Journal of Korea Water Resources Association (한국수자원학회논문집)

Korea Water Resources Association (한국수자원학회)
권호 표지
DOI QR코드

DOI QR Code

Development of the Bayesian method and its application to the water resources field

베이지안 기법의 발전 및 수자원 분야에의 적용

  • Na, Wooyoung (School of Civil, Environmental and Architectural Engineering, Korea University) ;
  • Yoo, Chulsang (School of Civil, Environmental and Architectural Engineering, Korea University)
  • 나우영 (고려대학교 공과대학 건축사회환경공학과) ;
  • 유철상 (고려대학교 공과대학 건축사회환경공학부)
  • Received : 2020.09.09
  • Accepted : 2020.10.05
  • Published : 2021.01.31
Download PDF
⟨ Previous Next ⟩

Abstract

The Bayesian method is a very useful statistical tool in various fields including water resources. Therefore, in this study, the background of the Bayesian statistics and its application to the water resources field are reviewed. First, the history of the Bayesian method from the birth to the present, and the achievements of Bayesian statisticians are summarized. Next, the derivation of the Bayes' theorem, which is the basis of the Bayesian method, is presented, and the roles of the three elements of the Bayes' theorem: priori distribution, likelihood function, and posteriori distribution are explained. In addition, the unique features and advantages of the Bayesian statistics are summarized. Finally, the cases in water resources where the Bayesian method is applied are summarized by dividing them into several categories. With a prevalence of information and big data in the future, the Bayesian method is expected to be used more actively in the water resources field.

베이지안 기법은 수자원을 포함한 다양한 분야에서 매우 유용한 통계적 도구로 이용되고 있다. 이에 본 연구에서는 베이지안 통계학에 대해 그 배경을 고찰하고, 수자원 분야에 적용된 사례를 소개하였다. 먼저, 베이지안 통계학의 탄생에서부터 현재에 이르기까지의 발전 과정과 이에 기여한 베이지안 통계학자들의 업적 등을 정리하였다. 다음으로 베이지안 기법의 근간이 되는 베이즈 정리의 유도 과정을 제시하고, 베이즈 정리의 세 요소인 사전분포, 우도함수, 사후분포의 역할에 대해 설명하였다. 또한, 베이지안 통계학이 가지는 고유한 특징과 장점에 대해 정리하였다. 마지막으로 수자원 분야에 베이지안 기법이 적용된 사례를 여러 범주로 나누어 정리하였다. 베이지안 기법은 정보 및 빅데이터의 활용이 커짐에 따라 수자원 분야에서 더욱 유용하게 적용될 것으로 전망된다.

