LING124 Acoustic models

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LING124 Acoustic models

• October 23, 2008

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Class outline

• Gaussian distribution
• Univariate Gaussian distribution
• Multivariate Gaussian distribution
• Mean vector
• Covariance matrix
• Gaussian mixture models

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Acoustic model

• In statistical approach to ASR, the acoustic
  model calculates the likelihood, P(OW)
• Among many different types of acoustic models, we
  focus on the most common method Gaussian mixture
  model (GMM)

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GMM Basic idea

• GMM is most often used with HMM
• A state represents a symbolic linguistic unit,
  most commonly a context-specific phone
• Each state emits a feature-vector, e.g. the
  feature vector consisting of MFCCs, deltas, and
  double-deltas
• The emission probability is calculated using a
  mixture of multivariate Gaussians

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Gaussian distribution
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Multivariate Gaussians

• A typical feature vector has 39 components
• MFCC(Energy), deltas, double-deltas
• A multivariate Gaussian is defined by
• a mean vector (µ)
• a covariance matrix (S)

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Covariance matrix (1)

• Variance of each vector component
• Covariance between any two components
• How much two variables change together
• e.g. Two variables X and Y have a positive
  covariance if X has a value above the mean, then
  Y also tends to have a value above the mean

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Covariance matrix (2)

• If you think about it, variance of a vector
  component is the covariance between the component
  and itself
• So all the necessary variances can be represented
  as a single covariance matrix
• M(i,j) covariance between the ith component and
  the jth component
• M(i,i) variance of a single component

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Diagonal matrix

• Covariance between two different components means
• More calculation
• More data to train the parameters
• Risking negative effect on performance,
  covariance between different components is often
  assumed to be zero
• The resulting covariance matrix is a diagonal
  matrix
• Except for the main diagonal of the matrix, all
  the elements of the matrix are zero

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Gaussian mixture model

• Using a multivariate Gaussian as observation pdf
  is assuming the vector components are normally
  distributed
• In reality, the normality assumption may not be
  true
• One way around this problem is to represent the
  pdf as a weighted sum of multiple Gaussians

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Gaussian mixture model (2)