LECTURE 2: ALGORITHMS AND APPLICATIONS

Transcription

1 LECTURE 2: ALGORITHMS AND APPLICATIONS

2 OUTLINE State Sequence Decoding Example: dice & coins Observation Sequence Evaluation Example: spoken digit recognition HMM architectures Other applications of HMMs HMM tools

3 STATE SEQUENCE DECODING The aim of decoding is to discover the hidden state sequence that most likely describes a given observation sequence. One solution to this problem is to use the Viterbi algorithm, which finds the single best state sequence for an observation sequence. Parameter δ: The probability of the most probable state path for the partial observation sequence:

4 VITERBI ALGORITHM:

5 First pass: π b (o ) δ, ψ () δ, ψ 2 () δ 3, ψ 3 () a b (o 2 ) a b (o 3 ) a b (o 4 ) δ 4, ψ 4 () b (o ) a 2 b (o 2 ) a 2 b (o 3 ) a 2 b (o 4 ) π 0 b 2 (o ) b 2 (o ) δ 2, ψ (2) a 2 b 2 (o 2 ) a 2 b 2 (o 3 ) a 2 b 2 (o 4 ) δ 2 2, ψ 2 (2) δ 3 2, ψ 3 (2) a 22 b 2 (o 2 ) a 22 b 2 (o 3 ) a 22 b 2 (o 4 ) δ 4 2, ψ 4 (2) Second pass (back track): ψ () ψ 2 () ψ 3 () δ 4, ψ 4 () 2 2 ψ (2) ψ 2 (2) ψ 3 (2) δ 4 2, ψ 4 (2)

6 DICE EXPERIMENT STATE SEQUENCE DECODING P H Red Coin = 0.9 P T Red Coin = 0. P H Green Coin = 0.95 State Red Die (6 sides) State2 Green Die (2 sides) 2 P T Green Coin = A = π = 0 B =

7 State Observation MATLAB Viterbi algorithm: obs = [2 6 ]; %Die outcomes states = [ 2 2]; %True state sequence Trial likelystates = hmmviterbi(obs,a,b) likelystates = Trial (From the dice experiment)

8 OBSERVATION SEQUENCE EVALUATION Imagine first we have L number of HMM models. This problem could be viewed as one of evaluating how well a model predicts a given observation sequence O = o,, o T ; and thus allows us to choose the most appropriate model λ l ( l L) from a set, i.e., P O λ l = P o,, o T λ l?

9 Remember that an observation sequence O depends on the state sequence Q = q,, q T of a HMM λ l. So, P O Q, λ l = P(o t q t, λ l ) T t= = b q o b q2 o 2 b qt o T For state sequence Q of the observation sequence O we have: P Q λ l = π q a q q 2 a q2 q 3 a qt q T

10 Finally, we can come up with the final evaluation of the observation sequence as: P O λ l = P O Q, λ l P(Q λ l ) Q = π q b q o q,,q T a q q 2 b q2 o 2 a qt q T b qt o T

11 OBSERVATION SEQUENCE EVALUATION There is an issue with the last expression: We would have to consider ALL possible state sequences for the observations evaluation (brute force). Solution: acknowledge there is redundancy in calculations Forward Backward Algorithm D A B C F-B Analogy: Obtain distance from city A to other three distant cities B, C, D.

12 OBSERVATION SEQUENCE EVALUATION Forward - Backward actually are two similar algorithms which compute the same thing (P O λ l ); it depends where calculations start. Either of the two can be use for the observation sequence evaluation. In the Forward case, we have a parameter α which represents the probability of the partial observation sequence o,, o t and state s i at time t, i. e., α t i = P(o,, o t, q t = s i λ l )

13 Forward Algorithm:

14 State Observation 6 4 O = {2, 6,,, } Trial 3 2 Q = {,,2, 2} Trial

15 π b (o ) α () a b (o 2 ) a b (o 3 ) a b (o 4 ) α 2 () α 3 () α 4 () b (o ) a 2 b (o 2 ) a 2 b (o 3 ) a 2 b (o 4 ) π 0 b 2 (o ) b 2 (o ) α (2) a 2 b 2 (o 2 ) a 2 b 2 (o 3 ) a 2 b 2 (o 4 ) α 2 (2) α 3 (2) a 22 b 2 (o 2 ) a 22 b 2 (o 3 ) a 22 b 2 (o 4 ) α 4 (2) t P o, o 2, o 3, o 4 λ = α 4 + α 4 2

16 Using MATLAB: obs = [2 6 ]; states = [ 2 2]; [PSTATES, LOGPSEQ, FORWARD, BACKWARD, S]= hmmdecode(obs,a,b); f = FORWARD.*repmat(cumprod(S),size(FORWARD,),); f = P o, o 2, o 3, o 4 λ = = 0.00 log [P o, o 2, o 3, o 4 λ ] = log ( ) = -3 This number would be significant if we can compare it with different HMMs.

17 Amplitude Amplitude Amplitude Amplitude DIGIT RECOGNITION EXAMPLE OBSERVATION SEQUENCE EVALUATION Recognize digits Zero and One from two speakers: SpkA - Zero 0.4 SpkA - One sec SpkB - Zero sec SpkB - One sec sec

18 Phonemes: /z/ /iy/ /r/ /ow/ /w/ /ax/ /n/

19 States: Abstract representation of the sounds /z/ /iy/ /r/ /ow/ State State2 State3 State4 /w/ /ax/ /n/ State State2 State3 State4

20 Amplitude Feature extraction of speech signals: SpkA - Zero sec Feature = 2 39-element feature vector per frame Divide the speech signal into frames using a 30ms window.