Keywords

References

  1. Ait-El-Fquih, B., Gharamti, M.E., and Hoteit, I. (2016). "A Bayesian consistent dual ensemble Kalman filter for state-parameter estimation in subsurface hydrology." Hydrology and Earth System Sciences, Copernicus, Vol. 20, No. 8, pp. 3289-3307. https://doi.org/10.5194/hess-20-3289-2016
  2. Ajami, N.K., Duan, Q., and Sorooshian, S. (2007). "An integrated hydrologic Bayesian multimodel combination framework: Confronting input, parameter, and model structural uncertainty in hydrologic prediction." Water Resources Research, Wiley, Vol. 43, No. 1, W01403.
  3. Ang, A.H.S., and Tang, W.H. (2007). Probability concepts in engineering planning and design: Emphasis on application to civil and environmental engineering. Wiley, New York, N.Y.
  4. Bates, B.C., and Campbell, E. (2001). "A markov chain monte carlo scheme for parameter estimation and inference in conceptual rainfall-runoff modeling." Water Resources Research, Wiley, Vol. 37, No. 4, pp. 937-947. https://doi.org/10.1029/2000wr900363
  5. Berger, J.O. (2000). "Bayesian analysis: A look at today and thoughts of tomorrow." Journal of the American Statistical Association, Taylor & Francis, Vol. 95, No. 452, pp. 1269-1276. https://doi.org/10.1080/01621459.2000.10474328
  6. Berry, D.A. (1987). "Interim analysis in clinical trials: The role of the likelihood principle." The American Statistician, Taylor & Francis, Vol. 41, No. 2, pp. 117-122. https://doi.org/10.2307/2684222
  7. Beven, K., and Binley, A. (1992). "The future of distributed models: Model calibration and uncertainty prediction." Hydrological Processes, Wiley, Vol. 6, No. 3, pp. 279-298. https://doi.org/10.1002/hyp.3360060305
  8. Chan, T.U., Hart, B.T., Kennard, M.J., Pusey, B.J., Shenton, W., Douglas, M.M., Valentine, E., and Patel, S. (2012). "Bayesian network models for environmental flow decision making in the Daly River, Northern Territory, Australia." River Research and Applications, Wiley, Vol. 28, No. 3, pp. 283-301. https://doi.org/10.1002/rra.1456
  9. Coles, S.G., and Powell, E.A. (1996). "Bayesian methods in extreme value modelling: A review and new developments." International Statistical Review, Wiley, Vol. 64, No. 1, pp. 119-136. https://doi.org/10.2307/1403426
  10. Coles, S.G., and Tawn, J.A. (1996). "A Bayesian analysis of extreme rainfall data." Journal of the Royal Statistical Society, Wiley, Vol. 45, No. 4, pp. 463-478. https://doi.org/10.2307/2986068
  11. Cornfield, J. (1967). "Bayes theorem." Review of the International Statistical Institute, Vol. 35, No. 1, pp. 34-49. https://doi.org/10.2307/1401634
  12. Dawsey, W.J., Minsker, B.S., and Amir, E. (2007). "Real time assessment of drinking water systems using a dynamic Bayesian network." Proceedings World Environmental and Water Resources Congress 2007: Restoring Our Natural Habitat, ASCE, Tampa, FL, pp. 1-6.
  13. Di, S., Kondo, D., and Cirne, W. (2014). "Google hostload prediction based on Bayesian model with optimized feature combination." Journal of Parallel and Distributed Computing, Elsevier, Vol. 74, No. 1, pp. 1820-1832. https://doi.org/10.1016/j.jpdc.2013.10.001
  14. Duan, Q., Ajami, N.K., Gao, X., and Sorooshian, S. (2007). "Multi-model ensemble hydrologic prediction using Bayesian model averaging." Advances in Water Resources, Elsevier, Vol. 30, No. 5, pp. 1371-1386. https://doi.org/10.1016/j.advwatres.2006.11.014
  15. Engeland, K., and Gottschalk, L. (2002). "Bayesian estimation of parameters in a regional hydrological model." Hydrology and Earth System Sciences, Copernicus, Vol. 6, No. 5, pp. 883-898. https://doi.org/10.5194/hess-6-883-2002
  16. Evin, G., Kavetski, D., Thyer, M., and Kuczera, G. (2013). "Pitfalls and improvements in the joint inference of heteroscedasticity and autocorrelation in hydrological model calibration." Water Resources Research, Wiley, Vol. 49, No. 7, pp. 4518-4524. https://doi.org/10.1002/wrcr.20284
  17. Fienberg, S.E. (2006). "When did Bayesian inference become "Bayesian"?" Bayesian Analysis, ISBA, Vol. 1, No. 1, pp. 1-40. https://doi.org/10.1214/06-BA101
  18. Fienen, M.N., Masterson, J.P., Plant, N.G., Gutierrez, B.T., and Thieler, E.R. (2013). "Bridging groundwater models and decision support with a Bayesian network." Water Resources Research, Wiley, Vol. 49, No. 10, pp. 6459-6473. https://doi.org/10.1002/wrcr.20496
