Farshad Fathian; Ahmad Fakheri-Fard; Yagob Dinpashoh; Seyed Saeid Mousavi Nadoushani
Abstract
Introduction: Time series models are generally categorized as a data-driven method or mathematically-based method. These models are known as one of the most important tools in modeling and forecasting of hydrological processes, which are used to design and scientific management of water resources projects. ...
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Introduction: Time series models are generally categorized as a data-driven method or mathematically-based method. These models are known as one of the most important tools in modeling and forecasting of hydrological processes, which are used to design and scientific management of water resources projects. On the other hand, a better understanding of the river flow process is vital for appropriate streamflow modeling and forecasting. One of the main concerns of hydrological time series modeling is whether the hydrologic variable is governed by the linear or nonlinear models through time. Although the linear time series models have been widely applied in hydrology research, there has been some recent increasing interest in the application of nonlinear time series approaches. The threshold autoregressive (TAR) method is frequently applied in modeling the mean (first order moment) of financial and economic time series. Thise type of the model has not received considerable attention yet from the hydrological community. The main purposes of this paper are to analyze and to discuss stochastic modeling of daily river flow time series of the study area using linear (such as ARMA: autoregressive integrated moving average) and non-linear (such as two- and three- regime TAR) models.
Material and Methods: The study area has constituted itself of four sub-basins namely, Saghez Chai, Jighato Chai, Khorkhoreh Chai and Sarogh Chai from west to east, respectively, which discharge water into the Zarrineh Roud dam reservoir. River flow time series of 6 hydro-gauge stations located on upstream basin rivers of Zarrineh Roud dam (located in the southern part of Urmia Lake basin) were considered to model purposes. All the data series used here to start from January 1, 1997, and ends until December 31, 2011. In this study, the daily river flow data from January 01 1997 to December 31 2009 (13 years) were chosen for calibration and data for January 01 2010 to December 31 2011 (2 years) were chosen for validation, subjectively. As data have seasonal cycles, statistical indices (such as mean and standard deviation) of daily discharge were estimated using Fourier series. Then ARMA and two- and three-regime SETAR models applied to the standardized daily river flow time series. Some performance criteria were used to evaluate the models accuracy. In other words, in this paper, linear and non-linear models such as ARMA and two- and three-regime SETAR models were fitted to observed river flows. The parameters associated to the models, e.g. the threshold value for the SETAR model was estimated. Finally, the fitted linear and non-linear models were selected using the Akaike Information Criterion (AIC), Root Mean Square (RMSE) and Sum of Squared Residuals (SSR) criteria. In order to check the adequacy of the fitted models the Ljung-Box test was used.
Results and Discussion: To a certain degree the result of the river flow data of study area indicates that the threshold models may be appropriate for modeling and forecasting the streamflows of rivers located in the upstream part of Zarrineh Roud dam. According to the obtained evaluation criteria of fitted models, it can be concluded the performance of two- and three- regime SETAR models are slightly better than the ARMA model in all selected stations. As well as, modeling and comparison of SETAR models showed that the three-regime SETAR model have evaluation criteria better than two-regime SETAR model in all stations except Ghabghablou station.
Conclusion: In the present study, we attempted to model daily streamflows of Zarrineh Rood Basin Rivers located in the south of Urmia Lake by applying ARMA and two- and three-regime SETAR models. This is mainly because very few efforts and rather less attention have been paid to this non-linear approach in hydrology and water resources engineering generally.
Therefore, two types of data-driven models were used for modeling and forecasting daily streamflow: (i) deseasonalized ARMA-type model, and (ii) Threshold Autoregressive model, including Self-Existing TAR (SETAR) model. Each ARMA and SETAR models were fitted to daily streamflow time series of the rivers located in the study area. In general, it can be concluded that the overall performance of SETAR model is slightly better than ARMA model. Furthermore, SETAR model is very similar AR model, therefor, it can be easily used in water resources engineering field. On the other hand, due to apply these non-linear models, the number of estimated parameters in comparison with linear models has decreased.
