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Projection pursuit

Projection pursuit models [60,102,50] are a generalization of additive models (391) (and a special case of models (393) additive in parameters) where the projections of $z$ = $(x,y)$ are also adapted

\begin{displaymath}
\phi (z) = \xi_0+\sum_l \phi_l (\xi_{0,l}+\xi_l \cdot z)
.
\end{displaymath} (408)

For such a model one has to determine one-dimensional `ridge' functions $\phi_l$ together with projections defined by vectors $\xi_l$ and constants $\xi_0$, $\xi_{0,l}$. Adaptive projections may also be used for product approaches
\begin{displaymath}
\phi (z) = \prod_l \phi_l (\xi_{0,l}+\xi_l \cdot z)
.
\end{displaymath} (409)

Similarly, $\phi $ may be decomposed into functions depending on distances to adapted reference points (centers). That gives models of the form
\begin{displaymath}
\phi (z) = \prod_l \phi_l (\vert\vert\xi_l -z\vert\vert),
\end{displaymath} (410)

which require to adapt parameter vectors (centers) $\xi_l$ and distance functions $\phi_l$. For high dimensional spaces the number of centers necessary to cover a high dimensional space with fixed density grows exponentially. Furthermore, as the volume of a high dimensional sphere tends to be concentrated near its surface, the tails become more important in higher dimensions. Thus, typically, projection methods are better suited for high dimensional spaces than distance methods [211].


next up previous contents
Next: Neural networks Up: Parameterizing likelihoods: Variational methods Previous: Decision trees   Contents
Joerg_Lemm 2001-01-21