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Mixture models

The function $\phi (z)$ can be approximated by a mixture model, i.e., by a linear combination of components functions

\begin{displaymath}
\phi(z) = \sum c_l B_l(\xi_l,z),
\end{displaymath} (390)

with parameter vectors $\xi_l$ and constants $c_l$ (which could also be included into the vector $\xi_l$) to be adapted. The functions $B_l(\xi_l,z)$ are often chosen to depend on one-dimensional combinations of the vectors $\xi_l$ and $z$. For example they may depend on some distance $\vert\vert\xi_l - z\vert\vert$ (`local or distance approaches') or the projection of $z$ in $\xi_l$-direction, i.e., $\sum_k \xi_{l,k} z_k$ (`projection approaches'). (For projection approaches see also Sections 4.5, 4.8 and 4.9).

A typical example are Radial Basis Functions (RBF) using Gaussian $B_l(\xi_l,z)$ for which centers (and possibly covariances and also number of components) can be adjusted. Other local methods include $k$-nearest neighbors methods ($k$NN) and learning vector quantizations (LVQ) and its variants. (For a comparison see [158].)



Joerg_Lemm 2001-01-21