which constrains are given on maximum values of input factors
and output variables.
In this subsection is presented, on example of Ukraina, the
general method of normative forecasting of processes in macroeconomy,
for observation and control aims. There is shown one variant
of full sets of input factors and output variables sets only.
No doubt it should be interesting too, same other variants,
which use other sets of factors and output variables.
Initial material characteristic and set of variables choice
Let us denote input factors of normative forecast by x_{i}
(1<i<M), output variables by y_{j} (1<j<L)
and external disturbances by z_{s} (1<s<K)
In we find that macroeconomic systems are characterised by
values of following basic variables:
y_{1}  Real Internal Whole Product (IWP in billions
of karbovanets of 1991};
y_{2}  Consumer prices inflation (CPI);
y_{3}  Budget deficit (in % of IWP);
y_{4}  Unemployment (thousands of people);
x_{1}  Monetary base increase (in % of IWP);
x_{2}  Number of privatised plants;
x_{3}  Consumer price index;
x_{4}  Monetary circulation rate
z_{1}  Gryvna course for nontrade operations (to US
dollar)
Averaged for quarters values of variables for 13 quarters of
19921995 are presented in table.
Year and
quarter

Y_{1}

Y_{2}

Y_{3}

Y_{4}

X_{1}

X_{2}

X_{3}

X_{4}

Z_{1}

1992

Q_{1}

86.4

3.0

24.1

13.2

13.2

5

75.4

65.1

0.002

Q_{2}

107.6

8.1

35.6

11.1

11.1

10

18.1

100

0.002

Q_{3}

97.5

9.3

60.7

18.5

18.5

20

19.2

123.3

0.003

Q_{4}

96.4

17.7

70.5

45.2

45.2

30

28.5

81

0.008

1993

Q_{1}

84.7

1.1

79.5

32.4

32.4

430

39.7

54.3

0.019

Q_{2}

78.9

13.2

73.3

35.9

35.9

830

39.4

38.2

0.031

Q_{3}

72.3

6.2

78.7

27.7

27.7

1685

44.5

38.9

0.083

Q_{4}

56.4

6.1

83.9

12.2

12.2

3585

66.4

20.6

0.267

1994

Q_{1}

38.8

6.4

98.6

11.8

11.8

5442

12.4

26.8

0.356

Q_{2}

42.4

8.9

92.8

12.9

12.9

8402

5.0

33.3

0.434

Q_{3}

46.9

19.8

88.9

22.6

22.6

10214

4.0

45.7

0.466

Q_{4}

44.5

8.1

82.2

7.3

7.3

11552

39.5

25.4

1.102

1995

Q_{1}

36.2

8.0

86.6

3.7

3.7

12802

16.8

22.8

1.427

It is convenient to transform data to dimensionfree form by
variables normalization formulae:
Y_{j} = (y_{j}y_{jmin})/(y_{jmax}y_{jmin});
X_{i}=(x_{i}x_{imin})/(x_{imax}x_{imin})
Difference forecasting optimal nonphysical models for explicit
patterns
By pattern is called
the graph which shows delayed arguments proposed to computer
selection choice. The use of implicit patterns leads to accuracy
increase, but connected with a system of linear equations solution
for each point of normative forecasting. For explicit patterns
forecasting accuracy is smaller, but forecasting models are
not linked, calculations are simpler. For first variable Y_{1(k)}
forecasting, using explicit pattern (b), Fig.1. we can complete
full regression equation in following form:
Y_{1}(k) = a_{00}+a_{11} Y_{1}(k1)
+ a_{12} Y_{1}(k2)+ a_{21} Y_{2}(k1)
+ a_{22} Y_{2}(k2) + a_{31} Y_{3}(k)
+ a_{32} Y_{3}(k2)
+ a_{41} Y_{4}(k1) + a_{42} Y_{4}(k2)+
a_{50} X_{1}(k) + a_{51} X_{1}(k1)
+ a_{52} X_{1}(k2)
+ a_{60}X_{2}(k) + a_{61} X_{2}(k1)
+ a_{62} X_{2}(k2) + a_{70} X_{3}(k)
+ a_{71} X_{3}(k1) + a_{72} X_{3}(k2)
+ a_{80} X_{4}(k) + a_{81} X_{4}(k1)
+ a_{82} X_{4}(k2) + a_{90} Z_{1}(k)
+ a_{91} Z_{1}(k1) + a_{92} Z_{1}(k2)
Fig. 1. Patterns:
 explicit, for variable Y_{1} stepwise forecasting;
 explicit, for normative Y_{1(k)} forecasting;
 implicit for normative of Y_{1(k)} Y_{2(k)}
Y_{3(k)} and Y_{4(k)} forecasting;
Full equations of analogical form were completed for output
variables Y_{2(k)}, Y_{3(k)} and Y_{4(k)}
too. Then, using Combinatorial GMDH algorithm, coefficients
for these equations were found. Forecastings were calculated
and their accuracy was evaluated.
