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Fig. 18.18
The first-order Takagi-Sugeno algorithm generates linear fitting functions for data
clusters (dotted lines). The output fitting (solid curve) is the weighted sum of these fittings.
If, in turn, non-spherical clusters are involved, we may specify such fittings for
these clusters which represent the cluster centers [8]. However, now we do not have
any dependent variables in our model. If we use such mathematical functions as lin-
ear functions for the cluster centers, we optimize a set of parameters for determining
these functions in practice. Our task may nevertheless be more challenging if the
appropriate cluster centers should be non-linear fittings, and then fuzzy systems may
provide us with a better resolution (Fig. 18.19).
Fig. 18.19
Two possible cluster centers for a non-spherical cluster in a 2-D space, a point (+)
and a curve (- -)
Within fuzzy systems we specify a fuzzy relation for each non-spherical cluster
and the maximal intensities of these relations are obtained in the cluster centers. In
other words, we may specify fittings for the clusters and the closer the data points
are to these fittings, the higher intensities are obtained. On the other hand, these
fittings should locate in the densest areas of the clusters. For example, in Fig. 18.20
the highest intensities are obtained close to the dotted curve.
