=Paper=
{{Paper
|id=Vol-1614/paper_91
|storemode=property
|title=Forecasting Economic Indices of Agricultural Enterprises Based on Vector Polynomial
Canonical Expansion of Random Sequences
|pdfUrl=https://ceur-ws.org/Vol-1614/paper_91.pdf
|volume=Vol-1614
|authors=Igor P. Atamanyuk,Yuriy P. Kondratenko,Natalia N. Sirenko
|dblpUrl=https://dblp.org/rec/conf/icteri/AtamanyukKS16a
}}
==Forecasting Economic Indices of Agricultural Enterprises Based on Vector Polynomial
Canonical Expansion of Random Sequences==
https://ceur-ws.org/Vol-1614/paper_91.pdf
Forecasting Economic Indices of Agricultural Enterprises
Based on Vector Polynomial Canonical Expansion
of Random Sequences
Igor P. Atamanyuk 1, Yuriy P. Kondratenko 2,3, Natalia N. Sirenko1
1
Mykolaiv National Agrarian University, Commune of Paris str. 9,
54010 Mykolaiv, Ukraine
atamanyuk_igor@mail.ru, sirenko@mdau.mk.ua
2
Petro Mohyla Black Sea State University, 68th Desantnykiv Str. 10,
54003 Mykolaiv, Ukraine
3
Cleveland State University, 2121 Euclid Av.,
44115, Cleveland, Ohio, USA
y.kondratenko@csuohio.edu, y_kondrat2002@yahoo.com
Abstract. Calculating method for forecasting economic indices of agricultural
enterprises on the basis of vector polynomial exponential algorithm of
extrapolation of the realizations of random sequences is worked out. The model
of prognosis allows estimate the results of enterprise functioning (to estimate
future gross profit, gross production) after its reorganization (change of land
resources, manpower resources, fixed assets). Prognostic model does not
impose any restrictions on the forecast random sequence (linearity, stationarity,
Markov behavior, monotonicity, etc.) and thus allows fully take into
consideration stochastic peculiarities of functioning of agricultural enterprises.
The simulation results confirm high efficiency of introduced calculating
method. The scheme reflecting the peculiarities of functioning of the forecast
model are also introduced in the work. The method can be realized in the
decision support systems for agricultural and non-agricultural enterprises with
various sets of economic indices.
Keywords. calculation method, random sequence, canonical decomposition,
forecasting economic indices
Key Terms. computation, mathematical model
1 Introduction
Many decision-making processes in different areas of economy (management of
enterprises, transport logistics, finance forecasting, investment under uncertainty and
so on) are based on the different mathematical models, experienced theoretical
ICTERI 2016, Kyiv, Ukraine, June 21-24, 2016
Copyright © 2016 by the paper authors
- 459 -
methods and modern intelligent algorithms [1-6]. For guaranteeing efficient
performance of an enterprise on the market, it is necessary to form the strategy and
tactics of enterprise development correctly, to ground the plans and management
decisions. To do this is possible only based on effective diagnostics and
prognostication of current and future economical situation at the enterprise. Western
specialists have the priority in the investigation of the possibilities of the management
on the basis of the forecasting of enterprise economic state. Bever (the USA) started
theoretical development and building of prognostic models, then it was continued in
the works of Altman (the USA) [7-8], Alberichi (Italy), Misha (France) and others [9-
10]. More contemporary trend in the building of the algorithms of economic indices
forecasting is the usage of stochastic methods of extrapolation. The relevance of such
approach is explained with the influence of great number of accidental factors on the
results of enterprise functioning (weather conditions, accidental variations of demand
and supply, inflation etc.), under the influence of which the change of economic state
indices obtains accidental character. It is especially important to take into account
stochastic peculiarities of economic indices during the solving of the problems of
prognostication of the state of agricultural enterprises.
But the existing models of prognosis impose considerable limitations on the
accidental sequence describing the change of economic indices [11-16] (Markovian
property, stationarity, monotony, scalarity etc.). Thereupon the problem of the
building of the forecast model under the most general assumptions about the
stochastic properties of the accidental process of the change of the indices of
enterprise economic state arises.