An example of normative forecasting
For example, let us consider normative forecasting of Ukraina
macroeconomy on second quarter of 1995 using data presented
in table. Variables having k index will be related to this quarter.
Following values of coefficients were received for the model
forecasting Y_{1(k)} :
a_{00}=0.823; a_{11}= 0; a_{12}2=
0; a_{20}= 0; a_{21}=1.527; a_{22}=1.539;
a_{30}= 0; a_{31}= 0.549; a_{32}=
0; a_{40}= 0 a_{41}=0.120 a_{42}=
0.798
a_{50}= 0 a_{51}= 0 a_{52}= 0.295;
a_{60}= 1.226; a_{61}=0.800; a_{62}=
0;
a_{70}= 1.311; a_{71}= 0; a_{72}=0.581;
a_{80}= 0.860; a_{81}= 0; a_{82}=
2.121;
a_{90}= 0.734; a_{91}=0.467; a_{92}=
2.364
Models accuracy can be characterised by minimal value of RR
criterion. For optimal nonphysical forecasting model of Y_{1(k)}
was RR_{min}=1.59e^{14} (MSE=0.0). For forecast
of variable Y_{2(k)} RR_{min}=7.55e^{14}
(MSE=3.92e^{15}); for forecast of Y_{3(k)}
RR_{min} = 3.63e^{5} (MSE=6.77e^{6});
and for forecast of Y_{4(k)} RR_{min}=0.0026
(MSE=1.05e^{4}). Small minimal value of criterion shows
high accuracy of variable Y_{1(k)} forecasting.
Arguments which have zero coefficient are excluded from model.
This means only that these arguments should be not used for
accurate forecasting. Conclusions about reasonable use of them
for control purposes are not true. For solution of questions
about control the analysis of other, physical model (which us
not considered here) is necessary.
Values of all variables for delayed moments k1 and k2 are
known. Substituting them into equations for variables Y_{1}(k),
Y_{2}(k), Y_{3}(k) and Y_{4}(k) for
explicit patterns we receive four separated calculating equations:
Y_{1(k)} = a_{0} + a_{1} X_{1(K)}
+ a_{2} X_{2(k)} + a_{3} X_{3(k)}
+ a_{4} X_{4(k)} + a_{5} Z_{1(k)}
(1)
Y_{2(k)} = b_{0} + b_{1} X_{1(k)}
+ b_{2} X_{2(k)} + b_{3} X_{3(k)}
+ b_{4} X_{4(k)} + b_{5} Z_{1(k)}
(2)
Y_{3(k)} = c_{0} + c_{1} X_{1(k)}
+ c_{2} X_{2(k)} + c_{3} X_{3(k)}
+ c_{4} X_{4(k)} + c_{5} Z_{1(k)}
(3)
Y_{4(k)} = d_{0} + d_{1} X_{1(k)}
+ d_{2} X_{2(k)} + d_{3} X_{3(k)}
+ d_{4} X_{4(k)} + d_{5} Z_{1(k)}
(4)
For equation (1) we receive:
a_{0}=2.213; a_{1}=0; a_{2}=1.226;
a_{3}=1.311; a_{4}=0.860; a_{5}=0.734
computer algorithm teaches us that to receive accurate normative
forecast of Y_{1(k)} variable is necessary to point
out the values of factors X_{2(k)} X_{3(k)}
and X_{4(k)}. For equation (2) we receive:
b_{0}=0.337; b_{1}=0.564; b_{2}=0;
b_{3}=0; b_{4}=1.542; b_{5}=0
For normative forecasting of Y_{2(k)} variable is necessary
to point out factors X_{1(k)} and X_{4(k)}.
For equation (3) we receive:
c_{0}=0.639; c_{1}=0.088; c_{2}=1.477;
c_{3}=0; c_{4}=0; c_{5}=0
For normative forecasting of Y_{3(k)} variable, factors
X_{1(k)} and X_{2(k) }should be pointed out.
At last, for equation (4) we find:
d_{o}=0.003; d_{1}=0; d_{2}=0; d_{3}=0.756;
d_{4}=0; d_{5}=0
For the most accurate normative forecasting of Y_{4(k)}
variable should be pointed out only one factor X_{3(k)}.
Graphical visualisation of variable Y1(k) normative forecasting
Let us show an example of output variable Y_{1(k)}
normative forecasting visualisation using example considered
above. Let give to variable X_{4(k)} two values:
X_{4(k)} = 0.25 and X_{4(k)} =0.75
The variable Y_{1(k)} change is shown by isolines on
the plane of two input factors X_{2(k)} and X_{3(k)}
(Fig.2a and Fig.2b).
Fig.2. Graphical presentation of variable Y_{1(k)}
forecast.
By comparison of Figures 2a and 2b is possible, particularly,
to conclude that increase of monetary circulation rate leads to
increase of Y_{1(k)}; Most important is that using Figures
2a and 2b is possible to establish quantitative connection between
output variable and their input factors. Similar figures can be
completed for each output variable.