改进先验概率的贝叶斯法在设计洪水中的应用
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摘要
贝叶斯模型是降低设计洪水计算过程中线型选择不确定性的主要方法,其中,先验概率的确定对贝叶斯模型的建立至关重要。为了提高先验概率确定方法的普适性,本文以模型评价准则为基础,多方面考虑模型拟合特征,提出综合指数量化线型的拟合情况,并将其作为先验信息应用于贝叶斯模型,以便合理确定各线型的先验概率,进而达到降低线型选择不确定性影响的目的。研究结果表明:基于综合指数值来挖掘样本的先验信息,可获得较为可靠的先验概率值来合理预估后验概率,使得拟合效果较好的线型获得更高的权重;有先验信息的贝叶斯模型相比无先验信息的贝叶斯模型结果更优;耦合多个评价准则的综合指数值为先验概率的求解提供了新思路。
Bayesian model is a powerful tool for reducing model uncertainty in flood frequency analysis,and the key is how to calculate prior probability. To improve the universality of the calculation, first we work out a formula for calculating the comprehensive index and quantifying the effect of fitting used by different models based on model evaluation criteria. Then, this index is used in calculation of the prior probability adopted in our Bayesian model so as to reduce the uncertainty in model selection. Results show that the comprehensive index calculations lead to more reliable values of the prior probability that help improve the estimation of posterior probability, thereby reducing the uncertainty in flood frequency calculations. Compared with Bayesian model without such prior information, the present results are much better. A comprehensive index coupled with multiple evaluation criteria provides a new idea in determining prior probability.
Bayesian model is a powerful tool for reducing model uncertainty in flood frequency analysis,and the key is how to calculate prior probability. To improve the universality of the calculation, first we work out a formula for calculating the comprehensive index and quantifying the effect of fitting used by different models based on model evaluation criteria. Then, this index is used in calculation of the prior probability adopted in our Bayesian model so as to reduce the uncertainty in model selection. Results show that the comprehensive index calculations lead to more reliable values of the prior probability that help improve the estimation of posterior probability, thereby reducing the uncertainty in flood frequency calculations. Compared with Bayesian model without such prior information, the present results are much better. A comprehensive index coupled with multiple evaluation criteria provides a new idea in determining prior probability.
引文
[1]水利水电工程设计洪水计算规范:SL 44-2006[S].北京:中国水利水电出版社,2006.Regulation for calculating design flood of water resources and hydropower projects:SL 44-2006[S].Beijng:China Water&Power Press,2006.(in Chinese)
[2]唐亦汉,陈晓宏,叶长青,等.考虑历史洪水的混合分布对不同尾型分布的应用对比研究[J].水力发电学报,2015,34(4):31-37.TANG Yihan,CHEN Xiaohong,YE Changqing,et al.Application of historical flood-concerned mixed distribution with different tail types of PDFs[J].Journal of Hydroelectric Engineering,2015,34(4):31-37.(in Chinese)
[3]叶长青,陈晓宏,张家鸣,等.变化环境下北江流域水文极值演变特征、成因及影响[J].自然资源学报,2012,27(12):2102-2112.YE Changqing,CHEN Xiaohong,ZHANG Jiaming,et al.Changing properties,causes and impacts of extreme stream flow under the changing environment in Beijiang River,China[J].Journal of Natural Resources,2012,27(12):2102-2112.(in Chinese)
[4]HOETING J A,MADIGAN D,VOLINSKY R C T.Bayesian model averaging:A tutorial[J].Statistical Science,1999,14(4):382-401.
[5]梁忠民,戴荣,王军,等.基于贝叶斯模型平均理论的水文模型合成预报研究[J].水力发电学报,2010,29(2):114-118.LIANG Zhongmin,DAI Rong,WANG Jun,et al.Study on forecast combination of different hydrological models by Bayesian model averaging[J].Journal of Hydroelectric Engineering,2010,29(2):114-118.(in Chinese)
[6]YAN H,MORADKHANI H.Bayesian Model Averaging for Flood Frequency Analysis[C].World Environmental and Water Resources Congress.2014:1886-1895.