2 Aim and the Raising of Problem
The aim of this work is the development of the efficient and robust method for
forecasting agricultural enterprise indices. The main requirement to the forecasting
method is the absence of any essential limitations on the stochastic properties of the
accidental process of economic indices change.
3 Theoretical Conception of the Proposed Forecasting Method
The most universal method (from the point of view of the requirements to the
investigated accidental sequence) is a method that based on the mechanism of
canonical expansions [17-18]. The main primary indices of the economic state of
agricultural enterprises are the gross profit, gross output, land resources, labour
resources, fixed assets that is why the object of the investigation is the vector
accidental sequence with five dependant constituents (if necessary the number of
figures and their qualitative composition may be changed). Preliminary investigations
(the check of dependence of accidental values on the basis of statistical data about the
work of agricultural enterprises in Nikolaev region) showed that the accidental
sequences describing the change of the economic state of the enterprises which relate
to the intensive [19] type of the development during the interval of eleven years that
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corresponds to the processing of twelve annual indices for the great number of the
enterprises of the mentioned type have the most stable and significant stochastic
relations. For such vector accidental sequence the canonical expansion has the
following look [20]:
i 5
X h i M X h i V( )h( ) i , i 1,12, h 1,5, (1)
1 1
where X 1 i , i 1,12 - gross profit;
X 2 i , i 1,12 - gross output;
X 3 i , i 1,12 - land resources;
X 4 i , i 1,12 - labour resources;
X 5 i , i 1,12 - fixed assets.
The elements of canonical expansion are the accidental coefficients
V( ) , 1,12, 1,5 and nonrandom coordinate functions h( ) i ,
1,12, 1,5 :
1 H 1
V( ) X M X V( j )
( j)
V( j )( j ) , 1,12; (2)
1 j 1 j 1
D M V( ) M X M 2 X
2 2
1 H 1 (3)
D j D j ( j ) , 1,12;
( j) 2 2
1 j 1 j 1
M V ( )
X i M [ X i ] 1
h( ) i ( M X X h i
h h
M V
( ) 2 D
1 H
M X M X h i D j ( )
( j)
( ) h( j ) (i ) (4)
1 j 1
1
D j ( )
( j)
( )h(j ) (i ), 1,5, 1, i.
j 1
Coordinate functions h i , h, 1,5, , i 1,12 have the following properties:
1, h & i;
h i (5)
0, i or h< & =i.
The algorithm of extrapolation on the basis of canonical expansion has the look
[20]:
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M [ X h (i )], 0,
( ,l 1)
mh i mh
( ,l )
i xl ml( ,l 1) h(l) i , l 1, (6)
( ,5)
mh i x1 m1 h(1) i , l 1,
( 1,5)
where mh( ,l ) (i ) M X h i / x , 1,5, 1, 1; x j ( ), j 1, l , h 1,5,
i k ,12 - is the linear optimal quantity by the criterion of the minimum of the average
square of the error of the prognosis is the estimation of the future values of the
investigated sequence under the condition that the values are known
x , 1,5, 1, 1; x j , j 1, l .
