Comparison of FIS and ARIMA models in runoff estimating

. The ability to surface runoff modelling plays an important role in the water resources management, and the possibility of estimating and predicting of runoff values takes on particular importance in the case of gaps in the recorded time series. Therefore, this study aims to compare between fuzzy inference system (FIS) models and autoregressive integrated moving average (ARIMA) models in estimating of the surface runoff at Al-Jawadiyah hydrometric station on the Orontes River in Syria. The MATLAB program was used to build the fuzzy inference models and the Minitab program to build the ARIMA models. A large number of fuzzy inference models were built with the change in the model parameters such as the type and number of membership functions and training algorithms. Likewise, a large number of ARIMA models were built with the change in autoregressive components, moving average components, and differences. The effect of seasonality on the model was also studied. Several criteria were used to compare the models and choose the best model, such as correlation coefficient and root mean square errors. The results showed that fuzzy inference models are superior to estimating surface runoff values with high reliability compared with ARIMA models. This study recommends creating complete databases for all factors related to water resources in the study area that can be relied upon in future studies.


Introduction
The topic of estimating and predicting surface runoff is considered one of the important topics in the allocation of irrigation water, hydroelectric power production and flood control [1], and its importance increases when there are gaps in the time series of surface runoff in the study area, which makes it hard to prepare accurate water balance studies and all studies related to water resources.
Artificial intelligence models have gained great interest in this field, as a number of researchers used artificial neural network models in the field of estimation and prediction of surface runoff and finding the rainfall-runoff relationship [2][3][4], and others were interested in fuzzy inference models, for example (Mahabir C. F. E. H. and Fayek A. R.,2003) studied the possibility of using fuzzy inference models in predicting water supplies, and the results showed the reliability of using these models [5].Also (Tayfur G. and Singh V. P., 2006) development of artificial neural network (ANN) and fuzzy logic (FL) models for predicting event-based rainfall runoff and tests these models against the kinematic wave approximation (KWA) [6], and (Şen Z. and Altunkaynak A., 2006) Ahmed conducted a study to estimate surface runoff using fuzzy inference models and using precipitation values [7].Others compared fuzzy inference models with other modelling methods, and fuzzy inference models showed good capabilities and high reliability in the process of estimating surface runoff and finding the rainfall-runoff relationship [8,9].(Nath A. et al. 2019) also used hybrid neural fuzzy models for surface runoff estimation, and the results showed high reliability of these models [10].
On the other hand, many studies have been conducted to use ARIMA models and Box-Jenkins models in the field of prediction and estimation of factors related to water resources, especially surface runoff, and compared with the results of other models.The results differed from one research to another according to the research area, the type of data, and the length of the time series used [11][12][13][14].
Therefore, his study aims to compare between Fuzzy inference system (FIS) models and autoregressive integrated moving average (ARIMA) models in estimating of the surface runoff at Al-Jawadiyah hydrometric station on the Orontes River in Syria.

Methods
Figures and tables, as originals of good quality and well contrasted, are to be in their final

Fuzzy inference system (FIS)
The idea of fuzzy logic is similar to the processes of feeling and reasoning in humans, that is, unlike the classical control strategy, which depends on the point-to-point system, the fuzzy inference system depends on the field-to-point.point or field-field, and therefore the outputs of the fuzzy inference system are derived from the fuzzy values of each of the inputs and outputs using membership functions that correlate with each other, and from this point of view, the output of the fuzzy inference system depends on the degree of its membership in different membership functions, which can be thought of as a range of input data [15].In the figure (1) comparison of a fuzzy system and a classical system in data representation.

Fig. 1.
Comparison of a fuzzy system and a classical system in data representation.
The fuzzy model is built through three steps: (Fuzzification, Fuzzy Inference Operations, Defuzzification).Membership functions are used to determine whether inputs belong to fuzzy groups.Gaussian curve, Triangular, trapezoidal, and bell-shaped membership functions are among the most commonly used types of membership functions [16].