[7]梁忠民,李磊,王军,等.考虑参数和线型不确定性的水文设计值估计的贝叶斯方法[J].天津大学学报,2010,43(5):379-384.LIANG Zhongmin,LI Lei,WANG Jun,et al.Bayesian method for hydrological frequency analysis considering uncertainties of parameter and model[J].Journal of Tianjin University,2010,43(5):379-384.(in Chinese)
[8]THEORELL A,KATHARINA N.Model uncertainty analysis for metabolic network inference:A case study in Bayesian model averaging[J].IFAC-PapersOnLine,2018,51(19):124-125.
[9]NG S H.A Bayesian model-averaging approach for multiple-response optimization[J].Journal of Quality Technology,2010,42(1):52-68.
[10]王斌,张洪波,辛琛,等.基于贝叶斯模型加权平均法的径流序列高频分量预测研究[J].水力发电学报,2016,35(5):75-83.WANG Bin,ZHANG Hongbo,XIN Chen,et al.Application of Bayesian model averaging method to prediction of high-frequency components in runoff series[J].Journal of Hydroelectric Engineering,2016,35(5):75-83.(in Chinese)
[11]董磊华,熊立华,万民.基于贝叶斯模型加权平均方法的水文模型不确定性分析[J].水利学报,2011,42(9):1065-1074.DONHG Leihua,XIONG Lihua,WAN Min.Uncertain-ty analysis of hydrological modeling using the Bayesian model averaging method[J].Journal of Hydraulic Engineering,2011,42(9):1065-1074.(in Chinese)
[12]WOOD E F,IGNACIO R.A Bayesian approach to analyzing uncertainty among flood frequency models[J].Water Resources Research,1975,11(11):839-843.
[13]TANG W H.Bayesian frequency analysis[J].Journal of the Hydraulics Division,1980,106(7):1203-1218.
[14]PERICCHI L R,RODRIGUEZ-ITURBE I.On some problems in Bayesian model choice in hydrology[J].Journal of the Royal Statistical Society:Series D(The Statistician),1983,32(1/2):273-278.
[15]KASS R E,RAFTERY A E.Bayes factors[J].Journal of the American Statistical Association,1995,90(430):773-795.
[16]VAN GELDER P,VAN NOORTWIJK J M,DUITS M T.Selection of probability distributions with a case study on extreme Oder river discharges[C]//Safety and Reliability.1999,2:1999.
[17]桑燕芳,王中根,刘昌明.水文频率线型选择与综合过程中贝叶斯因子求解新方法[J].自然资源学报,2013,28(4):687-695.SANG Yanfang,WANG Zhonggen,LIU Changming.New method of bayesian factor computation in hydrologic frequency model selections and averages[J].Journal of Natural Resources,2013,28(4):687-695.(in Chinese)
[18]刘攀,郭生练,田向荣,等.基于贝叶斯理论的水文频率线型选择与综合[J].武汉大学学报(工学版),2005,38(5):36-40.LIU Pan,GUO Shenglian,TIAN Xiangrong,et al.Selecting and averaging of flood frequency models based on Bayesian theory[J].Engineering Journal of Wuhan University,2005,38(5):36-40.(in Chinese)
[19]梁忠民,戴荣,雷杨,等.基于贝叶斯理论的水文频率分析方法研究[J].水力发电学报,2009,28(4):22-26.LIANG Zhongmin,DAI Rong,LEI Yang,et al.Hydrological frequency analysis based on Bayesian theory[J].Journal of Hydroelectric Engineering,2009,28(4):22-26.(in Chinese)
[20]黄伟军,丁晶.水文水资源系统贝叶斯分析现状与前景[J].水科学进展,1994,5(3):242-247.HUANG Weijun,DING Jing.Hydrology and water resources system Bayesian analysis about state-of-the-art and prospect[J].Advances in Water Science,1994,5(3):242-247.(in Chinese)
[21]WINKLER R L.The assessment of prior distributions in Bayesian analysis[J].Journal of the American Statistical Association,1967,62(319):776-800.
