Water Resources Management

, Volume 26, Issue 8, pp 2365–2382 | Cite as

Bayesian Neural Networks for Uncertainty Analysis of Hydrologic Modeling: A Comparison of Two Schemes

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Abstract

Bayesian Neural Networks (BNNs) have been shown as useful tools to analyze modeling uncertainty of Neural Networks (NNs). This research focuses on the comparison of two BNNs. The first BNNs (BNN-I) use statistical methods to describe the characteristics of different uncertainty sources (input, parameter, and model structure) and integrate these uncertainties into a Markov Chain Monte Carlo (MCMC) framework to estimate total uncertainty. The second BNNs (BNN-II) lump all uncertainties into a single error term (i.e. the residual between model prediction and measurement). In this study, we propose a simple BNN-II, which uses Genetic Algorithms (GA) and Bayesian Model Averaging (BMA) to calibrate Neural Networks with different structures (number of hidden units) and combine the predictions from different NNs to derive predictions and uncertainty estimation. We tested these two BNNs in two watersheds for daily and monthly hydrologic simulations. The BMA based BNNs (BNN-II) developed here outperforms BNN-I in the two watersheds in terms of both accurate prediction and uncertainty estimation. These results indicate that, given incomplete understanding of the characteristics associated with each uncertainty source and their interactions, the simple lumped error approach may yield better prediction and uncertainty estimation.

Keywords

Bayesian neural networks Bayesian model averaging Evolutionary Monte Carlo Hydrologic modeling Streamflow Uncertainty 
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Notes

Acknowledgements

We sincerely appreciate the constructive comments from the two anonymous reviewers, which highly improve the quality of this paper. This research is supported by US DOE Office of Science (DOE BER Office of Science DE-AC06-76RLO 1830).

