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American Review of Mathematics and Statistics June 2018, Vol. 6, No. 1, pp. 35-60
Published by American Research Institute for Policy Development DOI: 10.15640/arms.v6n1a3
URL: https://doi.org/10.15640/arms.v6n1a3
Three-Component Piecewise-Linear Economic-Mathematical Mode and Method of Multivariate Prediction of Economic Process with Regard to Unaccounted Factors
Influence in 3-Dimensional Vector Space
Azad Gabil oglu Aliyev1
Abstract
For the last 15 years in periodic literature there has appeared a series of scientific publications that has laid the foundation of a new scientific direction on creation of piecewise-linear economic-mathematical models at uncertainty conditions in finite dimensional vector space. Representation of economic processes in finite-dimensional vector space, in particular in Euclidean space, at uncertainty conditions in the form of mathematical models in connected with complexity of complete account of such important issues as: spatial in homogeneity of occurring economic processes, incomplete macro, micro and social-political information; time changeability of multifactor economic indices, their duration and their change rate. The above-listed one in mathematical plan reduces the solution of the given problem to creation of very complicated economic-mathematical models of nonlinear type. In this connection, it was established in these works that all possible economic processes considered with regard to uncertainty factor in finite-dimensional vector space should be explicitly determined in spatial-time aspect. Owing only to the stated principle of spatial-time certainty of economic process at uncertainty conditions in finite dimensional vector space it is possible to reveal systematically the dynamics and structure of the occurring process. In addition, imposing a series of softened additional conditions on the occurring economic process, it is possible to classify it in finite-dimensional vector space and also to suggest a new science-based method of multivariate prediction of economic process and its control in finite-dimensional vector space at uncertainty conditions, in particular, with regard to unaccounted factors influence.
Keywords: Finite-dimensional vector space; Unaccounted factors; Unaccounted parameters influence function; Principle of certainty of economic process in fine-dimensional space; Multi alternative forecasting; Principle of spatial-time certainty of economic process at uncertainty conditions in fine-dimensional space; Piecewise-linear economic-mathematical models in view of the factor of uncertainty in finite-dimensional vector space; Piecewise-linear vector-function; 3-Dimensional Vector Space; 3-Component Piecewise-Linear Economic-Mathematical Model in 3-Dimensional Vector Space; Hyperbolic surface.
I. Introduction. Formulation of the problem
In publications [1-5,12] theory of construction of piecewise-linear economic mathematical models with regard to unaccounted factors influence in finite-dimensional vector space was developed. A method for predicting economic process and controlling it at uncertainty conditions, and a way for defining the economic process control function in m-dimensional vector space, were suggested. In addition to this we should note that no availability of precise definition of the notion ―uncertainty‖ in economic processes, incomplete classification of display of this phenomenon, and also no availability of its precise and clear mathematical representation places the finding of the solution of problems of prediction of economic process and this control to the higher level by its complexity.
1 Doctor of Economical sciences (PhD), Assistant Professor, Tel.: (99470) 316-32-59 (mob), E-mail: [email protected], Azerbaijan State Oil and Industry University (ASOIU), www.asoiu.edu.az
36 American Review of Mathematics and Statistics, Vol. 6, No. 1, June 2018
Many-dimensionality and spatial in homogeneity of the occurring economic process, time changeability of multifactor economic indices and also their change velocity give additional complexity and uncertainty. Another complexity of the problem is connected with reliable construction of such a predicting vector equation in the
consequent small volume ),...,(211 mn
xxxV
of finite-dimensional vector-space that could sufficiently reflect the
state of economic process in the subsequent step. In other words, now by means of the given statistical points (vectors)
describing certain economic process in the preceding volume ),...,(∑1
21
N
N
mNxxxVV
of finite-dimensional vector
space m
R one can construct a predicting vector equation ),...,(211 mn
xxxZ
in the subsequent small volume
),...,(211 mn
xxxV
of finite-dimensional vector space. The goal of our investigation is to formulate the notion of
uncertainty for one class of economical processes and also to find mathematical representation of the predicting
function ),...,(211 mn
xxxZ
for the given class of processes depending on so-called unaccounted factors functions. In
connection with what has been said, below we suggest a method for constructing a predicting vector equation
),...,(211 mn
xxxZ
in the subsequent small volume ),...,(
211 mnxxxV
of finite-dimensional vector space [1-7, 14].
II. Materials and methods:
In these publications, the postulate spatial-time certainty of economic process at uncertainty conditions in finite-dimensional vector space‖ was suggested, the notion of piecewise-linear homogeneity of the occurring economic process at uncertainty conditions was introduced, and also a so called the unaccounted parameters influence function
),(n1,-n
k
nn
n influencing on the preceding volume ∑1
N
N
nVV
of economic process was suggested.
