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Gaussian Mixtures

The GM form of the PDF for ${\bf z}\in{\cal R}^{P}$ is given by

$\displaystyle p({\bf z}) = {\displaystyle \sum_{i=1}^{L}} \alpha_{i} \; {\cal N}({\bf z},$$\displaystyle \mbox{\boldmath$\mu$}$$\displaystyle _{i},$$\displaystyle \mbox{\boldmath$\Sigma$}$$\displaystyle _i)$ (13.1)

where

$\displaystyle {\cal N}({\bf z},$$\displaystyle \mbox{\boldmath$\mu$}$$\displaystyle _{i},$$\displaystyle \mbox{\boldmath$\Sigma$}$$\displaystyle _i) = (2\pi)^{-P/2} \; \vert$$\displaystyle \mbox{\boldmath$\Sigma$}$$\displaystyle _i\vert^{-1/2} \exp\left\{ -\frac{1}{2} \left({\bf z}- \mbox{\b... ...$\Sigma$}_i^{-1} \; \left({\bf z}- \mbox{\boldmath$\mu$}_{i}\right) \right\}. $

The $L$ mixture components are called modes. The GM parameters are denoted $\Lambda = \left\{ \alpha_{i}, \; \mbox{\boldmath $\mu$}_{i}, \; \mbox{\boldmath $\Sigma$}_i\right\}$. The most commonly used method for finding the maximum likelihood estimate of the parameters from a training set is the E-M algorithm [60], [61].