基于Box-Cox变换的贝叶斯概率水文预报效率
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摘要
针对贝叶斯概率预报模型(Bayesian processor of forecasts,BPF)中输入数据的正态转换问题,探讨了Meta-Gaussian模型(MG)和Box-Cox变换(BC)对BPF模型性能的影响。首先利用MG和BC分别对BPF模型输入数据进行正态转换,然后分别建立BPF-MG和BPF-BC模型进行概率预报,最后对BPF-MG和BPF-BC在不同预见期和不同数据样本条件下的预报能力进行了分析。结果表明,当数据样本较少时,BPF-MG具有较高的稳定性,但BC转换比MG更简单,BC变换系数非常敏感;当数据样本增多后,BC变换的转换系数稳定,BPFBC预报质量提高。
Aimed at the normal transformation of input data in the Bayesian processor of forecasts(BPF), the impacts of the meta-Gaussian model(MG) and Box-Cox transformation(BC) on the performance of a BPF model are compared and discussed. We make the normal transformation of the BPF model's input data using MG and BC separately, construct a BPF-MG model and a BPF-BC model for probability forecasting, and analyze the forecasting capabilities of these two models in the conditions of different forecast periods and different data sizes. Results indicate that when the number of data samples is small, BPF-MG can achieve a higher stability through its complicated MG procedure, but it involves more complicated transformation than the BPF-BC model that has a very sensitive transform coefficient. With an increasing data size, the BPF-BC model is improved in forecasting capability and its BC coefficient becomes more stable.
Aimed at the normal transformation of input data in the Bayesian processor of forecasts(BPF), the impacts of the meta-Gaussian model(MG) and Box-Cox transformation(BC) on the performance of a BPF model are compared and discussed. We make the normal transformation of the BPF model's input data using MG and BC separately, construct a BPF-MG model and a BPF-BC model for probability forecasting, and analyze the forecasting capabilities of these two models in the conditions of different forecast periods and different data sizes. Results indicate that when the number of data samples is small, BPF-MG can achieve a higher stability through its complicated MG procedure, but it involves more complicated transformation than the BPF-BC model that has a very sensitive transform coefficient. With an increasing data size, the BPF-BC model is improved in forecasting capability and its BC coefficient becomes more stable.
引文
[1]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].Advance in Water Science,2010,21(2):274-281.(in Chinese)
[2]Krzysztofowicz R.Bayesian model of forecasted time series[J].Water Resources Research,1985,21(5):805-814.
[3]黄伟军,赵永龙,丁晶.径流的最优组合预测及其贝叶斯分析[J].成都科技大学学报,1996(6):97-102.HUANG Weijun,ZHAO Yonglong,DING Jing.Optimal combination prediction and its Bayesian analysis of runoff[J].Journal of Chengdu University of Science and Technology,1996(6):97-102.(in Chinese)
[4]张铭,李承军,张勇传.贝叶斯概率水文预报系统在中长期径流预报中的应用[J].水科学进展,2009,20(1):40-44.ZHANG Ming,LI Chengjun,ZHANG Yongchuan.Application of the Bayesian statistic hydrological forecast system to middle and long term runoff forecast[J].Advance in Water Science,2009,20(1):40-44.(in Chinese)
[5]Ma Z,Li Z,Zhang M,et al.Bayesian statistic forecasting model for middle-term and long-term runoff of a hydropower station[J].Journal of Hydrologic Engineering,2013,18(11):1458-1463.
