Currency, international stock trading and derivatives contracts
play an increasing role for many investors. Commonly used a portfolio
consisting of a number of contracts. Assets returns must be predicted
and controlled by a prediction/control module. Control of risk via
prediction/control module of individual investments returns inside
the portfolio provides the most likely process.
It is known that in most economic applications i.e. financial
risk control, neural networks gives success only of 7080%. By means
of the new approach of GMDH twicemultilayered neural networks it
can be improved by 510%. Prediction accuracy for short and very
noised data also increases in short and longtime predictions by
1050% in comparison to statistical methods and neural networks,
especially for stochastic processes [30,31]. On the base of predictive
control it increases the results of a repetitive control.
Stock market indexes forecasting
As an example prediction of the activity on the stock exchange
in New York was considered in [10]. In the following on the base
of observations in the period of February 22 up to June 14, 1995
in seven periods 7 variables of the stock market (Dax, Dow Jones,
F.A.Z., Dollar and other) are predicted. In the information base
delays of all variables up to 35 are included. Also there were used
not only linear reference functions to describe the variables, but
also nonlinear. It was to model and to predict 7 time series not
independently as time series models but rather as highly interactions
network (input  output  model). Table 1 shows the accuracy of
predictions for all variables (mean MAD [%])
Table 1: Observation and prediction periods.

Observation period 
Longterm prediction period 
Model 
Prediction 

Period up to 
Days 
Begin 
End 
Days 
Max delay 
Mean MAD [%] 
a 
March, 17 
18 
March,20 
March, 31 
10 
5 
0.985 % 
b 
March, 31 
28 
April,3 
April, 18 
10 
10 
2.055 % 
c 
April, 18 
38 
April, 19 
May, 3 
10 
15 
0.809 % 
d 
April, 28 
46 
May, 2 
May, 15 
10 
20 
1.642 % 
e 
May, 15 
56 
May, 16 
May, 30 
10 
26 
1.217 % 
f 
May, 30 
66 
May, 31 
June ,14 
10 
30 
1.206 % 
g 
June, 14 
76 
June, 16 
June, 29 
10 
35 
0.760 % 
Using the results of model generation (at first
layer of neuronet) it is possible to improve the accuracy of models
in a second model generation, where the model outputs are used
as input variables for next layer. This procedure can be continued
up to increase accuracy of models.
Table 2 shows the resulting model error (MAPE [%]) and prediction
error (MAD [%]) of Dollar, Dax, F.A.Z., Dow Jones and the mean values
for all 7 variables obtained on 3 levels. It is shown that the repeated
application of selforganization gives more accurate approximation,
which results in better predictions in the second level. The models
obtained in the third level are overfitted, therefore the prediction
error increases.
Table 2: Multilevel application (model f).

MAPE [%]

MAD [%]

Level

1

2

3

1

2

3

Dollar

0.68

0.51

0.11

2.32

2.17

11.67

Dax

0.35

0.24

0.10

2.20

1.24

5.21

F.A.Z.

0.22

0.23

0.03

1.54

1.27

2.32

Dow Jones

0.27

0.16

0.06

2.15

0.84

4.84

Mean

0.267

0.184

0.051

1.43

0.98

3.67

The efforts in construction and using of the GMDH neural networks
are much less than in neural networks, where the architecture must
be chosen by trial and error. Only an adaptive synthesis of the
network structure allows an automatic model generation and therefore
applications in the fields where lots of decisions and forecasts
(monitoring of complex systems with many controlled variables) repeating
over short time periods are needed.
Objective selection of the best model
It is the aim of selforganizing modeling to get in an objective
way models of optimal complexity. But there is several freedom in
choice of class of systems to be model (linear/nonlinear), time
lag and in selection of appropriate parameters (number of best models,
complexity etc.). To reduce such a subjectivity it is recommended
to generate several alternative models (linear, nonlinear, with
several complexity and time lags) and in a second layer to select
the best model outputs or to generate there combination. Table 3
shows obtained results.
Table 3: Selection of best model results (model g):
prediction error MAD [%].

Linear

Nonlinear

Second
layer

Model

1

2

3

1

2

3

Dollar

2.88

2.1

0.89

1.25

1.41

1.4

1.55

F.A.Z.

1.22

1.45

1.01

0.82

1.12

1.57

0.88

Dax

1.36

2.41

1.51

1.69

2.43

4.54

1.94

Dow Jones

1.14

1.26

1.44

3.75

3.25

3.79

2.93

Mean

1.14

1.29

0.9

1.21

1.35

1.81

1.2

This results were obtained by Prof.J.A.Müller from Hochschule fur Technik und Wirtschaft (Dresden) and Dr.F.Lemke. We express our deep acknowledgments for their help.