∑21
2
111)()(11
n-
i=
i,i-
k
iin,n-nnnα,λω+α,λ+ ω+Az=z i
(1)
Here
ii
k
iii
k
iiсosii
,1-,1-),(
iik
k
k
ii
k
i-i
k
ii
k
k
iсos
azaa
azz
zzz
zazz
i
i-ii
,1-
1112
111
1-1-1
111-1-
11 --
)-(
)-(
--
- 1
1
1-
11-
1
(2)
2
1-1
1-12-i
11
)(
))(()μ-μ(μ
1-
1-i2-i
1
i
i-
k
ii
k
ii
k
ik
i-i-i
z-a
z-az-a=
, for 1-i
1-i≥1-i
k
(3)
nnnnnnn
сos,1-,1-
),(
nnk
k
k
nn
k
n-n
k
nn
k
nсos
azaa
azz
zzz
zazz
n
n-n
,1-
1112
111
1-1-1
111-1-
11 --
)-(
)-(
--
- 1
1
1-
11-
1
(3.1)
)-(
--
)-(
111
1112
111
1
1
azz
azaa
Ak
k
k
(4)
2
1-1
1-12-n
11
)(
))(()μ-μ(μ
1-
1-n2-n
1
n
n-
k
nn
k
nn
k
nk
n-n-n
z-a
z-az-a=
, 1-n
1-n≥1-n
k
(5)
Azad Gabil oglu Aliyev 37
On this basis, it was suggested the dependence of the n-th piecewise-linear function n
z
on the first
piecewise-linear function 1
z
and all spatial type unaccounted parameters influence function ),(n1,-n
k
nn
n
influencing on the preceding interval of economic process, in the form Eqs. (1)–(5):
∑21
2
111)()(11
n-
i=
i,i-
k
iin,n-nnnα,λω+α,λ+ ω+Az=z i
(6)
where
ii
k
iii
k
iiсosii
,1-,1-),(
iik
k
k
ii
k
i-i
k
ii
k
k
iсos
azaa
azz
zzz
zazz
i
i-ii
,1-
1112
111
1-1-1
111-1-
11 --
)-(
)-(
--
- 1
1
1-
11-
1
(7)
are unaccounted parameters influence functions influencing on the preceding 1
V , 2
V ,..i
V small volumes of
economic process;
2
11
112
1-1-
)z-(
)z-)(z-()-(
1
12
1-
i-
i-i-
i
k
i-i
k
i-i
k
i-ik
iii
a
aa
, for 1
11μμ i-
k
i-≥i- (8)
are arbitrary parameters referred to the i-th piecewise-linear straight line. And the parameters i
are connected
with the parameter 1-i
referred to the (i-1)-th piecewise-linear straight line, in the form Eq. (8);
)-(
--
)-(
111
1112
111
1
1
azz
azaa
Ak
k
k
(9)
is a constant quantity;
nnnnnnn
сos,1-,1-
),(
nnk
k
k
nn
k
n-n
k
nn
k
nсos
azaa
azz
zzz
zazz
n
n-n
,1-
1112
111
1-1-1
111-1-
11 --
)-(
)-(
--
- 1
1
1-
11-
1
(10)
is the expression of the unaccounted parameters influence function that influences on subsequent small volume
NV of finite-dimensional vector space. And the parameter
n referred to the n- piecewise-linear straight line is of
the form:
2
1-1
1-12-n
11
)(
))(()μ-μ(μ
1-
1-n2-n
1
n
n-
k
nn
k
nn
k
nk
n-n-n
z-a
z-az-a=
, 1-n
1-n≥1-n
k (11)
Here the parameter n
is connected with the parameter 1n
of the preceding (n-1)-th piecewise-linear vector
equation of the straightline in the form Eq. (11). Thus, in finite-dimensional vector space, the system of statistical points (vectors) is represented in the vector form
in the form of N piecewise-linear straight lines depending on the vector function of the first piecewise-linear straight-line
21111aaz
, and also on the unaccounted parameters influence function ),(n1,-nnn
in all the investigated
preceding volume of finite-dimensional vector space m
R .After that, in publications [6-11,13-15] a solution was found
of solve a problem on prediction of economic process and its control at uncertainty conditions in finite-dimensional vector space.
38 American Review of Mathematics and Statistics, Vol. 6, No. 1, June 2018
It became clear, that the unaccounted parameters influence functions ),(n1,-nnn
are integral characteristics
of influencing external factors occurring in environment that are not a priori situated in functional chain of sequence of the structured model but render very strong functional influence both on the function and on the results of prediction quantities Eq. (6). It is impossible to fix such a cause by statistical means. This means that the investigated this or other economic process in finite dimensional vector space directly or obliquely is connected with many dimensionality and spatial inhomogenlity of the occurring economic process, with time changeability of multifactor economic indices, vector and their change velocity. This in its turn is connected with the fact that the used statistical data of economic process in finite-dimensional vector space are of inhomogeneous in coordinates and time unstationary events character.
We assume the given unaccounted factors functions ),(n1,-nnn
hold on all the preceding interval of
finite-dimensional vector space, the uncertainty character of this class of economic process. In such a statement, the
problem on prediction of economic event on the subsequent small volume 1
N
V of finite-dimensional vector space
will be directly connected in the first turn with the enumerated invisible external facts fixed on the earlier stages and their
combinations, i.e., the functions ),(n1,-nnn
that earlier hold in the preceding small volumes N
VVV .....,,21
of finite-dimensional vector space. Therefore, by studying the problem on prediction of economic process on
subsequent small volume 1
N
V it is necessary to be ready to possible influence of such factors.
In connection with such a statement of the problem, let‘s investigate behavior of economic process in subsequent
small volume 1
N
V finite-dimensional vector space under the action of the desired unaccounted parameters function
),(n1,-nnn
that was earlier fixed by us in preceding small volumes n
V of finite-dimensional vector space, i.e.,
),(21,22
, ),(32,33
, …., ),(1,-N NNN
.
In connection with what has been said, the problem on prediction of economic process and its control in
finite-dimensional vector space may be solved by means of the introduced unaccounted parameters influence function
),(,1- nnnn
in the following way.
Construct the (N+1)-th vector equation of piecewise-linear straight line )(μ21N1
N
N
N
N
k
N
k
Nza+z=z
depending on the vector equation of the first piecewise-linear straight line 1
z
and the desired unaccounted parameter
influence function ),(1,-
that we have seen in one of the preceding small volumes N
VVV .....,,21
of
finite-dimensional vector space. For that in Eqs. (6)–(11) we change the index n by )1( N and get:
)()(11
111
2
111 ∑ NN,NN
N
i=
i,i-
k
iiNα,λωα,λω++Az=z i
(12)
Here
ii
k
iii
k
iiсosii
,1-,1-),(
iik
k
k
ii
k
i-i
k
ii
k
k
сos
azaa
azz
zzz
zazz
i
i-ii
i
,1-
1112
111
1-1-1
111-1-
11 --
)-(
)-(
--
- 1
1
1-
11-
1
(13)
2
1-1
1-12
11
)(
))(()μ-μ(μ
1-
1-i2
1
i
i-
i-
k
ii
k
ii
k
i-ik
i-i-i
z-a
z-az-a=
, 1
11
i-k
i-≥i- (14)
Azad Gabil oglu Aliyev 39
)-(
--
)-(
111
1112
111
1
1
azz
azaa
Ak
k
k
(15)
1,11,11
),(NNNNNNN
сos
1,N
1112
111
N1
2N
11 --
)-(
)-(
--
1
1
N1
1
Nk
k
k
N
k
NN
k
N
kсos
azaa
azz
zzz
zazz NN
N
(16)
2
2
211
1
)(
))(()μ-μ(μ
1
N
NN-
N
k
NN
k
NN
k
N-Nk
NNN
z-a
z-az-a=
, N
k
NN ≥ (17)
For the behavior of economic process on the subsequent small volume 1
N
V of finite-dimensional vector
space to be as in one of the desired preceding ones in small volume β
VΔ it is necessary that the vector equations of
piecewise-linear straight lines 1N
z
and β
z
to be situated in one of the planes of these vectors and to be parallel to one
another, i.e.
β1NzСz
(18)
In connection with what has been said, to 1
N
V finite-dimensional space there should be chosen such a
vector-point 2N
a
that the piecewise-linear straight lines )(21
Nk
NNNzaz
and )za(z1β
k
1β1ββ
could be
situated in the same plane of these vectors and at the same time be parallel to each other (Fig. 1).
Fig.1. The scheme of construction of prediction function of economic process )(1
NZ
at
uncertainty conditions in finite-dimensional vector spacem
R . Prediction function )(1
NZ
will lie in the same plane
with one of the desired preceding -th piecewise-linear straight line and will be parallel to it.
