بررسی کاربرد مدل‌های سری زمانی در پیش بینی جریان ماهانه ایستگاه هیدرومتری ارازکوسه

Introduction
Predicting an element is one of the most effective tools in management decisions, because the final efficiency of any decision depends on the sequence of events that will follow the initial predictions in the decision. In water resources management, knowledge of the status of water resources in an area has a very decisive role in water planning, agriculture, etc. Based on this, using statistical analysis, it is possible to predict the conditions of water resources in the future. Time series models are one of the efficient tools in predicting hydrological processes in water engineering. Due to the fact that the number of required data in time series models is less than other conceptual and physical models, it has led to the widespread use of these models in the field of hydrological and water resources engineering. Due to the high importance of modeling in water resources planning, the purpose of this study was to evaluate the Holt Winters exponential smoothing model (multiplication) and its integrated seasonal integrated moving average model and their application to predict The monthly flow is the Arazkuseh hydrometric station.
Methodology
In this research, the data of Arazkuseh hydrometric station on Chehelchai river, which is a tributary of Gorganrood, were used. The geographical position of this station is 55 degrees and 8 minutes east longitude and 37 degrees and 13 minutes north latitude. The height of this station from the surface of open waters is 34.5 meters and the area of its basin is 1.1678 square kilometers. Table 1 shows the discharge data of Arazkuse station. Figure 1 also shows the time series diagram of the average monthly flow of the Arazkuseh hydrometric station. The main purpose of time series analysis in hydrology is to describe the time history of motion of some variables such as the velocity of the flow in a river at a particular location. River flow and other hydrological sequences are characterized by variability and oscillating behavior. The purpose of hydrological studies is to understand and describe the quantitative statistical population as well as the process that creates this statistical population, based on a limited number of samples.

Table 1 - Flow data of Arazkuse station
Month average Flow
April 17.02
May 10.19
June 3.02
July 0.79
August 0.84
September 1.18
October 2.23
November 2.89
December 3.99
January 4.66
February 7.33
March 13.47
Annual average 5.63


Figure 1- Time series diagram of the average monthly flow of Arazkuseh hydrometric station

Conclusion
To examine the static static in the mean, the series diagram and its correlation graph can be used. If a sample degrades a lot and is cut off after a delay, it makes it necessary to differentiate. The instability in the mean is also evident from the time series diagram. It may be necessary to differentiate the initial data to eliminate the instability. Of course, experience has shown that it usually does not exceed 2.
The SARIMA model can be considered as a combination of a complex and simple model that can be used for unstable time series with seasonal changes. This model has been developed by Box et al. For seasonal time series. In the initial complex state, the self-correlated (AR) model with order p is used. Determining the appropriate value for p is determined by the function and the partial correlation coefficient. The Moving Average Model (MA) is also displayed with the parameter q. The autocorrelation function and graph are used to estimate this parameter. Also, the Integrated degree parameter is shown with the d parameter, which indicates the amount of the largest distance or difference that causes the time series to be static in the moving average smoothing method. Seasonal variation components include parameters. s Specifies the length of the seasons. On the other hand, the parameters P and Q, like p and q in the AR and MA models, are determined according to seasonal changes. Finally, parameter D is added to the model as a seasonal integration to eliminate seasonal changes.In this study, first the series autocorrelation and partial autocorrelation diagrams were examined. Due to the seasonal trend and instability in the series, the data were first differentiated and static. The normality of the data was also tested by drawing a normal distribution diagram. Then, the seasonal models of the Holt Winterzoo time series of their moving average seasonal integrated model (SARIMA) were used to predict the average monthly flow of the Arazkose hydrometric station. The results showed that the Holt Winters exponential smoothing model with three smoothing parameters, and, has the ability to modify outlier data and provides a more robust forecast.
The Holt Winters model is a way to examine time-dependent data. Predicting the behavior of random data requires a statistical model in which the parameters of this model are usually identified and estimated by the data. The Holt Winters model consists of three parts. The first part is called the mean (constant value), which shows the general behavior of the model and the values around it fluctuate. The second part is the trend (line slope), which is constant in time but is considered a multiple of the variable. The third section, which changes periodically, is also used to show seasonal changes. The prediction in Holt Winters method is done with the help of exponential smoothing, so the effect of data close to the prediction point is more than data that have been far away in the past. Because there are three axes or three components (characteristics, trends, and seasonal variations) in the Holt Winters time series model, it is sometimes referred to as triple exponential smoothing. In such a model, the future value is predicted by combining these three components. Such a model has several parameters. This group of parameters in this model are known as α, β and γ. Thus, the length of the seasonal change period and the number of seasonal change periods are also considered as parameters of the Holt Winters model.
Keywords: Time series; Prediction; Holt-Winters model; Integrated interconnected moving average model.