Crude oil
Gaussian Mixture Models∗
Douglas Reynolds
MIT Lincoln Laboratory, 244 Wood St., Lexington, MA 02140, USA dar@ll.mit.edu Synonyms
GMM; Mixture model; Gaussian mixture density
Definition
A Gaussian Mixture Model (GMM) is a parametric probability density function represented as a weighted sum of Gaussian component densities. GMMs are commonly used as a parametric model of the probability distribution of continuous measurements or features in a biometric system, such as vocal-tract related spectral features in a speaker recognition system. GMM parameters are estimated from training data using the iterative Expectation-Maximization (EM) algorithm or Maximum A
Posteriori (MAP) estimation from a well-trained prior model.
Main Body Text
Introduction
A Gaussian mixture model is a weighted sum of M component Gaussian densities as given by the equation,
M
wi g(x|µi , Σi ),
p(x|λ) =
(1)
i=1
where x is a D-dimensional continuous-valued data vector (i.e. measurement or features), wi , i = 1, . . . , M , are the mixture weights, and g(x|µi , Σi ), i = 1, . . . , M , are the component Gaussian densities. Each component density is a D-variate
Gaussian function of the form, g(x|µi , Σi ) =
1
1
−1 exp − (x − µi )′ Σi (x − µi ) ,
2
(2π)D/2 |Σi |1/2
(2)
M
with mean vector µi and covariance matrix Σi . The mixture weights satisfy the constraint that i=1 wi = 1.
The complete Gaussian mixture model is parameterized by the mean vectors, covariance matrices and mixture weights from all component densities. These parameters are collectively represented by the notation,
∗
This work was sponsored by the Department of Defense under Air Force Contract FA8721-05-C-0002. Opinions, interpretations, conclusions, and recommendations are those of the authors and are not necessarily endorsed by the United States Government.
2
Douglas Reynolds
λ = {wi , µi , Σi }
i = 1, . . . , M.
(3)
There are several variants on the GMM
Douglas Reynolds
MIT Lincoln Laboratory, 244 Wood St., Lexington, MA 02140, USA dar@ll.mit.edu Synonyms
GMM; Mixture model; Gaussian mixture density
Definition
A Gaussian Mixture Model (GMM) is a parametric probability density function represented as a weighted sum of Gaussian component densities. GMMs are commonly used as a parametric model of the probability distribution of continuous measurements or features in a biometric system, such as vocal-tract related spectral features in a speaker recognition system. GMM parameters are estimated from training data using the iterative Expectation-Maximization (EM) algorithm or Maximum A
Posteriori (MAP) estimation from a well-trained prior model.
Main Body Text
Introduction
A Gaussian mixture model is a weighted sum of M component Gaussian densities as given by the equation,
M
wi g(x|µi , Σi ),
p(x|λ) =
(1)
i=1
where x is a D-dimensional continuous-valued data vector (i.e. measurement or features), wi , i = 1, . . . , M , are the mixture weights, and g(x|µi , Σi ), i = 1, . . . , M , are the component Gaussian densities. Each component density is a D-variate
Gaussian function of the form, g(x|µi , Σi ) =
1
1
−1 exp − (x − µi )′ Σi (x − µi ) ,
2
(2π)D/2 |Σi |1/2
(2)
M
with mean vector µi and covariance matrix Σi . The mixture weights satisfy the constraint that i=1 wi = 1.
The complete Gaussian mixture model is parameterized by the mean vectors, covariance matrices and mixture weights from all component densities. These parameters are collectively represented by the notation,
∗
This work was sponsored by the Department of Defense under Air Force Contract FA8721-05-C-0002. Opinions, interpretations, conclusions, and recommendations are those of the authors and are not necessarily endorsed by the United States Government.
2
Douglas Reynolds
λ = {wi , µi , Σi }
i = 1, . . . , M.
(3)
There are several variants on the GMM
References: 1. Gray, R.: Vector Quantization. IEEE ASSP Magazine (1984) 4–29 2. Reynolds, D.A.: A Gaussian Mixture Modeling Approach to Text-Independent Speaker Identification. PhD thesis, Georgia Institute of Technology (1992) 3. Reynolds, D.A., Rose, R.C.: Robust Text-Independent Speaker Identification using Gaussian Mixture Speaker Models. IEEE Transactions on Acoustics, Speech, and Signal Processing 3(1) (1995) 72–83 4. McLachlan, G., ed.: Mixture Models. Marcel Dekker, New York, NY (1988) 5. Dempster, A., Laird, N., Rubin, D.: Maximum Likelihood from Incomplete Data via the EM Algorithm. Journal of the Royal Statistical Society 39(1) (1977) 1–38 6. Reynolds, D.A., Quatieri, T.F., Dunn, R.B.: Speaker Verification Using Adapted Gaussian Mixture Models. Digital Signal Processing 10(1) (2000) 19–41