40 American Review of Mathematics and Statistics, Vol. 6, No. 1, June 2018
In other words, they should satisfy the following parallelism condition:
)(2
Nk
NNza
= С )za(1β
k
1β1β
(19)
Here
m
M
m
mNNiaa
1
,22,
M
1m
mm1,β1βiaa
,
M
m
m
k
mN
k
Nizz NN
1
,
,
M
1m
m
k
m1,β
k
1βizz
1β1β
Excluding in Eq. (19) the parameter С, we get:
1βk
1,1β1,1β
1,1,2
za
Nk
NNza
=1β
k
1,2β1,2β
2,2,2
za
Nk
NNza
=.….=1β
N
k
M1,βM1,β
k
MN,M2,N
za
za
(20)
It is easy to define from system Eq. (20) the coefficients of the vector 2N
a
:
)(
za
zaza
1,1,2k
1,1β1,1β
k
1,2β1,2βk
N,22,2N1β
1β
N Nk
NNza
)z(aza
zaza N
1β
Nk
N,12,1N
1,1β1,1β
k
1,3β1,3βk
N,32,3N
—————————————————–
)z(a
za
zaza N
1β
1β
Nk
1MN,1M2,Nk
1M1,β1M1,β
k
M1,βM1,βk
MN,M2,N
(21)
In this case, the vector 2N
a
will have the following final form:
2Na
=MMNNNN
iaiaiaia
,233,222,211,2......
(22)
As the coordinates of the point (of the vector) 2N
a
now are determined by means of the piecewise-linear vector
1βk
1β1ββza
z taken from one of the preceding stage of economic process, it is appropriate to denote them in the
form )(2
Na
[8-13]. This will show that the coordinates of the point 2N
a
(3) were determined by means of
piecewise-linear straight line β
z
. In this case it is appropriate to represent Eq. (22) in the following compact form:
M
m
mmNNiaa
1
,22)()(
(23)
Azad Gabil oglu Aliyev 41
Now, in the system of Eqs. (12)–(17), instead of the vector 1N
a
we substitute the value of the vector
)(2
Na
, and also instead of ),(1,11 NNNN
introduce the denotation of the so-called predicting influence
function with regard to unaccounted parameters ),(1,11
NNNN
. In this case the prediction function of the
economic process )(1
NZ
with regard to influence of prediction function of unaccounted parameters
)(111
NN,NN
α,λ will take the following form:
)()(11)(
111
2
111 ∑ NN,NN
N
i=
i,i-
k
iiNα,λα,λω++Az=Z i
(24)
Here
ii
k
iii
k
iiсosii
,1-,1-),(
iik
k
k
ii
k
i-i
k
ii
k
k
сos
azaa
azz
zzz
zazz
i
i-ii
i
,1-
1112
111
1-1-1
111-1-
11 --
)-(
)-(
--
- 1
1
1-
11-
1
(25)
2
1-1
1-12
11
)(
))(()μ-μ(μ
1-
1-i2
1
i
i-
i-
k
ii
k
ii
k
i-ik
i-i-i
z-a
z-az-a=
, 1
11
i-k
i-≥i- (26)
)-(
--
)-(
111
1112
111
1
1
azz
azaa
Ak
k
k
(27)
and the prediction function of influence of unaccounted parameters ),(1,11
NNNN
will take the form:
1,11,11),(
NNNNNNNсos (28)
1N
1112
111
N1
2N
11 --
)-(
)-(
-)(-
1
1
N1
1
azaa
azz
zzz
zazz
k
k
k
N
k
NN
k
N
k
NN
N
(29)
2
2
211
1
))((
))()(()μ-μ(μ
1
N
NN-
N
k
NN
k
NN
k
N-Nk
NNN
z-a
z-az-a=
, N
k
NN ≥ (30)
Here the vector )(2
Na
is determined by Eq. (23).
Note the following points. It is seen from Eq. (11) that for Nk
NN the value of the parameter 0
1
N . By
this fact from Eq. (28) it will follow that the value of the predicting function of influence of unaccounted parameters
),(1,11
NNNN
will equal:
),(1,11
NNNN
=0 for 01
N
),(1,11 NNNN
0 for 01
N (31)
42 American Review of Mathematics and Statistics, Vol. 6, No. 1, June 2018
This will mean that the initial point from which the (N+1)-th vector equation of the prediction function of
economic process )(1
NZ
will enanimate, will coincide with the final point of the n-th vector equation of
piecewise-linear straight line N
z
and equal:
∑2
111)(11
N
i=
i,i-
k
iiNα,λω+Az=Z i
, for 0
1
N (32)
For any other values of the parameter 01
N the points of the )1( N -th vector equation will be
determined by Eq. (24).
It is seen from Eq. (28) that 01
N and 0);0(
1,11
NNNN will follow 0
1,
NNсos and
01
N . This will correspond to the case when the influence of external unaccounted factors on subsequent small
volume 1
N
V are as in the preceding small volume N
V of finite-dimensional vector space. In this case it suffices to
continue the preceding vector equation N
z
to the desired point Nk
NNN
1
*
1 of subsequent small volume of
finite-dimensional vector space. The value of the vector function ),;()(,1
*
1
*
11 NNNNNNNzZ
at the point
1
*
1
NN will be one of the desired prediction values of economic process in subsequent small volume
1
NV . In
this case, the value of the controlled parameter of unaccounted factors will be equal to zero, i.e.,
0)0;0сos;0;0(1,1NN,111
NNNNN
For any other value of the parameter 1N
, taken on the interval 1
*
10
NN and сos 0
1,
NN , the
corresponding prediction function of unaccounted parameters will differ from zero, i.e., 0),(1,11
NNNN
.
Thus, choosing by desire the numerical values of unaccounted parameters function
)α,λ;(μωβ1,ββ1Nβ
)α,(λΩ1NN,
*
1N1N corresponding to preceding small volumes
NVVV .....,,
21 and influencing by them beginning
with the point 01
N to the desired point
1
*
N , we get numerical values of predicting economic event
),;(1,
*
1
*
11 NNNNNZ
on subsequent step of the small volume 1
N
V (Fig. 2).
Fig.2. The graph of prediction of process and its control at uncertainty conditions in finite-dimensional vector
space. It is represented in the form of hypersonic surface whose points, of directory will form the line of economic process prediction.
Azad Gabil oglu Aliyev 43
Taking into account the fact that by desire we can choose the predicting influence function of unaccounted
parameters ),;(1,1
*
1
*
1
NNNNN , then this function will represent a predicting control function of unaccounted
factors, and its appropriate function ),;(1,1
*
1,
*
1
NNNNNNZ
will be a control aim function of economic event in
finite-dimensional vector space. Speaking about unaccounted parameters prediction function
),;(1,111
NNNNN
we should understand their preliminarily calculated values in previous small volumes
NVVV .....,,
21 of finite-dimensional vector space. Therefore, in Eq. (24) we used calculated ready values of the
function ),;(1,111
NNNNN
.
