Abstract
Despite significant research advances achieved during the last decades, seemingly inconsistent forecasting results related to stochastic, chaotic, and black-box approaches have been reported. Herein, we attempt to address the entropy/complexity resulting from hydrological and climatological conditions. Accordingly, mutual information function, correlation dimension, averaged false nearest neighbor with E1 and E2 quantities, and complexity analysis that uses sample entropy coupled with iterative amplitude adjusted Fourier transform were employed as nonlinear deterministic identification tools. We investigated forecasting of daily streamflow for three climatologically different Swedish rivers, Helge, Ljusnan, and Kalix Rivers using self-exciting threshold autoregressive (SETAR), k-nearest neighbor (k-nn), and artificial neural networks (ANN). The results suggest that the streamflow in these rivers during the 1957–2012 period exhibited dynamics from low to high complexity. Specifically, (1) lower complexity lead to higher predictability at all lead-times and the models’ worst performances were obtained for the most complex streamflow (Ljusnan River), (2) ANN was the best model for 1-day ahead forecasting independent of complexity, (3) SETAR was the best model for 7-day ahead forecasting by means of performance indices, especially for less complexity, (4) the largest error propagation was obtained with the k-nn and ANN and thus these models should be carefully used beyond 2-day forecasting, and (5) higher number input variables except for the dominant variables made insignificant impact on forecasting performances for ANN and k-nn models.
Similar content being viewed by others
References
Abarbanel H (1996) Analysis of observed chaotic data. Springer, New York
Abarbanel HDI, Brown R, Kadtke JB (1990) Prediction in chaotic nonlinear systems: methods for time series with broadband Fourier spectra. Phys Rev A 41(4):1782–1807
Adamowski J, Karapataki C (2010) Comparison of multivariate regression and artificial neural networks for peak urban water-demand forecasting: evaluation of different ANN learning algorithms. J Hydrol Eng 15(10):729–743. doi:10.1061/(ASCE)HE.1943-5584.0000245
Al-Awadhi S, Jolliffe I (1998) Time series modelling of surface pressure data. Int J Climatol 18(4):443–455
Aqil M, Kita I, Yano A, Nishiyama S (2007) Neural networks for real time catchment flow modeling and prediction. Water Resour Manag 21(10):1781–1796. doi:10.1007/s11269-006-9127-y
Badrzadeh H, Sarukkalige R, Jayawardena AW (2013) Impact of multi-resolution analysis of artificial intelligence models inputs on multi-step ahead river flow forecasting. J Hydrol 507:75–85. doi:10.1016/j.jhydrol.2013.10.017
Ben Taieb S, Bontempi G, Atiya AF, Sorjamaa A (2012) A review and comparison of strategies for multi-step ahead time series forecasting based on the NN5 forecasting competition. Exp Syst Appl 39(8):7067–7083. doi:10.1016/j.eswa.2012.01.039
Birikundavyi S, Labib R, Trung HT, Rousselle J (2002) Performance of neural networks in daily streamflow forecasting. J Hydrol 75:392–398
Cao L (1997) Practical method for determining the minimum embedding dimension of a scalar time series. Physica D 110(1):43–50. doi:10.1016/S0167-2789(97)00118-8
Chan WS, Wong ACS, Tong H (2004) Some nonlinear threshold autoregressive time series models for actuarial use. N Am Actuar J 8(4):37–61
Chang F-J, Chiang Y-M, Chang L-C (2007) Multi-step-ahead neural networks for flood forecasting. Hydrol Sci J 52(1):114–130
Chen C-S, Liu C-H, Su H-C (2008) A nonlinear time series analysis using two-stage genetic algorithms for streamflow forecasting. Hydrol Process 22(18):3697–3711. doi:10.1002/hyp.6973
Chou C-M (2014) Complexity analysis of rainfall and runoff time series based on sample entropy in different temporal scales. Stoch Environ Res Risk Assess 28(6):1401–1408. doi:10.1007/s00477-014-0859-6
