Three-Component Piecewise-Linear Economic-Mathematical...

American Review of Mathematics and Statistics June 2018, Vol. 6, No. 1, pp. 35-60

ISSN: 2374-2348 (Print), 2374-2356 (Online) Copyright © The Author(s).All Rights Reserved.

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

mailto:[email protected]

http://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.


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)


)-(

--

)-(

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.


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)


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)


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.


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)


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:


)-(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

2

k and

3 , and also on

unaccounted parameters spatial influence functions ),(2,122

2 k

and ),(3,233

in the form (Fig. 3):

)]}()(1[1{3,233212213

2 α,λ+ωα,λ+ω+Az=z,

k

for 2

11

k , 2

22

k (45)

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)


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)


∑

∑

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):


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)


,)(

))(()(

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):


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)


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)


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 .


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)


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


)(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


)(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)


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)


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


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


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)


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)

References

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Azerbaijan-№ 9, ISSN 0207-3188, Baku-1998, pp. 195–203. Aliyev Azad, G., Ekonomi Meselelerin Cozumune Iliskin Matematik Dinamik Bir Model. Publishing house-‗Bilgi ve

Toplum, Turkey, National publication, 99-34-Y-0147, ISBN 975-478-120-5, Istanbul-1999, pp.83–106. Aliyev Azad, G., Экономико-математические методы и модели с учетом неполной информации, National

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