[6]张洪刚,郭生练,刘攀,等.基于贝叶斯分析的概率洪水预报模型研究[J].水电能源科学,2004,22(1):22-25.ZHANG Honggang,GUO Shenglian,LIU Pan,et al.Probabilistic flood forecasting model based on Bayesian method[J].Water Resources and Power,2004,22(1):22-25.(in Chinese)
[7]钱名开,徐时进,王善序,等.淮河息县站流量概率预报模型研究[J].水文,2004,24(2):23-25.QIAN Mingkai,XU Shijin,WANG Shanxu,et al.Study on the probability flood forecasting model at Xixian Gauging Station of Huaihe River basin[J].China Journal of Hydrology,2004,24(2):23-25.(in Chinese)
[8]张冬冬.洪水频率分析与预报中的不确定性问题研究[D].北京:中国水利水电科学研究院,2015.ZHANG Dongdong.Research on uncertainty analysis of flood frequency and flood forecast[D].Beijing:China Institute of Water Resources&Hydropower Research,2015.(in Chinese)
[9]张洪刚.贝叶斯概率水文预报系统及其应用研究[D].武汉:武汉大学,2005.ZHANG Honggang.Probabilistic hydrologic forecasting system based on the Bayesian method[D].Wuhan:Wuhan University,2005.(in Chinese)
[10]周娅,郭萍,古今今.基于BP神经网络的概率径流预测模型[J].水力发电学报,2014,33(2):45-50.ZHOU Ya,GUO Ping,GU Jinjin.Probabilistic runoff forecasting model based on BP artificial neural network[J].Journal of Hydroelectric Engineering,2014,33(2):45-50.(in Chinese)
[11]李向阳,程春田,林剑艺.基于BP神经网络的贝叶斯概率水文预报模型[J].水利学报,2006,37(3):354-359.LI Xiangyang,CHENG Chuntian,LIN Jianyi.Bayesian probabilistic forecasting model based on BP ANN[J].Journal of Hydraulic Engineering,2006,37(3):354-359.(in Chinese)
[12]邢贞相.确定性水文模型的贝叶斯概率预报方法研究[D].南京:河海大学,2007.XING Zhengxiang.Study on Bayesian forecasting system of deterministic hydrologic models[D].Nanjing:Hohai University,2007.(in Chinese)
[13]Zhang X,Liang F,Yu B,et al.Explicitly integrating parameter,input,and structure uncertainties into Bayesian Neural Networks for probabilistic hydrologic forecasting[J].Journal of Hydrology,2011,409(3/4):696-709.
[14]Krzysztofowicz R,Kelly K S.Hydrologic uncertainty processor for probabilistic river stage forecasting[J].Water Resources Research,2000,36(11):3265-3277.
[15]Krzysztofowice R.Bayesian system for probabilistic river stage forecasting[J].Journal of Hydrology,2002,268(1):16-40.
[16]Biondi D,Versace P,Sirangelo B.Uncertainty assessment through a precipitation dependent hydrologic uncertainty processor:An application to a small catchment in southern Italy[J].Journal of Hydrology,2010,386(1):38-54.
[17]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.DONG Leihua,XIONG Lihua,WAN Min.Uncertainty analysis of hydrological modeling using the Bayesian model averaging method[J].Journal of Hydraulic Engineering,2011,42(9):1065-1074.(in Chinese)
[18]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):114-188.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)
[19]Box G E P,Cox D R.An analysis of transformations[J].Journal of the Royal Statistical Society,1964,26(2):211-252.
[20]刘章君,郭生练,李天元,等.贝叶斯概率洪水预报模型及其比较应用研究[J].水利学报,2014,45(9):1019-1028.LIU Zhangjun,GUO Shenglian,LI Tianyuan,et al.Comparative study of Bayesian probabilistic flood forecasting models[J].Journal of Hydraulic Engineering,2014,45(9):1019-1028.(in Chinese)
[21]陈璐,卢韦韦,周建中,等.水文预报不确定性对水库防洪调度的影响分析[J].水利学报,2016,47(1):77-84.CHEN Lu,LU Weiwei,ZHOU Jianzhong,et al.Effect of streamflow forecast uncertainty on reservoir operation[J].Journal of Hydraulic Engineering,2016,47(1):77-84.(in Chinese)
[22]李明亮,杨大文,陈劲松.基于采样贝叶斯方法的洪水概率预报研究[J].水力发电学报,2011,30(3):27-33.LI Mingliang,YANG Dawen,CHEN Jinsong.Probabilistic flood forecasting by a sampling-based Bayesian model[J].Journal of Hydroelectric Engineering,2011,30(3):27-33.(in Chinese)
[23]Chiang Y M,Chang L C,Chang F J.Comparison of staticfeedforward and dynamic-feedback neural networks for rainfall-runoff modeling[J].Journal of Hydrology,2004,290(3/4):297-311.
[24]Santhosh D,Srinivas V V.Bivariate frequency analysis of floods using a diffusion based kernel density estimator[J].Water Resources Research,2013,49(12):8328-8343.
[25]Chen X,Tung Y K.Investigation of polynomial normal transform[J].Structural Safety,2003,25(4):423-445.
[26]Hamasaki T,Kim S Y.Box and Cox power-transformation to confined and censored non-normal responses in regression[J].Computational Statistics&Data Analysis,2007,51(8):3788-3799.
