General Concept About Correlation Relationships.
Istamjon Karshiev
Tashkent University of Applied Sciences, Gavhar Str. 1, Tashkent 100149, Uzbekistan
https://doi.org/10.5281/zenodo.10471195
Keywords:
correlation
,
determination coefficient
,
regression equation
,
Fisher's criterion
,
determination coefficient
,
multiple correlation,
Kramer,
constant coefficients,
Student criterion.
Abstract:
This annotation provides a comprehensive insight into the general concept of correlation relationships. It
explores the fundamental principles and applications of correlations within various fields, emphasizing their
significance in statistical analysis, predictive modeling, and decision-making processes. The annotation
delves into the types of correlations, their mathematical representation, and the interpretation of correlation
coefficients. Additionally, it discusses the strengths and limitations of correlational studies, shedding light
on their role in establishing connections between variables and elucidating associations within datasets. The
annotation aims to offer a foundational understanding of correlation relationships, serving as a valuable
resource for researchers, analysts, and students exploring this critical statistical concept.
1 INTRODUCTION
Topic objective. To illustrate the relationship
(dependence) between economic indicators.
In economics, technology, and other fields, it is
appropriate to study the relationships between various
numerical and qualitative indicators. Among the
indicators, there are two types of relationships -
functional and correlation (or statistical) relationships.
In functional relationships, for any arbitrary value of
one variable, there is a specific value of another
variable that corresponds to it. Such relationships are
specifically studied in subjects such as mathematics,
physics, and chemistry.
Topic objective. To illustrate the relationship
(dependence) between economic indicators.
In economics, technology, and other fields, it is
appropriate to study the relationships between various
numerical and qualitative indicators. Among the
indicators, there are two types of relationships -
functional and correlation (or statistical) relationships.
In functional relationships, for any arbitrary value of
one variable, there is a specific value of another
variable that corresponds to it. Such relationships are
specifically studied in subjects such as mathematics,
physics, and chemistry.
If there is such a relationship (correlation) between
two random variables, and the average value of one
variable changes in a regular manner with the change
in the average value of the other variable, then such a
relationship is called a statistical or correlation
relationship. Correlation models are very important
models in the group of economic-statistical models.
In the simplest case, the relationship between two
indicators or variables is tested. In this case, it is
called a simple correlation. If there is a relationship
between two or more indicators, then it is called a
multiple correlation.
If an indicator is related to other indicators, it is called
a dependent indicator and is identified with it, as well
as with the variables that are dependent on it, such as
argument or factor variables or simply factors.
In this case, the correlation model is written as
follows:
~
у
a bx
i
i
= +
The regression equation is a formula that expresses the
relationship between the dependent variable, which is
found through testing, the independent variables
obtained from experiments, and the constant
coefficients. They have the following economic
meanings: a represents the significance of the
unaccounted factors, b indicates how much y will
increase when x increases by one unit. The correlation
is represented graphically as follows. [
1
]
у
a bx
= +
O
correlation field
x
(1)
Creating correlation models consists of several stages:
1) Formulating the problem;
2) Collecting statistical data;
3) Determining the form of the regression equation;
4) Finding the strength of the relationship;
5) Finding finite values for the parameters in the
regression equation;
6) Applying the obtained result to economics.
Let's say we have a regression equation with an
unknown variable
~
у
a bx
i
i
= +
where the size of the
sample is N. To find the equation a and b its
coefficients, we will use the method of least squares:
(
)
S
y
y
i
i
i
N
=
−
=
=
~
min
1
2
(2)
Here we
у
i
-
present the positive indicators that can
be obtained from experience: (1) and (2).
(
)
S
y
a bx
i
i
i
N
=
− −
=
=
1
2
min
We obtain special properties with respect to the
normal distribution, i.e.
(
)
(
)
S
a
y
a bx
S
b
x
y
a bx
i
i
i
N
i
i
i
i
N
= −
− −
=
= −
− −
=
=
=
2
0
2
0
1
1
(3)
(3) Let's multiply the base of the triangle by (-2) and
divide the system into both sides.
(
)
(
)
y
a bx
x
y
a bx
i
i
i
N
i
i
i
i
N
− −
=
− −
=
=
=
1
1
0
0
(4)
(4) We open the locks and simplify.
