American Journal of Applied Science and Technology
36
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VOLUME
Vol.05 Issue 06 2025
PAGE NO.
36-43
10.37547/ajast/Volume05Issue06-09
Statistical Analysis and Forecasting Of Cotton Yield
Dynamics In Region
Fayziyev Axtam Asraevich
Candidate Of Physical And Mathematical Sciences, Acting Professor, Tashkent University Of Economics And Pedagogy, Republic Of
Uzbekistan
Received:
14 April 2025;
Accepted:
10 May 2025;
Published:
17 June 2025
Abstract:
Observations of a certain phenomenon, the nature of which changes over time, give rise to an ordered
sequence, which is called a time series. In the article, using the method of statistical time series analysis, statistical
pattern of time series ((y_t )
̅
) -average yield of cotton in 5-regions of the Republic of Uzbekistan(on materials of
CSB of the Republic of Uzbekistan for 20-30 years). Point and interval estimates for the average cotton yield were
constructed with a 95% guarantee, explicit types of trends were determined, and the yield in the region was
predicted for subsequent years. Using statistical criteria of Durbin-Watson, it was found that the average yield of
cotton in the region has an autocorrelation dependence. The used methods of processing and analysis of dynamic
series after testing can be used in the research of masters and researchers.
Keywords:
Discrete, dynamic, series, trend, seasonality, component, linear, smallest, hypothesis, autocorrelation,
asymmetry, kurtosis.
Introduction:
In almost every field, there are phenomena that are
important to study in terms of their development and
changes over time. For example, one may strive to
predict the future based on knowledge of the past,
manage a process, or describe the key features of a
data series based on a limited amount of information.
When processing time series data, the methods
largely
rely
on
techniques
developed
by
mathematical statistics for distribution series. At
present, statistics offers a wide variety of time series
analysis methods, ranging from the simplest to quite
complex ones ([1,2,3,4,5]).
In general, a time series yt,t
∈
T{y_t, t
∈
T}yt,t
∈
T
consists of four components: trend; fluctuations
around the trend; seasonal effect; and random
component ([1,2,3,4,5]).
In this study, cotton yield data over the past 20
–
30
years in five regions of the Republic of Uzbekistan is
processed and analyzed as a discrete time series.
Using statistical methods of time series analysis, point
and interval estimates of the average cotton yield
were constructed, explicit types of trends were
identified, forecasts for future yields were made, and
various statistical hypotheses were tested.
The study and analysis of dynamic series have been
addressed in the works of Anderson [1], Kendall [2],
Tikhomirov [3], Vainu [4], Sulaimanov [5], and others.
Analysis of Results and Examples.
A graphical representation of the observed data
(Table 1, column 3, showing the statistical analysis of
cotton yield dynamics for the Bukhara region) and the
coordinate system provide a basis to preliminarily
assume the hypothesis that the trend component of
the process follows a linear dependence (Figure 1) of
the form:
yt=a+bty_t = a + btyt=a+bt
where the unknown parameters are determined
using the least squares method, i.e., based on the
empirical data, by solving the following system of
normal equations:
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American Journal of Applied Science and Technology (ISSN: 2771-2745)
{
=
+
t
y
t
a
T
a
1
0
t
y
t
a
t
a
t
=
+
2
1
0
(1)
а)
в) с)
Figure 1. Diagram of the Time Series.
Using the calculations from Table 1, we obtain:
∑ 𝑦
𝑡
= 564 , 𝑎
0
=
1
Т
∑ 𝑦
𝑡
=
564
19
= 29,68
,
𝑎
1
=
1
∑ 𝑡
2
∑ 𝑦
𝑡
𝑡 =
19,7
570
= 0,035
.
From this, the linear trend (tendency) equation for cotton yield in the region is determined as:
y(t)=0.035t+29.68(2)y(t) = 0.035t + 29.68 \tag{2}y(t)=0.035t+29.68(2)
Using statistical criteria, it was established that in this equation the null hypothesis
H0:a1=0H_0: a_1 = 0H0a1=0
is rejected, and the alternative hypothesis
H1:a
1≠0H_1: a_1
\ne 0H1:a1 =0
is accepted at the significance level of α=0.05
\
alpha = 0.05α=0.05.