Coulibaly P, Anctil F, Bobée B (2000) Daily reservoir inflow forecasting using artificial neural networks with stopped training approach. J Hydrol 230(3):244–257. doi:10.1016/S0022-1694(00)00214-6
Dahlqvist R, Andersson K, Ingri J, Larsson T, Stolpe B, Turner D (2007) Temporal variations of colloidal carrier phases and associated trace elements in a boreal river. Geochim Cosmochim Acta 71(22):5339–5354
Daliakopoulos IN, Coulibaly P, Tsanis IK (2005) Groundwater level forecasting using artificial neural networks. J Hydrol 309:229–240
Darshana PA, Pandey RP (2013) Analysing trends in reference evapotranspiration and weather variables in the Tons River Basin in Central India. Stoch Environ Res Risk Assess 27:1407–1421. doi:10.1007/s00477-012-0677-7
Durdu Ö (2010) Application of linear stochastic models for drought forecasting in the Büyük Menderes river basin, western Turkey. Stoch Environ Res Risk Assess 24:1145–1162. doi:10.1007/s00477-010-0366-3
Elshorbagy A, Panu US, Simonovic SP (2001) Analysis of cross-correlated chaotic streamflows. Hydrol Sci J 46:781–793
Farmer DJ, Sidorowich JJ (1987) Predicting chaotic time series. Phys Rev Lett 59:845–848
Firat M (2008) Comparison of artificial intelligence techniques for river flow forecasting. Hydrol Earth Syst Sci 12:123–139. doi:10.5194/hess-12-123-2008
Fraser AM, Swinney HL (1986) Independent coordinates for strange attractors from mutual information. Phys Rev A 33:1134–1140
Gonçalves R, Pinto A, Calheiros F (2007) Comparison of methodologies in river flow prediction. The Paiva river case. Math Methods Eng pp 381–390
Grassberger P, Procaccia I (1983a) Characterization of strange attractors. Phys Rev Lett 50:346–349
Grassberger P, Procaccia I (1983b) Measuring the strangeness of strange attractors. Phys D 9:189–208
Grayson R, Blöschl G (2001) Spatial patterns in catchment hydrology: observations and modelling. Cambridge University Press, Cambridge
Hagan MT, Menhaj MB (1994) Training feedforward networks with the Marquardt algorithm. IEEE Trans Neural Netw 5(6):989–993
Hagan MT, Demuth HB, Beale MH (1996) Neural network design. PWS Publishing Company, Boston
Hassan SA, Ansari MRK (2010) Nonlinear analysis of seasonality and stochasticity of the Indus River. Hydrol Sci J 55(2):250–265. doi:10.1080/02626660903546167
Hunt BR, Kennedy JA, Li T-Y, Nusse HE (2004) The theory of chaotic attractors. Springer, New York
Ingri J, Torssander P, Andersson P, Mörth C-M, Kusakabe M (1997) Hydrogeochemistry of sulfur isotopes in the Kalix River catchment, northern Sweden. Appl Geochem 12(4):483–496
Islam MN, Sivakumar B (2002) Characterization and prediction of runoff dynamics: a nonlinear dynamical view. Adv Water Resour 25(2):179–190
Jothiprakash V, Magar RB (2012) Multi-time-step ahead daily and hourly intermittent reservoir inflow prediction by artificial intelligent techniques using lumped and distributed data. J Hydrol 450:293–307. doi:10.1016/j.jhydrol.2012.04.045
Kalteh AM, Berndtsson R (2007) Interpolating monthly precipitation by self-organizing map (SOM) and multilayer perceptron (MLP). Hydrol Sci J 52:305–317
Kantz H, Schreiber T (1997) Nonlinear time series analysis. Cambridge University Press, Cambridge
Karunasinghe DSK, Liong S-Y (2006) Chaotic time series prediction with a global model: artificial neural network. J Hydrol 323(1):92–105. doi:10.1016/j.jhydrol.2005.07.048
Kędra M (2014) Deterministic chaotic dynamics of Raba River flow (Polish Carpathian Mountains). J Hydrol 509:474–503. doi:10.1016/j.jhydrol.2013.11.055
Khatibi R, Sivakumar B, Ghorbani MA, Kisi O, Koçak K, Farsadi Zadeh D (2012) Investigating chaos in river stage and discharge time series. J Hydrol 414:108–117. doi:10.1016/j.jhydrol.2011.10.026
