The American Journal of Applied Sciences
18
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TYPE
Original Research
PAGE NO.
18-25
10.37547/tajas/Volume07Issue06-04
OPEN ACCESS
SUBMITED
14 April 2025
ACCEPTED
10 May 2025
PUBLISHED
12 June 2025
VOLUME
Vol.07 Issue06 2025
CITATION
Fayziyev Axtam Asraevich. (2025). Statistical Analysis and Forecasting of
The Dynamics of Pollutant Emissions into The Atmosphere in The Republic
of Uzbekistan. The American Journal of Applied Sciences, 7(06), 18
–
25.
https://doi.org/10.37547/tajas/Volume07Issue06-04
COPYRIGHT
© 2025 Original content from this work may be used under the terms
of the creative commons attributes 4.0 License.
Statistical Analysis and
Forecasting of The
Dynamics of Pollutant
Emissions into The
Atmosphere in The
Republic of Uzbekistan
Fayziyev Axtam Asraevich
Candidate of Physical and Mathematical Sciences, Acting Professor,
Tashkent University of Economics and Pedagogy, Republic of Uzbekistan
Abstract:
Air pollution is one of the most serious
environmental threats to human health. In this article,
using the statistical analysis method of time series, the
statistical
regularity
of
the
dynamic
series
ytˉ
\
bar{y_t}ytˉ —
the average amount of pollutants
emitted into the atmosphere in the Republic of
Uzbekistan
—
is studied (based on data from the State
Statistics Committee of the Republic of Uzbekistan for
the period 2011
–
2022). With a 95% confidence level,
point and interval estimates of the average amount of
atmospheric pollutant emissions in Uzbekistan are
constructed, obvious trends are identified, and
forecasts are made for the following years. Using the
Durbin-Watson statistical criterion, it is established that
the average amount of pollutants emitted into the
atmosphere has autocorrelation dependence.
The applied methods of processing and analyzing
dynamic series, after testing, can be used in the research
of graduate students and scientific researchers.
Keywords:
Discrete,
pollutants,
substances,
atmosphere, dynamic, seasonality, component, linear,
least squares, normal, hypothesis, autocorrelation,
skewness, kurtosis.
Introduction:
In almost every field, there are
phenomena that are important to study in their
development and change over time. For example, one
may seek to predict the future based on past data,
control a process, or describe the characteristic features
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of a series based on a limited amount of information.
In time series processing, the methods largely rely on
those developed by mathematical statistics for
dynamic distribution series. Currently, statistics has a
wide variety of time series analysis methods ranging
from the most elementary to quite complex ones ([1,
2, 3, 4]).
METHODS
This study involves the processing and analysis of the
amount of pollutants emitted into the atmosphere in
the Republic of Uzbekistan during the observation
period from 2011 to 2022, treated as a discrete time
series. The study of the dynamics of atmospheric
pollutant emissions globally, and in particular in
Uzbekistan, plays an important role in environmental
science.
In the general case, a time series
𝑦
𝑡
̅
—
consists of four
components:
1.
Trend
2.
Fluctuations around the trend
3.
Seasonal effect
4.
Random component
This study uses time series processing and analysis
methods such as: trend determination methods,
normality and randomness tests, autocorrelation
check, moving average method, finite differences
method, least squares method, Durbin-Watson
criterion, and others.
Using time series statistical analysis methods, the
quantity of atmospheric pollutant emissions in the
Republic of Uzbekistan is estimated and forecasted.
The study and analysis of dynamic series has been
addressed in the works of Anderson [1], Kendall [2],
Tikhomirov [3], Sulaimanov [4], Fayziyev [5], and others.
RESULTS AND DISCUSSION
Let us assume that the quantity of pollutants emitted
into the atmosphere in the Republic of Uzbekistan over
the observation period 2011
–
2022 forms a discrete time
series. Using the above-mentioned time series statistical
analysis methods, we construct point and interval
estimates for the amount of atmospheric pollutants,
determine the evident trend type of this process, and
forecast it for subsequent years. We also test various
statistical hypotheses related to this process.
In Figure 1, based on empirical data (Table 1, Column 3),
the dynamics of atmospheric pollutant emissions in the
Republic of Uzbekistan are geometrically represented as
follows:
a) point plot,
b) histogram,
c) pie chart,
d) radar (spider) chart.
а)
b)
с) d)
Figure 1.