Thus, influencing by the unaccounted parameters influence functions of the form ),;(1,111
NNNNN
or by their combinations from the end of the vector equation of piecewise-linear straight line ),;(1,- NN
k
N
k
N
k
N
NNNz
situated on the boundary of the small volume )(1
NZ
),;(
1,111
NNNNNZ
there will originate the vectors
NV and
1
NV , lying on the subsequent small volume
1
NV .
These vectors will represent the generators of hyperbolic surface of finite-dimensional vector space. The values of
this series vector-functions for small values of the parameter 1
*
1
NN , i.e., ),;(
1,1
*
11 NNNNNZ
will
represent the points directrix of hypersonic surface of finite-dimensional vector space (Fig. 2). The series of the values of the points of directory hypersonic surface will create a domain of change of predictable values of the function of
),;(1,1
*
1
*
1
NNNNNZ
in the further step in the small volume1
N
V .
This predictable function will have minimum and maximum of its values min1,1
*
1
*
1)],;([
NNNNNZ
and
х1,1
*
1
*
1)],;([
maNNNNNZ
.
Thus, the found domain of change of predictable function of economic process in the form
),;(1,1
*
11
NNNNNZ
, or in other words, the points of directrix of hyperbolic surface will represent the domain of
economic process control in finite-dimensional vector-space.
III. 3-Component Piecewise-Linear Economic-Mathematical Model and Method of Multivariate Prediction of Economic Process With Regard to Unaccounted Factors Influence in 3-Dimensional Vector Space
In this article we give a number of practically important piecewise-linear economic-mathematical models with regard to unaccounted parameters influence factor in their-dimensional vector space. And by means of three-component piecewise-linear models suggest an appropriate method of multivariant prediction of economic process in subsequent stages and its control then at uncertainty conditions in 3-dimensional vector space [5-16]. In this section, we have given numerical construction of three-component piecewise-linear economic model with regard to unaccounted factors influence in 3-dimensional vector space, and construct appropriate vector-functions on the
subsequent 4
V small volume of 3-dimensional space.
Given a statistical table describing some economic process in the form of the points (vector) set }{n
a
of
3-dimensional vector space 3
R . Here the numbers ni
a are the coordinates of the vector n
a
(an1, an2, an3, …… ani). With
the help of the points (vectors) n
a
represent the set of statistical points in the vector form in the form of 3-component
piecewise-linear function [5-16]:
)-(12111
aaaz
(33)
)-(23222
aaaz
(34)
)-(34333
aaaz
(35)
44 American Review of Mathematics and Statistics, Vol. 6, No. 1, June 2018
Here the vectors ),,(13121111
zzzzz
, ),,(23222122
zzzzz
and ),,(33323133
zzzzz
with the
coordinates ij
z are given in the form of linear vector functions for the first, second and third piecewise-linear straight
lines in 3-dimensional vector-space; the vectors (points) ),,(1312111
aaaa
, ),,(23222122
aaaaa
,
),,(33323133
aaaaa
and ),,(43424144
aaaaa
are the given of 3-dimensional vector
space 3
R ; 0≥1
, 0≥2
and 0≥3
are arbitrary parameters of the first, second and third piecewise-linear
straight lines. It holds the equality 111 , 1
22 and 1
33 ;
2,1 and
3,2 are the adjacent
angles between the first and second and also between the second and third piecewise-linear straight lines; 1
k and 2
k
are the intersection points between the piecewise-linear straight lines (Fig. 3).
Fig. 3. Compact form of representation of numerical expression of the predicting vector function )(3Z
constructed
on the base of 2-component model n 3-dimensional vector space 3
R .
As the intersection point between the second and third piecewise-linear straight lines in 3-dimensional
vector space may also not coincide with the point 3
a
, we denote this intersection point by 2
2
kz
(Fig. 3). With regard to
this factor, according to Eq. (1-11), an equation for the third piecewise-linear straight line is written in the form:
)-( 22
24323
kkzazz
(36)
Here 2
2
kz
is the value of the point (vector) of the second piecewise-linear straight line at the 2
k -th
intersection point, represented by Eq. (1-11), and calculated for the value of the parameter 2
11
k , i.e., at the second
intersection point 2
k and equal:
)]},(1)((12,122112
2222 kkkk
Azz
(37)
where the parameter 2
1
k is calculated:
))((
)(
1213
2
13
2111
1
212
aaza
za
k
k
kkk
(38)
and the vector 2
1
kz
is calculated by means of Eq. (33) at the point 2
11
k in the form:
Azad Gabil oglu Aliyev 45
)-(12111
22 aaazkk
(39)
and the coefficients )( 2
1
kA and ),(
2,122
2 k
are calculated by Eqs. (1-11) at the point 2
11
k in the form:
)-)((
--
)-()(
1111
1112
11112
1
212
azz
azaa
Akk
k
kkk
(40)
1112
1111
11111
13111
11
2
2
--
)-)((
)-)()((
--)(
- 1
12
122
112
21
2
2
azaa
azz
zzz
zazz
k
kk
kkk
kkk
kk
k
k
(41)
112
112
13111
13111
2,1
)(
))()((
kkk
kkk
zazz
zazzсos
(42)
2,122,122
22 ),( сoskk
(43)
Here the value of the parameter at the second intersection point 2k
2μ corresponding to the final point of the
second piecewise-linear straight line is connected with the appropriate value of the parameter 2k
1μ acting on the first
straight line in the form Eq. (38):
2
13
1213
1
k
1
k
2
) (
))( ()μ-μ(μ
1
1
122
k
k
k
z-a
a-az-a=
(44)
Thus, giving the values of the parameter 1k
1μ and 2
k
2μ at the intersection point
1k and
2k by Eq. (42) or Eq.