Na b
x
y
a
x
b
x
x y
i
i
N
i
i
N
i
i
i
N
i
N
i i
i
N
+
=
+
=
=
=
=
=
=
1
1
2
1
1
1
(5)
(5) Kramer left with a sad face.
a
y
x
x y
x
N
x
x
x
i
i
N
i
i
N
i i
i
N
i
i
N
i
i
N
i
i
N
i
i
N
=
=
=
=
=
=
=
=
1
1
1
2
1
1
1
2
1
b
N
y
x
x y
N
x
x
x
i
i
N
i
i
N
i i
i
N
i
i
N
i
i
N
i
i
N
=
=
=
=
=
=
=
1
1
1
1
1
2
1
[
2
]
The correlation coefficient is used to determine the
strength of the relationship.
(
)(
)
(
)
r
x
x y
y
N
yx
i
i
i
N
x
y
=
−
−
−
=
1
1
(
)
(
)
x
i
i
N
x
x
N
=
−
−
=
2
1
1
(
)
(
)
y
i
i
N
y
y
N
=
−
−
=
2
1
1
If
r
yx
is close to 1, it is called a strong connection,
otherwise it is called a weak connection, and it varies
in the range of
0
1
r
yx
. [
3
]
Analyse the correlation relationship between the cost
incurred by the British government for road
construction from 2013 to 2023, denoted as Y (in
million pounds sterling), and the time. The results
required to find the coefficients of the regression
equation based on the economic information provided
in the table are presented.
Year
t
t
2
y
i
y
i
2
t
y
i
2013
1
1
560
313600
560
2014
2
4
608
369664
1216
2015
3
9
685
469225
2055
2016
4
16
807
651249
3228
2017
5
25
839
703921
4195
2018
6
36
914
935396
5484
2019
7
49
1100
1210000
7700
2020
8
64
1196
1430416
9568
2021
9
81
1490
2247001
13491
2022
10
100
1544
2477476
15740
2023
11
121
1513
2289169
16643
66
506
11295
12997117
79880
b
=
−
11 79880
11 506 66
2
=
−
−
878680 745470
5566 4356
=
=
133210
1210
110 09
,
a
=
−
=
−
=
11295
11
110 09 66
11
1026 81 660 54 366 27
,
,
,
,
The equation of regression in the result.
~
,
,
у
t
i
=
+
366 27 110 09
It will be seen in the future. This discovered
regression equation provides the opportunity to predict
for the next year, i.e., when t=12.
~
,
,
,
у
12
366 27 110 09 12 1687 35
=
+
=
The correlation coefficient of
r
=
0 95
,
indicates
that there is a strong positive correlation between the
variables. [
4
]
The Student criterion is used to determine the
consistency of coefficients.
t
b
S
j
j
b
j
=
In this equation,
b
j
is the coefficient of the
b
j
-
regression;
S
b
j
is the average quadratic deviation of
the coefficient
j
.
S
b
y
S
b
j
j
i
i
N
j
=
=
2
2
1
(
)
S
y
y
N
i
i
i
i
N
2
2
1
1
=
−
−
=
We use the Fisher criterion to determine the stability
of the regression equation.
F
S
S
y
=
2
2
(
)
S
y
y
N
y
i
i
N
2
2
1
1
=
−
−
=
𝑆̄
𝑦̃
2
=
∑
(𝑦̃
𝑖
−𝑦̃̄)
2
𝑁
𝑖=1
𝑁−1
When these criteria are taken into account, the larger
the result is compared to the result in the table, the
more stable the regression equation becomes. [
5
]
2. Multiple factor correlation relationship.
When testing practical problems, the correlation
relationship is usually dependent on factors that have a
significant impact, and in this case, the regression
equation takes the following form: In this case, the
regression equation is:
𝑦̃
𝑖
= 𝑏
0
+ 𝑏
1
𝑥
1𝑖
+
𝑏
2
𝑥
2𝑖
+. . . +𝑏
𝑘
𝑥
𝑘
Let's write the given statistical data in the form of a
table: (1)
№
𝑥
1𝑖
𝑥
2𝑖
𝑥
3𝑖
…
𝑥
𝑘𝑖
𝑦
𝑖
1
𝑥
11
𝑥
21
𝑥
31
…
𝑥
𝑘1
𝑦
2
𝑥
12
𝑥
22
𝑥
32
…
𝑥
𝑘2
𝑦
2
3
𝑥
13
𝑥
23
𝑥
33
…
𝑥
𝑘3
𝑦
3
⋮
⋮
⋮
⋮
⋮
⋮
⋮
N
𝑥
1𝑁
𝑥
2𝑁
𝑥
3𝑁
…
𝑥
𝑘𝑁
𝑦
𝑁
Let's convert the measurement from the old unit to the
new unit using the following formulas:
𝑦
𝑖
0
=
𝑦
𝑖
−𝑦̄
𝑆𝑦
𝑥
𝑗𝑖
0
=
𝑥
𝑗𝑖
−𝑥̄
𝑗
𝑆𝑥
𝑗
𝑖 = 1,2, . . . , 𝑁
𝑗 = 1,2, . . . , 𝑘
Here,
𝑦
𝑖
0
and
𝑥
𝑗𝑖
0
are the normalized values of the
factors;
𝑦̄
and
𝑥̄
𝑗
are the average values of the factors;
𝑆
𝑦
and
𝑆
𝑥
𝑗
are the average quadratic deviations of the
factors.