Proceeding to the calculation of the data for
determining the time series trend.
Table 1
1
2
3
4
5
6
7
N
Years of Observation
𝑦
𝑡
c/ha
t
t
2
𝑦
𝑡
∙
t
𝑦
𝑡
∙ 𝑡
2
1
2001
27,1
-9
81
-243,9
2195,1
2
2002
28,2
-8
64
-225,6
1804,8
3
2003
29,3
-7
49
-205,1
1435,7
4
2004
30,6
-6
36
-183,6
1101,6
5
2005
30,9
-5
25
-154,5
772,5
6
2006
29,4
-4
16
-117,6
470,4
7
2007
30,3
-3
9
-90,9
272,7
8
2008
25,8
-2
4
-51,6
103,2
0
10
20
30
40
1 4 7 10 13 16 19
27.1 28.2
29.3
30.6
30.9
29.4
30.3
25.8
28.6
31
32.6
31.5
31.4
31.3
32.6
30.2
29.3
28.3 25.6
0
10
20
30
40
1
4
7 10 13 16 19
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American Journal of Applied Science and Technology (ISSN: 2771-2745)
9
2009
28,6
-1
1
-28,6
28,6
10
2010
31
0
0
0
0
11
2011
32,6
1
1
32,6
32,6
12
2012
31,5
2
4
63
126
13
2013
31,4
3
9
94,2
282,6
14
2014
31,3
4
16
125,2
500,8
15
2015
32,6
5
25
163
815
16
2016
30,2
6
36
181,2
1087,2
17
2017
29,3
7
49
205,1
1435,7
18
2018
28,3
8
64
226,4
1811,2
19
2019
25,6
9
81
230,4
2073,6
Total
564
0
570
19,7
16349,3
By substituting t=2t = 2t=2 into equation (2), the expected cotton yield in the Bukhara region for the year 2024 is
found to be, on average, 29.82 centners per hectare (c/ha).
Results of the Statistical Analysis of Cotton Yield Dynamics in 5 Regions of the Republic of Uzbekistan (Table 2):
Table 2
№
Area
Trend part of the time series
Forecast average yields
2025 c/ha
1
Andijan
y(t) = 0,30t + 29,47
30,67
2
Bukhara
𝑦(𝑡) = 0,035𝑡 + 29,68
29,82
3
Samarkand
𝒚
(
𝒕
)=
𝟎
,
𝟏𝟏𝒕
+
𝟐𝟑
,
𝟖𝟐
24,37
4
Fergana
𝑦(𝑡) = 0,096𝑡 + 26,46
26,94
5
Khorezm
𝒚
(
𝒕
)=
𝟎
,09
𝟏𝒕
+
𝟐
5,54
26
6
By Republic
𝒚
(
𝒕
)=
𝟎
,58
𝒕
+
𝟐
4,64
26,96
For further statistical analysis, we calculate the finite differences (Table 2):
Δ
y1,
Δy2,…
\Delta y_1, \Delta y_2,
\ldots
Δ
y1,
Δ
y2,
…
. Then, we proceed to compute the higher-order finite differences (Table 3).