(44), it is easy to define the appropriate value of the parameter 2k
1μ . Using [10,11,13(Capter 2),14,16] in 3-dimensional
vector space write an equation for the points of the third piecewise-linear straight line depending on the vector equation
of the first piecewise-linear straight line, spatial form of unaccounted parameters 2
Here the unaccounted parameter influence function ),(2,122
2 k
is calculated by means of Eq. (11), the
unaccounted parameter influence function ),(3,233
is calculated in the form [10,11,13(Capter 2),14,16]:
3,233,233),( сos for ,0
3 2
22
k
1112
111
221
2422
11
3
3
--
)-(
)-(
--
- 1
1
2
22
1
azaa
azz
zzz
zazz
k
k
k
kk
k
for 2
22
k , 0
3 (46)
22
22
24222
24222
3,2
--)(
)-)(-)((
kk
kk
zazz
zazzсos
, for 2
22
k (47)
46 American Review of Mathematics and Statistics, Vol. 6, No. 1, June 2018
2
24
2413
223
)-(
)-)(-()-(
2
21
2
k
kk
k
za
zaza
for 2
22
k (48)
2
13
1213
112
)-(
)-)(-()-(
1
1
1
k
k
k
za
aaza
for 1
12≥
k (49)
2
13
1213
112
)-(
)-)(-()-(
1
1
122
k
k
kkk
za
aaza
for 12
11≥
kk (50)
)-(
--
)-(
111
1112
111
1
1
azz
azaa
Ak
k
k
(51)
Thus, giving the vectors (point) ,,,,4321
aaaa
1
1
kz
, 2
2
kz
and )(22
z
Eq. (45) will represent a vector
equation for the third piecewise-linear straight line ),(3133
zz
in 3-dimensional vector space depending on the
parameter 2
11≥
k (i.е., for 2
22≥
k ) and unaccounted parameters influence functions ),(
2,122
2 k
and
),(3,233
.Note that Eq. (45) defines all the points of the third piecewise-linear straight line in
3-dimensional space. To the case 03 there will correspond the value of the initial point of the third straight line
that will be expressed by the vector-function of the first piecewise-linear straight line 1
z
, by the value of the parameters
of intersection points of piecewise-linear straight lines 1
1
k and 2
2
k , and also
2,1Cos generated between the first
and second piecewise-linear straight lines. It will equal:
1)(
)(1
111
1112
111031
1
1
3 azz
azaa
zzk
k
k
2,1
1112
111
111
1311
11
2
1
1
1
11
1
2 )(
)(
сos
azaa
azz
zzz
zazz
k
k
k
kk
k
k
(52)
Write the coordinate form of vector equation Eq. (45). Therefore, we have to take into account that in
3-dimensional vector space ∑3
1
33
m
mmizz
and ∑3
1
11
m
mmizz
. In this case, the coordinates of the vector 3
z
i.e.,
mz
3, will be expressed by the coordinates of the first piecewise-linear straight line
mz
1, spatial form of unaccounted
parameters 2
2
k and
3 , and also on unaccounted parameters influence functions ),(
2,122
2 k
and ),(3,233
in the form:
)]}()(1[1{3,233212213
2 α,λ+ωα,λ+ω+A=zz,
k
m, )3,2,1( m (53)
Here
)]-([)-(
)-(
)-(
1211
3
1
12
3
1
2
12
11
∑
∑1
ii
i
ii
i
ii
k
aaaaa
aa
A
(54)
Azad Gabil oglu Aliyev 47
∑
∑
3
1
2
12
2
1211
3
1
3
11
2
2
)-(
})]-([-{
-
1
21
2
2
i
ii
ii
k
i
i
i
kk
k
k
aa
aaaa
(55)
2,122,122
22 ),( сoskk (56)
3,233,233
),( сos
3
1
22
3
1
2
24
3
1
2
22
11
3
)(
)()(
2
22
1
i
k
ii
i
k
ii
i
k
ii
k
zz
zazz
3,23
1
2
12
3
1
121
)(
)(
сos
aa
aaz
i
ii
i
iii
(57)
3
1
2
2i4
3
1
2i41i3
223
)-(
)-)(-(
)-(
2
21
2
i
k
i
i
k
i
k
i
k
za
zaza
, for 2
22
k (58)
∑
∑
3
1
2
1i3
3
1
121i3
112
)-(
)-)(-(
)-(
1
1
1
i
k
i
i
ii
k
i
k
za
aaza
, for 1
11≥
k (59)
∑
∑
3
1
2
1i3
3
1
121i3
112
)-(
)-)(-(
)-(
1
1
122
i
k
i
i
ii
k
i
kkk
za
aaza
, for 12
11≥
kk (60)
Now for the case of economic process represented in the form of three-component piecewise-linear economic-mathematical model investigate the prediction and control of such a process on the subsequent
)х,х,х(3214
V small volume of 3-dimensional vector space with regard to unaccounted parameters influence
functions ),(2,122
2 k
and ),(3,233
3 k
. The values of the unaccounted parameters functions ),(2,122
and
),(3,233
are unknown [5-16]. In [10,11,13(Capter 2),14,16], we developed a method for constructing an economic
process predicting vector function )(1
NZ
with regard to the introduced unaccounted parameter influence
predicting function ),(1,11
NNNN
in the m-th vector space that found its reflection in Eqs. (51)–(57). Apply this
method to the case of the given three-component piecewise-linear economic process in 3-dimensional vector space. In
this case, the predicting vector function )(4Z
will be of the form (Fig. 4):
48 American Review of Mathematics and Statistics, Vol. 6, No. 1, June 2018
Fig.4. Construction of predicting vector functions )(Z4
with regard to unaccounted parameter influence
predicting function ),(4,344
on the base of 3-component piecewise-linear economic-mathematical model in
3-dimensional vector space R3.
