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 [52], [53].



Baggenstoss 2017-05-19