0
500
1000
1500
1
2
3
4
5
6
7
8
9 10 11 12
0
500
1000
1500
2011 2013 2015 2017 2019 2021
788.2817.6
855.2
1162.1
975.1
1008.2
853.5
883.7
952.8
924.4
909
874
2011
2012
2013
2014
2015
2016
2017
2018
2019
2020
2021
2022
788.2
817.6
855.2
1162.1
975.1
1008.2
853.5
883.7
952.8
924.4
909
874
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The geometric representation of the empirical data in
a coordinate system provides grounds, at the first
approximation, to hypothesize that the trend
component of this process (the general direction of its
development) has a linear dependence of the form:
yt=a+bty_t = a + btyt=a+bt
where the unknown parameters aaa and bbb are
determined by the
method of least squares
, i.e., based
on empirical data by solving the following system of
normal equations:
{
=
+
t
y
t
a
T
a
1
0
t
y
t
a
t
a
t
=
+
2
1
0
(1)
By solving the system of equations (1) and using the calculations based on Table 1, we obtain:
∑ 𝑦
𝑡
11003,8
thousand tons,
𝑎
0
=
1
Т
∑ 𝑦
𝑡
=
11003,8
12
= 916,98
thousand tons
𝑎
1
=
1
∑ 𝑡
2
∑ 𝑦
𝑡
𝑡 =
5889,6
146
= 40, 34
.From this, we determine the
linear trend equation
(tendency) for the
amount of pollutants emitted:
This equation serves as the
estimated trend of the time series
for the pollutant emissions in the Republic of
Uzbekistan.
These calculations form the basis for determining the time series trend.
Table -1
1
2
3
4
5
6
7
N
Year
of
Observation
𝑦
𝑡
thousand
tons
t
t
2
𝑦
𝑡
𝑡
𝑦
𝑡
𝑡
2
1
2011
788,2
-5
25
-3941
19705
2
2012
817,6
-4
16
-3270,4
13081,6
3
2013
855,2
-3
9
-2565,6
7696,8
4
2014
1162,1
-2
4
-2324,2
4648,4
5
2015
975,1
-1
1
-975,1
975,1
6
2016
1008,2
0
0
0
0
7
2017
853,5
1
1
853,5
853,5
8
2018
883,7
2
4
1767,4
3534,8
9
2019
952,8
3
9
2858,4
8575,2
10
2020
924,4
4
16
3697,6
14790,4
11
2021
909
5
25
4545
22725
12
2022
874
6
36
5244
31464
Total
11003,8
6
146
5889,6
128050
Pollutants Emitted into the Atmosphere of the Republic of Uzbekistan [1, 2, 3, 4, 5]::
𝑦(𝑡) = 40, 34 𝑡 + 916,98
(2
In particular, by substituting t=3t = 3t=3 into Equation (2), we obtain the
expected amount of pollutants emitted
into the atmosphere
of the Republic of Uzbekistan in the year
2025
, which is on average
1,038 thousand tons
.
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Using statistical criteria ([1]
–
[5]), it was established that in Equation (2), the
null hypothesis
H0:a1=0H_0:
a_1 = 0
0
1
)
(
a
t
a
t
y
+
=
is rejected and the
alternative hypothesis
𝐻
0
∶ 𝑎
1
= 0
отвергается
и
принимается
альтерна
-
тивная
гипотеза
𝐻
1
∶ 𝑎
1
≠ 0
с
is accepted at a
significance level of
α
=0.05\alpha = 0.05
α
=0.05.
Therefore, the amount of pollutants emitted into the atmosphere in the Republic of Uzbekistan exhibits a
linear
trend
.
Autocorrelation
represents the correlation dependence between subsequent and preceding members of a time
series. We test for the presence of
autocorrelation
in the average amount of pollutants emitted into the
atmosphere in the Republic of Uzbekistan, that is:
Yt=ρYt−1+e
t,Y_t = \rho Y_{t-1} + e_t,
Y
t
=
Y
t-1
+
t
,
where
𝜌
= Cov(Y
t
,Y
t+1
) = M
[(𝑌
𝑡
− 𝑦
𝑡
̅ )( 𝑌
𝑡+1
− 𝑦
𝑡
̅ )]
.
For further analysis, it is necessary to calculate the following
finite differences
based on the observed data. We
denote:,
,
,
.
According to Table 2, the following are calculated:
(3)
The
coefficients of variation of the differences
are calculated, and it is established that
V1≈V2≈V3.V_1
\approx V_2 \approx V_3.V1
≈
V2
≈
V3.
Consequently, the
first-order finite differences eliminate the linear trend
.
Let us now construct a table for the calculation of finite differences.