Table 3
Data Calculations for Determining Finite Differences
1
2
3
4
5
6
7
8
9
Years of
Observation
Y(t)
c/ha
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
2001
27,1
734,41
2002
28,2
795,24
1,1
1,21
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American Journal of Applied Science and Technology (ISSN: 2771-2745)
2003
29,3
858,49
1,1
1,21
2,2
4,84
2004
30,6
936,36
1,3
1,69
2,4
5,76
0,2
0,04
2005
30,9
954,81
0,3
0,09
1,6
2,56
-0,8
0,64
2006
29,4
864,36
-1,5
2,25
-1,2
1,44
-2,8
7,84
2007
30,3
918,09
0,9
0,81
-0,6
0,36
0,6
0,36
2008
25,8
665,64
-4,5
20,25
-3,6
12,96
-3
9
2009
28,6
817,96
2,8
7,84
-1,7
2,89
1,9
3,61
2010
31
961
2,4
5,76
5,2
27,04
6,9
47,61
2011
32,6
1062,76
1,6
2,56
4
16
-1,2
1,44
2012
31,5
992,25
-1,1
1,21
0,5
0,25
-3,5
12,25
2013
31,4
985,96
-0,1
0,01
-1,2
1,44
-1,7
2,89
2014
31,3
979,69
-0,1
0,01
-0,2
0,04
1
1
2015
32,6
1062,76
1,3
1,69
1,2
1,44
1,4
1,96
2016
30,2
912,04
-2,4
5,76
-1,1
1,21
-2,3
5,29
2017
29,3
858,49
-0,9
0,81
-3,3
10,89
-2,2
4,84
2018
28,3
800,89
-1
1
-1,9
3,61
1,4
1,96
2019
25,6
655,36
-2,7
7,29
-3,7
13,69
-1,8
3,24
Total
564
16816,56
-1,5
61,45
-1,4
106,42
-5,9
103,97
According to Table 3, the coefficients of variation of the finite differences are calculated, and it is established that
V1≈V2≈V3.V_1
\approx V_2 \approx V_3.V1
≈
V2
≈
V3.
Therefore, the first-order finite differences eliminate the linear trend.
The presence of autocorrelation in the cotton yield time series is tested using the
Durbin
–
Watson criterion
:
d=∑t=2n(et−et−1)2∑t=1net2(3)d =
\frac{\sum_{t=2}^{n} (e_t - e_{t-1})^2}{\sum_{t=1}^{n} e_t^2}
\tag{3}d=
∑
t=1net2
∑
t=2n(et
−
et
−
1)2(3)
Using formula (3), the computed value is:
dobserved=0.0026d_{\text{observed}} = 0.0026dobserved=0.0026
This is compared with the
critical value
from the table:
dcritical=1.08([5], p. 120).d_{\text{critical}} = 1.08 \quad \text{([5], p. 120)}.dcritical=1.08([5], p. 120).
Since
dobserved=0.0026<dcritical=1.08,d_{\text{observed}} = 0.0026 < d_{\text{critical}} =
1.08,dobserved=0.0026<dcritical=1.08,
it follows that the average cotton yield in the region exhibits
autocorrelation dependence
:
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American Journal of Applied Science and Technology (ISSN: 2771-2745)
Yt=ρYt−1+et,Y_
t = \rho Y_{t-1} + e_t,Yt=
ρYt−
1+et,
where
ρ=Cov(Yt,Yt+1)=E[(Yt−yˉt)(Yt+1−yˉt)].
\rho = \text{Cov}(Y_t, Y_{t+1}) = \mathbb{E}[(Y_t - \bar{y}_t)(Y_{t+1} -
\bar{y}_t)].
ρ
=Cov(Yt,Yt+1)=E[(Yt
−yˉt
)(Yt+1
−yˉt
)].
Using Table 4 and the formulas from sources [1, 2, 3, 4, 5], the
autocorrelation coefficients
RLR_LRL are
determined for
L=1,2,3,4,5,L = 1, 2, 3, 4, 5,L=1,2,3,4,5,
where LLL represents the
lag
, i.e., the time shift or delay between the interrelated phenomena.