)]},(),(),(1[1{)(4,3443,2332,12214
32 kk
AzZ
for 2,1 (61)
where
4,344,344),( сos (62)
1112
111
331
3533
11
4
41
1
3
33
1
)(
)(
)(
azaa
azz
zzz
zazz
k
k
k
kk
k
for 2,1 , 3
31≥
k , 0
4 (63)
,))((
))()(()(
2
35
3524
3343
32
3
k
kk
k
za
zaza
3
33
k , 0
4 for 2,1 (64)
The expressions of the unaccounted parameters functions ),(2,122
2 k
and ),(3,233
3 k
have the form
Eq. (53)–(55) and
2,122,122
22 ),( сoskk
=2,1
1112
111
111
1311
11
2
1
1
1
11
1
2 )(
)(
сos
azaa
azz
zzz
zazz
k
k
k
kk
K
k
(65)
Azad Gabil oglu Aliyev 49
,)(
))(()(
2
13
1312
1121
1
1
k
k
k
za
zaaa
1
11
k (66)
)()(
111
1112
111
1
1
azz
azaa
Ak
k
k
(67)
Here the vector )(5a
for each value of 2,1 , according [13(Capter 2), 15], is of the form:
3
1
53532521515)()()()()(
m
mmiaiaiaiaa
for 2,1 (68)
And by means of Eq. (37), the coordinates of the vector )(a
will be expressed by the coordinates of the vectors
1a
, 1-
1-
k
z
and 3
3
kz
in the form:
1
3
1
3
1
3
3,13,1
3353
2,12,1
3252
1,11,1
3151
-
-)(
-
-)(
-
-)(
k
k
k
k
k
k
za
za
za
za
za
zaC
(69)
Hence, by Eq. (69), the coordinates )(52
a and )(53
a will be expressed by the arbitrarily given coordinate
3
3151)(
kza , in the form:
1-
1-
33
1,1-1,1
2,1-2,1
31513252
-
-)-)(()(
k
k
kk
za
zazaza
1-
1-
33
1,1-1,1
3,1-3,1
31513353
-
-)-)(()(
k
k
kk
za
zazaza
(70)
Here the coefficients m
a,1
and 1-
,1-
k
mz are the coordinates of the vectors
1-a
and 1-
z
in
3-dimensional vector space and equal:
∑3
1
,1-1-
m
mmiaa
, ∑
3
1
,1-1-
m
mmizz
(71)
Note that in the vectors )(4Z
and )(
5a
the index ( ) in the parenthesis means that the vector )(
4Z
is
parallel to the -th piecewise-linear vector-function
z
. This will mean that the occurring economic process,
beginning with the point 3
3
kz
will occur by the scenario of the -th piecewise-linear equation. In our example
2,1 . In our case, there will be three predicting functions, i.e., ),1(4
Z
)2(
4Z
and the case when the influence of
unaccounted factors )0(4
Z
will not be available. In all these cases, the predicting vector-functions )(
Z
will
emanate from one point 3
3
kz
, and the predicting vector-function )1(4
Z
will be parallel to the first piecewise-linear
straight line; )2(4
Z
will be continuation of the third vector straight line )0(
4Z
, and all of them will emanate from one
point 3
3
kz
. The expression 4,3
сos corresponding to the cosine of the angle between the third piecewise-linear
straight line 3
z
and the predicting fourth vector straight line )(
Z
for each value of on the base of scalar
product of two vectors is represented in the form (Fig. 4):
50 American Review of Mathematics and Statistics, Vol. 6, No. 1, June 2018
33
33
3533
3533
4,3
-)(-
)-)()(-(
kk
kk
zazz
zazzсos
(72)
IV. Results Method of Numerical Calculation of Three-Component Economic-Mathematical Model and Definition of Predicting Vector Function with Regard to Unaccounted Factors Influence in 3-Dmensional Vector Space
In this section, we have given numerical calculates of three-component piecewise-linear economic-mathematical model with regard to unaccounted parameters influence in
3-dimensional vector space, and construct appropriate predicting vector functions )1(4
Z
, )2(
4Z
and )0(
4Z
on
subsequent 4
V small volume of 3-dimensional space [5-16].
Consider the case of economic process given in the form of the statistical points (vectors) set {n
a } in
3-dimensional vector space 3
R represented in the form of three-component piecewise-linear function of the form Eqs.
(33)–(35). The vectors ),,(321 iiiii
aaaaa
(where 3,2,1i ) are the given points of 3-dimensional
vector space 3
R and have the form:
3211iiia
,
32125,423 iiia
,
3213746 iiia
,
32141098 iiia
(73)
Below, by means of these vectors we have showed a method for calculating a chain form of each piecewise-linear
vector equation depending on the first piecewise-linear vector straight line 1
z
, cosines of the angles 2,1
сos and
3,2сos generated between the adjacent first and second and also third and fourth piecewise-linear vector lines, and
also on the parameter 1
corresponding to the first vector line [5-16].
Substituting Eq. (73) in Eq. (33), the equation of the first straight line in the coordinate form will be of the form:
3111111)5,31()1()21( iiiz
(74)
Giving the value of the parameter 1
for the intersection point 1
k between the first and second
piecewise-linear straight lines of the form 5,11
1
k , the coordinate form of the intersection point 1
1
kz
is defined from
Eq. (74) in the form:
311
5,1
1125,65,24
1
1 iiizzk
(75)
By means of intersection point Eq. (75) and the given point 3a
on the second straight line, by Eq. (49) set up a
numerical relation between the parameters 1
and 2
in the form:
)5,1-(1927,112
, for 1
5,1≥ , 0≥2
(76)
Hence:
218384,05,1 (77)
Azad Gabil oglu Aliyev 51
Eq. (76) means that on the second piecewise-linear straight line, to the value of the parameter 2
there will be
determined appropriate value of the operator 1
by Eq. (77). For the given value of the parameter 2
2
k ,
corresponding to the intersection point between the second and third piecewise-linear straight lines equal 2, i.е.,
22
2
k from Eq. (77) or Eq. (50) the appropriate value of the parameter 2
1
k will equal:
1768,32
1
k (78)
This means that when the parameter 2
corresponding to the points of the second piecewise-linear straight line
will change within 2≤≤02
, then the appropriate value of the parameter 1
will change in the interval:
1768,3≤≤5,11
(79)
This case will correspond to the case of the segment of the second straight line. For the value of the parameter
2≥2
the appropriate value of the parameter 1
, will be 1768,3≥1
. This case will correspond to the vector
equation of the second straight line restricted from one end. Now establish the form of the vector equation of the second piecewise-linear straight line depending on the
vector equation of the first piecewise-linear straight line 1
z
, 2,1
сos and the parameter 1
:
)]},(1[1{2,12212
Azz
(80)
where the coefficient A , the unaccounted factor parameter 2
and the unaccounted parameters function
),(2,122
will be of the form:
)-(
--
)-(
111
1211
111
1
1
azz
aaaz
Ak
k
k
(81)
1112
111
111
1311
11
2
2
--
)-(
)-(
--
- 1
1
1
11
1
azaa
azz
zzz
zazz
k
k
k
kk
k
(82)
2,122,122),( сos (83)
112
112
13111
13111
2,1
)(
))()((
kkk
kkk
zazz
zazzсos
(84)
Note that by Eq. (80) we must carry out numerical calculation for the values of the parameter 1
. In conformity
to our problem, we should use the range of the parameter 1
given in Eq. (79).
Determine the numerical values of the coefficients A , 2
, 2,1
сos and ),(,2,122
. For that substitute
Eqs. (73–76) and the value of the parameter 5,15,1
11
1 k
in Eq. (81)–(84), and get:
)(875,2510
A , for 1768,35,11 (85)
)(1203,0112
, for 1768,35,11 (86)
where
1
1
10
875,2575,9
5,1)(
(86а)
52 American Review of Mathematics and Statistics, Vol. 6, No. 1, June 2018
2
1
1
2
11
1
11
25,17
75,518125,38
25,17375,1975,9
875,2575,9)(
(86b)
7494,02,1сos (87)
)(0901,0),(112,122
(88)
Substituting the numerical expressions of A , 2
, 2,1
сos and ),(,2,122
Eqs. (85)–(88) in Eq. (80), find
the final form of the vector function of the second piecewise-linear straight line depending on the first piecewise-linear
straight line 1
z
, and 2,1
сos in the form:
)(1212
zz
for 1768,35,11 (89)
where
)](0901,01[)(8751,251)(111012
(90)
Note that the obtained Eq. (89) is a vector equation of the second straight line where the value of the parameter
5,11 . When we impose on the parameter
1 the condition 1768,35,1
1 , Eq. (89) will represent a vector
equation of the second piecewise-linear segment. Calculate the value of the intersection point of the second and third piecewise-linear straight lines, i.e., at the point
2к . Therefore, according to approximation of piecewise-linear straight lines, for the intersection point accept the value
of the parameter 22
2
к , and the approximate value of the parameter 2
1
к calculated earlier will correspond to the
upper value of inequality Eq. (79), i.e., 1768,32
1
к . In this case, we find the value of the vector function 2
2
kz
in the
coordinate form from Eq. (89) in the following form (Fig. 5):
Fig. 5. Numerical construction of three-component piecewise-linear economic-mathematical model in 3-dimensional
vector space 3
R .