Khatibi R, Ghorbani MA, Naghipour L, Jothiprakash V, Fathima TA, Fazelifard MH (2014) Inter-comparison of time series models of lake levels predicted by several modeling strategies. J Hydrol 511:530–545. doi:10.1016/j.jhydrol.2014.01.009
Khokhlov V, Glushkov A, Loboda N, Serbov N, Zhurbenko K (2008) Signatures of low-dimensional chaos in hourly water level measurements at coastal site of Mariupol, Ukraine. Stoch Environ Res Risk Assess 22(6):777–787
Kisi O (2010) Wavelet regression model for short-term streamflow forecasting. J Hydrol 389(3):344–353. doi:10.1016/j.jhydrol.2010.06.013
Kolmogorov AN (1959) Entropy per unit time as a metric invariant of automorphisms. Dokl Akad Nauk SSSR 124:754–755
Komorník J, Komornikova M, Mesiar R, Szökeova D, Szolgay J (2006) Comparison of forecasting performance of nonlinear models of hydrological time series. Phys Chem Earth 31(18):1127–1145
Kozhevnikova IA, Shveikina VI (2008) Nonlinear dynamics of level variations in the Caspian Sea. Water Resour 35(3):297–304. doi:10.1134/s0097807808030056
Laio F, Porporato A, Revelli R, Ridolfi L (2003) A comparison of nonlinear flood forecasting methods. Water Resour Res 39(5):1129. doi:10.1029/2002WR001551
Lall U, Sharma A (1996) A nearest neighbor bootstrap for resampling hydrologic time series. Water Resour Res 32(3):679–693
LeBaron B, Weigend AS (1998) A bootstrap evaluation of the effect of data splitting on financial time series. IEEE Trans Neural Netw 9(1):213–220
Lee T, Ouarda TBMJ (2011) Identification of model order and number of neighbors for k-nearest neighbor resampling. J Hydrol 404(3):136–145. doi:10.1016/j.jhydrol.2011.04.024
Li Z, Zhang Y-K (2008) Multi-scale entropy analysis of Mississippi River flow. Stoch Environ Res Risk Assess 22(4):507–512
Lisi F, Villi V (2001) Chaotic forecasting of discharge time series: a case study. J Am Water Resour Assoc 37(2):271–279
Lohani AK, Goel N, Bhatia K (2011) Comparative study of neural network, fuzzy logic and linear transfer function techniques in daily rainfall-runoff modelling under different input domains. Hydrol Process 25(2):175–193
Lohani AK, Kumar R, Singh RD (2012) Hydrological time series modeling: A comparison between adaptive neuro-fuzzy, neural network and autoregressive techniques. J Hydrol 442:23–35. doi:10.1016/j.jhydrol.2012.03.031
Maier HR, Dandy GC (2000) Neural networks for the prediction and forecasting of water resources variables: a review of modelling issues and applications. Environ Model Softw 15(1):101–124. doi:10.1016/S1364-8152(99)00007-9
McLeod AI, Li WK (1983) Diagnostic checking ARMA time series models using squared residual autocorrelations. J Time Ser Anal 4(4):269–273
Melesse AM, Ahmad S, McClain ME, Wang X, Lim YH (2011) Suspended sediment load prediction of river systems: An artificial neural network approach. Agric Water Manag 98(5):855–866. doi:10.1016/j.agwat.2010.12.012
Menezes JMP Jr, Barreto GA (2008) Long-term time series prediction with the NARX network: an empirical evaluation. Neurocomputing 71(16):3335–3343. doi:10.1016/j.neucom.2008.01.030
Mishra AK, Desai VR (2005) Drought forecasting using stochastic models. Stoch Environ Res Risk Assess 19(5):326–339. doi:10.1007/s00477-005-0238-4
Moosavi V, Vafakhah M, Shirmohammadi B, Behnia N (2013) A wavelet-ANFIS hybrid model for groundwater level forecasting for different prediction periods. Water Resour Manag 27(5):1301–1321. doi:10.1007/s11269-012-0239-2
Packard NH, Crutchfield JP, Farmer JD, Shaw RS (1980) Geometry from a time series. Phys Rev Lett 45(9):712–716
Palani S, Liong S-Y, Tkalich P (2008) An ANN application for water quality forecasting. Marin Poll Bull 56(9):1586–1597
Patel S, Ramachandran P (2015) A comparison of machine learning techniques for modeling river flow time series: the case of Upper Cauvery River Basin. Water Resour Manag 29(2):589–602. doi:10.1007/s11269-014-0705-0