Essential deficiency of the forecast model (6) is the assumption of existence of
only linear stochastic relations in the sequence X h i , h 1,5, i 1,12, describing the
process of change of economic indices of agricultural enterprises. The analysis of
statistical data about the work of agricultural enterprises of Nikolaev region showed
that the stochastic relations till the fourth order
o o
M X hl X ms i 0, , i 1,12, l s 4, h, m 1,5 are essential for such a
sequence. Non-linear canonical model of the investigated sequence with taking
account of non-linear relations takes the form [21]:
i 1 5 3 h 1 3
X h (i ) M X h (i ) W(l ) l(h ,1) ( , i ) Wil( ) l(h ,1) (i, i ) Wih(1) , i 1,12. (7)
1 l 1 1 l 1 1
Random coefficients W(l ) , =1,12, l=1,5, =1,3 and nonrandom coordinate
functions l(h , s ) , i , , i =1,12, l, h=1,5, ,s=1,3 are determined with the help of
expressions:
1 5 3
W(l ) X l M X l W(mj ) mj(l , ) , (8)
1 m 1 j 1
l 1 3 1
W(mj ) mj(l , ) , W(l j ) lj(l , ) , , 1,12;
m 1 j 1 j 1
Dl , ( ) M W M X M X
( ) 2 2 2
(9)
l l l
1 5 3 l 1 3
Dmj mj(l , ) , Dmj mj(l , ) ,
2 2
1 m 1 j 1 m 1 j 1
1
Dlj lj(l , ) , , 1,12;
2
j 1
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M W (l ) X hs i M [ X hs i ]
l(h , s ) , i (10)
M W (l )
2
1
( M X l X hs i M X l M X hs i
D l ( )
1 5 3
D mj ( ) mj
(l , )
, mj( h , s ) , i
1 m 1 j 1
l 1 3
D mj ( ) mj
(l , )
, mj( h , s ) , i
m 1 j 1
1
D lj ( ) lj( l , ) , lj( h , s ) , i , 1, h , i 1,12, 1, i.
j 1
Vector algorithm of extrapolation [22-24] for the considered quantity of the
components and order of stochastic relations on the basis of canonical expansion (7)
takes the form:
M X h i , 0;
(11)
( , l 1)
m j,h s, i x j m j, j l , (j ,hl , s ) , i , l 1, j 5;
l ( , l 1)
m (j , h, l ) s , i
( ,3)
m j , h s , i x j 1 m j , j 1 3, j 1,1 , i , l 1, j 5;
( ,1) (h,s )
m 5,( h,3) s , i x1 1 m 5,1( ,3)
3, 1 1,1( h , s ) 1, i , l 1, j 5.
m(j , h,l ) 1, i M X h i / xn , 1,5, n 1,3, 1, 1; xn ( ), 1, j , n 1, l is
optimal by the criterion of minimum of mean-square error of prognosis estimation of
future values of economic index with ordinal number h provided that for the
prognosis values xn , 1,5, n 1,3, 1, 1; xn , 1, j , n 1, l are
used.
Altogether 165 values xh i , h 1,5, i 1,11, =1,3 and 5220 not equal to zero
weight coefficients l(h, s ) , i , , i =1,12, l, h=1,5, ,s=1,3 are used in the algorithm
of prognosis (11).
The expression for mean-square error of extrapolation with the help of algorithm
(11) by known values x nj , =1, k ; j 1,5; n =1,3 is in the form:
5 3
Eh( k ,3) i M X h2 i M 2 X h i D jn ( ) (jnh ,1) , i ,i k 1,12.
k 2
(12)
1 j 1 n 1
This expression is equal to dispersion of a posteriori casual sequence
X i / x , 1,5, n 1,3, 1, 1; x ( ), 1, j, n 1, l .
h
n
n
In Fig. 1 the scheme reflecting the peculiarities of functioning of the forecast
model (11) is represented.
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Fig. 1. Scheme of functioning of the forecast model (11) ( N 3, H 5)
- 464 -
Method of prognostication of future values of economic indices on the basis of the
forecast model (11) presupposes realization of the following stages:
Stage 1. Gathering of statistical data about the results of enterprises functioning;
Stage 2. Estimation of moment functions M X s i , M X lj X hs i on the
basis of cumulated realizations of random sequence describing the process of change
of economic indices;
Stage 3. Calculating of the parameters of the algorithm of extrapolation (11);
Stage 4. Estimation of future values of economic indices on the basis of the
forecast model (11);
Stage 5. Estimation of the quality of the solving of the forecast problem for
investigated sequence with the help of the expression (12).
