {1, 2, 2}: P (O, X λ) = P (X λ) P (O X, λ) = π 1 b 1 (o 1 ) a 12 b 2 (o 2 ) a 22 b 2 (o 3 ) η 2 = = given observations O = {R, G, G}. 23

24 HMM Recognition & Training Three tasks within HMM framework 1. Compute likelihood of a set of observations with a given model, P (O λ) 2. Decode a test sequence by calculating the most likely path, X 3. Optimise pattern templates by training parameters in the models, Λ = {λ} State Observations 24

25 Normalisation Probability fundamentals Discrete: probability of all possibilities sums to one: all X P (X) = 1. (15) Continuous: integral over entire probabilty density function (pdf) comes to one: p(x) dx = 1. (16) Joint probability The joint probability that two independent events occur is the product of their individual probabilities: P (A, B) = P (A) P (B). (17) 25

26 Conditional probability If two events are dependent, we need to determine their conditional probabilities. The joint probability is now P (A, B) = P (A) P (B A), (18) where P (B A) is the probability of event B given that A occurred; conversely, taking the events the other way P (A, B) = P (A B) P (B). (19) P (A, B) A Ā B B These expressions can be rearranged to yield the conditional probabilities. Also, we can combine them to obtain the theorem proposed by Rev. Thomas Bayes (C.18th). 26

27 Bayes theorem Equating the RHS of eqs. 18 and 19 gives P (B A) = P (A B) P (B). (20) P (A) For example, in a word recognition application we have P (w O) = which can be interpreted as p(o w) P (w), (21) p(o) posterior = likelihood prior evidence (22) posterior probability gives basis for Bayesian inference, likelihood describes how likely are data for given class, prior incorporates other knowledge (e.g., language model), evidence normalises and is often discarded (same for all classes). 27

28 Marginalisation Discrete: probability of event B, which depends on A, is the sum over A of all joint probabilities: P (B) = all A P (A, B) = all A P (B A) P (A). (23) Continuous: similarly, the nuisance factor x can be eliminated from its joint pdf with y: p(y) = p(x, y) dx = p(y x)p(x) dx. (24) 28

29 Task 1: Computing P (O λ) So far, we calculated the joint probability of observations and state sequence, for a given model λ, P (O, X λ) = P (X λ) P (O X, λ) For the total probability of the observations, we marginalise the state sequence by summing over all possible X: P (O λ) = P (O, X λ) = P (o T 1, xt 1 λ) (25) all X all x T 1 Now, we define forward likelihood for state j as α t (j) = P (o t 1, x t = j λ) = {x t 1 1, x t=j} P (o t 1, xt 1 λ) (26) and apply the HMM s simplifying assumptions to yield α t (j) = N i=1 α t 1 (i) P (x t = j x t 1 = i, λ) P (o t x t = j, λ) (27) as current state x t depends only on previous state x t 1, and observation o t on current state (Gold & Morgan, 2000). 29

30 Forward procedure To calculate forward likelihood, α t (i) = P (o t 1, x t = i λ): 1. Initialise at t = 1, α 1 (i) = π i b i (o 1 ) for 1 i N 2. Recur for t = {2, 3,..., T }, α t (j) = [ Ni=1 α t 1 (i) a ij ] bj (o t ) for 1 j N (28) 3. Finalise, P (O λ) = N i=1 α T (i) η i Thus, we can solve Task 1 efficiently by recursion. 30

31 Worked example of the forward procedure state transition matrix A = output matrix B = [ ] State Time α 1 (1) = = 0.5 α 1 (2) = 0 0 = 0 α 2 (1) = = 0.08 α 2 (2) = = 0.09 α 3 (1) = = α 3 (2) = ( ) 0.9 = P (O λ) = =

32 Backward procedure We define backward likelihood, β t (i) = P (o T t+1 x t = i, λ), and calculate: 1. Initialise at t = T, β T (i) = η i for 1 i N 2. Recur for t = {T 1, T 2,..., 1}, β t (i) = N j=1 a ij b j (o t+1 ) β t+1 (j) for 1 i N (29) 3. Finalise, P (O λ) = N i=1 π i b i (o 1 ) β 1 (i) This is an equivalent way of computing P (O λ) recursively. 32