  19. Freer, J., Beven, K., and Ambroise, B. (1996). "Bayesian estimation of uncertainty in runoff prediction and the value of data: An application of the GLUE approach." Water Resources Research, Wiley, Vol. 32, No. 7, pp. 2161-2173. https://doi.org/10.1029/95WR03723
  20. Gelman, A., Carlin, J.B., Stern, H.S., Dunson, D.B., Vehtari, A., and Rubin, D.B. (2013). Bayesian data analysis. CRC press, Boca Raton, F.L., U.S.
  21. Han, S., and Coulibaly, P. (2017). "Bayesian flood forecasting methods: A review." Journal of Hydrology, Elsevier, Vol. 551, pp. 340-351. https://doi.org/10.1016/j.jhydrol.2017.06.004
  22. Hoeting, J.A., Madigan, D., Raftery, A.E., and Volinsky, C.T. (1999). "Bayesian model averaging: A tutorial." Statistical Science, IMS, Vol. 14, No. 4, pp. 382-401. https://doi.org/10.1214/ss/1009212519
  23. Insua, D.R., Diez, R.M., and Palomo, J. (2002) "Bayesian methods in Hydrology: A review." Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas, Springer, Vol. 96, No. 3, pp. 461-479.
  24. Ji, L., Zhi, X., Zhu, S., and Fraedrich, K. (2019). "Probabilistic precipitation forecasting over East Asia using Bayesian model averaging." Weather and Forecasting, AMS, Vol. 34, No. 2, pp. 377-392. https://doi.org/10.1175/WAF-D-18-0093.1
  25. Kaheil, Y.H., Gill, M.K., McKee, M., and Bastidas, L. (2006). "A new Bayesian recursive technique for parameter estimation." Water Resources Research, Wiley, Vol. 42, No. 8, W08423.
  26. Kavetski, D., Kuczera, G., and Franks, S.W. (2006). "Bayesian analysis of input uncertainty in hydrological modeling: 1. Theory." Water Resources Research, Wiley, Vol. 42, No. 3, W03407.
  27. Khan, M.S., and Coulibaly, P. (2010). "Assessing hydrologic impact of climate change with uncertainty estimates: Bayesian neural network approach." Journal of Hydrometeorology, AMS, Vol. 11, No. 2, pp. 482-495. https://doi.org/10.1175/2009JHM1160.1
  28. Kim, Y.O., and Palmer, R.N. (1997). "Value of seasonal flow forecasts in Bayesian stochastic programming." Journal of Water Resources Planning and Management, ASCE, Vol. 123, No. 6, p. 335.
  29. Krzysztofowicz, R. (1999). "Bayesian theory of probabilistic forecasting via deterministic hydrologic model." Water Resources Research, Wiley, Vol. 35, No. 9, pp. 2739-2750. https://doi.org/10.1029/1999WR900099
  30. Kuczera, G. (1982). "Combining site-specific and regional information: An empirical Bayes approach." Water Resources Research, Wiley, Vol. 18, No. 2, pp. 306-314. https://doi.org/10.1029/WR018i002p00306
  31. Kuczera, G. (1999). "Comprehensive at-site flood frequency analysis using Monte Carlo Bayesian inference." Water Resources Research, Wiley, Vol. 35, No. 5, pp. 1551-1557. https://doi.org/10.1029/1999WR900012
  32. Kuczera, G., Kavetski, D., Franks, S., and Thyer, M. (2006). "Towards a Bayesian total error analysis of conceptual rainfall-runoff models: Characterising model error using storm-dependent parameters." Journal of Hydrology, Elsevier, Vol. 331, No. 1-2, pp. 161-177. https://doi.org/10.1016/j.jhydrol.2006.05.010
  33. Kuczera, G., and Parent, E. (1998). "Monte Carlo assessment of parameter uncertainty in conceptual catchment models: the Metropolis algorithm." Journal of Hydrology, Elsevier, Vol. 211, No. 1-4, pp. 69-85. https://doi.org/10.1016/S0022-1694(98)00198-X
  34. Kwon, H.H., Brown, C., and Lall, U. (2008). "Climate informed flood frequency analysis and prediction in Montana using hierarchical Bayesian modeling." Geophysical Research Letters, Wiley, Vol. 35, No. 5, L05404.
  35. Lazar, N.A. (2003). "Bayesian empirical likelihood." Biometrika, Oxford Academic, Vol. 90, No. 2, pp. 319-326. https://doi.org/10.1093/biomet/90.2.319
  36. Lee, J., Lee, K., and Lee, Y. (2014). "History and future of Bayesian statistics." The Korean Journal of Applied Statistics, KJAS, Vol. 27, No. 6, pp. 855-863. https://doi.org/10.5351/KJAS.2014.27.6.855
  37. Lee, K.S., and Kim, S.U. (2008). "Identification of uncertainty in low flow frequency analysis using Bayesian MCMC method." Hydrological Processes: An International Journal, Wiley, Vol. 22, No. 12, pp. 1949-1964. https://doi.org/10.1002/hyp.6778