𝑆
𝑦
= √
∑
(𝑦
𝑖
−𝑦̄
𝑖
)
2
𝑁
𝑖=1
𝑁−1
𝑆
𝑥
𝑗
= √
∑
(𝑥
𝑗𝑖
−𝑥̄
𝑗
)
2
𝑁
𝑖=1
𝑁−1
[
6
]
We will present statistical data in the 2nd table with
new measurements.
№
𝑥
1
0
𝑥
1
0
𝑥
1
0
…
𝑦
𝑖
1
𝑥
11
0
𝑥
21
0
𝑥
31
0
…
𝑦
𝑖
0
2
𝑥
12
0
𝑥
22
0
𝑥
32
0
…
𝑦
2
0
3
𝑥
13
0
𝑥
23
0
𝑥
33
0
…
𝑦
3
0
⋮
⋮
⋮
⋮
⋮
⋮
N
𝑥
1𝑁
0
𝑥
2𝑁
0
𝑥
3𝑁
0
…
𝑦
𝑁
0
On a new scale
𝑥̄
𝑗
0
= 0,
𝑦̄
0
= 0
𝑆
𝑥
𝑗
0
= 1,
𝑆
𝑦
0
= 1
(3)
The coefficient of determination is given by the
following formula:
𝑟
𝑦
0
𝑥
𝑗
0
∗
=
1
𝑁 − 1
∑ 𝑦
𝑖
0
𝑥
𝑗𝑖
0
𝑁
𝑖=1
𝑟
𝑥
𝑖
0
𝑥
𝑚
0
∗
=
1
𝑁 − 1
∑ 𝑥
𝑗𝑖
0
𝑥
𝑚𝑖
0
𝑁
𝑖=1
}
𝑙 > 𝑚
𝑙, 𝑚 = 1,2, . . . , 𝑘
(4)
In general, the error term in the regression equation is
not independent and follows the following pattern:
𝑦̃
0
= 𝑎
1
𝑥
1
0
+ 𝑎
2
𝑥
2
0
+. . . +𝑎
𝑘
𝑥
𝑘
0
(5)
(5) The coefficients are found from this condition:
𝑆 = ∑(𝑦
𝑖
0
− 𝑦̃
𝑖
0
)
2
𝑁
𝑖=1
= 𝑚𝑖𝑛
The minimum condition of the function is found in
only one variable.
𝜕𝑆
𝜕𝑎
1
= 0
𝜕𝑆
𝜕𝑎
2
= 0. . .
𝜕𝑆
𝜕𝑎
𝑘
= 0
(6)
𝑎
1
∑
(𝑥
1𝑖
0
)
2
𝑁
𝑖=1
+ 𝑎
2
∑
𝑥
1𝑖
0
𝑥
2𝑖
0
𝑁
𝑖=1
+. . . +𝑎
𝑘
∑
𝑥
1𝑖
0
𝑥
𝑘𝑖
0
𝑁
𝑖=1
= ∑
𝑥
1𝑖
0
𝑦
𝑖
0
𝑁
𝑖=1
𝑎
1
∑
𝑥
2𝑖
0
𝑥
1𝑖
0
𝑁
𝑖=1
+ 𝑎
2
∑
(𝑥
2𝑖
0
)0,2
𝑁
𝑖=1
+. . . +𝑎
𝑘
∑
𝑥
2𝑖
0
𝑥
𝑘𝑖
0
𝑁
𝑖=1
= ∑
𝑥
2𝑖
0
𝑦
𝑖
0
𝑁
𝑖=1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
𝑎
1
∑
𝑥
𝑘𝑖
0
𝑥
1𝑖
0
𝑁
𝑖=1
+ 𝑎
2
∑
𝑥
𝑘𝑖
0
𝑥
2𝑖
0
𝑁
𝑖=1
+. . . +𝑎
𝑘
∑
(𝑥
𝑘𝑖
0
)
2
𝑁
𝑖=1
= ∑
𝑥
𝑘𝑖
0
𝑦
𝑖
0
𝑁
𝑖=1
}
(7)
We increase the left and right sides of the system by
1
𝑁−1
. As a result, the determination coefficient
𝑟
∗
is
mainly produced in front of the
𝑎
𝑗
coefficients, which
is (7).
