APPLICATION OF TIME SERIES MODELS IN FORECASTING PRODUCTION ACTIVITY IN AGRICULTURE

Аннотация

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:timeseries, economic processes, trend, mathematical modeling, regression Login 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 andfuture predictions are made. In most cases, thesetrends 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 orother forms, for example: a. Types ofnon-linear trends b. Mathematical Modeling (WithTable 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 ofnon-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 anexponential equation: (1.1) w h=ere : -initialvalue, -rate ofincrease ordecrease, -time (2.718)- baseof natural logarithm Applica tion:It isused i nthe areas ofpopulation growt h, infla tion rate, and price growth model. b)Logarithmictrend If thetimeseries grows rapidly and then stabilizes, then thelogarithmic modelis used: (1.2) w h=ere : + ln- (n a)tural logarithm , -parameters Application: The relationship between income and consumption is applied in the areas of the ln ( ) , https://ijmri.de/index.php/jmsi volume 4,issue5,2025 836 speed of adoption ofnew technologies [1]. c)Polynomial trend If a timeseries has complex fluctuations, itcan beexpressed byan n-degree polynomial: (1.3) 2 3 A p=pli ca+tio n1: +It is2 pe+rf o3rm e+d ⋯in+th e areas of stock indices, unemployment rate dynamics, economicgrowth forecasting, mathematical modeling, and tabulation [2]. Example: Monthly economic indicators show the changes in a certain variable over five months as shownin Table1below. Table1.Parameters of economicindicators 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 inFigure 2. No. True y ExponentialY Logarithmic Polynomial Y No. Y 1 100 72.42242261034382 100.0 111.5837060063560 1 2 150 179.2560081050147 150.0 150.2129357352261 5 3 210 210.0 202.2152415417167 241.7496494369847 2 8 4 280 280.0 272.2202566086027 286.0095935996855 7 4 5 360 366.4603495912352 320.4823262479711 360.0 The application of nonlinear trend models based on the parameters of economic indicator https://ijmri.de/index.php/jmsi volume 4,issue5,2025 837 changes is illustrated in Figure 1. Figure 1.Nonlinear trend models ofeconomic indicators change parameters Based on thetable and graph above, thefollowing results can beobtained: 1. Bluedots are real timeseries values (sales volume). 2. Red line(exponential model)- used when thedata has exponentialgrowth. 3. The green line (logarithmic model) represents a process that initially grows rapidly and then slows down. 4. Theyellowline(polynomial model)- identifies thecomplex dynamics of thetimeseries. You can check which of these models fits better by evaluating the error. If you need to compare byMSE(Mean Squared Error) orother evaluation criteria, you can say [3]. Timeseries smoothing isa technique usedto reduce random variations in timeseries and tomore clearly see theunderlying trends. Smoothing methods are divided intothe following: a. MovingAverage (MA) b. Exponential smoothing (ExponentialSmoothing, ES) c. Holt-Winters model d. MovingAverage In this method, the average value for each point in the time series is taken over a certain interval. Theformulafor a simple3-period movingaverage is: (1.4) ' −1+ + +1 Fi g = ure 2be3lowshows atimeseries smoothingscheme: https://ijmri.de/index.php/jmsi volume 4,issue5,2025 838 Figure 2.Movingaverage when smoothingtimeseries As can be seen, the time series is smoothed and the volatility is reduced with the help of the movingaverage. Now itis possibletoapply theexponentialsmoothingmethod [4]. Simple Exponential Smoothing Figure 3.Timeseries smoothingscheme Based on the exponential smoothing in the table and graph above, the following results can be obtained: 1. Bluedots andline- real timeseries (sales volume). 2. there