3.2.2 Alternative distance measure

For some applications, it proved to be of advantage to modify the error measure E($\bf x$). The error (3.2) depends on the size of the hyper-ellipsoid. Its volume is proportional to V = $\sqrt{{\det {\mathbf\Lambda}}}$ $\sigma^{{d-q}}_{}$. For a big ellipsoid, (3.2) has therefore a huge bias, namely 2 ln V. Thus, if the other ellipsoids are much smaller, this huge ellipsoid often has a high rank, independent of the Euclidean distance of $\bf x$ to the center of the ellipsoid. Since the weight $\alpha$ decreases exponentially with the rank, $\alpha$ is almost zero, and the ellipsoid cannot change its size anymore, it is `dead' (for example, figure 3.8.A). Thus, to avoid dead units, we tested an alternative error measure that is independent of the volume of the ellipsoid. This can be achieved by normalizing $\lambda^{l}_{}$ and $\sigma$ such that the resulting volume $\tilde{V}$ is one. Let $\tilde{\lambda}^{l}_{}$ and $\tilde{\sigma}$ be the normalized values. They can be computed by

$$\tilde{\lambda}^{l}_{} {\frac{{\lambda^l}}{{\sqrt[d]{V^2}}}}$$ =

$$\tilde{\sigma} {\frac{{\sigma}}{{\sqrt[d]{V}}}}$$ = (3.9)

Using these substitutions, it can be straight-forwardly verified that the new volume $\tilde{V}$ = $\tilde{\sigma}^{{d-q}}_{}$$\prod_{{l=1}}^{q}$$\tilde{\lambda}^{{l/2}}_{}$ fulfills $\tilde{V}$ = 1. To compute the modified error, we replace $\lambda^{l}_{}$ and $\sigma$ by their normalized values, and then we use the substitution (3.9) to obtain the error again in the original variables,

$$\tilde{E} \bf x \left(\vphantom{\boldsymbol{\xi}^T {\bf W}{\mathbf\Lambda}^{-1} {... ...ymbol{\xi}- \boldsymbol{\xi}^T {\bf W}{\bf W}^T \boldsymbol{\xi}\right)}\right. \xi \bf W \Lambda \bf W^{T}_{} \xi {\frac{{1}}{{\sigma^2}}} \left(\vphantom{\boldsymbol{\xi}^T \boldsymbol{\xi}- \boldsymbol{\xi}^T {\bf W}{\bf W}^T \boldsymbol{\xi}}\right. \xi \xi \xi \bf W \bf W^{T}_{} \xi \left.\vphantom{\boldsymbol{\xi}^T \boldsymbol{\xi}- \boldsymbol{\xi}^T {\bf W}{\bf W}^T \boldsymbol{\xi}}\right) \left.\vphantom{\boldsymbol{\xi}^T {\bf W}{\mathbf\Lambda}^{-1} {... ...ymbol{\xi}- \boldsymbol{\xi}^T {\bf W}{\bf W}^T \boldsymbol{\xi}\right)}\right)$$ (3.10)

Here, the logarithm terms can be omitted because they are the same for all units. The modified algorithm uses (3.10) instead of (3.2) for the competition of the units. Everything else stays unchanged. The ellipsoids are still allowed to change their size according to the local PCA. In the following, this algorithm will be called `NGPCA-constV', in contrast to NGPCA, which uses (3.2) for the competition.