References

  1. Ajami K, Duan Q, Gao X, Sorooshian S (2006) Multi-model combination techniques for hydrological forecasting: application to distributed model intercomparison project results. J Hydrometeor 8:755–768CrossRefGoogle Scholar
  2. Akkoyunlu A, Akiner ME (2010) Feasibility assessment of data driven models in predicting pollution trends of Omerli Lake, Turkey. Water Resour Manag 24(13):3419–3436CrossRefGoogle Scholar
  3. Aqil M, Kita I, Yano A, Nishiyama S (2007) Neural networks for real time catchment flow modeling and prediction. Water Resour Manag 21(10):1781–1796CrossRefGoogle Scholar
  4. ASCE Task Committee on Application of Artificial Neural Networks in Hydrology (2000a) Artificial neural networks in hydrology. I: Preliminary concepts. J Hydrol Eng 5(2):115–123CrossRefGoogle Scholar
  5. ASCE Task Committee on Application of Artificial Neural Networks in Hydrology (2000b) Artificial neural networks in hydrology. II: Hydrologic applications. J Hydrol Eng 5(2):124–137CrossRefGoogle Scholar
  6. Bosch DD, Sheridan JM, Lowrance RR, Hubbard RK, Strickland TC, Feyereisen GW, Sullivan DG (2007) Little river experimental watershed database. Water Resour Res 43:W09470. doi: 10.1029/2006WR005844 CrossRefGoogle Scholar
  7. Chen C, Chou FN, Chen BP (2010) Spatial information-based back-propagation neural network modeling for outflow estimation of ungauged catchment. Water Resour Manage 24(14):4175–4197CrossRefGoogle Scholar
  8. Duan Q, Ajami NK, Gao X, Sorooshian S (2007) Multi-model ensemble hydrologic prediction using Bayesian model averaging. Adv Water Resour 30(5):1371–1386CrossRefGoogle Scholar
  9. Gelman A, Carlin JB, Stern HS, Rubin DB (2004) Bayesian data analysis, 2nd edn. Chapman & Hall/CRC, Boca RatonGoogle Scholar
  10. Goldberg D (1989) Genetic algorithms in search, optimization and machine learning. Addison-Wesley, ReadingGoogle Scholar
  11. Güldal V, Tongal H (2010) Comparison of recurrent neural network, adaptive neuro-fuzzy inference system and stochastic models in Eğirdir Lake Level Forecasting. Water Resour Manage 24(1):105–128CrossRefGoogle Scholar
  12. Hersbach H (2000) Decomposition of the continuous ranked probability score for Ensembler Prediction Systems. Weather Forecast 15(5):559–570CrossRefGoogle Scholar
  13. Holland J (1975) Adaptation in natural and artificial systems. University of Michigan Press, Ann ArborGoogle Scholar
  14. Kass RE, Carlin BP, Gelman A, Neal RM (1998) Markov chain Monte Carlo in practice: a roundtable discussion. Am Stat 52(2):93–100Google Scholar
  15. Kavetski D, Kuczera G, Franks SW (2006) Bayesian analysis of input uncertainty in hydrological modeling: 1. Theory. Water Resour Res 42:W03407. doi: 10.1029/2005WR004368 CrossRefGoogle Scholar
  16. Khan MS, Coulibaly P (2006) Bayesian neural network for rainfall-runoff modeling. Water Resour Res 42:W07409. doi: 10.1029/2005WR003971 CrossRefGoogle Scholar
  17. Kingston GB, Lambert MF, Maier HR (2005) Bayesian training of artificial neural networks used for water resources modeling. Water Resour Res 41:W12409. doi: 10.1029/2005WR004152 CrossRefGoogle Scholar
  18. Laio F, Tamea S (2007) Verification tools for probabilistic forecasts of continuous hydrological variables. Hydrol Earth Syst Sci 11:1267–1277CrossRefGoogle Scholar
  19. Lampinen J, Vehtari A (2001) Bayesian approach for neural networks—reviews and case studies. Neural Netw 14(3):7–24CrossRefGoogle Scholar
  20. Legates DR, McCabe GJ (1999) Evaluating the use of “goodness of fit” measures in hydrologic and hydroclimatic model validation. Water Resour Res 35(1):233–241CrossRefGoogle Scholar
  21. Liang F (2005) Bayesian neural networks for non-linear time series forecasting. Stat Comput 15:13–29CrossRefGoogle Scholar
  22. Liang F, Wong WH (2001) Real-parameter evolutionary sampling with applications in Bayesian Mixture Models. J Am Stat Assoc 96:653–666CrossRefGoogle Scholar
  23. Maier HR, Dandy GC (2000) Neural networks for the prediction and forecasting of water resources variables: a review of modeling issues and applications. Environ Model Software 15:101–123CrossRefGoogle Scholar
  24. Matheson JE, Winkler RL (1976) Scoring rules for continuous probability distributions. Manage Sci 22:1087–1095CrossRefGoogle Scholar