33 Task 2: finding the best path Given observations O = {o 1,..., o T }, find the HMM state sequence X = {x 1,..., x T } that has greatest likelihood where X = arg max X P (O, X λ), (30) P (O, X λ) = P (O X, λ)p (X λ) = π x1 b x1 (o 1 ) T t=2 a xt 1 x t b xt (o t ) η xt (31) Viterbi algorithm is an inductive method to find optimal state sequence X efficiently, similar to forward procedure. It computes maximum cumulative likelihood δ t (i) up to current time t for each state i: δ t (i) = max P (o t {x t 1 1, x 1, xt 1 1, x t = i λ) (32) t=i} 33

34 Viterbi algorithm To compute the maximum cumulative likelihood, δ t (i): 1. Initialise at t = 1, δ 1 (i) = π i b i (o 1 ) ψ 1 (i) = 0 2. Recur for t = {2, 3,..., T }, δ t (j) = max i [ δt 1 (i)a ij ] bj (o t ) ψ t (j) = arg max i [ δt 1 (i)a ij ] for 1 i N for 1 j N 3. Finalise, P (O, X λ) = max i [δ T (i)η i ] x T = arg max i [δ T (i)η i ] 4. Trace back, for t = {T, T 1,..., 2}, x t 1 = ψ t ( x t ), and X = {x 1, x 2,..., x T } (33) 34

35 Illustration of the Viterbi algorithm 1. Initialise, δ 1 (i) = π i b i (o 1 ) ψ 1 (i) = 0 State Recur for t = 2, δ 2 (j) = max i [ δ1 (i)a ij ] bj (o 2 ) ψ 2 (j) = arg max i [ δ1 (i)a ij ] Recur for t = 3, δ 3 (j) = max i [ δ2 (i)a ij ] bj (o 3 ) ψ 3 (j) = arg max i [ δ2 (i)a ij ] State 1 2 State Time Time Time 3. Finalise, P (O, X λ) = max i [δ 3 (i)η i ] x 3 = arg max i [δ 3 (i)η i ] State Trace back for t={3..2}, x 2 =ψ ( ) 3 x 3 ( x 2 ) x 1 =ψ 2 X ={x 1, x 2, x 3 } State Time Time 35

36 Worked example of the Viterbi algorithm state transition matrix A = output matrix B = [ ] State Time δ 1 (1) = = 0.5 ψ 1 (1) = 0 δ 1 (2) = 0 0 = 0 ψ 1 (2) = 0 δ 2 (1) = = 0.08 ψ 2 (1) = 1 δ 2 (2) = = 0.09 ψ 2 (2) = 1 δ 3 (1) = = ψ 3 (1) = 1 δ 3 (2) = max( , ) 0.9 = ψ 3 (2) = 2 P (O, X λ) = =

37 Isolated word recognition application The problem is to find ŵ = arg max i where, according to Bayes, {P (w i O)}, (34) P (w i O) = P (O w i) P (w i ). (35) P (O) Concept: a single word w Speech Waveform Parameterise Speech Vectors Recognise Isolated word problem (Young et al., 2009) w 37

38 Isolated word recognition application In Viterbi decoding, P (O w i ) P (O λ i ) P (O, X λ i ) and the best path X identifies the recognition output. Task grammar: $word = ONE TWO THREE FOUR FIVE SIX SEVEN EIGHT NINE OH ZERO; ( SENT-START $word SENT-END ) one E.g.: five, oh, six one two two... zero Isolated word recognition trellis diagram 38

39 Task 3: training the models Motivation for the most likely model parameters Given two different probability density functions (pdfs), p(o) 1 2 o how would you classify observations in regions 1 and 2 by least-squares, and compare this to their relative likelihoods? 39

40 Maximum likelihood training In general, we want to find the value of some model parameter c that is most likely to give our set of training data O train This maximum likelihood (ML) estimate ĉ is obtained by setting the derivative of P (O train c) w.r.t. c equal to zero, which is equivalent to: ln P (O train ĉ) c = 0 (36) Solving this likelihood equation tells us how to optimise the model parameters in training. 40

41 Re-estimating the parameters of the model λ As a preliminary approach, let us use the optimal path X computed by the Viterbi algorithm with some initial model parameters λ = {A, B}. In so doing, we approximate the total likelihood of the observations: P (O λ) = all X P (O, X λ) P (O, X λ) (37) Using X, we make a hard binary decision about the state occupation, q t (i) {0, 1}, and train the parameters of our model accordingly. 41