21 OBSERVATION SEQUENCE EVALUATION For a word (zero/one) it has several feature vectors: For a HMM these are the observations!

22 Basic idea: HMM Zero Score HMM One Score Classification Max {Score}

23 HMM architecture: Left-Right Architecture S S 2 S 3 S 4 Create a HMM for word Zero and One. Both HMM have the same number of states (4). Model the emission probabilities with 2 Gaussian Mixtures. States will be an abstract representation of the features. 3 utterances of each word (both speakers) will be used as training data. 2 utterances of each word (both speakers) will be used as test data.

24 Build the training data for each word by concatenation: zero_data: one_data:

25 We start by estimating the transition matrix for both HMMs (HMM Toolbox, Kevin Murphy, 998) M = 2; %mixtures Q = 4; %states O = size(one_data,2); %dimension T = size(one_data,); nex = ; data = zeros(o,t,nex); data(:,:,nex) = one_data'; prior0 = normalise(rand(q,)); transmat0 = mk_stochastic(rand(q,q)); [mu0, Sigma0] = mixgauss_init(q*m,reshape(data, [O T*nex]), 'diag'); mu0 = reshape(mu0, [O Q M]); Sigma0 = reshape(sigma0, [O O Q M]); mixmat0 = mk_stochastic(rand(q,m)); [LL, prior, transmat, mu, Sigma, mixmat] = mhmm_em(data, prior0, transmat0, mu0, Sigma0, mixmat0, 'max_iter', 20); onea = transmat; omu = mu; osigma = Sigma; oprior = prior; omixmat = mixmat; Parameters

26 Transition probability matrices: za = oa =

27 Now, we can evaluate the test data by feeding to each of the HMM models, compute the log likelihood score, and assigned to a HMM based on the max of score. LogLikScore = zeros(4,2); LogLikScore(,) = mhmm_logprob(ts_zeromfcc', zprior, za, zmu, zsigma, zmixmat); LogLikScore(,2) = mhmm_logprob(ts_zeromfcc', oprior, oa, omu, osigma, omixmat); LogLikScore(2,) = mhmm_logprob(ts_zeromfcc2', zprior, za, zmu, zsigma, zmixmat); LogLikScore(2,2) = mhmm_logprob(ts_zeromfcc2', oprior, oa, omu, osigma, omixmat); LogLikScore(3,) = mhmm_logprob(ts_onemfcc', zprior, za, zmu, zsigma, zmixmat); LogLikScore(3,2) = mhmm_logprob(ts_onemfcc', oprior, oa, omu, osigma, omixmat); LogLikScore(4,) = mhmm_logprob(ts_onemfcc2', zprior, za, zmu, zsigma, zmixmat); LogLikScore(4,2) = mhmm_logprob(ts_onemfcc2', oprior, oa, omu, osigma, omixmat); Results = { }; for i=:size(loglikscore,) if(loglikscore(i,)>loglikscore(i,2)) Results = [Results;{'Zero'}]; else Results = [Results;{'One'}]; end end LogLikScore =.0e+03 * Inf Inf Results = 'Zero' 'Zero' 'One' 'One'

28 HMM ARCHITECTURES Left-Right HMM Ergodic HMM Parallel HMM

29 OTHER APPLICATIONS OF HMMS Due to the powerfulness that Markov models provide, it can be used anywhere sequential information exists: Finance Biology Tracking systems Speech processing Image processing Communication systems Many more

30 Model non-stationary and non-linearity of financial data to predict the direction of the time series. Zhang, Y.; Prediction of Financial Time Series with Hidden Markov Models; Shandong University, China, 200.

31 DNA is composed if 4 bases (A, G, T, C) which pair together form nucleotides. Markov models can compute likelihoods of an DNA sequence. Nelson, R.; Foo, S.; Weatherspoon, M.;, "Using Hidden Markov Modeling in DNA Sequencing," System Theory, SSST th Southeastern Symposium on, vol., no., pp.25-27, 6-8 March 2008

32 Markov models can be used to estimate the position in a tracking system. Hieu and Thanh, Ground Mobile Target Tracking by Hidden Markov Models

33 Speech recognition has been the most exploited area for use of Markov models. Christopher M. Bishop. Pattern recognition and machine learning. Springer, st ed corr. 2nd printing edition, October 2006.

34 Human action recognition can be modeled with Markov models Brand, M.; Oliver, N.; Pentland, A.;, "Coupled hidden Markov models for complex action recognition," Computer Vision and Pattern Recognition, 997. Proceedings., 997 IEEE Computer Society Conference on, vol., no., pp , 7-9 Jun 997

35 HMM TOOLS Hidden Markov Toolkit (HTK) - Cambridge University : MATLAB functions: hmmtrain, hmmgenerate, hmmdecode, hmmestimate, hmmviterbi HMM Matlab Toolbox (Kevin Murphy, 998): MATLAB functions for training and evaluating HMMs and GMMs (Ron Weiss, Columbia University): Sage (open-source mathematics software ): HMM: statistics package Online Sage Notebook (Gmail account)

36 REFERENCES Rabiner, L.R.;, "A tutorial on hidden Markov models and selected applications in speech recognition," Proceedings of the IEEE, vol.77, no.2, pp , Feb 989 John R. Deller, John, and John H. L. Hansen. Discrete-Time Processing of Speech Signals. Prentice Hall, New Jersey, 987. Barbara Resch (modified Erhard and Car Line Rank and Mathew Magimai-doss); Hidden Markov Models A Tutorial for the Course Computational Intelligence. Henry Stark and John W. Woods. Probability and Random Processes with Applications to Signal Processing (3 rd Edition). Prentice Hall, 3 edition, August 200. HTKBook: