Skip to main content
SpringerLink
Log in

Impact of complexity on daily and multi-step forecasting of streamflow with chaotic, stochastic, and black-box models

  • Original Paper
  • Published:
Stochastic Environmental Research and Risk Assessment Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  • Abarbanel H (1996) Analysis of observed chaotic data. Springer, New York

    Book  Google Scholar 

  • 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

    Article  CAS  Google Scholar 

  • 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

    Article  Google Scholar 

  • Al-Awadhi S, Jolliffe I (1998) Time series modelling of surface pressure data. Int J Climatol 18(4):443–455

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Birikundavyi S, Labib R, Trung HT, Rousselle J (2002) Performance of neural networks in daily streamflow forecasting. J Hydrol 75:392–398

    Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  CAS  Google Scholar 

  • Daliakopoulos IN, Coulibaly P, Tsanis IK (2005) Groundwater level forecasting using artificial neural networks. J Hydrol 309:229–240

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Elshorbagy A, Panu US, Simonovic SP (2001) Analysis of cross-correlated chaotic streamflows. Hydrol Sci J 46:781–793

    Article  Google Scholar 

  • Farmer DJ, Sidorowich JJ (1987) Predicting chaotic time series. Phys Rev Lett 59:845–848

    Article  CAS  Google Scholar 

  • 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

    Article  Google Scholar 

  • Fraser AM, Swinney HL (1986) Independent coordinates for strange attractors from mutual information. Phys Rev A 33:1134–1140

    Article  CAS  Google Scholar 

  • 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

    Article  Google Scholar 

  • Grassberger P, Procaccia I (1983b) Measuring the strangeness of strange attractors. Phys D 9:189–208

    Article  Google Scholar 

  • Grayson R, Blöschl G (2001) Spatial patterns in catchment hydrology: observations and modelling. Cambridge University Press, Cambridge

    Google Scholar 

  • Hagan MT, Menhaj MB (1994) Training feedforward networks with the Marquardt algorithm. IEEE Trans Neural Netw 5(6):989–993

    Article  CAS  Google Scholar 

  • Hagan MT, Demuth HB, Beale MH (1996) Neural network design. PWS Publishing Company, Boston

    Google Scholar 

  • 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

    Article  CAS  Google Scholar 

  • Hunt BR, Kennedy JA, Li T-Y, Nusse HE (2004) The theory of chaotic attractors. Springer, New York

    Book  Google Scholar 

  • 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

    Article  CAS  Google Scholar 

  • Islam MN, Sivakumar B (2002) Characterization and prediction of runoff dynamics: a nonlinear dynamical view. Adv Water Resour 25(2):179–190

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Kalteh AM, Berndtsson R (2007) Interpolating monthly precipitation by self-organizing map (SOM) and multilayer perceptron (MLP). Hydrol Sci J 52:305–317

    Article  Google Scholar 

  • Kantz H, Schreiber T (1997) Nonlinear time series analysis. Cambridge University Press, Cambridge

    Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Kolmogorov AN (1959) Entropy per unit time as a metric invariant of automorphisms. Dokl Akad Nauk SSSR 124:754–755

    Google Scholar 

  • 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

    Article  Google Scholar 

  • Kozhevnikova IA, Shveikina VI (2008) Nonlinear dynamics of level variations in the Caspian Sea. Water Resour 35(3):297–304. doi:10.1134/s0097807808030056

    Article  CAS  Google Scholar 

  • 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

    Article  Google Scholar 

  • Lall U, Sharma A (1996) A nearest neighbor bootstrap for resampling hydrologic time series. Water Resour Res 32(3):679–693

    Article  Google Scholar 

  • 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

    Article  CAS  Google Scholar 

  • 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

    Article  Google Scholar 

  • Li Z, Zhang Y-K (2008) Multi-scale entropy analysis of Mississippi River flow. Stoch Environ Res Risk Assess 22(4):507–512

    Article  Google Scholar 

  • Lisi F, Villi V (2001) Chaotic forecasting of discharge time series: a case study. J Am Water Resour Assoc 37(2):271–279

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • McLeod AI, Li WK (1983) Diagnostic checking ARMA time series models using squared residual autocorrelations. J Time Ser Anal 4(4):269–273

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Packard NH, Crutchfield JP, Farmer JD, Shaw RS (1980) Geometry from a time series. Phys Rev Lett 45(9):712–716

    Article  Google Scholar 

  • Palani S, Liong S-Y, Tkalich P (2008) An ANN application for water quality forecasting. Marin Poll Bull 56(9):1586–1597

    Article  CAS  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  CAS  Google Scholar 

  • Porporato A, Ridolfi L (2001) Multivariate nonlinear prediction of river flows. J Hydrol 248(1):109–122

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Regonda SK, Sivakumar B, Jain A (2004) Temporal scaling in river flow: can it be chaotic? Hydrol Sci J 49(3):373–385

    Article  Google Scholar 

  • 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

    CAS  Google Scholar 

  • Schreiber T (1999) Interdisciplinary application of nonlinear time series method. Phys Rep 308(1):1–64

    Article  Google Scholar 

  • Schreiber T, Schmitz A (1996) Improved surrogate data for nonlinearity tests. Phys Rev Lett 77(4):635–638

    Article  CAS  Google Scholar 

  • 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

    Article  Google Scholar 

  • Sharma A, Tarboton DG, Lall U (1997) Streamflow simulation: A nonparametric approach. Water Resour Res 33(2):291–308

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Google Scholar 

  • 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

    Article  CAS  Google Scholar 

  • Sivakumar B (2000) Chaos theory in hydrology: important issues and interpretations. J Hydrol 227(1):1–20

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Sivakumar B (2008) Dominant processes concept, model simplification and classification framework in catchment hydrology. Stoch Environ Res Risk Assess 22(6):737–748

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  CAS  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Google Scholar 

  • Tong H (1978) On a threshold model. Sijhoff & Noordhoff, Amsterdam

    Book  Google Scholar 

  • Tong H (1983) Threshold models in nonlinear time series analysis. Springer, New York

    Book  Google Scholar 

  • Tongal H (2013a) Nonlinear dynamical approach and self-exciting threshold model in forecasting daily stream-flow. Fresenius Environ Bull 10:2836–2847

    Google Scholar 

  • Tongal H (2013b) Nonlinear forecasting of stream flows using a chaotic approach and artificial neural networks. Earth Sci Res J 17(2):119–126

    Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Wu CL, Chau KW (2010) Data-driven models for monthly streamflow time series prediction. Eng Appl Artif Intel 23(8):1350–1367

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  CAS  Google Scholar 

  • 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

    Google Scholar 

  • Yakowitz SJ (1973) A stochastic model for daily river flows in an arid region. Water Resour Res 9(5):1271–1285

    Article  Google Scholar 

  • 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

    Google Scholar 

Download references

Acknowledgments

We would like to thank anonymous reviewers for their invaluable comments to improve this paper.

Author information

Authors and Affiliations

  1. Department of Civil Engineering, Engineering Faculty, Süleyman Demirel University, 32260, Isparta, Turkey

    Hakan Tongal

  2. Center for Middle Eastern Studies & Department of Water Resources Engineering, Lund University, 221 00, Lund, Sweden

    Ronny Berndtsson

Authors
  1. Hakan Tongal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ronny Berndtsson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hakan Tongal.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (DOCX 4822 kb)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

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

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00477-016-1236-4

Keywords

Search

Navigation