  38. Liedloff, A.C., Woodward, E.L., Harrington, G.A., and Jackson, S. (2013). "Integrating indigenous ecological and scientific hydrogeological knowledge using a Bayesian network in the context of water resource development." Journal of Hydrology, Elsevier, Vol. 499, pp. 177-187. https://doi.org/10.1016/j.jhydrol.2013.06.051
  39. Lima, C.H., Kwon, H.H., and Kim, Y.T. (2018). "A local-regional scaling-invariant Bayesian GEV model for estimating rainfall IDF curves in a future climate." Journal of Hydrology, Elsevier, Vol. 566, pp. 73-88. https://doi.org/10.1016/j.jhydrol.2018.08.075
  40. Madadgar, S., and Moradkhani, H. (2013). "A Bayesian framework for probabilistic seasonal drought forecasting." Journal of Hydrometeorology, AMS, Vol. 14, No. 6, pp. 1685-1705. https://doi.org/10.1175/JHM-D-13-010.1
  41. Maranzano, C.J., and Krzysztofowicz, R. (2004). "Identification of likelihood and prior dependence structures for hydrologic uncertainty processor." Journal of Hydrology, Elsevier, Vol. 290, No. 1-2, pp. 1-21. https://doi.org/10.1016/j.jhydrol.2003.11.021
  42. Marshall, L., Nott, D., and Sharma, A. (2007). "Towards dynamic catchment modelling: A Bayesian hierarchical mixtures of experts framework." Hydrological Processes: An International Journal, Wiley, Vol. 21, No. 7, pp. 847-861. https://doi.org/10.1002/hyp.6294
  43. McGrayne, S.B. (2011). The theory that would not die: how Bayes' rule cracked the enigma code, hunted down Russian submarines, & emerged triumphant from two centuries of controversy. Yale University Press, New Haven, C.T. U.S.
  44. Micevski, T., and Kuczera, G. (2009). "Combining site and regional flood information using a Bayesian Monte Carlo approach." Water Resources Research, Wiley, Vol. 45, No. 4, W04405.
  45. Mount, N., and Stott, T. (2008). "A discrete Bayesian network to investigate suspended sediment concentrations in an Alpine proglacial zone." Hydrological Processes: An International Journal, Wiley, Vol. 22, No. 18, pp. 3772-3784. https://doi.org/10.1002/hyp.6981
  46. Najafi, M.R., and Moradkhani, H. (2014). "A hierarchical Bayesian approach for the analysis of climate change impact on runoff extremes." Hydrological Processes, Wiley, Vol. 28, No. 26, pp. 6292-6308. https://doi.org/10.1002/hyp.10113
  47. O'Connell, D.R., Ostenaa, D.A., Levish, D.R., and Klinger, R.E. (2002). "Bayesian flood frequency analysis with paleohydrologic bound data." Water Resources Research, Wiley, Vol. 38, No. 5, pp. 16-1-16-13.
  48. Ouarda, T.B.M.J., and El‐Adlouni, S. (2011). "Bayesian nonstationary frequency analysis of hydrological variables." Journal of the American Water Resources Association, Wiley, Vol. 47, No. 3, pp. 496-505. https://doi.org/10.1111/j.1752-1688.2011.00544.x
  49. Perelman, L., and Ostfeld, A. (2010). "Bayesian networks for estimating contaminant source and propagation in a water distribution system using cluster structure." Proceedings of the 12th Annual Conference on Water Distribution Systems Analysis, Tucson, A.Z., U.S., pp. 426-435.
  50. Prochazka, A., Vysata, O., Valis, M., Tupa, O., Schatz, M., and Marik, V. (2015). "Bayesian classification and analysis of gait disorders using image and depth sensors of Microsoft Kinect." Digital Signal Processing, Elsevier, Vol. 47, pp. 169-177. https://doi.org/10.1016/j.dsp.2015.05.011
  51. Puga, J.L., Krzywinski, M., and Altman, N. (2015). "Points of significance: Bayes' theorem." Nature Methods, Nature Publishing Group, Vol. 12, No. 4, pp. 277-278. https://doi.org/10.1038/nmeth.3335
  52. Raftery, A.E., Gneiting, T., Balabdaoui, F., and Polakowski, M. (2005). "Using Bayesian model averaging to calibrate forecast ensembles." Monthly Weather Review, AMS, Vol. 133, No. 5, pp. 1155-1174. https://doi.org/10.1175/MWR2906.1
  53. Reis Jr., D.S., and Stedinger, J.R. (2005). "Bayesian MCMC flood frequency analysis with historical information." Journal of Hydrology, Elsevier, Vol. 313, No. 1, pp. 97-116. https://doi.org/10.1016/j.jhydrol.2005.02.028
  54. Seo, D.J., and Smith, J.A. (1991). "Rainfall estiation using raingages and radar - a Bayesian approach: 1. Derivation of estimators." Stochastic Hydrology and Hydraulics, Springer, Vol. 5, No. 1, pp. 17-29. https://doi.org/10.1007/BF01544175
