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Hidden Markov Model in Automatic Speech Recognition Z. Fodroczi Pazmany Peter Catholic. Univ.

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Outline Discrete Markov process Hidden Markov Model Vitterbi algorithm Forward algorithm Parameter estimation Type of them HMMs in ASR Systems Popular models State of the art and limitation

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Discrete Markov Processes S1S1 S2S2 S3S3 a 11 a 22 a 33 a 12 a 21 a 23 a 32 States={S 1,S 2,S 3 } P(q t =S i |q t-1 =S j )=a ij Sum (j=1..N) (aij)=1 a 31 a 13 Consider the following model of weather: S1: rain or snow S2: cloudy S3: sunny What is the probability that the weather for the next 3 days will be “sun,sun,rain”. O={S3,S3,S1} – observation sequence P(O|Model)=P(S3,S3,S1|Model)=P(S3)*P(S3|S3)*P(S3|S1)=1*0.8*0.3=0.24

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Hidden Markov Model The observation is a probabilistic function of the state. Teacher-mood-model Situation: Your school teacher gave three dierent types of daily homework assignments: A: took about 5 minutes to complete B: took about 1 hour to complete C: took about 3 hours to complete Your teacher did not reveal openly his mood to you daily, but you knew that your teacher was either in a bad, neutral, or a good mood for a whole day. Mood changes occurred only overnight. Question: How were his moods related to the homework type assigned that day?

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Hidden Markov Model Model parameters: S - states {good, neutral, bad} a kl,- probability that state l is followed by k Σ – alphabet {A,B,C} e k (x) –probability that state k emit symbol x€Σ

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Hidden Markov Model One week, your teacher gave the following homework assignments: Monday: A Tuesday: C Wednesday: B Thursday: A Friday: C Questions: What did his mood curve look like most likely that week? (Searching for the most probable path – Viterbi algorithm) What is the probability that he would assign this order of homework assignments? (Probability of a sequence - Forward algorithm) How do we adjust the model parameters λ(S,aij,ei(x)) to maximize P(O| λ) (create a HMM for a given sequence set)

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HMM – Viterbi algorithm Given: Hidden Markov model: S, a kl,Σ,e k (x) Observed symbol sequence E = x 1 x 2, … x n. Most probable path of states that resulted in symbol sequence E. Let v k (i) be the probability of the most probable path of the symbol sequence x 1, x 2, …. x i ending in state k. Then:

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HMM – Viterbi algorithm Matrix v k (i), where k€S and 1 <= I <= n. Initialization: v k (1) = e k (x1)/#states for all states k€S. v l (i) = e l (x i )max k (v k (i - 1)akl) for all states k€S, i >=2. Algorithm Iteratively build up matrix v k (i). Store pointers to chosen path. Probability of most probable path in maximum entry in last column. Reconstruct path along pointers.

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HMM – Viterbi algorithm Empty table

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HMM – Viterbi algorithm

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Question: What did his mood curve look like most likely that week? Answer: Most probable mood curve: Day: Mon Tue Wed Thu Fri Assignment: A C B A C Mood: good bad neutral good bad

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HMM – Forward algorithm Used to test the probability of a sequence. Given: Hidden Markov model: S, akl,, el(x). Observed symbol sequence E = x1; : : : ; xn. What is the probability of E. Let f k (i) be the probability of the symbol sequence x1, x2, …. xi ending in state k. Then:

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HMM – Forward algorithm Matrix f k (i), where k€S and 1 <= i <= n. Initialization: f k (1) = e k (x 1 )=#states for all states k€S. f l (i) = e l (x i )P k (f k (i - 1)a kl ) for all states k€S, i <=2. Algorithm Iteratively build up matrix f k (i). Probability of symbol sequence is sum of entries in last column.

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HMM – Forward algorithm

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HMM – Parameter estimation

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Type of HMMs Till now I was speaking about HMMs with discrete output (finite |Σ|). Extensions: continuous observation probability density function mixture of Gaussian pdfs

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HMMs in ASR Separation Feature extraction HMM decoder Understanding Application

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HMMs in ASR How HMM can used to classify feature sequences to known classes. Make a HMM to each class. By determineing the probability of a sequence to the HMMs, we can decide which HMM could most probable generate the sequence. There are several idea what to model: Isolated word recognition ( HMM for each known word) Usable just on small dictionaries. (digit recognition etc.) Number of states usually >=4. Left-to-rigth HMM Monophone acoustic model ( HMM for each phone) ~50 HMM Triphone acoustic model (HMM for each three phone sequence) 50^3 = 125000 triphones each triphone has 3 state

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HMMs in ASR Hierarchical system of HMMs HMM of a triphone Higher level HMM of a word Language model

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ASR state of the art Typical state-of-the-art large-vocabulary ASR system: - speaker independent - 64k word vocabulary - trigram (2-word context) language model - multiple pronunciations for each word - triphone or quinphone HMM-based acoustic model - 100-300X real-time recognition - WER 10%-50%

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HMM Limitations Data intensive Computationally intensive 50 phones = 125000 possible triphones 3 states per triphone 3 Gaussian mixture for each state 64k word vocabulary 262 trillion trigrams 2-20 phonemes per word in 64k vocabulary 39 dimensional feature vector sampled every 10ms 100 frame per second

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Hidden Markov Models. A Hidden Markov Model consists of 1.A sequence of states {X t |t T } = {X 1, X 2,..., X T }, and 2.A sequence of observations.

Hidden Markov Models. A Hidden Markov Model consists of 1.A sequence of states {X t |t T } = {X 1, X 2,..., X T }, and 2.A sequence of observations.

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