=Paper=
{{Paper
|id=Vol-2300/Paper3
|storemode=property
|title=Construction of Two-Dimensional Correlation Models in a Cartesian and Spherical Coordinate System
|pdfUrl=https://ceur-ws.org/Vol-2300/Paper3.pdf
|volume=Vol-2300
|authors=Andriy Segin,Alina Davletova,Ihor Havryshchak
|dblpUrl=https://dblp.org/rec/conf/acit4/SeginDH18
}}
==Construction of Two-Dimensional Correlation Models in a Cartesian and Spherical Coordinate System==
https://ceur-ws.org/Vol-2300/Paper3.pdf
10
Construction of Two-Dimensional Correlation Models in
a Cartesian and Spherical Coordinate System
Andriy Segin1, Alina Davletova1, Ihor Havryshchak2
1. Department of Specialized Computer Systems, Ternopil National Economic University, UKRAINE, Ternopil, 8 Chehova str., email:
ase@tneu.edu.ua, a90f@meta.ua
2. Department of the Ukrainian Language, I. Horbachevsky Ternopil State Medical University, UKRAINE, Ternopil, 1 Voli sq.
e-mail: havruchukii@tdmu.edu.ua
Abstract: This paper presents a methodology for the form of the Earth in the form of a ball, the satellite's orbit
constructing and visualizing correlation models of two- is stationary in the form of a circle over a certain latitude of
dimensional signals in a rectangular and spherical the Earth. To assess the adequacy of the model an analysis of
coordinate system. its results over the elementary types of signals presented in the
Keywords: correlation models, spherical coordinate form of sinusoidal.
system, two-demensional models. II.TWO-DIMENSIONAL PROCESSES IN CARTESIAN
I. INTRODUCTION AND SPHERICAL COORDINATE SYSTEMS
The mathematical apparatus of correlation analysis in the The definition of correlation in a spherical coordinate
Cartesian coordinate system remains a powerful tool for system, by the way, as in the Cartesian or D-denier coordinate
researching technological processes and natural phenomena system, requires a clear definition whether it is necessary to
[1]. Areas of its application are extremely diverse both in the make correlations of figures or correlation of processes.
fundamental directions of science and applied science. In the first case, under the correlation of the figures it is
Correlation analysis is successfully used in the study of necessary to understand the correlation of the coordinates of
processes in atomic and nuclear physics, energy, electronics, the points belonging to a bulk figure with the corresponding
radio engineering, astronomy, astrophysics, economics and coordinates of the points of the standard figure. In this case, it
other fields of science [2-4]. is sufficiently to consider the functions describing the studied
Correlation models give an opportunity to investigate both and standard figures given in parametric or normal form,
one-dimensional and multidimensional deterministic and while the functions depend on time, so the time coordinate is
stochastic processes [5]. It is clear that the correlation analysis not taken into account. This means that spatial figures remain
is not universal, has its own limitations and possibilities of use unchanged in time, therefore, it is sufficient to determine the
for a certain class of tasks. However, this is a convenient and
correlation of only spatial coordinates, in the Cartesian
reliable method for solving a wide range of tasks, which needs
coordinate system or, - in a spherical coordinate system. Then
further development and improvement for its greater
efficiency and extending its use. the investigated and standard objects are described in the
At the present stage of microprocessor development and Cartesian coordinate system by the equations in the normal
computer technology, new perspectives of correlation models form:
development are opened, which will allow them to be used in z = f ( x, y ) ,
real-time systems and create more complex models. z e = f e ( x, y ) .
The article presents mathematical expressions for In a spherical coordinate system:
constructing correlation models in a spherical coordinate
system and proposes to take into account the effect of "aging" r = ϕ (Θ, λ )
information and transport delays in correlation models. The re = ϕ e (Θ, λ )
construction of correlation models in a spherical coordinate As a rule, objects describing closed figures in the
system opens up new prospects for their use in the study of corresponding space are described, or they specify certain
processes and phenomena, which are simpler described in finite surfaces, although they can also describe infinite
spherical coordinates than in Cartesian space. Such processes surfaces that are investigated on a finite spatial range.
include the determination of orbits and figures of celestial One of the least complicated figures in the spherical
bodies, the determination of objects’ movement in space, coordinate system is the sphere with the center at the origin
radar, and propagation of waves of various nature in space, point, which is described by the known equation
trajectories of motion of elemental particles in atoms, rotating
processes in mechanics, determination of precise coordinates r (λ , Θ) = R .
in cartography, and many others.