Azad Gabil oglu Aliyev 53
3212615,93139,38343,52 iiiz
k
(91)
Now construct a vector equation for the third piecewise-linear straight line depending on the vector equation of
the first piecewise-linear function 1
z
, 2,1
сos , 3,2
сos and also parameter 1
corresponding to the parameter 3
. For that we use the following defining Eqs. (33), (45)–(48):
)]},(),(1[1{3,2332,12213
2 k
Azz
for 2
11
k , 2
22
k (92)
Here the unaccounted parameters influence function ),(2,122
2 k
is calculated by means of Eq. (43), the
unaccounted parameter influence function ),(3,233
is calculated from Eqs. (46)–(48) in
the form:
3,233,233),( сos (93)
1112
111
221
2422
11
3
31
1
2
22
1
)(
)( azaa
azz
zzz
zazz
k
k
k
kk
k
(94)
22
22
24222
24222
3,2
)(
))()((
kk
kk
zazz
zazzсos
, for 2
22
k (95)
2
24
2413
223
)(
))(()(
2
21
2
k
kk
k
za
zaza
, for 2
22
k (96)
)()(
111
1112
111
1
1
azz
azaaA
k
k
k
(97)
2,122,122
22 ),( сoskk (98)
Note that calculation of the function ),(2,122
2 k
is simplified owing to expression Eq. (88), where
instead of the parameter 1
we should use its value corresponding to the second intersection point, i.e.,
1768,32
11
k . In this case, we get:
1768,3
2,1222,1221
2 ),(),(
k
5613,0)(0901,01768,311
1
(99)
The mathematical Eq. of the relation of the parameter 3
with the parameter 1
corresponding to the points
of the first piecewise-linear straight line, will look like as follows. We have the condition of relation of the parameter 2
and the parameter 1
in the form Eq. (76). Therefore, substituting Eqs. (76) in Eq. (96), we get:
2
24
2413
213
)(
))((])5,1(1927,1[
2
21
2
k
kk
k
za
zaza
(100)
Taking into account Eqs. (73), (75), (78), (91) for 22
2
k , Eq. (100) accepts the form:
3394,14216,013 for 0
3 , 1768,32
11
k (101)
54 American Review of Mathematics and Statistics, Vol. 6, No. 1, June 2018
Thus, Eq. (101) establishes the numerical relation between the parameters 3
and 1
.
For 3
=0, 1768,32
3101
k
, i.е., coincides with the value of the parameter 2
1
k . For any values of the
parameter 03 , the appropriate value of the parameter
1 will be greater than 3,1768, i.е., 1768,3
1 .
Substituting Eqs. (73), (75), (90), (91), (101) in Eq. (94), we get the numerical dependence of the unaccounted factors
parameter on the third vector straight line 3
depending on the parameter 1
for 1768,32
11
k in the form:
)(2356,0133
, for 1768,32
11
k (102)
where the expression )(13
is of the form:
)(
3394,14216,0)(
10
1
13
763,1848,635-3)13(17,25)(
137,469218,785]
)[48,635(2-3]13[17,25)(
11
2
112
1121
2
11
2
2
(103)
Substituting Eqs. (73), (90), (91) in Eq. (95), we get the numerical dependence of 3,2
сos generated between the
second and third piecewise-linear straight lines depending on the parameter 1 for the values 1768,32
11
k in
the form: )(1640,0143,2
сos , for 1768,31 (104)
Where the expression )(14
is of the form:
)(14
2
12
2
112
2
112
12
112
112
]615,9)3,5(1[
]3139,3))(1([
]8343,5)2)(1([
]615,9)3,5(1[385,0
]3139,3))(1(5,6861[
]8343,5)2)(1([1657,2
(105)
As the angle between the two straight lines is a constant quantity, we calculate the numerical value of 3,2
сos for
51 . In this case, we have (Fig. 5):
434,03,2сos (106)
Substituting Eqs. (102) and (106) in Eq. (93), the numerical value of the parameter ),(3,233
will equal:
)(-0,1022),(133,233
for 1768,31 (107)
Now calculate the coefficient A Eq. (97) for 1768,32
11
k . For that we substitute Eqs. (99), (107), and
(108) in Eq. (97) and get the following numerical expression of the coefficient A :
)(875,25875,2575,9
5,1875,25
10
1
1
А , for 1768,32
11
k (108)
Or
Azad Gabil oglu Aliyev 55
)(875,2510
А (109)
Substituting the numerical values Eqs. (99), (107) and (108) in Eq. (92), we get a vector equation for the third piecewise-linear straight line, expressed by the vector equation of the first piecewise-linear straight-line and the
parameter 1
in the form (Fig. 5):
)(1513
zz
, for 1768,32
11
k (110)
where
)](0,233-[1)(11,3514-1)(131015
(111)
or in the form:
]i)3,5(1i)(1i)2[(1)(312111153
z (112а)
Now investigate the prediction of economic process and its control on the subsequent ),,(3214
xxxV small
volume of 3-dimensional vector space with regard to unaccounted parameter factors ),(2,122
2 k
and
),(3,233
3 k
that hold on the preceding stages of the process [5-16].
And the numerical values of these unaccounted parameters functions ),(2,122
, ),(2,122
2 k
and
),(3,233
are assumed to be known and are given by Eqs. (99), (107), and (88), having the following numerical
expressions:
)(0,0901),(112,122
(113)
-0,5613),(2,122
2 k
(114)
)(-0,1022),(133,233
for 1768,31 (115)
where the expressions )(11
and )(13
are represented by Eqs. (86.b) and (103).