(4)
Data for Calculating Autocorrelation Coefficients
Table 4
T
𝑌
𝑡
𝑌
𝑡
∙ 𝑌
𝑡+1
𝑌
𝑡
∙ 𝑌
𝑡+2
𝑌
𝑡
∙ 𝑌
𝑡+3
𝑌
𝑡
∙ 𝑌
𝑡+4
𝑌
𝑡
∙ 𝑌
𝑡+5
2001
27,1
2002
28,2
764,22
2003
29,3
826,26
794,03
2004
30,6
896,58
862,92
829,26
2005
30,9
945,54
905,37
871,38
837,39
2006
29,4
908,46
899,64
861,42
829,08
796,74
2007
30,3
890,82
936,27
927,18
887,79
854,46
2008
25,8
781,74
758,52
797,22
789,48
755,94
2009
28,6
737,88
866,58
840,84
883,74
875,16
2010
31
886,6
799,8
939,3
911,4
957,9
2011
32,6
1010,6
932,36
841,08
987,78
958,44
2012
31,5
1026,9
976,5
900,9
812,7
954,45
2013
31,4
989,1
1023,64
973,4
898,04
810,12
2014
31,3
982,82
985,95
1020,38
970,3
895,18
2015
32,6
1020,38
1023,64
1026,9
1062,76
1010,6
2016
30,2
984,52
945,26
948,28
951,3
984,52
−
−
−
−
−
−
=
+
=
+
=
−
=
−
=
−
=
−
=
+
=
+
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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American Journal of Applied Science and Technology (ISSN: 2771-2745)
2017
29,3
884,86
955,18
917,09
920,02
922,95
2018
28,3
829,19
854,66
922,58
885,79
888,62
2019
25,6
724,48
750,08
773,12
834,56
801,28
Сумма
564
8029,92
15270,4
14390,33
13462,13
12466,36
A significant deviation of the autocorrelation
coefficient RLR_LRL from zero provides grounds to
assume that there is a substantial
autocorrelation
dependence
in cotton yield. Consequently, the cotton
yield in the Bukhara region this year depends on the
yields of previous and subsequent years.
Based on sample data and using the software packages
EV
М
x7.2019
and
Excel
, the numerical characteristics
yty_tyt for the
average cotton yield in the Bukhara
region
are calculated (see
Table 5
):
Estimation of the Main Parameters of the Time Series Table 5
Sample Characteristics
Sample Estimates of Characteristics
Average cotton yield
𝑦̅
Т
c/ha
29,68
Variance
4,15
Standard deviation
отклонение
𝜎
𝑇
2,04
Coefficient of variation
𝑣
(%)
6,87 %
Skewness А
-0,60
Kurtosis
𝐸
𝐾
-0,25
Standard error of the mean
𝑦̅
Т
,
𝑚
у
m
у
=
𝜎
у
√𝑛
= 0,58
Maximum error
𝑚
у
′
m’
у
= t m
у
= 2,06
∙ 0
,47 = 0,97
Standard error of the standard
deviation
𝜎
𝑇
m
𝜎
=
𝜎
√2𝑛
=
2,04
6,16
= 0,33
Confidence interval (95%)
:
𝑦̅
Т
± 𝑡𝑚
у
for cotton yield
𝑦̅
𝑇
± t m
у
= 29,68 ± 0,97
(28,71; 30,65 )
ц
/
га
Statistical hypothesis testing
:
Н
0
: 𝑃(𝑋 < 𝑥) = Ф
а,𝜎
(х)
95%
гарантий гипотезы
Н
0
принимается
Results of the Statistical Analysis of Cotton Yield Dynamics in 5 Regions and in the Republic of Uzbekistan
Table 6
Parameter
Estimation
Andijan
Bukhara
Samarkand
Fergana
Khorezm
Republic-
wide or
Across the
Republic
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American Journal of Applied Science and Technology (ISSN: 2771-2745)