42 Viterbi training (hard state assignment) Model parameters can be re-estimated using the Viterbi alignment to assign observations to states (a.k.a. segmental k-means training). State-transition probabilities, â ij = Tt=2 q t 1 (i) q t (j) Tt=1 q t (i) where state indicator q t (i) = Discrete output probabilities, ˆb j (k) = Tt=1 q t (j) ω t (k) Tt=1 q t (j) { 1 0 for 1 i, j N for i = x t otherwise for 1 j N and 1 k K where event indicator ω t (k) = { 1 0 for k = o t otherwise 42

43 Maximum likelihood training by EM Baum-Welch re-estimation (occupation) Yet, the hidden state occupation is not known with absolute certainty. So, the expectation maximisation (EM) method optimises the model parameters based on soft assignment of observations to states via the occupation likelihood, γ t (i) = P (x t = i O, λ) (38) relaxing the assumption previously made in eq. 37. We can rearrange, using Bayes theorem, to obtain γ t (i) = P (O, x t = i λ) P (O λ) = α t(i) β t (i) P (O λ) (39) where α t, β t and P (O λ) are computed by the forward and backward procedures, which also give P (O λ). 43

44 Baum-Welch re-estimation (transition) Similarly, we also define a transition likelihood, ξ t (i, j) = P ( x t 1 = i, x t = j O, λ ) = P ( O, x t 1 = i, x t = j λ ) P (O λ) = α t 1(i) a ij b j (o t ) β t (j) P (O λ) (40) α t ( i) β t ( i) ξ t ( i, j ) γ t ( i) α t 1 a ij b j ( o t ) ( i) β ( j) t Trellis depiction of (left) occupation and (right) transition likelihoods 44

45 Baum-Welch training (soft state assignment) State-transition probabilities, â ij = Tt=2 ξ t (i,j) Tt=1 γ t (i) for 1 i, j N Discrete output probabilities, ˆb j (k) = Tt=1 γ t (j) ω t (k) Tt=1 γ t (j) for 1 j N and 1 k K Re-estimation increases the likelihood over the training data for the new model ˆλ P (O train ˆλ) P (O train λ) although it does not guarantee a global maximum. 45

46 Use of HMMs with training and test data (a) Training Training Examples one two three Initial HMM Forward/Backward Algorithm Update HMM Parameters Estimate Models Converged? Yes No M 1 M 2 M 3 Estimated HMM (b) Recognition Unknown O = P( O M 1 ) P( O M 2 ) P( O M 3 ) Choose Max Isolated word training and recognition (Young et al., 2009) 46

47 HMMs for continuous feature spaces Parameters of a continuous HMM, λ State-transition probabilities, A = {π j, a ij, η i } = {P (x t = j x t 1 = i)} where N is the number of states for 1 i, j N Continuous output probability densities, B = {b i (o t )} = {p(o t x t = i)} for 1 i N where the output pdf for each state i can be Gaussian b i (o t ) = N (o t ; µ i, Σ i ) ( 1 (o µi ) 2 = exp 2πΣi 2Σ i ) (41) evaluated at o t with mean µ i and variance Σ i 47

48 Univariate Gaussian (scalar observations) For a given state i, b i (o t ) = [ 1 exp (o t µ i ) 2 ]. 2πΣi 2Σ i p(o) b (o) 1 b (o) 2 o Multivariate Gaussian (vector observations) b i (o t ) = 1 exp (2π) K Σ i [ 1 2 (o t µ i )Σ 1 i (o t µ i ) T ], where the dimensionality of the observation space is K. 48

49 Baum-Welch training of Gaussian state parameters For observations produced by an HMM with a continuous multivariate Gaussian distribution, i.e.: b i (o t ) = N (o t ; µ i, Σ i ) (42) we can again make a soft (i.e., probabilitistic) allocation of the observations to the states. Thus, if γ t (i) denotes the likelihood of occupying state i at time t then the ML estimates of the Gaussian output pdf parameters become weighted averages, ˆµ i = Tt=1 γ t (i)o t Tt=1 γ t (i) (43) ˆΣ i = Tt=1 γ t (i)(o t µ i )(o t µ i ) T Tt=1 γ t (i) (44) normalised by a denominator which is the total likelihood of all paths passing through state i. 49

