High-Dimensional Mapping - High Dimensional Neurocomputing: Growth, Appraisal and Applications - page 111

5.2.1.1 Transformation of Disk

Example 5.3 This example considers an experiment in which a set of points on a
disk in a complex plane 'z' get mapped to points on an image disk in other complex
plane ' zz '. This kind of mapping is used to study viscous flow across the bodies with
different cross section. In this example, different networks (2-M-2) are trained up
to 2,000 epochs using C RPROP ( μ − ,μ + 10 ( − 6 ) , max = 0 . 5 = 1 . 2 , min = =

0 001). The hidden layer of considered networks contain one C RSP

or C RSS or two C RPN or three conventional neurons, respectively. The learning
patterns form a set of 36 points, which are on a circle

. 04 , 0 = 0 .

| z |= 0 . 62 referenced at

origin. Output points are corresponding value of defined by Eq. ( 5.4 ). The

input-output mapping of training patterns is shown in Fig. 5.4 a.

3 ( z )

) = (

0 . 2 z + 0 . 2 + 0 . 3 j )

zz = 3 ( z (5.4)

( j + 0 . 4 )

The trained network is able to generalize this mapping from z -plane to zz -plane for
circles of varying radius. Testing input patterns contain six circles of varying radius
from 0.2 to 0.9 at regular interval and each circle has 72 points on its circumference.
The input-output mapping of test patterns in shown in Fig. 5.4 b. Figure 5.4 c-f present
the transformation results of these test patterns with different neurons. C RSP neuron
shows best accuracy among all the generalization results presented.

5.2.2 Bilinear Transformation

Bilinear transformation is an important class of elementary mapping studied by Au-
gustus Ferdinand Mobius (1790-1868). The Bilinear transformation ( Mobius Trans-
formation ) is considered as a linear transformation followed by a reciprocal transfor-
mation, commonly known as linear fractional transformation. It conformally maps
one-one from a complex plane ' z ' onto other complex plane ' zz '.
Example 5.4 Following mapping maps a disk D z :

| | <

z 1 one to one and onto the

( )>

upper half plane Im zz 0.

( − )

1 z

zz = 4 ( z ) = j × (5.5)

( + )

1 z

Different networks of novel neurons are trainedwith 108 points (First input) on the
circumference of three concentric disk (36 points on each) with radius 0 5.

All disks are referenced at origin (Second input). The output patterns are corre-
sponding values of zz defined by Eq. ( 5.5 ). All points in a data set are normal-
ized in between

. 1 , 0 . 3 , 0 .

1 to 1. The hidden layer of considered networks contain four
C RSP or C RSS, or eight C RPN or conventional neurons, respectively. Normal-
ized input-output mapping for training patterns is shown in Fig. 5.5 a. Learning

−