1
𝑁−1
∑
(𝑥
𝑗𝑖
0
)
2
= 𝑆
𝑥
𝑗
0
2
= 1
𝑁
𝑖=1
[
7
]
Let's create a normal equations system:
𝑎
1
+ 𝑎
2
𝑟
𝑥
1
𝑥
2
∗
+ 𝑎
3
𝑟
𝑥
1
𝑥
3
∗
+. . . +𝑎
𝑘
𝑟
𝑥
1
𝑥
𝑘
∗
= 𝑟
𝑦𝑥
1
∗
𝑎
1
𝑟
𝑥
2
𝑥
1
∗
+ 𝑎
2
+ 𝑎
3
𝑟
𝑥
2
𝑥
3
∗
+. . . +𝑎
𝑘
𝑟
𝑥
2
𝑥
𝑘
∗
= 𝑟
𝑦𝑥
2
∗
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
𝑎
1
𝑟
𝑥
𝑘
𝑥
1
∗
+ 𝑎
2
𝑟
𝑥
𝑘
𝑥
2
∗
+ 𝑎
3
𝑟
𝑥
𝑘
𝑥
3
∗
. . . +𝑎
𝑘
= 𝑟
𝑦𝑥
𝑘
∗
}
If we calculate the correlation coefficients in
𝑟
𝑥
𝑙
𝑥
𝑚
∗
=
𝑟
𝑥
𝑚
𝑥
𝑖
∗
, they are found to be zero.
(8) It is possible to calculate the correlation coefficient
R for a large number of factors.
𝑅 = √𝑎
1
𝑟
𝑦𝑥
1
∗
+ 𝑎
2
𝑟
𝑦𝑥
2
∗
+. . . +𝑎
𝑘
𝑟
𝑦𝑥
𝑘
∗
The correlation coefficient R varies in the following
range:
0 ≤ 𝑅 ≤ 1
If the value of R is close to 1, the correlation
relationship is considered strong or stable. To use (5)
in practice, we need to convert it to the natural scale
using these formulas:
𝑏
𝑗
= 𝑎
𝑗
𝑆
𝑦
𝑆
𝑥
𝑦
𝑗 = 1,2, . . . , 𝑘
𝑗 ≠ 0
𝑏
0
= 𝑦̄ − ∑ 𝑏
𝑗
𝑥̄
𝑗
𝑅
𝑗=1
An algorithm has been written in "Pascal" language
for the mentioned algorithm, making it convenient to
solve practical problems. [
8
]
Years
Indicators
Labor
productivity
Armed
forces
fund
x
1
Specialization
level
x
2
Workload
x
3
1
100
100
100
100
2
109
108
140
109
3
119
130
104
123
4
132
135
99
139
5
139
212
97
153
6
148
152
102
157
7
156
158
120
164
8
165
265
147
171
9
173
288
151
179
10
188
305
155
195
11
192
325
171
204
12
197
350
199
215
13
202
360
226
228
14
206
400
226
236
15
211
392
238
256
CONCLUSIONS
The regression equation for this issue
~
y
b
b x
b x
b x
i
i
i
i
= +
+
+
0
1 1
2 2
3 3
(1)
Need to find Z from (1)
j
=
0 1 2,3
, ,
We use the formula of correlation to find unknown
b
j
, with
j
=
0 4
,
coefficients, according to the
instructions of the program: refer to the results
obtained from the computer.
b
0
43 9
=
,
b
1
0 157
=
,
b
2
0 026
= −
,
b
3
0 467
=
,
.
If we substitute the obtained results into (1)
~
,
,
,
,
y
x
x
x
i
i
i
i
=
+
−
+
43 9 0157
0 026
0 467
1
2
3
The stability of the relationship found in (2) is
indicated by the correlation coefficient.
R
=
0 985
,
Fisher's criterion
𝐹 = 182,3 (𝐹
𝑗𝑎𝑑𝑣
= 6,54)
When
F=1, it indicates the correctness of equation (2).
ACKNOWLEDGMENTS
The author thanks Kayumov E. and Ataniyazova M.,
professors and teachers of Tashkent University of
Applied Sciences, for their scientific and practical
help in writing this article.
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математик
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К.Д.Льюнс.
Методы
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