The article presents the first theoretical stage of the study of The sphere in the Cartesian coordinates is described by a
the satellite's interconnection in a stationary orbit with points more complex expression:
on the Earth's surface. It consists in the development of a
x2 + y2 + z 2 = R ,
mathematical apparatus for constructing a correlation model
in a spherical coordinate system. For the formation of the where R is the radius of the sphere in the polar and Cartesian
model were adopted simplifications, which consist in taking coordinate systems.
ACIT 2018, June 1-3, 2018, Ceske Budejovice, Czech Republic
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It is obvious that manipulation with such figure, including
the calculation of correlation estimates, is easier to carry out in
a spherical coordinate system.
There is a number of figures that are described simply in a
spherical system, for example, almost all rotation functions:
ellipsoids, paraboloids, hyperboloids, and others.
In the second case when processes are being investigated,
while constructing both processes itself and their correlation
functions, it is necessary to take into account the time
coordinate, since at different times the process may have
different spatial coordinates. Therefore, processes need to be
investigated in the spatial-temporal coordinate system.
Accordingly, it is convenient to represent such processes in a
parametric form:
x(t ) = f 1(t ),
y (t ) = f 2(t ), (1)
z (t ) = f 3(t ). Fig. 1. Graph of the function r = R given in the polar
coordinate system.
In a spherical coordinate system, the process given Another example of a function that describes a number of
parametrically, respectively, is determined by a system of processes is: r = R , Θ = α , where R – radius, α –
equations: coordinate angle of latitude.
This function describes the circle (fig. 2) and in practice, it
λ (t ) = ϕ1(t ), can determine the satellite's orbit, which moves over a certain
Θ(t ) = ϕ 2(t ), (2) parallel of the Earth. If the center of the Earth is taken as the
r (t ) = ϕ 3(t ). coordinates’ center, then R- determines the height of the orbit
above the surface of the Earth, and the angle of coordinate of
Then the spherical coordinates in the parametric form have latitude is the parallel along which the satellite will move. If,
the following form: α = 0 , then the orbit will pass over the equator. If
Θ = α while 0 < α < π / 2 the orbit is passing over a certain
r= ( f1 (t ) )2 + ( f 2 (t ) )2 + ( f 3 (t ) )2 parallel in the North hemisphere. If Θ = α while
−π / 2 < α < 0 the orbit is passing over a certain parallel in
Θ = arcctg
f 3 (t )
(3) the South hemisphere. If α = π / 2 or α = −π / 2 , then the
f1 (t ) + f 2 (t ) satellite will "hang" over the north or south pole respectively
(fig. 2-3).
f (t ) If the process is periodic with a period T , then you can
ϕ = arcctg 2
determine the periodicity of the coordinates λ and Θ , having
f1 (t ) their own periods of repetition: λ – 2π - from 0 to 2π ; Θ -
also 2π , from −π to π . Consequently, the coordinate period
From (1), (2) and (3) it can be seen that the complexity of
λ and Θ finally will be determined from the functions ϕ1(t )
expressions will depend on the functions.
We consider a function that describes the sphere in for the coordinate λ and the function ϕ 2(t ) - for the
spherical coordinates. This means that, depending on the coordinate Θ . It is obvious that when changing the
moment of time, the value of the function will correspond to a coordinates of time t on the segment [0, T 1] , [T 1, T 2] etc.,
certain point on the sphere. This function describes processes respectively, coordinate λ and Θ should change for the
in astronomy, in the description of orbits of planets, satellites
period [0,2π ] , [2π ,4π ] , and [−π , π ] , [π ,3π ] , etc.
and other cosmic bodies, the position of objects in radar,
guidance systems and a number of other cases. In this case,
the expressions will have the simplest form when choosing a
coordinate system with the beginning in the scope of the
sphere. Such function is given in the polar coordinate system
by the equation, where is the radius of the sphere. (fig. 1).
When transitioning to the Cartesian coordinate system we use
the well-known expressions:
x = R sin Θ cos λ ,
y = R sin Θ sin λ , (4)
z = R cos Θ .
Fig. 2. Schedule of function r = R , Θ = α
ACIT 2018, June 1-3, 2018, Ceske Budejovice, Czech Republic
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1 1 N − P −1 N −Q −1
K Rre ( p, q) = ⋅ ∑ ∑
m n i =0 j =0
ϕ (λi , Θ j ) ⋅ ϕ e (λi + p , Θ j + q ) (6).
Fig. 3. Graph of an example of a satellite's motion in orbit
over a given parallel of the Earth.