4 Results of Numerical Experiment
Method is approbated on the basis of statistical data of functioning of agricultural
enterprises in Nikolaev region during the period 2004-2015 (74 enterprises with
gross profit 200-900 thousands grivnas). Moment functions M X hs i ,
M X lj X hs i were estimated by known formulae of mathematical statistics for
sections 2004, 2005, …, 2014. Data about the work of the enterprises for 2015 were
supposed to be unknown and the estimation of moment functions M X hs 12 ,
M X lj X hs 12 for the last section (corresponding to 2015) was carried out on
the basis of determinate models with the use of four previous years (2011-2014) in
tabular processor Microsoft Excel (instrument “Search for solutions”). For example,
o o
in Table 1 the values of autocorrelated function M X 1 X 1 i , 1,12, i=1,12
for the component X 1 i , i=1,12 (gross profit) are represented.
o o
For 2015 values M X h X h 12 , 1,11 are obtained on the basis of
determinate model:
o o
o o
o o
M X 1 X 1 12 0, 718M X 1 X 1 11 -0, 053M X 1 X 1 10 (13)
o o
o o
0, 2128M X 1 X 1 9 0,105M X 1 X 1 8 , 1,11,
Coordinate function 11(1,1) , i , i 1,12 (Table 2) corresponds to correlated
o o
function M X 1 X 1 i , 1,12, i 1,12 .
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Table 1. Autocorrelated function of the component X 1 i , i=1,12
2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015
2004 1 0,99 0,70 0,42 0,79 0,74 0,49 0,72 0,63 0,46 0,55 0,43
2005 0,99 1 0,72 0,42 0,74 0,74 0,52 0,70 0,64 0,48 0,59 0,46
2006 0,70 0,72 1 0,57 0,67 0,58 0,701 0,69 0,70 0,66 0,78 0,60
2007 0,42 0,42 0,57 1 0,38 0,36 0,45 0,21 0,41 0,36 0,19 0,18
2008 0,79 0,74 0,67 0,38 1 0,81 0,55 0,91 0,80 0,72 0,53 0,41
2009 0,74 0,74 0,58 0,36 0,81 1 0,72 0,73 0,92 0,81 0,51 0,44
2010 0,49 0,52 0,70 0,45 0,55 0,72 1 0,51 0,74 0,73 0,49 0,41
2011 0,72 0,70 0,69 0,21 0,91 0,73 0,51 1 0,77 0,80 0,74 0,55
2012 0,63 0,64 0,70 0,41 0,80 0,92 0,74 0,77 1 0,91 0,60 0,59
2013 0,46 0,48 0,66 0,36 0,72 0,81 0,73 0,80 0,91 1 0,71 0,46
2014 0,55 0,59 0,78 0,19 0,53 0,51 0,49 0,74 0,60 0,71 1 0,71
2015 0,43 0,46 0,60 0,18 0,41 0,44 0,41 0,55 0,59 0,46 0,71 1
Table 2. Coordinate function 11(1,1) , i , i 1,12
2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015
2004 1 0,89 0,54 0,55 0,62 0,43 0,45 0,89 0,858 0,90 2,36 2,65
2005 0 1 2,25 -1,46 -2,40 0,27 5,47 -2,71 3,55 1,83 2,85 4,70
2006 0 0 1 5,09 1,17 -1,53 -0,03 -2,77 -0,23 -5,52 2,34 5,07
2007 0 0 0 1 0,17 0,26 0,94 0,18 0,77 1,05 -0,57 -1,17
2008 0 0 0 0 1 0,48 1,27 1,06 1,05 2,01 -2,37 0,69
2009 0 0 0 0 0 1 -1,81 0,74 3,53 0,37 9,31 2,86
2010 0 0 0 0 0 0 1 -0,68 1,44 3,18 -6,74 -3,39
2011 0 0 0 0 0 0 0 1 1,29 2,21 -3,30 0,93
2012 0 0 0 0 0 0 0 0 1 3,88 0,19 -8,44
2013 0 0 0 0 0 0 0 0 0 1 1,99 -4,96
2014 0 0 0 0 0 0 0 0 0 0 1 0,50
2015 0 0 0 0 0 0 0 0 0 0 0 1
In Table 3 weight coefficients 13(1,1) , i , 1,11, i 2,12 determining the
influence of values x13 i , i 1,11 of gross profit in high-order third degree on future
values of this parameter are represented.
