Testing for Stationarity and Nonlinearity of Daily Streamflow Time Series Based on Different Statistical Tests (Case Study: Upstream Basin Rivers of Zarrineh Roud Dam)

Document Type : Research Article

Authors

• Farshad Fathian ¹
• Ahmad Fakheri Fard ¹
• Yagob Dinpashoh ¹
• Seyed Saeid Mousavi Nadoshani ²

¹ University of Tabriz

² Abbaspour School of Engineering, Shahid Beheshti University

https://doi.org/10.22067/jsw.v30i4.44103

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 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.

Keywords

• Hydrological processes
• Nonlinearity tests
• Stationary tests
• Time series modeling
• Urmia Lake

20.1001.1.20084757.1395.30.4.25.1

References

1- Brock W.A., Dechert W.D., Scheinkman J.A., and LeBaron B. 1996. A test for independence based on the correlation dimension, Econ Rev, 15 (3): 197- 235.

2- Chen H.L., and Rao A.R. 2003. Linearity analysis on stationarity segments of hydrologic time series, Journal Hydro, 277: 89- 99.

3- Cryer J.D., and Chan K.S. 2008. Time series analysis: with applications in R. Springer.

4- Dickey D.A., and Fuller W.A. 1979. Distribution of the estimators for autoregressive time series with a unit root, Journal Am. Stat. Assoc., 74: 423– 431.

5- Fathian F., Dehghan Z., Bazrkar M.H., and Eslamian S. 2014. Trends in hydrologic and climatic variables affected by four variations of Mann-Kendall approach in Urmia Lake basin, Iran, Hydrological Sciences Journal, doi: 10.1080/02626667. 2014.932911

6- Fathian F., Morid S., and Kahya E. 2015. Identification of trends in hydrological and climatic variables in Urmia Lake basin, Iran, Theoretical and Applied Climatology, 119(3-4): 443- 464.

7- Gimeno R., Manchado B., and Minguez R. 1999. Stationary tests for financial time series, Phys. A, 269: 72-78.

8- Grassberger P., and Procaccia I. 1983. Measuring the strangeness of strange attractors, Phys D, 9: 189- 208.

9- Hinich M.J. 1982. Testing for Gaussianity and linearity of a stationary time series, J Time Ser Anal, 3(3): 169-176.

10- Järas J., and Gishani A.M. 2010. Threshold Detection in Autoregressive Non-linear Models. Doctoral dissertation, Master Thesis: 15 ECTS, Lund University Department of Statistics.

11- Keenan D. 1985. A Tukey nonlinear type test for time series nonlinearities, Biometrika, 72: 39– 44.

12- Khalili K., Fakheri-Fard A., Dinpashoh Y., and Ghorbani M.A. 2011. Nonlinearity Testing of Stream Flow Processes by BDS Test (Case study: Shaharchi River in Urmia), Journal of Soil and Water, 21(2): 26-37. (In Persian)

13- Kim H.S., Kang D.S., and Kim J.H. 2003. The BDS statistic and residual test, Stochast Environ Res Risk Assess, 17: 104- 115.

14- Kwiatkowski D., Phillips P.C.B., Schmidt P., and Shin Y. 1992. Testing the null of stationarity against the alternative of a unit root: How sure are we that economic time series have a unit root? J. Econometrics, 54: 159– 178.

15- Lettenmaier D.P., Wood A.W., Palmer R.N., Wood E.F., and Stakhiv E.Z. 1999. Water resources implications of global warming: A U.S. regional perspective, Clim. Change, 43: 537- 579.

16- Modarres R., and Ouarda T.B.M.J. 2012. Generalized autoregressive conditional heteroscedasticity modelling of hydrologic time series, Hydrological Processes, 27(22): 3174- 3191.

17- Modarres R., and Ouarda T.B.M.J. 2013. Testing and Modelling the Volatility Change in ENSO, Atmosphere-Ocean, 51(5): 561- 570.

18- Rao A.R., Yu G.H. 1990. Gaussianity and linearity tests of hydrologic time series, Stoch Hydrol, 4: 121- 134.

19- Rogres W.F., and Zia, H.A. 1982. Linear and nonlinear runoff from large drain basins, J Hydrol, 55: 267- 278.

20- Said S.E., and Dickey D. 1984. Testing for unit roots in autoregressive moving-average models with unknown order, Biometrika, 71: 599- 607.

21- Trapletti A., and Hornik K. 2014. tseries: Time series analysis and computational finance. R package version 0.10- 32.

22- Wang W., Van Gelder P.H.A.J.M., and Vrijling J.K. 2005. Trend and stationarity analysis for streamflow processes of rivers in Western Europe in the 20th century, In Proceedings: IWA International Conference on Water Economics, Statistics, and Finance Rethymno, Greece (pp. 8-10).

23- Wang W., Vrijling J.K., Van Gelder P.H.A.J.M, and Ma J. 2006. Testing for nonlinearity of streamflow processes at different timescales, Journal of Hydrology, 322(1): 247- 268.

24- Wiener N. 1958. Nonlinear Problems in Random Theory, Cambridge, MA: MIT Press.