Chap 8. Hidden Markov Model

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Chap 8. Hidden Markov Model

• HMM was the most powerful statistical method for
  speech.
• If we treat the speech recognition problem as a
  template (pattern) matching problem, the most
  difficult problem in speech recognition is the
  Time alignment and normalization of speech
  signal.
• Dynamic Time Warping (DTW), dynamic programming

Test template
Reference templates
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DTW

• Let
• The global pattern dissimilarity measure
• Find the optimal path

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• DTW ?Viterbi search in Trellis
• Complexity
• Search constraint in DTW.

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Markov chain

• If X is first order Markov chain, ie.
• Then

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• Probabilistic finite state machine.
• Example
• Find the probability of
• a specified sequence

Average state duration
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• Example 1
• Example 2

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Hidden Markov Model

• In the Markov chain, each state correspond to a
  deterministic observation.
• HMM the observation is probabilistic function
  of state
• Example

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• Notations in HMM

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• Using HMM for speech production-
• Signal in same state have the same statistics
  phone in speech
• The state machine template structure of speech
• 3 problems for HMM
• (1) The evaluation problem (Recognition)
• When the models are known, for a observation
  sequence
• Find
• (2) The decoding problem (Segmentation)
• When the models are known, for a observation
  sequence
• Find
• (3) The learning problem (Training)
• Find

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• Problem 1
• The evaluation problem (Recognition)
• How many computation need?
• Is there a more efficient way to find the
  likelihood function?
• ? Forward-Backward method!

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• Forward algorithm

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• Problem 2
• The decoding problem (Segmentation)

survivor
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• Leave survivor in each node
• Complexity ? no. of states in trellis time

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• Problem 3
• The learning problem (Training)
• estimated the parameter in HMM, Baum-Welch
  algorithm

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• Find the prob. transition from state i to j at
  time t
• State occupied prob.

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• Parameter estimation using EM algorithm

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• And, the A and B have the constraints

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• Multiple Training data
• Multiple Models
• The same

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Continuous HMM

• The observation prob. is continuous mixture
  Gaussian pdf.
• Similar,

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• And,
• where

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Semicontinuous HMM (SCHMM)

• Continuous observation pdf mixture Gaussian.
• Same mixtures were used for all states, but
  different weights.
• Parameter estimation will be more accuracy.
• Observation pdf
• Parameters estimation

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ASR for continuous speech

• One-state (one-pass) Algorithm
• Only 1 survivor
• leave in each frame
• ? Frame synchronous

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• Algorithm for one-state recognizer

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Segmental K-mean

• If we change the state occupy prob. into delta
  function
• And, use force alignment (Viterbi decoding) to
  find the state sequence
• ? segmental K-mean

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Implementation of HMM

• HMM Types left-to-right one step forward only
• or allow 1-state skip
• word graph
• Parameters in HMM Models
• (1) How many words need to recognize?
• (2) Training data?
• (3) Speaker dependent/independent.
• (3) Sub-word model less training data need!
• (4) Unit selection
• phone for spelling language like English,
• initial/final for Mandarin.
• (5) Number of states in sub-unit 3 states for
  phone-level unit,
• 3/5 for initial/final.

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• Initial model of HMM
• (1) Hand labeling (to word or phone level)
• (2) Uniform segmentation (inside word/phone
  level)
• (3) Using old HMM model to do force alignment
• do the decoding problem - find the segmentation
• Number of mixtures in CHMM
• (1) the inside recognition rate will increase
  when the number of mixtures
• increased. (overfit)
• (2) the state number will depends on the number
  of training samples
• (3) degenerate case if the number of training
  samples is too low
• combined the similar model/state
• example zh(?),z(?)using the same models
• model/state tying
• (4) Gender dependent model.
• (5) left/right dependent models (di/tri-phone
  models).

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Normalization in Baum-Welch algorithm

• In Baum-Welch algorithm
  may overflow.
• Normalization -

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Duration model in HMM

• In fact, in HMM model, the observation
  probability B is the most important item. The
  transition probability A is usually been ignored,
  because the dynamic range of b() is usually much
  larger than a.
• And, the effect of transition probability A can
  be replace by the duration model.
• Multiply a Gamma duration pdf in state transition
• The log(?) can be found numerically.
• The parameter of the Gamma pdf