Comparing The Models SARIMA, ANFIS And ANFIS-DE In Forecasting Monthly Evapotranspiration Rates Under Heterogeneous Climatic Conditions

24 Reference crop evapotranspiration (ET0) is one of the most important hydro-climatological 25 components which directly affects agricultural productions, and its forecasting is critical for water 26 managers and irrigation planners. In this study, adaptive neuro-fuzzy inference system (ANFIS) 27 model has been hybridized by differential evolution (DE) optimization algorithm as a novel 28 approach to forecast monthly ET0. Furthermore, this model has been compared with the classic 29 stochastic time series model. For this, the ET0 rates were calculated on monthly scale during 1995- 30 2018, based on FAO-56 Penman-Monteith equation and meteorological data including: minimum 31 air temperature, maximum air temperature, mean air temperature, minimum relative humidity, 32 maximum relative humidity & sunshine duration. The investigation was performed on 6 stations 33 in different climates of Iran, including: Bandar Anzali & Ramsar (per-humid), Gharakhil (sub- 34 humid), Shiraz (semi-arid), Ahwaz (arid) and Yazd (extra- arid). The models’ performances were 35 evaluated by the criteria percent bias (PB), root mean squared error (RMSE), normalized RMSE 36 (NRMSE) and Nash-Sutcliff (NS) coefficient. Surveys confirm the high capability of the hybrid 37 ANFIS-DE model in monthly ET0 forecasting; so that the DE algorithm was able to improve the 38 accuracy of ANFIS, by 16% on average. Seasonal autoregressive integrated moving average 39 (SARIMA) was the most suitable pattern among the time series stochastic models, and superior 40 compared to its other competitors. Consequently, due to the simplicity and parsimony, the 41 SARIMA was suggested more appropriate for monthly ET0 forecasting in all the climates. 42 Comparison between the different climates confirmed that the climate type significantly affects 43 the forecasting accuracies: it’s revealed that all the models work better in extra -arid, arid and semi- 44 arid climates, than the humid and per-humid areas.


Introduction
The process of water parting the surface of moist soil is called evaporation, whereas this 49 phenomenon from leaves' pores is called transpiration. Since recognizing these two phenomena 50 on farms is not easy, they are to be considered as one integrated single variable referred to as  Reference crop evapotranspiration (ET0) is one of the main components of the hydrological cycle 77 associated with agricultural systems. Accurate estimation and prediction of ET0 is very important 78 in water resources management, irrigation planning, and determining the water needs of plants. 79 Forecasting the evapotranspiration rates, through providing information on the future status of

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The present study intends to use the ANFIS model to predict the reference evapotranspiration and 120 compare it with the classical SARIMA stochastic model. Moreover, as a novelty, the Differential 121 Evolution (DE) algorithm (a bio-inspired algorithm) which is hybridized with the ANFIS model, 122 has been used as ANFIS-DE to optimize and improve the ANFIS's prediction accuracy. In this 123 study, stations from different climates (from extra-arid to per-humid) are studied and for the first 124 time, the effect of climate type is also investigated on the accuracy of the models predicting ET0; 125 which is another novelty aspect of the current research.  Table 1.
The data used in this paper include monthly meteorological data and belong to the period 1995-151 2018. These data include minimum air temperature (Tmin), maximum air temperature (Tmax), 152 mean air temperature (Tmean), minimum relative humidity (RHmin), maximum relative humidity 153 (RHmax) and sunshine duration (SSD), which are prepared on a monthly scale of the Iranian Meteorological Organization (IRIMO). Using these data and FAO-56 PM model, the amount of 155 monthly evapotranspiration was estimated in the 6 mentioned stations. The "Evapotranspiration" 156 package in R software was used to estimate the evapotranspiration rates, based on the FAO-56 PM 157 method. For modeling, the period under study was divided into two parts of training and testing, 158 which include 75% (the first 18 years) and 25% (the remaining 6 years), respectively. The 159 characteristics of the meteorological data as well as the estimated evapotranspiration data are 160 shown in Table 2.  Eq. 2 In this research, the Minitab software and the SARIMA model have been used to simulate and 188 predict evapotranspiration time series.

2.3.Adaptive Neuro-Fuzzy Inference System (ANFIS)
190 ANFIS model has the ability to make relationships between input and output data using fuzzy rules 191 and to learn from a neural network in order to generate input structure for a system. ANFIS model 192 designs and creates non-linear maps to define relationships between input and output spaces by 193 employing Artificial Neural Network (ANN) and fuzzy logic, which is known as a neuro-fuzzy To implement the ANFIS model, MATLAB software is used in this study.

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To summarize, the ANFIS model contains two sets of parameters: premise parameters and 222 consequence parameters. Premise parameters are input parameters of MFs and their aim is to specify the shape and the location of the input MFs (parameters of input MFs). Consequence 224 parameters are output parameters of MFs (parameters of output MFs) (Jang, 1993  In this paper, the DE algorithm is implemented by coding in MATLAB software's environment.

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The trial and error method is used to choose the best operators of DE to optimize the ANFIS model.

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They are illustrated in Table 3. Eq. 19 shows the amount of observed evapotranspiration of the i th month, is the amount of  In Table 4, the predictions of all three models were evaluated by the mentioned evaluation metrics.

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Since the test section actually shows the validity of the models, the test section is also discussed 307 in the interpretations of this section. At first, it can be seen that in all stations, the R coefficients 308 are very high, which indicates the optimal performance of the models in predicting monthly ET0

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The authors declare that they have no conflicts of interest.