Phoon K, Islam M, Liaw C, Liong S (2002) Practical inverse approach for forecasting nonlinear hydrological time series. J Hydrol Eng 7(2):116–128
Pincus SM (1991) Approximate entropy as a measure of system complexity. Proc Nat Acad Sci 88(6):2297–2301. doi:10.1073/pnas.88.6.2297
Porporato A, Ridolfi L (2001) Multivariate nonlinear prediction of river flows. J Hydrol 248(1):109–122
Pramanik N, Panda RK (2009) Application of neural network and adaptive neuro-fuzzy inference systems for river flow prediction. Hydrol Sci J 54(2):247–260. doi:10.1623/hysj.54.2.247
Regonda SK, Sivakumar B, Jain A (2004) Temporal scaling in river flow: can it be chaotic? Hydrol Sci J 49(3):373–385
Richman JS, Moorman JR (2000) Physiological time-series analysis using approximate entropy and sample entropy. Am J Physiol Heart Circ Physiol 278(6):H2039–H2049
Schreiber T (1999) Interdisciplinary application of nonlinear time series method. Phys Rep 308(1):1–64
Schreiber T, Schmitz A (1996) Improved surrogate data for nonlinearity tests. Phys Rev Lett 77(4):635–638
Sen KA (2009) Complexity analysis of riverflow time series. Stoch Environ Res Risk Assess 23(3):361–366. doi:10.1007/s00477-008-0222-x
Sharma A, Tarboton DG, Lall U (1997) Streamflow simulation: A nonparametric approach. Water Resour Res 33(2):291–308
Siek M, Solomatine DP (2010) Nonlinear chaotic model for predicting storm surges. Nonlinear Process Geophys 17(5):405–420. doi:10.5194/npg-17-405-2010
Singh T (2014) On the regime-switching and asymmetric dynamics of economic growth in the OECD countries. Res Econ 68(2):169–192. doi:10.1016/j.rie.2013.12.004
Singh KP, Basant A, Malik A, Jain G (2009) Artificial neural network modeling of the river water quality: a case study. Ecol Model 220(6):888–895
Sivakumar B (2000) Chaos theory in hydrology: important issues and interpretations. J Hydrol 227(1):1–20
Sivakumar B (2007) Nonlinear determinism in river flow: prediction as a possible indicator. Earth Surf Proces Landf 32(7):969–979. doi:10.1002/esp.1462
Sivakumar B (2008) Dominant processes concept, model simplification and classification framework in catchment hydrology. Stoch Environ Res Risk Assess 22(6):737–748
Sivakumar B, Jayawardena AW (2002) An investigation of the presence of low-dimensional chaotic behaviour in the sediment transport phenomenon. Hydrol Sci J 47(3):405–416
Sivakumar B, Singh VP (2012) Hydrologic system complexity and nonlinear dynamic concepts for a catchment classification framework. Hydrol Earth Syst Sci 16(11):4119–4131. doi:10.5194/hess-16-4119-2012
Sivakumar B, Liong S-Y, Liaw C-Y (1998) Evidence of chaotic behavior in Singapore rainfall. J Am Water Resour Assoc 34(2):301–310. doi:10.1111/j.1752-1688.1998.tb04136.x
Sivakumar B, Jayawardena AW, Fernando TMKG (2002) River flow forecasting: use of phase-space reconstruction and artificial neural networks approaches. J Hydrol 265(1):225–245
Sivakumar B, Wallender WW, Horwath WR, Mitchell JP, Prentice SE, Joyce BA (2006) Nonlinear analysis of rainfall dynamics in California’s Sacramento Valley. Hydrol Proces 20(8):1723–1736
Sivakumar B, Woldemeskel F, Puente C (2013) Nonlinear analysis of rainfall variability in Australia. Stochastic Environmental Research and Risk Assessment pp 1–11. doi: 10.1007/s00477-013-0689-y
Sprott JC (2003) Chaos and time-series analysis. Oxford University Press, Oxford
Srivastav RK, Sudheer KP, Chaubey I (2007) A simplified approach to quantifying predictive and parametric uncertainty in artificial neural network hydrologic models. Water Resour Res 43(10):W10407. doi:10.1029/2006WR005352
St-Hilaire A, Ouarda TBMJ, Bargaoui Z, Daigle A, Bilodeau L (2012) Daily river water temperature forecast model with a k-nearest neighbour approach. Hydrol Proces 26(9):1302–1310
