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Fig. 3.8.
Highest possible number of terms in the polynomial equation.
3.4.3.2.
Number of terms in the polynomials
For construction of PDs of first layer in the PNN model the biquadratic
polynomials (5) used consists of 6 terms. The second layer takes two such
inputs from the first layer, so the maximum number of terms possible in
layer 2 is 6
6 i.e. 36. In general the maximum number of terms that can be
generated in any layer is , where l is the number of layer. Figure 3.8 shows
the possibility of generations of maximum number of terms at different
layers. However, we know that if all the features belong to unique categories,
then generation of maximum terms may be possible. For example, let us
consider
a, b, c
and
d
are the four unique features, then multiplication of
polynomial of two terms generate four terms i.e.:
∗
(
a
+
b
)
∗
(
c
+
d
)=
ac
+
ad
+
bc
+
bd
But if we consider only
a
and
b
, it will generate three terms i.e.:
(
a
+
b
)=
a
2
+
b
2
+2
ab
(
a
+
b
)
∗
In our PNN models
111
each PDs from layer 2 onwards get inputs which
are combinations of outputs from PDs of first layer and original inputs
given to layer 1. For example in layer 2, if the output of one such PDs is
produced by taking features
x
1
and
x
2
i.e.
c
0
+
c
1
x
i
+
c
2
x
j
+
c
3
x
i
x
j
+
c
4
x
i
+
c
5
x
j
and the other input is feature, then the polynomial equation generated out
of it after ignoring the coecients is as follows:
