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6.7 Summary 91

bi-variate splines within these regions. His best results for 400 data points
were reported as a NRMS of 0.069 for the impedance value Z.
In contrast the PSOM approach constructs (here) a 4-dimensional pa-
rameterized manifold throughout. Fig. 6.10 visualizes this underlying map-
ping manifold in 3 D isometric views of various 2 D slices. Shown are
,

.
z
Z
F f LZ H F L C , Z H H R C , and Z H
f CAll

views are obtained from the same PSOM, here a n , n L-PSOM, in

the following also denoted (3-of-5) L-PSOM.
The mapping accuracy depends crucially on the faithfulness of the
parametric model – and the fit to the given mapping task. Fig. 6.11 de-
picts one situation in different views: Drawn are the change in impedance
and phase lag with L and C, given R and f, and three PSOM networks
z
(here left: Z
H L C , right:
H L C . Note, the reference
z
vectors do not lie within the visualized surfaces, instead they lay “below”
and “above” in the 4-dimensional mapping manifold, embedded in the
6 D space X).
The first row depicts the target mapping situation at the RLC-circuit,

the others present a 3 PSOM, (3-of-5) L-PSOM, and 5 PSOM (3 is the
short form notation of 3 3 3 3 nodes etc.). The impedance Z plot is

marked by a curved valley, which is insufficiently picked up by the case,
but well approximated by the latter two PSOM types. The phase diagram
in the right column shows a plateau with a steep slope at the left and front

side. Again, the type approximates a too smooth curve, because the
network does not represent more detailed information. The PSOM with
five nodes per axis exhibits at the right side little “overshots”. Here, the
contrast to the Local-PSOM is again apparent: the L-PSOM represents the
flat plateau without long-range interferences of the “step”.

6.7 Summary

We proposed the multi-start algorithm to face the problem of obtaining
several solutions to the given mapping task. The concept of cost function
modulation was introduced. It allows to dynamically add optimization
goals which are of lower priority and possibly conflict with the primary
goal.
Furthermore, we presented two extensions to the PSOM algorithm which