Autoregressive integrated moving average (ARIMA)
In this study, the SARIMA and ARIMA models were used for runoff estimating where: Autoregressive integrated moving average ARIMA (p,d,q) as p is the order of the nonseasonal autoregressive model, q is the order of non-seasonal moving average model and d is the number of non-seasonal differences.Also Seasonal autoregressive integrated moving average SARIMA (p,d,q) (P,D,Q)s where P is the order of seasonal autoregressive model, Q is the order of seasonal moving average model, D is the number of seasonal differences and s is the periodic term [13].
The ARIMA models were built on the Minitab program, while the fuzzy models were built using the MATLAB.A comparison was made between the models using the correlation coefficient (R) and root mean square errors (RMSE).

Results
In order to build fuzzy inference models, 266 monthly values of surface runoff were used at Al-Jawadiyah station (entrance to Lake Qattinah) and al-Amiri station on the Syrian-Lebanese border from december 1978 to September 2011 with some missing data.These data were divided into three datasets for training, validation, and testing as (70:15:15) % respectively.The data was divided according to the divide block function in order to save the dataset in each group during the training process and thus increase the reliability of the comparison process between the results.A large number of fuzzy inference models were built with the change in the model parameters such as the type and number of membership functions and training algorithms and the table (1) presents values of the correlation coefficient (R) and root mean square error (RMSE) obtained by the best FIS models.For ARIMA models, it was initially searched for a complete and long data series as possible to be reliable in building the model, as the presence of any missing value in the data series leads to the inability to build the model.The data series used consisted of 237 monthly runoff values from February 1989 to October 2008.The figure (5-a-) shows the general trend chart for this series, in which it appears that this series is stable on average with a slight downward slope.It was then verified that the series is stable in the variance and with normal distribution of the data as shown in the figure(5-b-).Autocorrelation and partial autocorrelation were also plotted as shown in Figure (6).Several attempts were made to reduce the effect of seasonality, such as first-or second-degree differentiation, etc., but this did not affect the autocorrelation and partial autocorrelation of the data.Therefore, we use the same basic series without any changes.After that, a very large number of ARIMA models were built without the seasonal components or with them, and the results were compared and the table shows the results of the best five models that were reached.

Discussion
A large number of fuzzy inference models were built with the change in the model parameters such as the type and number of membership functions and training algorithms.Likewise, a large number of ARIMA models were built with the change in autoregressive components, moving average components, and differences.The effect of seasonality on the model was also studied.Several criteria were used to compare the models and choose the best model, such as correlation coefficient and root mean square errors.
In comparison between the results of fuzzy inference models and ARIMA models, fuzzy models generally show preference in terms of comparison values, in addition to fuzzy models were characterized by the ability to build the model even if there are some gaps in the data series used, as well as the possibility of studying the effect of other factors that could have an impact on estimating surface runoff values, such as studying the effect of runoff at a nearby hydrometric station or studying the effect of other climatic factors affecting surface runoff, which can be a helpful factor in reaching to the best model for estimating and predicting.

Conclusions
In this study, a comparison was made between fuzzy inference models and ARIMA models in estimating surface runoff values, and the results showed that the fuzzy models are superior in terms of evaluation criteria or in terms of speed to reach the best model, especially if building fuzzy inference models depends on programming.This study recommends creating complete databases for all factors related to water resources in the study area that can be relied upon in future studies.

figure ( 2 )
figure(2) presents structure of the best fuzzy inference model, and the figure(3) presents correlation between measured runoff and runoff computed with FIS model during all periods, and the figure(4) presents comparison between real values and values computed with FIS model during the validation and testing periods.

Fig. 2 .
Fig. 2. Correlation between measured runoff and runoff computed with FIS model during all periods.

Fig. 5 .
Fig. 5. Verifying the stability of average and normal distribution of the time series (-a-Trend analysis plot, -b-Probability plot)

Fig. 6 .
Fig. 6.Check the stability of the time series seasonally (-a-Autocorrelation function with 5% significance limits, -b-Partial autocorrelation function with 5% significance limits)

Fig. 7 .Fig. 8 .
Fig. 7. Comparison between the measured values and the values estimated by the ARIMA model

Table 1 .
Correlation coefficient (R) and root mean square error (RMSE) obtained by the best FIS models.

Table 2 .
Root mean square error (RMSE) obtained by the best ARIMA models.