Takens F (1981) Detecting strange attractors in turbulence. In: Rand DA, Jung LS (eds) Dynamical systems and turbulence. Lecture notes in mathematicsSpringer, Berlin, pp 366–381
Tong H (1978) On a threshold model. Sijhoff & Noordhoff, Amsterdam
Tong H (1983) Threshold models in nonlinear time series analysis. Springer, New York
Tongal H (2013a) Nonlinear dynamical approach and self-exciting threshold model in forecasting daily stream-flow. Fresenius Environ Bull 10:2836–2847
Tongal H (2013b) Nonlinear forecasting of stream flows using a chaotic approach and artificial neural networks. Earth Sci Res J 17(2):119–126
Tongal H, Berndtsson R (2014) Phase-space reconstruction and self-exciting threshold modeling approach to forecast lake water levels. Stoch Environ Res Risk Assess 28(4):955–971. doi:10.1007/s00477-013-0795-x
Tongal H, Demirel MC, Booij MJ (2013) Seasonality of low flows and dominant processes in the Rhine River. Stoch Environ Res Risk Assess 27(2):489–503. doi:10.1007/s00477-012-0594-9
Toth E, Brath A (2007) Multistep ahead streamflow forecasting: role of calibration data in conceptual and neural network modeling. Water Resour Res 43(11):W11405. doi:10.1029/2006WR005383
Valipour M, Banihabib ME, Behbahani SMR (2013) Comparison of the ARMA, ARIMA, and the autoregressive artificial neural network models in forecasting the monthly inflow of Dez dam reservoir. J Hydrol 476:433–441. doi:10.1016/j.jhydrol.2012.11.017
Wang W, Vrijling JK, Van Gelder PHAJM, Mac J (2006) Testing for nonlinearity of streamflow processes at different timescales. J Hydrol 322(1):247–268
Wang W-C, Chau K-W, Cheng C-T, Qiu L (2009) A comparison of performance of several artificial intelligence methods for forecasting monthly discharge time series. J Hydrol 374(3):294–306. doi:10.1016/j.jhydrol.2009.06.019
Wu CL, Chau KW (2010) Data-driven models for monthly streamflow time series prediction. Eng Appl Artif Intel 23(8):1350–1367
Wu CL, Chau KW (2013) Prediction of rainfall time series using modular soft computing methods. Eng Appl Artif Intel 997(3):997–1007. doi:10.1016/j.engappai.2012.05.023
Wu CL, Chau KW, Fan C (2010) Prediction of rainfall time series using modular artificial neural networks coupled with data-preprocessing techniques. J Hydrol 389(1):146–167. doi:10.1016/j.jhydrol.2010.05.040
Xie H-B, Guo J-Y, Zheng Y-P (2010) Using the modified sample entropy to detect determinism. Phys Lett A 374(38):3926–3931. doi:10.1016/j.physleta.2010.07.058
Xu J, Li W, Ji M, Lu F, Dong S (2010) A comprehensive approach to characterization of the nonlinearity of runoff in the headwaters of the Tarim River, western China. Hydrol Proces 24(2):136–146. doi:10.1002/hyp.7484
Yakowitz SJ (1973) A stochastic model for daily river flows in an arid region. Water Resour Res 9(5):1271–1285
Yan K, Huanjie C, Songbai S (2011) A measure of hydrological system complexity based on sample entropy. IEEE international symposium on water resource and environmental protection (ISWREP) 2011, pp 470–473
Yevjevich V (1972) Stochastic processes in hydrology. Water Resources Publications, LLC
Yu XY, Liong SY, Babovic V (2004) EC-SVM approach for real-time hydrologic forecasting. J Hydroinform 6(3):209–233
Acknowledgments
We would like to thank anonymous reviewers for their invaluable comments to improve this paper.
Author information
Authors and Affiliations
Corresponding author
Electronic supplementary material
Below is the link to the electronic supplementary material.
Rights and permissions
About this article
Cite this article
Tongal, H., Berndtsson, R. Impact of complexity on daily and multi-step forecasting of streamflow with chaotic, stochastic, and black-box models. Stoch Environ Res Risk Assess 31, 661–682 (2017). https://doi.org/10.1007/s00477-016-1236-4
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00477-016-1236-4