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ANALYSIS OF TIME SERIES USING COMPLEX ECONOMIC-MATHEMATICAL
METHODS
1,2,3
Tashkent State University of
Economics
Bozorov Askar Khaitmurotovich
Toshpulov Begzod Sultanmurodovich
Otaboyeva Dildora Ilxomboy kizi
Abstract:
This article covers the stages of determining nonlinear trends in time series,
mathematical modeling, finding trend equations using regression methods, and forecasting. In
particular, taking into account the complex dynamics of economic processes, the advantages of
exponential, logarithmic, and polynomial models are analyzed. The accuracy and error criteria of
forecast results are considered through practical examples.
Keywords:
time series, economic processes, trend, mathematical modeling, regression
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Time series are an important tool in modern economic analysis. Time series analysis is used to
observe and analyze how most economic indicators change over time. Through this analysis,
trends are identified and future predictions are made. In most cases, these trends are not linear, so
it is necessary to use complex economic and mathematical methods. The trend of time series is
not always linear. Many economic processes can have non-linear trends, that is, exponential,
logarithmic, polynomial or other forms, for example:
a.
Types of non-linear trends
b.
Mathematical Modeling (With Table and Practical Examples)
c.
Finding trend equations using regression methods
d.
We will explain these steps in detail in nonlinear trend prediction and error analysis .
a.
Types of non-linear trends
If the time series does not fit a precise linear model, one of the following nonlinear trend models
is
used:
a) Exponential trend
Some economic processes have a tendency to grow or decline rapidly and can be represented by
an exponential equation:
�
�
= ��
��
(1.1)
where:
�
-initial value,
�
- rate of increase or decrease,
�
-time
�
(2.718)- base of natural logarithm
Application: It is used in the areas of population growth, inflation rate, and price growth model.
b) Logarithmic trend
If the time series grows rapidly and then stabilizes, then the logarithmic model is used:
�
�
= � + �ln (�)
(1.2)
where:
ln (�)
- natural logarithm ,
�, �
- parameters
Application: The relationship between income and consumption is applied in the areas of the
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speed of adoption of new technologies [1].
c) Polynomial trend
If a time series has complex fluctuations, it can be expressed by an n-degree polynomial:
�
�
= � + �
1
� + �
2
�
2
+ �
3
�
3
+ ⋯ + �
�
�
�
(1.3)
Application: It is performed in the areas of stock indices, unemployment rate dynamics,
economic growth forecasting, mathematical modeling, and tabulation [2].
Example: Monthly economic indicators show the changes in a certain variable over five months
as shown in Table 1 below.
Table 1. Parameters of economic indicators change
Moon
�
Sales volume
�
�� (�)
�
�
�
�
January
1
100
0.000
1
2.718
February
2
150
0.693
4
7.389
March
3
210
1.099
9
20,085
April
4
280
1.386
16
54,598
May
5
360
1.609
25
148,413
Using the table above, the parameters of the equations corresponding to each trend model are
shown in Figure 2.
No.
No.
True
y
Y
Exponential Y
Logarithmic
�
Polynomial Y
1
100
111.5837060063560
1
72.42242261034382
100.0
2
150
150.2129357352261
5
179.2560081050147
150.0
3
210
202.2152415417167
2
241.7496494369847
8
210.0
4
280
272.2202566086027
7
286.0095935996855
4
280.0
5
360
366.4603495912352
320.4823262479711
360.0
The application of nonlinear trend models based on the parameters of economic indicator
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changes is illustrated in Figure 1.
Figure 1. Nonlinear trend models of economic indicators change parameters
Based on the table and graph above, the following results can be obtained:
1.
Blue dots are real time series values (sales volume).
2.
Red line (exponential model) - used when the data has exponential growth.
3.
The green line (logarithmic model) represents a process that initially grows rapidly and
then slows down.
4.
The yellow line (polynomial model) - identifies the complex dynamics of the time series.
You can check which of these models fits better by evaluating the error. If you need to compare
by MSE (Mean Squared Error) or other evaluation criteria, you can say [3].
Time series smoothing is a technique used to reduce random variations in time series and to more
clearly see the underlying trends. Smoothing methods are divided into the following:
a.
Moving Average (MA)
b.
Exponential smoothing (Exponential Smoothing, ES)
c.
Holt-Winters model
d.
Moving Average
In this method, the average value for each point in the time series is taken over a certain interval.
The formula for a simple 3-period moving average is:
�
�
'
=
�
�−1
+�
�
+�
�+1
3
( 1.4)
Figure 2 below shows a time series smoothing scheme:
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Figure 2. Moving average when smoothing time series
As can be seen, the time series is smoothed and the volatility is reduced with the help of the
moving average. Now it is possible to apply the exponential smoothing method [4].
Simple
Exponential
Smoothing
Figure 3. Time series smoothing scheme
Based on the exponential smoothing in the table and graph above, the following results can be
obtained:
1.
Blue dots and line - real time series (sales volume).
2.
the results obtained using exponential smoothing .
(� = 0.3)
Exponential smoothing adapts to new data faster than a moving average.
Conclusion
Nonlinear time series analysis provides a deeper analysis of economic processes. Since each
economic indicator has its own trend, it is important to choose an appropriate trend model. These
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processes are accurately represented using exponential, logarithmic or polynomial regression
models. Using the equations determined by regression, it is possible to make reliable predictions
and make decisions.
Literature
1.
Tursunov BX Modeling of economic processes. – Tashkent: Economics, 2017. – 280 p.
2.
Tokhtasinov Sh.T. Statistics and economic analysis. – Tashkent: Fan, 2020. – 320 p.
3.
Gujarati, DN Basic Econometrics . – 5th Edition. - McGraw-Hill, 2009. - 924 p.
4.
Box, GEP, Jenkins, GM, & Reinsel, GC Time Series Analysis: Forecasting and Control .
– 5th Edition. – Wiley, 2015. – 712 p.
