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For the ringlinesquare distribution, figure 5.3 shows the isopotential curves of a spherical potential field and a cylindrical potential field with q = 40. The 40 principal components explained 94.5% of the variance of the distribution in feature space. The cylindrical field shows a more balanced potential field, having valleys of almost the same depth. This difference is also reflected in the quality measure (93.7% compared to 68.2%).
Figure 5.3:
Isopotential curves in the original space of a spherical potential field (top row) and a cylindrical potential field (bottom row) with 40 principal components in feature space. The right pictures show the isopotential curves enclosing an area of same size as the distribution (top: covering 68.2% of the data points, bottom: covering 93.7% of the data points).

On the vortex distribution, the cylindrical potential with q = 40 could also follow the shape of the distribution better then the spherical field (figure 5.4). The 40 principal components explained 99.0% of the variance.
Figure 5.4:
Isopotential curves in the original space of a spherical potential field (top row) and a cylindrical potential field (bottom row) with 40 principal components in feature space. The right pictures show the isopotential curves enclosing an area of same size as the distribution (top: covering 25.6% of the data points, bottom: covering 95.0% of the data points).

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Heiko Hoffmann
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