Farshad Fathian; Ahmad Fakheri Fard; Yagob Dinpashoh; Seyed Saeid Mousavi Nadoshani
Abstract
Introduction: Time series models are one of the most important tools for investigating and modeling hydrological processes in order to solve problems related to water resources management. Many hydrological time series shows nonstationary and nonlinear behaviors. One of the important hydrological modeling ...
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Introduction: Time series models are one of the most important tools for investigating and modeling hydrological processes in order to solve problems related to water resources management. Many hydrological time series shows nonstationary and nonlinear behaviors. One of the important hydrological modeling tasks is determining the existence of nonstationarity and the way through which we can access the stationarity accordingly. On the other hand, streamflow processes are usually considered as nonlinear mechanisms while in many studies linear time series models are used to model streamflow time series. However, it is not clear what kind of nonlinearity is acting underlying the streamflowprocesses and how intensive it is.
Materials and Methods: Streamflow time series of 6 hydro-gauge stations located in the upstream basin rivers of ZarrinehRoud dam (located in the southern part of Urmia Lake basin) have been considered to investigate stationarity and nonlinearity. All data series used here to startfrom January 1, 1997, and end on December 31, 2011. In this study, stationarity is tested by ADF and KPSS tests and nonlinearity is tested by BDS, Keenan and TLRT tests. The stationarity test is carried out with two methods. Thefirst one method is the augmented Dickey-Fuller (ADF) unit root test first proposed by Dickey and Fuller (1979) and modified by Said and Dickey (1984), which examinsthe presence of unit roots in time series.The second onemethod is KPSS test, proposed by Kwiatkowski et al. (1992), which examinesthestationarity around a deterministic trend (trend stationarity) and the stationarity around a fixed level (level stationarity). The BDS test (Brock et al., 1996) is a nonparametric method for testing the serial independence and nonlinear structure in time series based on the correlation integral of the series. The null hypothesis is the time series sample comes from an independent identically distributed (i.i.d.) process. The alternative hypothesis arenot specified. Keenan test has also been proposed for assessing the linearity or nonlinearitybehavior of a time series in time series analysis. Keenan (1985) derived a test for nonlinearity analogous to Tukey’s degree of freedom for nonadditivity test. Keenan’s test is motivated by approximation a nonlinear stationary time series by a second-order Volterra expansion. While Keenan’s test for nonlinearity is designed for detecting quadratic nonlinearity, it may not be sensitive to threshold nonlinearity. Here, we applied the likelihood ratio test (TLRT) with the threshold model as the specific alternative.The null hypothesis of the TLRT approach for threshold nonlinearity is the fitted model to the series is an AR (p) model, and the alternative hypothesis is the fitted model to the series is a threshold autoregressive (TAR) model with autoregressive order p in each regime.
Results and Discussion: Because both the ADF and KPSS tests are based on linear regression, which has the normal distribution assumption, logarithmization can convert exponential trend possibly present in the data into a linear trend. In the case of stationary analysis, the results showed the standardized daily streamflow time series of all stations are significantly stationary. According to KPSS stationary test, the daily standardized streamflow time series are stationary around a fixed level, but they are not stationary around a trend stationaryin low lag values. Based on the BDS test, the results showed the daily streamflowseries have strong nonlinear structure, but based on the Keenan test, it can be seen the linear structure in thembyusing logarithmization and deseasonalization operators, and it means the coefficients of the double sum part are zero. It should be considered the Keenan test is used to detect quadratic nonlinearity, and it cannot be adequatelyfor threshold autoregressive models since they are linear in each regime.
Conclusion: Streamflow processes of main rivers at 6 stations located in the southern partof Urmia Lake basin were investigated for testingthenonstationarity and nonlinearity behaviors. In general, streamflowprocesses have been considered as nonlinear behaviors. But, the type and intensity of nonlinearity have not been detected at different time scale due to the existence of several evaluation tests. In this study, all daily streamflow series appear to be significantly stationary and have the nonlinearity behavior. Therefore, to model the daily streamflow time series, linear and nonlinear models can be used and their results can be evaluated.