Above for the three-component piecewise-linear economic process we have constructed the third
piecewise-linear straight line Eq. (110) depending on an arbitrary parameter 1 and unaccounted parameters influence
spatial functions ),(2,122
and ),(3,233
. On the other hand, by Eq. (61) we suggested for the
three-component case the economic process predicting vector function )(4Z
with regard to the introduced
unaccounted parameters predicting influence function ),(3,444
[5-16]:
)]},(
),(),(A[1{1z)(
3,444
3,2332,12214
32
kk
Z
for 2,1 (116)
4,343,444),( сos (117)
56 American Review of Mathematics and Statistics, Vol. 6, No. 1, June 2018
1112
111
331
3533
11
4
41
1
3
33
1
)(
)(
)(
azaa
azz
zzz
zazz
k
k
k
kk
k
, for 2,1 , 0,
433
3 k
(118)
2
35
3524
314
))((
))()(a()(
3
32
3
k
kk
k
za
zaz
, for 2,1 , 0,
433
3 k
(119)
Here, ),(2,122
2 k
has numerical expression Eq. (114), the function ),(3,233
3 k
for the final point of the
third piecewise-linear straight line for 3
33
k and its appropriate values 3
11
k is calculated by means of Eq.
(115). As the intersection points of the straight lines are given, accept the value of the intersection point 3
k between the
third and fourth predicting straight lines in the form 33
3
k . And define the appropriate value of the parameter 3
1
k
from the Eq. connecting the parameters 1
and 3 in the form Eq. (101):
3394,14216,013 , for 0
3 , 1768,32
11
k (120)
Hence
1768,33719,231 (121)
Substituting the value 33
33
k in Eq. (121), define the numerical value of the parameter 3
1
k corresponding to
the value of the parameter 33
3
k at the intersection point of the third piecewise-linear straight line with the
predicting fourth straight line in the form: 2926,103
1
k
(122)
Substituting the numerical value of 3
1
k Eq. (122) in Eq. (115), define the numerical value of the unaccounted
parameters function ),(3,233
3 k
at the intersection point between the third piecewise-linear straight line and
predicting fourth vector straight line. For that as preliminarily, by Eqs. (86.a, b), (90) calculate the functions )(10
,
)(11
and )(22
for 2926,103
11
k , and get:
03185,0)(2926,1010
1
(123)
23,6)(2926,1011
1
(124)
639,0)(2926,1012
1
(125)
Now, substituting the numerical values Eqs. (123)–(125) in Eq. (168), take into account 2926,101 and
define the numerical value of the unaccounted parameter function ),(3,233
3 k
at the third intersection point 3
k in
the form: -0,2172),(3,233
3 k
(126)
Substituting Eq. (123)–(125) in Eq. (103) allowing for 2926,101 we define the function
2926,10131
)(
in the
form: 125,2)(2926,1013
1
(127)
Substituting Eqs. (123) and (127) in Eq. (111), where we accept 2926,101 , find the numerical value of
2926,10151
)(
in the form:
8175,0)(2926,1015
1
(128)
Azad Gabil oglu Aliyev 57
Substituting Eqs. (128) and (74) in Eq. (110) or Eq. (112), where it is accepted 2926,101 , find the
coordinate expression of the vector point 3
3
kz
in the form (Fig. 6):
Fig. 6. Numerical construction of the predicting vector function )(Z4
with regard to unaccounted parameter
influence factor ),(4,344
on the base of three-component piecewise-linear economic-mathematical model
in3-dimensional vector space 3
R .
32132672,302317,96459,173 iiiz
k (129)
Now, by Eq. (117) calculate the unaccounted parameters predicting function ),(3,444
. For defining it, as
preliminarily we find the numerical dependence of the parameter 4
on the parameter 1
, 4
, 4,3
сos , and also on
the vector )(5a
for 2,1 . Therewith we note that the vector )(
5a
for the values 2,1 has coordinate
form Eq. (69). Here the coordinate )(5
i
a
in 3-dimensional space are determined by Eq. (170). Substituting Eqs. (70),
(73), (91), (120) in Eq. (119) for 33
3
k establish the numerical dependence of the parameter
4 on the parameter
1 in the form:
),(4,3393)-4216,0(1614
for 2926,10
1 , 2,1 (130)
where
),(16
2
1553
2
1552
2
1551
1553
15521551
)]3,5(1-)([a
)](1-)([a)]2(1-)([a
)]3,5(1-)([a385,0
)](1-)(5,6861[a)]2(1-)(2,1657[a
(131)
Substitute Eqs. (168), (73)–(75), (110), (129), and (130) allowing for Eq. (70) in Eq. (118) and define the predicting
58 American Review of Mathematics and Statistics, Vol. 6, No. 1, June 2018
parameter 4
in the form:
),(),()(1716184 (132)
where
2
53
2
51
2
51
17
30,2672]-)([a
9,2317]-)([a17,6459]-)([a),(
(133)
)(18
)]3,530,2672(1)9,2317(1
)2[17,6459(1-
-])3,5(1)(1)2[(1
30,2672]-)3,5[(1
9,2317]-)[(1
17,6459]-)2[(1
11
1
2
1
2
1
2
15
2
51
2
51
2
51
(134)
Now define the numerical value of 4,3
сos generated between the third piecewise-linear function 3
z
and
predicting fourth vector function )(44
Z
(Fig. 6) [5-16]:
33
33
3533
3533
4,3
)(
))()((
kk
kk
zazz
zazzсos
(135)
For that substitute Eqs. (69), (74), (110), and (129) in Eq. (135) and get:
),(),(
),(
111110
19
4,3
сos (136)
where
17,6459]-)2)[(1(a),(515119
9,2317]-))[(1(a5152
-30,2672]-)3,5)[(1(a5153
)9,2317(1)2[17,6459(1-115
1312,7085)]5,330,2672(11
(137)
),(110
7055,1312)]5,330,2672(1)9,2317(1
)2[17,6459(12
])3,5(1)(1)2[(1
11
15
2
1
2
1
2
1
2
5
(138)
2
53
2
51
2
51
111
30,2672]-)([a
9,2317]-)([a17,6459]-)([a),(
(139)
Azad Gabil oglu Aliyev 59
Substituting Eqs. (132) and (136) in Eq. (117), establish the numerical representation of the unaccounted
parameter predicting influence function ),(3,444
in the form:
),(),(1123,444 (140)
where
),(112
),(),(
),()(),(),(
111110
19
181716
(141)
Representing the numerical values Eqs. (99), (108), (126), and (140) in Eq. (116), define the concrete form of the
predicting vector function on the fourth small volume of 3-dimensional space for 2,1 in the form (Fig. 6):
)]},(-0,2172-0,5613-)[1(25,875-{1z)(1121014
Z
for 2,1
or
)]},(-)[0,2215(25,875-{1z)(1121014
Z
for 2,1 (142)
or in the coordinate form:
-)[0,2215(25,875-1{)(104
Z
]i)3,5(1i)(1i)2[(1)]},(-312111112
for 2926,101 , 2,1 (143)
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