Average cotton
yield
𝑦̅
Т
c/ha
29,47
29,68
23,82
26,46
25,54
24,64
Variance
6,70
4,15
4,72
9,31
15,66
3,09
Standard
deviation
𝜎
𝑇
2,58
2,04
2,17
3,05
3,96
1,76
Coefficient of
variation
𝑣
(%)
8,75
6,87
19,81
11,52
15,48
7,14
Skewness А
0,17
0,60
-1,81
0,66
-0,99
-1,49
Kurtosis
𝐸
𝐾
0,23
0,25
2,68
0,18
2,17
1,65
Standard error
of the mean
𝑦̅
Т
,
𝑚
у
m
у
=
𝜎
у
√𝑛
=
0,58
m
у
=
𝜎
у
√𝑛
=
0,58
m
у
=
𝜎
у
√𝑛
=
0,40
m
у
=
𝜎
у
√𝑛
=
0,58
M
у
=
𝜎
у
√𝑛
= 0,75
M
у
=
𝜎
у
√𝑛
=
0,44
Maximum error
𝑚
у
′
m’
у
= t m
у
=
2,09
∙ 0
,58 =
1,21
m’
у
= t
m
у
= 2,06
∙
0
,47 = 0,97
m’
у
= t m
у
=
2,06
∙ 0
,40
=0,82
m’
у
= t
m
у
= 2,06
∙
0
,58 = 1,20
m’
у
=tm
у
=
2,06
∙ 0
,747 =
1,54
m’
у
=tm
у
=
2,06
∙ 0
,44
=0,94
Standard error
of the standard
deviation
𝜎
𝑇
m
𝜎
=
𝜎
√2𝑛
=
2,58
6,33
=
0,41
m
𝜎
=
𝜎
√2𝑛
=
2,04
6,16
=
0,33
m
𝜎
=
𝜎
√2𝑛
=
2,17
6,16
= 0,35
m
𝜎
=
𝜎
√2𝑛
=
3,05
7,48
=
0,41
m
𝜎
=
𝜎
√2𝑛
=
3,956
7,483
= 0,53
m
𝜎
=
𝜎
√2𝑛
=
1,76
7,483
=
0,24
Confidence
interval (95%)
for cotton yield
:
𝑦̅
Т
± 𝑡𝑚
у
𝑦̅
𝑇
± t m
у
=
29,47 ± 1,21
(28,26; 30,68
)
ц
/
га
𝑦̅
𝑇
± t m
у
=
29,68 ± 0,97
(28,71; 30,65
)
ц
/
га
𝑦̅
𝑇
± t m
у
=
23,82 ± 0,82
(23,00; 23,64
)
ц
/
га
𝑦̅
𝑇
± t m
у
=
26,46 ± 1,20
(25,26; 27,66
)
ц
/
га
𝑦̅
𝑇
± tm
у
=
25,54 ± 1,54,
(24,00; 27,08)
ц
/
га
𝑦̅
𝑇
± tm
у
=
24,64 ± 0,94,
(23,7; 25,6)
ц
/
га
Statistical
hypothesis test
:
Н
0
: 𝑃(𝑋 < 𝑥)
= Ф
а,𝜎
(х)
95%
Guarantee
hypothesis
Н
0
accepted
95%
Guarantee
hypothesis
Н
0
accepted
95%
Guarantee
hypothesis
Н
0
accepted
95%
Guarantee
hypothesis
ы
Н
0
accepted
95%
Guarantee
hypothesis
Н
0
accepted
95%
Guarantee
hypothesis
Н
0
accepted
CONCLUSIONS
Based on the above statistical analyses of the dynamics
of the average cotton yield
yˉt
\bar{y}_t
yˉt
in five
regions of the Republic of Uzbekistan as a time series,
with a confidence level of
γ
=0.95\gamma = 0.95
γ
=0.95,
the following conclusions can be drawn:
•
Point and interval statistical estimates for the
average cotton yield have been constructed;
•
Clear types of trends have been identified, and
their linearity has been established;
•
The average cotton yield has been forecasted
for the coming years;
•
Using statistical criteria, it has been proven
that the average cotton yield in the region exhibits
autocorrelation. Consequently, the cotton yield in a
American Journal of Applied Science and Technology
43
https://theusajournals.com/index.php/ajast
American Journal of Applied Science and Technology (ISSN: 2771-2745)
given year depends on the yields of previous and
subsequent years.
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