autocorrelation function

The present study sought to investigate precipitation variability and prediction the annual rainfall in central and southeast (SE) Iran using stochastic modeling (time series analysis). For this, the annual rainfall data gathered from 15 synoptic stations in the center and SE of Iran were obtained from Iran Meteorological Organization (IMO). Then, different autoregressive integrated moving average (ARIMA) models were used to fit the time series of annual rainfall in the studied stations. Based on the results, ARMA (2, 1) was found to be the best model fitted to the annual rainfall time series of the stations under study. Excluding the Ferdows and Iranshahr stations with an uptick in the rate of precipitation, the trend of predicted annual precipitation until 2027 was slipping for all the studied stations. The validation indices including MAD (mean absolute deviation), MSE (mean square error), RMSE (root mean square error), and MAE (mean absolute error) were employed to measure the accuracy of annual rainfall forecasts until 2027. According to these indices, the forecasts had substantial errors and were of low accuracy. Such a high rate of errors may be due to the short-term statistics obtained from synoptic stations in these regions. Hence, for modeling in central and SE Iran, future studies are recommended to use different networks of rainfall databases with long-term statistics.

Temperature is the most important climatic element and also one of the main factors in climatic zonation and classification and accordingly fluctuations and significant variations in the temperature of the globe or global warming have been considered as the most important phenomena of climate change in the present century. Therefore, prediction of climatic elements will necessary give planners more time to plan and provide necessary measures. To predict data series, stochastic statistical models have been used extensively in hydrological, weather and climate themes, including models such as Time Series or Box-Jenkins models. In this research, SARIMA seasonal stochastic model is used for modeling and forecasting the average monthly temperature of 5 synoptic stations from 4 different climates of Iran, to examine the model accuracy for estimating average monthly temperature of different climates. Five synoptic stations were selected in the cities Abadan, Isfahan, Anzali, Tabriz and Mashhad which were placed in 4 climatic classes; warm and super dry, cold and super dry, temperate and humid, cold and semi-arid, cold and semi-arid by De Martonne method’s evaluating and had long-term data on monthly temperature over the years 1951-2014.
The time series model, which is also called as the Box-Jenkins model, is a model commonly used to measure time-based data. This model which was introduced for the first time by Box and Jenkins in 1976, is intended for numerical simulation as well as prediction of the variables sorted by time that are recorded at the same time intervals. Among the time series models, the SARIMA model has been used in this research, which can be used to simulate the stochastic behavior of seasonal time series.
Autocorrelation function (ACF): This function is a very important function in the analysis of time series modeling, especially periodic time series. Among ACF’s usages, displaying and analyzing seasonal trends in data, and assessing return period of the series can be mentioned.
Model evaluation criteria: In order to ensure the accuracy of modeling and prediction, the outputs of the model should be compared with the same times’ actual values. For this, Schwarz Bayesian criterion, Root Mean Squared Error and coefficient of determination () have been used in this study. The temperature series were measured by ACF and a seasonal trend was confirmed with a 12 return period in each series and after 4 degrees of seasonal differencing, it was found that the best removing of the seasonal trend, is in the first degree in all series. Data were divided into two sections: 61 years old for calibration and 3 years old for validation that the first 61 years, by entering seasonal and non-seasonal autoregressive and moving average model from 0 to 3, in total 256 models for temperature series of each synoptic station were extracted, their outputs were measured by the evaluation criteria and the best models of each series were used to predict a long step in the next 3 years or 36 months. Best model for Abadan station was SARIMA(1,0,1)(1,1,1)12, Isfahan station was SARIMA(2,0,2)(3,1,1)12, Anzali station SARIMA(1,0,0)(1,1,1)12, Tabriz station SARIMA(1,0,2)(1,1,1)12 and Mashhad station was SARIMA(0,0,1)(0,1,1)12 and their errors was evaluated during 6, 12, 18, 24, 30 and 36 months forecasting horizons which expressed the remarkable accuracy of these models to forecast the monthly temperature time series. During the evaluations, SARIMA model in order to accuracy, in Abadan synoptic station with modeling Root Mean Squared Error=1.23 and predicting Root Mean Squared Error =0.97 degrees of centigrade had the best performance among the 5 synoptic stations’ temperature series and after that the stations Anzali, Isfahan, Tabriz and Mashhad had the best results with the Root Mean Squared Error in order 1.36, 1.44, 1.81 and 1.90 degrees of centigrade for modeling, and 1.58, 1.06, 1.86, 1.46 degrees of centigrade for predicting. Estimating average monthly temperature during the similar statistical period, At these stations, the model shows that the model has the highest accuracy in estimating and predicting the temperature of hot and super dry climate of Khuzestan province, then in the temperate and humid climate of the north of the country, then in the cold and dry climate of Isfahan province, after that in the cold and semi-arid climate of the northwest and at last in the cold and semi-arid climate of Iran’s northwest.