[22]LINDLEY D V.The use of prior probability distributions in statistical inference and decision[C]//Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability.1961,1:453-468.
[23]梁忠民,戴荣,李彬权.基于贝叶斯理论的水文不确定性分析研究进展[J].水科学进展,2010,21(2):274-281.LIANG Zhongmin,DAI Rong,LI Binquan.A review of hydrological uncertainty analysis based on Bayesian theory[J].Advances in Water Science,2010,21(2):274-281.(in Chinese)
[24]张立杰,蒙潇然.线型选择对西江流域水文频率计算不确定性的影响[J].人民珠江,2017,38(2):37-41.ZHANG Lijie,MENG Xiaoran.Influence of model selection on uncertainties of hydrological frequency analysis in Xijiang River basin[J].Pearl River,2017,38(2):37-41.(in Chinese)
[25]DI BALDASSARRE G,LAIO F,MONTANARI A.Design flood estimation using model selection criteria[J].Physics and Chemistry of the Earth,Parts A/B/C,2009,34(10/11/12):606-611.
[26]舒晓娟,陈洋波,任启伟.模型选择准则在洪水频率分析中的应用[J].水利学报,2010,41(1):80-85.SHU Xiaojuan,CHEN Yangbo,REN Qiwei.Model selection criteria for flood frequency analysis[J].Journal of Hydraulic Engineering,2010,41(1):80-85.(in Chinese)
[27]MOHAMMADPOUR O,HASSANZADEH Y,KHODADADI A,et al.Selecting the best flood flow frequency model using multi-criteria group decisionmaking[J].Water Resources Management,2014,28(12):3957-3974.
[28]CHEN X H,SHAO Q X,XU C Y,et al.Comparative study on the selection criteria for fitting flood frequency distribution models with emphasis on upper-tail behavior[J].Water,2017,9(5):320.
[29]HOSKING J R M,WALLIS J R.Regional frequency analysis:An approach based on L-moments[M].New York:Cambridge University Press,1997.
[30]RAFTERY K A E.Bayes factors[J].Journal of the American Statistical Association,1995,90(430):773-795.
[31]李晓伟,王宏声.喜河水电站设计洪水及地区组成分析[J].西北水利发电,2007,23(3):20-23.LI Xiaowei,WANG Hongsheng.On the design flood and its regional composition of the Xihe hydropower station[J].Journal of Northwest Hydroelectric Power,2007,23(3):20-23.(in Chinese)
[32]梁忠民,钟平安,华家鹏.水文水利计算[M].北京:中国水利水电出版社,2006.LIANG Zhongmin,ZHONG Pin’an,HUA Jiapeng.Calculation of hydrology and water conservancy[M].Beijng:China Water&Power Press,2006.(in Chinese)
[33]杨之麟.汉江上游“83.7”特大洪水与安康水电站设计洪水复核[J].水力发电,1987(4):4-7.YANG Zhilin."83.7"great flood in the upper reaches of Hanjiang River and review of design flood of Ankang Hydropower Station[J].Water Power,1987(4):4-7.(in Chinese)
[2]唐亦汉,陈晓宏,叶长青,等.考虑历史洪水的混合分布对不同尾型分布的应用对比研究[J].水力发电学报,2015,34(4):31-37.TANG Yihan,CHEN Xiaohong,YE Changqing,et al.Application of historical flood-concerned mixed distribution with different tail types of PDFs[J].Journal of Hydroelectric Engineering,2015,34(4):31-37.(in Chinese)
[3]叶长青,陈晓宏,张家鸣,等.变化环境下北江流域水文极值演变特征、成因及影响[J].自然资源学报,2012,27(12):2102-2112.YE Changqing,CHEN Xiaohong,ZHANG Jiaming,et al.Changing properties,causes and impacts of extreme stream flow under the changing environment in Beijiang River,China[J].Journal of Natural Resources,2012,27(12):2102-2112.(in Chinese)
[4]HOETING J A,MADIGAN D,VOLINSKY R C T.Bayesian model averaging:A tutorial[J].Statistical Science,1999,14(4):382-401.
