Evaluation of the Efficiency of Time Series and Fuzzy Models in Monthly Discharge Modeling (Case Study: Kohsukhteh Watershed)

Streamflow modeling is an important attitude generally considered for planning and management of water resources as well as watershed management practices. Thus it is highlighted as one of the fundamental issues in applied hydrology. It is also one of the methods used for modelling streamflow processing methods, which is considered as one of the common black–box methods that correlates input and output data. Regression techniques and time series models have been derived from data processing methods. Since 1962, use have been made of hydrologic as well as stochastic methodsfor flow discharge modeling asAutoregression (AR),  Moving Average (MA),  Autoregressive Moving Average (ARMA) and Autoregressive Integrated Moving Average (ARIMA) statistical models were introduced. Application of both fuzzy and time series models in discharge modeling of conducted studies, have shown the appropriate results of discharge estimation. The purpose of this study wasto evaluate the performance of two time series models (ARIMA) and fuzzy least squares regression with symmetric triangular membership function in streamflow modeling. Consideringthe background of the research, the least squares fuzzy regression method had not been used and no comparisons have been made between this model and time series model. Since both models arefrom black-boxed family, the present study have used both models and their efficacy have been evaluated.

This study aimed atsimulating the average monthly discharge of KoheSokhteh Watershed. This region has been located inShahrekord, Boroujen and Kiar in Chaharmahal and Bakhtiari province.The monthly data and an average of 25 yearshave been used.  The ARIMA model investigates the modelling of auto-correlated and random components shown as .It has to be noted that the values cannot be negative.-Identification of the type and rank of model for investigating timeseries models The autocorrelation function (ACF) and the partial autocorrelation function (PACF) were used in order to determine the type and rank of the time series model. Having diagnosed model`s ranks, a modal of model`s figure is identified and model`s parameters are appointed through obtained correlational function. ACF charts are obtained with k delay.  The other way for expressing the time dependency of the time series data is to define the PACF function. If beconsidered as partial autocorrelation function with k delay,…. -Parameters` estimation and model adequacy After choosing an appropriate model along with its order, the next step is to estimate the parameters of the selected model. We should calculate the remaining values that follow the normal distribution with a zero  mean using multi-year data for simulated discharge values. Before starting simulation, data is transformed to a normal data set. -Calculating the relationship between fuzzy least squares` regression model  In this section, the dependent variable of discharge is considered as a fuzzy dataset and observations related to independent variables (precipitation, evaporation, squer root, lag time) are consideredas non-fuzzy variables.  Based on this type of data and taking the 20 percent as theallowance error for the measured data, a model with fuzzy coefficients is fitted to the data. -Data used in the model The data was provided based onobserved data of observation matrices as well as matrix A. The matrices s and y, and also the matrices a and σ, were calculated. If Rank (x)=n+1, then the matrix A would bea definite positive. If A wasa definite positive, it would havea reverse A-1, andthe relation between σ and α would haveunique answers.-Description of the performance of least squares fuzzy regression model in watersheds With regards to the matrix X, the observation matrix (matrix X) was calculated for watersheds. Then the calculation proceeds using the matrix of observations and matrix A. Matrix A was calculated for three of the watersheds. After calculating the matrix A, the y vector (discharge observation values) was calculated for watersheds. Then the s vector wascalculated for each watershed. Finally, the optimal fitted model applied to the data as well as α and σ matrices were obtained.

In order to develop time series models, it was identified in the initial analysis that once applying differentials forconverting an unstable time series to a static time series was sufficient. Nextstep wasidentifying the order of the model. ACF as well asPACF graphshave been utilized. ACF suggested MA and PACF suggested AR. Subsequently a combination of MA and AR was proposed for modeling this series. The parameters of the selected model were calculated using the MINITAB software based on the information extracted from PACF and ACF. Standard errors of the parameters for the selected model were relatively small, indicatingthe applicability of the parameters in modeling. -Fuzzy least squares regression -Estimated discharge The values of estimated discharge of watersheds were calculated using fuzzy least squares regression. The results of the estimated discharge have been shown in Table 2.

To investigate the efficiency of the ARIMA time series model and fuzzy model at monthly forecasting scale, ARIMA model as well as fuzzy least squares regression models were utilized. Both models properly predictedmonthly discharge. The predicted discharge values using ARIMA model were lower than the observed discharge values. In the fuzzy least-squares regression model, peak discharge was well modelled but discharge with low values had further differences with the observations. During the calibration phase, the fuzzy least squares regression model showed a Nash-Sutcliff coefficient of 88% and ARIMA (211)(111) model had a Nash-Sutcliff coefficient of 84%. The least squares fuzzy regression model with low difference showed a superior to ARIMA model. The fuzzy least squares regression model`s superiority showed less effect of seasonal changes in predicting discharge, which was in contrast to Moayeni et al. (18) research.