Table 2
–
Calculation of Finite Differences
2
3
4
5
6
7
8
9
Year of
Observ
ation
Y(t)
thousand
tons
Y
t
2
ΔY
t
ΔY
t
2
Δ
2
Y
t
Δ
2
Y
t
2
Δ
3
Y
t
Δ
3
Y
t
2
2011
788,2
621259,2
2012
817,6
668469,8
29,4
864,36
2013
855,2
731367
37,6
1413,76
8,2
67,24
2014
1162,1
1350476
306,9
94187,6
1
269,3
72522,4
9
261,1
68173,21
2015
975,1
950820
-187
34969
-493,9
243937,
2
-
763,2
582474,2
2016
1008,2
1016467
33,1
1095,61
220,1
48444,0
1
714
509796
2017
853,5
728462,3
-154,7
23932,0
9
-187,8
35268,8
4
-
407,9
166382,4
2018
883,7
780925,7
30,2
912,04
184,9
34188,0
1
372,7
138905,3
2019
952,8
907827,8
69,1
4774,81
38,9
1513,21
-146
21316
2020
924,4
854515,4
-28,4
806,56
-97,5
9506,25
-
18604,96
t
t
t
Y
Y
Y
−
=
+
1
t
t
t
Y
Y
Y
−
=
+
1
2
t
t
t
Y
Y
Y
2
1
2
3
−
=
+
(
)
(
)
C
k
k
T
k
t
t
k
k
k
T
y
V
2
2
−
=
=
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136,4
2021
909
826281
-15,4
237,16
13
169
110,5
12210,25
2022
874
763876
-35
1225
-19,6
384,16
-32,6
1062,76
Total
11003,8
10200748
85,8
164418
-64,4
446000,
4
-27,8
1518925
Table 3
–
Data for Calculating Autocorrelation Indicators
1
2
3
4
4
5
6
7
N
п/п
Year of
Obser
vation
thousand
tons
1
2011
788,2
2
2012
817,6
644432,3
3
2013
855,2
699211,5
674068,6
4
2014
1162,1
993827,9
950133
915967,2
5
2015
975,1
1133164
833905,5
797241,8
768573,8
6
2016
1008,2
983095,8
1171629
862212,6
824304,3
794663,2
7
2017
853,5
860498,7
832247,9
991852,4
729913,2
697821,6
8
2018
883,7
754238
890946,3
861695,9
1026948
755740,2
9
2019
952,8
841989,4
813214,8
960613
929075,3
1107249
10
2020
924,4
880768,3
816892,3
788975,4
931980,1
901382,4
11
2021
909
840279,6
866095,2
803283,3
775831,5
916453,8
12
2022
874
794466
807925,6
832747,2
772353,8
745959
totla
11003,8
9425971
8657058
7814589
6758980
5919269
Using Table 3
and formulas from the literature [1, 2, 3, 4, 5], the
autocorrelation coefficients RLR_LRL
are
determined for
lags L=1,2,3,4,5L = 1, 2, 3, 4, 5L=1,2,3,4,5
(where
LLL
is the
lag
, i.e., the time shift
—
the time
interval by which one phenomenon lags behind another related to it):
(4)
A
significant deviation of the autocorrelation
coefficient RLR_LRL
from zero provides grounds to
assume that there is a
substantial autocorrelation
dependence
in the average amount of pollutants
emitted into the atmosphere in the Republic of
Uzbekistan.
On the other hand, let us test the
hypothesis of the
existence of autocorrelation
in the average amount of
atmospheric pollutant emissions in the Republic of
Uzbekistan using the
Durbin
–
Watson test
:
−
−
−
−
−
−
=
+
=
+
=
−
=
−
=
−
=
−
=
+
=
+
N
L
t
N
L
t
t
t
L
N
t
L
N
t
t
t
L
N
t
L
N
t
N
L
t
t
t
L
t
t
L
L
N
Y
Y
L
N
Y
Y
L
N
Y
Y
Y
Y
R
1
2
1
2
1
2
1
2
1
1
1
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.
(5)
Using formula (5), we calculate: (5)
𝑑
наб
= 0, 006
and
compare it with the
critical value
с 𝑑
крит
= 1,22
from the table values [1], [2], [5]. Since
𝑑
наб
=0,006
<
𝑑
крит
= 1,22
. it follows that, with
95% confidence
, the
Durbin
–
Watson test
confirms that the
average amount
of pollutants emitted into the atmosphere in the
Republic of Uzbekistan
exhibits an
autocorrelated
dependence
: Y
t
=
Y
t-1
+
t
. Therefore, the average
amount of pollutants emitted into the atmosphere in
the Republic of Uzbekistan
this year depends on the
emissions from previous and subsequent years
.
Testing the Statistical Hypothesis of Normality For the
variable ,
𝑦
𝑡
̅ the average amount of pollutants emitted into the atmosphere in the Republic of Uzbekistan ([1]– [5]):
Н
0
: 𝑃(𝑦
𝑡
̅ < 𝑥) = Ф
а,𝜎
(х), Н
1
: 𝑃(𝑦
𝑡
̅ < 𝑥) ≠ Ф
а,𝜎
(х)
The
significance level
is chosen as
α
=0.05\alpha = 0.05
α
=0.05 (see Table 4).
