Databases Reference
In-Depth Information
Pseudocode
(1)
Determine system's input variables
.
(2)
Form training and testing data.
(3)
Initialize the weights to the nets in the swarm.
(4)
Calculate the output of the nets and determine their error.
(5)
Update gbest and pbest if required.
(6)
If stopping criterion not met go to Step 4.
9.5. Polynomial Neural Network
Group Methods of Data Handling (GMDH) is the realization of inductive
approach for mathematical modeling of complex systems.
115,116
It belongs
to the category of self-organization data driven approaches. With small
data samples, GMDH is capable of optimizing structures of models
objectively.
117
The relationship between input-output variables can be approximated
by Volterra functional series, the discrete form of which is Kolmogorov-
Gabor Polynomial:
118
y
=
c
0
+
k
1
c
k
1
x
k
1
+
k
1
k
2
c
k
1
k
2
x
k
1
x
k
2
+
k
1
k
2
k
3
c
k
1
k
2
k
3
x
k
1
x
k
2
x
k
3
+
ยทยทยท
(9.13)
where
c
k
denotes the coecients or weights of the Kolmogorov-Gabor
polynomial and
x
vector is the input vector. This polynomial can
approximate any stationary random sequence of observations and it can
be solved by either adaptive methods or by Gaussian equations.
119
This
polynomial is not computationally suitable if the number of input variables
increase and there are missing observations in input dataset. Also it takes
more computation time to solve all necessary normal equations when the
input variables are large.
A new algorithm called GMDH is developed by Ivakhnenko
118,120,121
which is a form of Kolmogorov-Gabor polynomial. He proved that a second
order polynomial i.e.:
y
=
a
0
+
a
1
x
i
+
a
2
x
j
+
a
3
x
i
x
j
+
a
4
x
i
+
a
5
x
j
(9.14)
which takes only two input variables at a time and can reconstruct the
complete Kolmogorov-Gabor polynomial through an iterative procedure.
The GMDH method belongs to the category of heuristic self-organization