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Table 3. Values of coordinate function 13(1,1) , i , 1,11, i 2,12
2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015
-6
2004 9*10 -1*10 -3*10 -5*10 -3*10 -2*10 -8*10 -3*10 -1*10 -3*10 -4*10-7
-7 -7 -7 -7 -7 -6 -6 -6 -7
2005 0 -1*10-4 -1*10-4 1*10-5 -3*10-4 3,5*10-5 7*10-5 10-4 6*10-5 -2*10-4 8*10-5
2006 0 0 -1*10-5 2*10-6 -3*10-6 -9*10-7 2*10-8 6*10-6 -6*10-6 -9*10-6 -7*10-7
2007 0 0 0 2*10-8 -9*10-8 2*10-9 3*10-8 7*10-8 8*10-9 4*10-9 3*10-7
2008 0 0 0 0 -8*10-8 2*10-8 2*10-8 7*10-7 3*10-7 4*10-7 3*10-8
2009 0 0 0 0 0 -8*10-7 -6*10-7 -7*10-7 -4*10-8 -3*10-7 -9*10-8
2010 0 0 0 0 0 0 8*10-9 8*10-6 5*10-7 -10-8 -4*10-7
2011 0 0 0 0 0 0 0 2*10-5 8*10-7 -2*10-6 8*10-6
2012 0 0 0 0 0 0 0 0 2*10-8 4*10-7 -3*10-6
2013 0 0 0 0 0 0 0 0 0 3*10-8 -7*10-6
2014 0 0 0 0 0 0 0 0 0 0 9*10-6
As it can be seen in Table 3 values 11(3) i , i 1,11 are relatively small but this
doesn’t mean that given weight coefficients don’t influence on the forming of the
estimation of future value as 11(3) i , i 1,11 are multiplied in the process of
calculations by values x13 i , i 1,11 (values of the sixth-seventh order).
For functioning of the forecast model (11) on the basis of statistical data 25 tables
of weight coefficients analogous to Tables 2-3 were calculated.
During the application of the method of economic indices prognostication for 2016
optimal order of non-linear relations of the investigated random sequence is unknown.
But taking into consideration that N=4 is invariable during 11 years there is quite high
probability that given parameter will remain on the same level.
Values in Table 4 reflects the change of relative error of prognostication of gross
profit of enterprise (component X 1 i , i 1,12 ) during 2015 depending on the order
of stochastic relations used in model (11).
Table 4. Relative error of prognostication of gross profit
Order of stochastic relations 2 3 4
Relative error 6,9 % 3,3 % 1,5 %
Thus the results of the experiment showed (Table 4) that application of nonlinear
relations in the forecast model allows increase considerably the quality of economic
indices prognostication.
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5 Conclusion
Calculating method of the estimation of future values of economic indices of
agricultural enterprises functioning is obtained in the work. The algorithm of
extrapolation of vector random sequence based on nonlinear polynomial canonical
expansion is assumed as a basis of the method. The optimal algorithm of the
extrapolation of the economic indices of agricultural enterprises, which as well as
canonical expansion put into its base doesn’t impose any essential limitations on the
stochastic properties of economic indices. In addition to pre-aggregate indicators
(gross output, land resources, manpower, plant and equipment) a range of parameters
can be used as the components of investigated random sequence (weather conditions,
prices of resources, etc) influencing the effectiveness of the functioning of
agricultural enterprises. The results of the numerical experiment showed that the
forecast model possesses high accuracy characteristics at the expense of maximal
taking into consideration of stochastic qualities of random sequence of economic
indices change. Schemes of calculation of the parameters of the forecast model and
estimations of future values of economic indices on its basis are introduced in the
work. Expression for the mean-square error of extrapolation allows to estimate the
quality of the forecast problem solving.
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