Then, using the following formula, we construct the
confidence interval estimate
for the
average amount of
pollutants emitted into the atmosphere in the Republic of Uzbekistan
:
(6)
were
The
critical value
𝑡
крит.
= 𝑡(𝑇 − 2; 𝛼)
is determined from the
Student’s t
-distribution table
(see [5], p. 181).
Using
formula (6)
, we determine the
confidence interval estimate
for
𝑦
𝑡
̅
—
the
average annual amount of
pollutants emitted into the atmosphere
in the Republic of Uzbekistan,
with a probability of 0.95
:
:
(853,44; 980,52 )
thousand tons
Based on sample data and using the
software package X7.2019
and
Microsoft Excel
([4], [5], [6]), we compute
the
statistical characteristics
of
𝑦
𝑡
̅
—
the
average annual amount of pollutants emitted into the atmosphere in
the Republic of Uzbekistan
(see
Table 4
).
Estimation of the Main Parameters of the Time Series
Table 4
Выборочные характеристики
Оценки выборочные характеристик
Average annual amount of pollutants
emitted
𝑦
𝑡
̅
–
thousand tons
916,98
≈ 917
Variance
10040,60
Standard deviation
𝜎
𝑇
100,20
Coefficient of variation
𝑣
(%)
10,93 %
Skewness А
1,30
Kurtosis
𝐸
𝐾
2,38
Standard error of the mean
𝑦̅
Т
,
𝑚
у
m
у
=
𝜎
у
√𝑛
= 28,88
Maximum error of the mean
𝑚
у
′
m’
у
= t m
у
=
2,20 ∙ 28,88 = 63,54
Error of the standard deviation
𝜎
𝑇
m
𝜎
=
𝜎
√2𝑛
=
100,20
4,90
= 20,45
Confidence interval estimate (95% )
𝑦̅
Т
±
𝑦̅
𝑇
± t m
у
=
916,98
± 63,54
−
=
=
+
−
=
1
1
1
2
2
1
/
)
(
T
t
T
t
t
t
t
Y
Y
Y
d
(
)
(
)
(
)
y
i
T
y
i
T
T
t
Y
i
T
a
a
T
t
Y
;
2
;
2
1
0
−
+
+
+
−
−
+
+
;
)
(
2
1
1
5
.
0
1
2
2
−
+
−
+
=
=
i
y
t
t
i
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The American Journal of Applied Sciences
𝑡𝑚
у
(853,44; 980,52 )
тысяча тонна
Test of statistical hypothesis
Н
0
: 𝑃(𝑦̅
𝑡
< 𝑥) = Ф
а,𝜎
(х)
95% The null hypothesis
Н
0
is
accepted with a given level of confidence.
CONCLUSIONS
Based on the above statistical analyses, the dynamics
of
𝑦
𝑡
̅ −
the
average annual amount of pollutants
emitted into the atmosphere
in the Republic of
Uzbekistan
—
considered as a
discrete time series
,
with
95% confidence
(Table 4), the following
conclusions can be drawn:
1.
Point and interval statistical estimates
have been
constructed for
𝑦
𝑡
̅ −
—
the average annual amount of
pollutants emitted into the atmosphere in the Republic
of Uzbekistan. In particular, the forecasted average
annual amount of atmospheric pollutants for the year
2023
is expected, with
95% confidence
, to fall within
the interval:
(853,44; 980,52 )
тысяча тонна
;
2. The
explicit type of trend
has been determined and
its
linearity established
, with the following equation:
𝒚(𝒕) = 𝟒𝟎, 𝟑𝟒 𝒕 +
𝟗𝟏𝟔, 𝟗𝟖
;
3. Based on the
Durbin
–
Watson criterion
, it was
established that the
average annual amount of
pollutants emitted into the atmosphere
in the
Republic of Uzbekistan exhibits an
autocorrelated
dependence
:Y
t
=
Y
t-1
+
t
, где
𝜌
= Cov(Y
t
,Y
t+1
) =
M
[(𝑌
𝑡
− 𝑦
𝑡
̅ )( 𝑌
𝑡+1
− 𝑦
𝑡
̅ )]
i.e., the volume of pollutants
emitted this year depends on the amounts from
previous and following years.
4. It has been shown that the
dynamics of the average
amount of pollutants
in Uzbekistan
form a non-
stationary time series
.;
5. The
high level of atmospheric pollution
globally,
and in particular in the Republic of Uzbekistan, warns
humanity of the urgent need to
take immediate action
to reduce pollution levels. This requires
coordinated
efforts
from
local, national, and regional authorities
in sectors such as
energy, transportation, waste
management, urban planning, and agriculture
.
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