[5]梁忠民,戴荣,王军,等.基于贝叶斯模型平均理论的水文模型合成预报研究[J].水力发电学报,2010,29(2):114-118.LIANG Zhongmin,DAI Rong,WANG Jun,et al.Study on forecast combination of different hydrological models by Bayesian model averaging[J].Journal of Hydroelectric Engineering,2010,29(2):114-118.(in Chinese)
[6]YAN H,MORADKHANI H.Bayesian Model Averaging for Flood Frequency Analysis[C].World Environmental and Water Resources Congress.2014:1886-1895.
[7]梁忠民,李磊,王军,等.考虑参数和线型不确定性的水文设计值估计的贝叶斯方法[J].天津大学学报,2010,43(5):379-384.LIANG Zhongmin,LI Lei,WANG Jun,et al.Bayesian method for hydrological frequency analysis considering uncertainties of parameter and model[J].Journal of Tianjin University,2010,43(5):379-384.(in Chinese)
[8]THEORELL A,KATHARINA N.Model uncertainty analysis for metabolic network inference:A case study in Bayesian model averaging[J].IFAC-PapersOnLine,2018,51(19):124-125.
[9]NG S H.A Bayesian model-averaging approach for multiple-response optimization[J].Journal of Quality Technology,2010,42(1):52-68.
[10]王斌,张洪波,辛琛,等.基于贝叶斯模型加权平均法的径流序列高频分量预测研究[J].水力发电学报,2016,35(5):75-83.WANG Bin,ZHANG Hongbo,XIN Chen,et al.Application of Bayesian model averaging method to prediction of high-frequency components in runoff series[J].Journal of Hydroelectric Engineering,2016,35(5):75-83.(in Chinese)
[11]董磊华,熊立华,万民.基于贝叶斯模型加权平均方法的水文模型不确定性分析[J].水利学报,2011,42(9):1065-1074.DONHG Leihua,XIONG Lihua,WAN Min.Uncertain-ty analysis of hydrological modeling using the Bayesian model averaging method[J].Journal of Hydraulic Engineering,2011,42(9):1065-1074.(in Chinese)
[12]WOOD E F,IGNACIO R.A Bayesian approach to analyzing uncertainty among flood frequency models[J].Water Resources Research,1975,11(11):839-843.
[13]TANG W H.Bayesian frequency analysis[J].Journal of the Hydraulics Division,1980,106(7):1203-1218.
[14]PERICCHI L R,RODRIGUEZ-ITURBE I.On some problems in Bayesian model choice in hydrology[J].Journal of the Royal Statistical Society:Series D(The Statistician),1983,32(1/2):273-278.
[15]KASS R E,RAFTERY A E.Bayes factors[J].Journal of the American Statistical Association,1995,90(430):773-795.
[16]VAN GELDER P,VAN NOORTWIJK J M,DUITS M T.Selection of probability distributions with a case study on extreme Oder river discharges[C]//Safety and Reliability.1999,2:1999.