50 Gaussian mixture pdfs Univariate Gaussian mixture b i (o t ) = M m=1 c im N (o t ; µ im, Σ im ) (45) where M is the number of mixture components (M-mix), and the mixture weights sum to one: M m=1 c im = 1. Multivariate Gaussian mixture b i (o t ) = M m=1 c im N (o t ; µ im, Σ im ), (46) where N ( ) is the multivariate normal distribution with vector mean µ im and covariance matrix Σ im, evaluated at o t. 50

51 Univariate mixture of Gaussians Probability density, p(o) Observation variable, o Mixture of two univariate Gaussian components 51

52 Multivariate mixtures of Gaussians Examples of two multivariate Gaussian pdfs (upper), and two Gaussian mixtures of them with different weights (lower) 52

53 Baum-Welch training with mixtures We define the mixture-occupation likelihood: where γ t (j, m) = α t(j, m) β t (j) p(o λ) γ t (j) = M m=1 γ t (j, m), (47) { πj b jm (o t ) for t=1 α t (j, m) = Ni=1 α t 1 (i) a ij b jm (o t ) otherwise b j,m (o t ) = c jm N ( ) o t ; µ jm, Σ jm a ij M-component Gaussian mixture j a ij c j1 a ij c j2 Single Gaussians j 1 j 2 a ij c jm... j M Occupations of a mixture (Young et al., 2009) 53

54 Baum-Welch re-estimation of mixture parameters Using soft assignment of observations to mixture components given by mixture-occupation likelihood γ t (j, m), we train our parameters with revised update equations. Mean vector: ˆµ jm = Tt=1 γ t (j, m)o t Tt=1 γ t (j, m) (48) Covariance matrix: Tt=1 γ t (j, m)(o t µ jm )(o t µ jm ) ˆΣ T jm = Tt=1 γ t (j, m) (49) Mixture weights: ĉ jm = Tt=1 γ t (j, m) Tt=1 γ t (j) (50) 54

55 Practical issues Decoding Probabilities stored as log probs to avoid underflow Paths propagated through trellis by token passing Search space kept tractable by beam pruning Model initialisation 1. Random 2. Flat start Proto HMM Definition HCompV Sample of Training Speech 3. Viterbi alignment (supervised/ unsupervised) Identical ih eh b d etc Flat start (Young et al., 2009) 55

56 Re-estimation and embedded re-estimation Labelled Utterances th ih s ih s p iy t sh sh t iy s z ih s ih th Transcriptions th ih s ih s p iy t sh sh t iy s z ih s ih th Unlabelled Utterances HInit HRest HCompV HERest HHEd Sub-Word HMMs HMM training with variously labelled data (Young et al., 2009) 56

57 Number of parameters & Regularisation Context sensitivity Size of database Parsimonious models Occam s razor variance floor parameter tying agglomerative clustering decision trees Related topics for further reading Discriminative training Language modeling Noise robustness factorial HMMs loosely-coupled HMMs Model adaptation MLLR/MAP Probabilistic modeling Graphical model Bayesian network Markov random field Markov decision process Markov chain Monte Carlo 57

58 Relationship of HMM to other estimation methods Discrete classifier Discrete HMM Key: Dependency Discrete hidden Discrete observed Continuous hidden Continuous observed Gaussian classifier Continuous HMM Kalman filter Mixture of Gaussians HMM GMM Probabilistic Data Association Filter Conditional dependencies for various probabilistic methods, drawn using graphical model notation 58

59 Summary Introduction to discrete and continuous HMMs Pattern recognition by DTW Probabilistic time series modeling by MM Probabilistic observations within HMM likelihood computation Viterbi decoding Baum-Welch training Discrete and continuous HMMs Mixtures of Gaussians Practical decoding, initialisation and regularisation 59

60 References B. Gold, N. Morgan & D. Ellis, Speech and Audio Signal Processing, New York: Wiley-Blackwell, 2nd ed., 2011 [ISBN-13: ]. J. N. Holmes & W. J. Holmes, Speech Synthesis and Recognition, CRC Press, 2nd ed., 2001 [ISBN-13: ]. F. Jelinek, Statistical Methods for Speech Recognition, MIT Press, 1998 [ISBN-13: ]. D. Jurafsky & J. H. Martin, Speech and Language Processing, Prentice Hall, 2nd ed., 2003 [ISBN-13: ] L. R. Rabiner. A tutorial on HMM and selected applications in speech recognition. In Proc. IEEE, Vol. 77, No. 2, pp , S. J. Young, et al., The HTK Book, Cambridge Univ. Eng. Dept. (v3.4), 2009 [ 60