At early stages of mathematical modeling scientists believed that
for solution of interpolation problems probabilistic concept should
be used [40]: optimal value of output variable should correspond
to maximum of a posteriori probability calculated by Bayes formula
or to the rules of Walds' statistical decision theory. Soon it was
established that we have not in our disposition initial data samplings
sufficient for reliable calculation of empirical probabilities necessary
for these calculations. Now we can say, that it was historically
strategic mistake to refuse of probabilistic algorithms usage because
except of usually used ordinary probabilities can be calculated
pair probabilities which demands a very small number of data sampling
lines for their calculation. Decision about optimal value of output
variable should correspond to maximum of sum of ordinary and pairs
empirical probabilities. To the number of ordinary random events
E(y,xi) should be added the number of pairs random events E(y,xi,xj).
The requirements to input data sample are decreased significantly:
it is enough to have one observation for each pattern or discrete
level of output variable. Algorithm is recommended for recognition
of binary objects and for the veritability of input data control
to avoid the possible experts errors in it.
Example: Recognition and forecast of situation with spring
wheat crop.
The problem of wheat crop recognition and forecasting for field
with constant cultivation order is solved. In series of years only
fertilizations are changed. In table: X1  length of bright day;
X2  contents of nitrogen in fertilization; X3  contents of sodium;
X4  contents of potassium.
Years
(marked as
letters) 
Factors

Class of crop 
x_{1}

x_{2}

x_{3}

x_{4}

A 
1

1

1

1

R1 ("Average") 
F 
1

2

1

2

L 
1

2

2

3

D 
1

3

4

3

K 
1

3

5

3

R2 ("Good") 
B 
2

3

5

4

G 
1

5

5

2

H 
1

5

5

2

I 
5

4

4

2

R3 ("Excellent") 
J 
3

5

4

3

E 
5

5

3

2

C 
5

5

3

3

New 
5

4

3

2

? 
Using this data sampling we calculate single and pair empirical
probabilities. Sum of single empirical random events, proportional
to ordinary empirical probabilities, are shown in table 2.
Table 2. Number of single probability events calculation.
New> 
5

4

3

2


R1 
0

1

3

4

Sum = 8 
R2 
6

1

3

3

Sum = 13 
R3 
6

3

4

2

Sum = 15 <max 
Sum of pairs of empirical random events are shown in table 3. There
are shown all possible pairs of variables: their discrete values
(called discreets) and sums of random events pairs.
Table 3. Sums of pair random events to reach possible discreets.
x_{i}  x_{j} 
54

53

52

43

42

32

Sum 
R1 
0

0

0

3

4

12

19 
R2 
6

18

18

3

3

9

51 
R3 
18

24

12

12

6

8

80 <max 
Criterion for choice of next crop estimation:
where: P_{single} = E_{single} / N; P_{pair}
= E_{pair} / N; E_{pair}(ij) = E_{single}(i)
* E_{single}(j)
N  number of possible events.
For output vector X=(5,4,3,2) we found that crop will be "Excellent"
(Sum=80).
Taking prehistory of process into account
Important moment is that discretization and normalizing according
to Widrow rule (the same number of points for each level) can be
applied to initial data. Forecast should be calculated for all levels
of output variable R_{i} and the most probable levels should
be chosen out as the most probable forecast for each given output
vector of factors. To get high accuracy of forecast there is necessary
to take into account random events calculated for each delay time
separately. For example for Dollar rate forecast it is necessary
to take into account several delayed values of Dollar rate. Criterion
in following form would be calculated:
Lead time of forecast can be estimated approximately by formulae:
T = h / RR
where: h  time step given in data sample, RR  error variance criterion
(when RR = 0.01 lead time is about T_{L} = 100 time steps).
When we have up to 100 factors in the sampling Combinatorial algorithm
is recommended. When number of factors is equal from 100 to about
500 lines the multilayered GMDH algorithms can be used. In case
when number of observations is big the algorithm of data sampling
clusterization should be used, to transform initial data sampling
into sampling of centers of clusters of physical clusterization
coordinates.
Genetic types of algorithms, with crossover recombination of "mother"
and "father" partial descriptions and random mutation
of new modelscandidates are recommended in cases when data sampling
contains more than 500 variables. If we say that it is enough to
have one line of data for each level of output variable in this
algorithm it means according to genetic approach that we use algorithm
of fern plants, having only "mother."