[17]桑燕芳,王中根,刘昌明.水文频率线型选择与综合过程中贝叶斯因子求解新方法[J].自然资源学报,2013,28(4):687-695.SANG Yanfang,WANG Zhonggen,LIU Changming.New method of bayesian factor computation in hydrologic frequency model selections and averages[J].Journal of Natural Resources,2013,28(4):687-695.(in Chinese)
[18]刘攀,郭生练,田向荣,等.基于贝叶斯理论的水文频率线型选择与综合[J].武汉大学学报(工学版),2005,38(5):36-40.LIU Pan,GUO Shenglian,TIAN Xiangrong,et al.Selecting and averaging of flood frequency models based on Bayesian theory[J].Engineering Journal of Wuhan University,2005,38(5):36-40.(in Chinese)
[19]梁忠民,戴荣,雷杨,等.基于贝叶斯理论的水文频率分析方法研究[J].水力发电学报,2009,28(4):22-26.LIANG Zhongmin,DAI Rong,LEI Yang,et al.Hydrological frequency analysis based on Bayesian theory[J].Journal of Hydroelectric Engineering,2009,28(4):22-26.(in Chinese)
[20]黄伟军,丁晶.水文水资源系统贝叶斯分析现状与前景[J].水科学进展,1994,5(3):242-247.HUANG Weijun,DING Jing.Hydrology and water resources system Bayesian analysis about state-of-the-art and prospect[J].Advances in Water Science,1994,5(3):242-247.(in Chinese)
[21]WINKLER R L.The assessment of prior distributions in Bayesian analysis[J].Journal of the American Statistical Association,1967,62(319):776-800.
[22]LINDLEY D V.The use of prior probability distributions in statistical inference and decision[C]//Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability.1961,1:453-468.
[23]梁忠民,戴荣,李彬权.基于贝叶斯理论的水文不确定性分析研究进展[J].水科学进展,2010,21(2):274-281.LIANG Zhongmin,DAI Rong,LI Binquan.A review of hydrological uncertainty analysis based on Bayesian theory[J].Advances in Water Science,2010,21(2):274-281.(in Chinese)
[24]张立杰,蒙潇然.线型选择对西江流域水文频率计算不确定性的影响[J].人民珠江,2017,38(2):37-41.ZHANG Lijie,MENG Xiaoran.Influence of model selection on uncertainties of hydrological frequency analysis in Xijiang River basin[J].Pearl River,2017,38(2):37-41.(in Chinese)
[25]DI BALDASSARRE G,LAIO F,MONTANARI A.Design flood estimation using model selection criteria[J].Physics and Chemistry of the Earth,Parts A/B/C,2009,34(10/11/12):606-611.
[26]舒晓娟,陈洋波,任启伟.模型选择准则在洪水频率分析中的应用[J].水利学报,2010,41(1):80-85.SHU Xiaojuan,CHEN Yangbo,REN Qiwei.Model selection criteria for flood frequency analysis[J].Journal of Hydraulic Engineering,2010,41(1):80-85.(in Chinese)
[27]MOHAMMADPOUR O,HASSANZADEH Y,KHODADADI A,et al.Selecting the best flood flow frequency model using multi-criteria group decisionmaking[J].Water Resources Management,2014,28(12):3957-3974.
[28]CHEN X H,SHAO Q X,XU C Y,et al.Comparative study on the selection criteria for fitting flood frequency distribution models with emphasis on upper-tail behavior[J].Water,2017,9(5):320.
[29]HOSKING J R M,WALLIS J R.Regional frequency analysis:An approach based on L-moments[M].New York:Cambridge University Press,1997.
[30]RAFTERY K A E.Bayes factors[J].Journal of the American Statistical Association,1995,90(430):773-795.
[31]李晓伟,王宏声.喜河水电站设计洪水及地区组成分析[J].西北水利发电,2007,23(3):20-23.LI Xiaowei,WANG Hongsheng.On the design flood and its regional composition of the Xihe hydropower station[J].Journal of Northwest Hydroelectric Power,2007,23(3):20-23.(in Chinese)
[32]梁忠民,钟平安,华家鹏.水文水利计算[M].北京:中国水利水电出版社,2006.LIANG Zhongmin,ZHONG Pin’an,HUA Jiapeng.Calculation of hydrology and water conservancy[M].Beijng:China Water&Power Press,2006.(in Chinese)
[33]杨之麟.汉江上游“83.7”特大洪水与安康水电站设计洪水复核[J].水力发电,1987(4):4-7.YANG Zhilin."83.7"great flood in the upper reaches of Hanjiang River and review of design flood of Ankang Hydropower Station[J].Water Power,1987(4):4-7.(in Chinese)