If it is difficult or impossible to describe the reference
objects or the reference objects functionally, then they are
given by an array of coordinate triples, x , y , z – in the
Cartesian coordinate system or r , Θ , λ – in the spherical
coordinate system of the discrete points of the figures and a)
approximated by certain methods.
III. THEORETICAL FOUNDATIONS OF CONSTRUCTING
CORRELATION MODELS IN A SPHERICAL
COORDINATE SYSTEM
Having considered the peculiarities of constructing
functions in a spherical coordinate system, we turn to the
study of the construction of correlation models in a spherical
coordinate system and the representation of their graphs in
this coordinate system.
Consequently, for the construction of a correlation model of
discrete spatial figures [6], the expression for the Cartesian
coordinate system will have the form:
N −Q −1
1 1 N − P −1
Kxe( p, q ) = ⋅ ⋅ ∑ ∑ f ( xi , y j ) ⋅ fe( xei + p , ye j +q )
n m i =0
(5)
j =0
where i , j - coordinate indices x , y working sample b)
points;
Fig. 4. a) - Charts functions that correlate
n , m – the volumes of working samples according to
sin x + sin y sin 2 x + sin 2 y
coordinates x , y ; z1( x, y ) = and z 2( x, y ) = 5 + ;
2 2
p , q – indices of displacement by the corresponding b) - Charts their covariance function.
coordinates, x , y ;
To represent correlation models in a spherical coordinate
P , Q – the maximum value of the displacement by the system we use formulas (3) to translate Cartesian coordinates
corresponding coordinates x , y ; into spherical ones.
N –the total number of discrete points of an object. Most often, correlation models are built for time processes.
That is F ( x, y, z ) , the function is not a spatial coordinate x ,
We construct graphs and the corresponding function of
y , z , but a time-dependent t function f (t ) . In this case,
correlation (fig. 4 ), according to (5) for the functions:
processes in Cartesian coordinates, are set, as a rule,
sin x + sin y parametrically (1).
z1( x, y ) = and Then spatial coordinates are determined at each time point.
2
sin 2 x + sin 2 y In this perspective, it is expedient to consider the time-spatial
z 2( x, y ) = 5 + . coordinate system. As already noted, there is a certain
2 category of processes that are much more convenient to
A similar expression for determining the correlation is represent in a spherical coordinate system. In addition, the
obtained for a spherical coordinate system: analysis of such processes, including correlation, is also
ACIT 2018, June 1-3, 2018, Ceske Budejovice, Czech Republic
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simpler and more convenient in this coordinate system, since
mathematical expressions are less complex.
Consider a function that describes the sphere in spherical
coordinates. This means that, depending on the moment of
time, the value of the function will correspond to a certain
point on the sphere. This function describes the processes in
astronomy, in the description of orbits of planets, satellites and
other cosmic bodies, the position of objects in radar, guidance
systems and in a number of other cases. In this case, the
expressions will have the simplest form when choosing a
coordinate system with the beginning in the center of the
sphere. Such functions are given in the polar coordinate
system by the equation r = R = const , where R – is the
radius of the sphere (Fig. 5).
Fig. 6. Representation of a covariance function in a polar
coordinate system.
As expected, the covariance function looks like a sphere
with a radius R = R1 ⋅ R 2 = 2 ⋅ 5 = 10 .
Correlation functions in the polar coordinate system
require further research and are an effective research tool.
VI. CONCLUTION
The method of constructing correlation models in a
spherical coordinate system proposed in the article is a very
useful tool for the investigation of two-dimensional stochastic
and deterministic signals in radar, astronomy, radio
engineering, and other spheres. A number of phenomena and
processes in a spherical coordinate system are described by
simpler mathematical equations, which simplifies their further
analysis. The results of the correlation analysis reflected in the
spherical coordinate system are often more visible and
understandable.
REFERENCES
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Physics, Chemistry and the Natural Sciences (Springer
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Fig. 5. Charts of functions of the sphere with radii Telecommunications and Computer Science, 2004.
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Using the expression of the covariance function (6), for [6] Nykolaychuk Y., Segin A. Information source models and
which it is not necessary to center the values of the random methods of there building. Methods and equipment of
process, unlike the correlation function, we will construct a
quality valuation. Ivano-Frankivsk, 1998, № 2, PP. 80 –
covariance model, the graph of which is presented in fig. 6.
84.
ACIT 2018, June 1-3, 2018, Ceske Budejovice, Czech Republic