  25. Mohanty S, Jha MK, Kumar A, Sudheer KP (2010) Artificial neural network modeling for groundwater level forecasting in a River Island of Eastern India. Water Resour Manage 24(9):1845–1865CrossRefGoogle Scholar
  26. Müller P, Insua DR (1998) Issues in Bayesian analysis of neural network models. Neural Comput 10:749–770CrossRefGoogle Scholar
  27. Neal RM (1996) Bayesian learning for neural networks. Springer, New YorkCrossRefGoogle Scholar
  28. Raftery AE, Gneiting T, Balabdaoui F, Polakowski M (2005) Using bayesian model averaging to calibrate forecast ensembles. Mon Weather Rev 113:1155–1174CrossRefGoogle Scholar
  29. Rezaeian Zadeh M, Amin S, Khalili D, Singh VP (2010) Daily outflow prediction by multi layer perceptron with logistic sigmoid and tangent sigmoid activation functions. Water Resour Manage 24(11):2673–2688CrossRefGoogle Scholar
  30. Safavi HR, Darzi F, Marino MA (2010) Simulation-optimization modeling of conjunctive use of surface water and groundwater. Water Resour Manage 24(10):1965–1988CrossRefGoogle Scholar
  31. Seyfried MS, Harris RC, Marks D, Jacob B (2000) A geographic database for watershed research, Reynolds Creek Experimental Watershed, Idaho, USA, USDA ARS Technical Bulletin NWRC-2000-3Google Scholar
  32. Sheridan JM (1997) Rainfall-streamflow relations for coastal plain watersheds. Trans ASAE 13(3):333–344Google Scholar
  33. Shirsath PB, Singh AK (2010) A comparative study of daily pan evaporation estimation using ANN, regression and climate based models. Water Resour Manage 24(8):1571–1581CrossRefGoogle Scholar
  34. Singh P, Deo MC (2007) Suitability of different neural networks in daily flow forecasting. Appl Soft Comput 7:968–978CrossRefGoogle Scholar
  35. Singh KK, Pal M, Singh VP (2010) Estimation of mean annual flood in indian catchments using backpropagation neural network and M5 model tree. Water Resour Manage 24(10):2007–2019CrossRefGoogle Scholar
  36. Thyer M, Renard B, Kavetski D, Kuczera G, Franks SW, Srikanthan S (2009) Critical evaluation of parameter consistency and predictive uncertainty in hydrological modeling: a case study using Bayesian total error analysis. Water Resour Res 45:W00B14. doi: 10.1029/2008WR006825 CrossRefGoogle Scholar
  37. Vrugt JA, Robinson BA (2007) Treatment of uncertainty using ensemble methods: comparison of sequential data assimilation and Bayesian model averaging. Water Resour Res 43:W01411. doi: 10.1029/2005WR004838 CrossRefGoogle Scholar
  38. Wang Y, Change J, Huang Q (2010) Simulation with RBF neural network model for reservoir operation rules. Water Resour Manage 24(11):2597–2610CrossRefGoogle Scholar
  39. Zhang B, Govindaraju RS (2000) Prediction of watershed runoff using Bayesian concepts and modular neural networks. Water Resour Res 36(3):753–762CrossRefGoogle Scholar
  40. Zhang X, Srinivasan R (2009) GIS-based spatial precipitation estimation: a comparison of geostatistical approaches. J Am Water Resour Assoc 45(4):894–906CrossRefGoogle Scholar
  41. Zhang X, Srinivasan R (2010) GIS-based spatial precipitation estimation using next generation radar and raingauge data. Environ Model Softw 25(12):81–1788Google Scholar
  42. Zhang X, Liang F, Srinivasan R, Van Liew M (2009a) Estimating uncertainty of streamflow simulation using Bayesian neural networks. Water Resour Res. doi: 10.1029/2008WR007030
  43. Zhang X, Srinivasan R, Bosch D (2009b) Calibration and uncertainty analysis of the SWAT model using Genetic Algorithms and Bayesian Model Averaging. J Hydrol. doi: 10.1016/j.jhydrol.2009.06.023
  44. Zhang X, Srinivasan R, Zhao K, Van Liew M (2009c) Evaluation of global optimization algorithms for parameter calibration of a computationally intensive hydrologic model. Hydrol Process 23(3):430–441CrossRefGoogle Scholar
  45. Zhang X, Liang F, Yu B, Zong Z (2011) Explicitly integrating parameter, input, and structure uncertainties into Bayesian Neural Networks for probabilistic hydrologic forecasting. J Hydrol 409(3–4):696–709CrossRefGoogle Scholar
  46. Zhao K, Popescu S, Zhang X (2008) Bayesian learning with Gaussian processes for supervised classification of hyperspectral data. Photogramm Eng Remote Sens 74:1223–1234Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Joint Global Change Research InstitutePacific Northwest National Laboratory and University of MarylandCollege ParkUSA
  2. 2.Department of Biology & Center on Global ChangeDuke UniversityDurhamUSA

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