  55. Shenton, W., Hart, B.T., and Chan, T.U. (2014). "A Bayesian network approach to support environmental flow restoration decisions in the Yarra River, Australia." Stochastic Environmental Research and Risk Assessment, Springer, Vol. 28, No. 1, pp. 57-65. https://doi.org/10.1007/s00477-013-0698-x
  56. Shin, J.Y., Ajmal, M., Yoo, J., and Kim, T.W. (2016). "A Bayesian network-based probabilistic framework for drought forecasting and outlook." Advances in Meteorology, Hindawi, London, UK.
  57. Smith, M. (2006). "Dam risk analysis using Bayesian networks." Proceedings of Geohazard, Engineering Conference International, Lillehammer, Norway.
  58. Smith, T.J., and Marshall, L.A. (2008). "Bayesian methods in hydrologic modeling: A study of recent advancements in Markov chain Monte Carlo techniques." Water Resources Research, Wiley, Vol. 44, No. 12, W00B05.
  59. Stedinger, J.R., and Kim, Y.O. (2010). "Probabilities for ensemble forecasts reflecting climate information." Journal of Hydrology, Elsevier, Vol. 391, No. 1-2, pp. 9-23. https://doi.org/10.1016/j.jhydrol.2010.06.038
  60. Tajiki, M., Schoups, G., Hendricks Franssen, H.J., Najafinejad, A., and Bahremand, A. (2020). "Recursive Bayesian estimation of conceptual rainfall‐runoff model errors in real‐time prediction of streamflow." Water Resources Research, Wiley, Vol. 56, No. 2, WR025237.
  61. Tang, Y., Marshall, L., Sharma, A., and Smith, T. (2016). "Tools for investigating the prior distribution in Bayesian hydrology." Journal of Hydrology, Elsevier, Vol. 538, pp. 551-562. https://doi.org/10.1016/j.jhydrol.2016.04.032
  62. Thiemann, M., Trosset, M., Gupta, H., and Sorooshian, S. (2001). "Bayesian recursive parameter estimation for hydrologic models." Water Resources Research, Wiley, Vol. 37, No. 10, pp. 2521-2535. https://doi.org/10.1029/2000WR900405
  63. Thyer, M., Renard, B., Kavetski, D., Kuczera, G., Franks, S.W., and Srikanthan, S. (2009). "Critical evaluation of parameter consistency and predictive uncertainty in hydrological modeling: A case study using Bayesian total error analysis." Water Resources Research, Wiley, Vol. 45, No. 12, W00B14.
  64. Vicens, G.J., Rodriguez-Iturbe, I., and Schaake Jr, J.C. (1975a). "A Bayesian framework for the use of regional information in hydrology." Water Resources Research, Wiley, Vol. 11, No. 3, pp. 405-414. https://doi.org/10.1029/WR011i003p00405
  65. Vicens, G.J., Rodriguez-Iturbe, I., and Schaake Jr, J.C. (1975b). "Bayesian generation of synthetic streamflows." Water Resources Research, Wiley, Vol. 11, No. 6, pp. 827-838. https://doi.org/10.1029/WR011i006p00827
  66. Viglione, A., Merz, R., Salinas, J.L., and Bloschl, G. (2013). "Flood frequency hydrology: 3. A Bayesian analysis." Water Resources Research, Wiley, Vol. 49, No. 2, pp. 675-692. https://doi.org/10.1029/2011WR010782
  67. Vrugt, J.A., Ter Braak, C.J., Gupta, H.V., and Robinson, B.A. (2009). "Equifinality of formal (DREAM) and informal (GLUE) Bayesian approaches in hydrologic modeling?" Stochastic Environmental Research and Risk Assessment, Springer, Vol. 23, No. 7, pp. 1011-1026. https://doi.org/10.1007/s00477-008-0274-y
  68. Wellen, C., Arhonditsis, G.B., Long, T., and Boyd, D. (2014). "Accommodating environmental thresholds and extreme events in hydrological models: A Bayesian approach." Journal of Great Lakes Research, Elsevier, Vol. 40, pp. 102-116.
  69. Yang, J., Reichert, P., and Abbaspour, K.C. (2007). "Bayesian uncertainty analysis in distributed hydrologic modeling: A case study in the Thur River basin (Switzerland)." Water Resources Research, Wiley, Vol. 43, No. 10, W10401.
  70. Zhang, X., Liang, F., Yu, B., and Zong, Z. (2011). "Explicitly integrating parameter, input, and structure uncertainties into Bayesian neural networks for probabilistic hydrologic forecasting." Journal of Hydrology, Elsevier, Vol. 409, No. 3-4, pp. 696-709. https://doi.org/10.1016/j.jhydrol.2011.09.002
  71. Zhang, X., and Zhao, K. (2012). "Bayesian neural networks for uncertainty analysis of hydrologic modeling: A comparison of two schemes." Water Resources Management, Springer, Vol. 26, No. 8, pp. 2365-2382. https://doi.org/10.1007/s11269-012-0021-5