[27]徐炜,张驰,彭勇,等.基于降雨预报信息的梯级水电站不确定优化调度研究I:聚合分解降维[J].水利学报,2013,44(8):924-933.XU Wei,ZHANG Chi,PENG Yong,et al.Stochastic optimization operation for cascade hydropower reservoirs by using precipitation forecasts I.Using aggregation decomposition methodology to reduce high computational costs[J].Journal of Hydraulic Engineering,2013,44(8):924-933.(in Chinese)
[28]韩义超,徐炜,张弛,等.径流贝叶斯概率预报在水库发电优化调度中的应用[J].水利水电科技进展,2014,34(4):39-45.HAN Yichao,XU Wei,ZHANG Chi,et al.Building optimization model for reservoir hydropower generation using Bayesian forecasting system and stochastic dynamic programming[J].Advances in Science and Technology of Water Resources,2014,34(4):39-45.(in Chinese)
[29]Li L,Xu C Y,Xia J,et al.Uncertainty estimates by Bayesian method with likelihood of AR(1)plus Normal model and AR(1)plus Multi-Normal model in different time-scales hydrological models[J].Journal of Hydrology,2011,406(1):54-65.
[2]Krzysztofowicz R.Bayesian model of forecasted time series[J].Water Resources Research,1985,21(5):805-814.
[3]黄伟军,赵永龙,丁晶.径流的最优组合预测及其贝叶斯分析[J].成都科技大学学报,1996(6):97-102.HUANG Weijun,ZHAO Yonglong,DING Jing.Optimal combination prediction and its Bayesian analysis of runoff[J].Journal of Chengdu University of Science and Technology,1996(6):97-102.(in Chinese)
[4]张铭,李承军,张勇传.贝叶斯概率水文预报系统在中长期径流预报中的应用[J].水科学进展,2009,20(1):40-44.ZHANG Ming,LI Chengjun,ZHANG Yongchuan.Application of the Bayesian statistic hydrological forecast system to middle and long term runoff forecast[J].Advance in Water Science,2009,20(1):40-44.(in Chinese)
[5]Ma Z,Li Z,Zhang M,et al.Bayesian statistic forecasting model for middle-term and long-term runoff of a hydropower station[J].Journal of Hydrologic Engineering,2013,18(11):1458-1463.
[6]张洪刚,郭生练,刘攀,等.基于贝叶斯分析的概率洪水预报模型研究[J].水电能源科学,2004,22(1):22-25.ZHANG Honggang,GUO Shenglian,LIU Pan,et al.Probabilistic flood forecasting model based on Bayesian method[J].Water Resources and Power,2004,22(1):22-25.(in Chinese)
[7]钱名开,徐时进,王善序,等.淮河息县站流量概率预报模型研究[J].水文,2004,24(2):23-25.QIAN Mingkai,XU Shijin,WANG Shanxu,et al.Study on the probability flood forecasting model at Xixian Gauging Station of Huaihe River basin[J].China Journal of Hydrology,2004,24(2):23-25.(in Chinese)
[8]张冬冬.洪水频率分析与预报中的不确定性问题研究[D].北京:中国水利水电科学研究院,2015.ZHANG Dongdong.Research on uncertainty analysis of flood frequency and flood forecast[D].Beijing:China Institute of Water Resources&Hydropower Research,2015.(in Chinese)
[9]张洪刚.贝叶斯概率水文预报系统及其应用研究[D].武汉:武汉大学,2005.ZHANG Honggang.Probabilistic hydrologic forecasting system based on the Bayesian method[D].Wuhan:Wuhan University,2005.(in Chinese)
[10]周娅,郭萍,古今今.基于BP神经网络的概率径流预测模型[J].水力发电学报,2014,33(2):45-50.ZHOU Ya,GUO Ping,GU Jinjin.Probabilistic runoff forecasting model based on BP artificial neural network[J].Journal of Hydroelectric Engineering,2014,33(2):45-50.(in Chinese)
[11]李向阳,程春田,林剑艺.基于BP神经网络的贝叶斯概率水文预报模型[J].水利学报,2006,37(3):354-359.LI Xiangyang,CHENG Chuntian,LIN Jianyi.Bayesian probabilistic forecasting model based on BP ANN[J].Journal of Hydraulic Engineering,2006,37(3):354-359.(in Chinese)
[12]邢贞相.确定性水文模型的贝叶斯概率预报方法研究[D].南京:河海大学,2007.XING Zhengxiang.Study on Bayesian forecasting system of deterministic hydrologic models[D].Nanjing:Hohai University,2007.(in Chinese)
[13]Zhang X,Liang F,Yu B,et al.Explicitly integrating parameter,input,and structure uncertainties into Bayesian Neural Networks for probabilistic hydrologic forecasting[J].Journal of Hydrology,2011,409(3/4):696-709.
[14]Krzysztofowicz R,Kelly K S.Hydrologic uncertainty processor for probabilistic river stage forecasting[J].Water Resources Research,2000,36(11):3265-3277.
[15]Krzysztofowice R.Bayesian system for probabilistic river stage forecasting[J].Journal of Hydrology,2002,268(1):16-40.
[16]Biondi D,Versace P,Sirangelo B.Uncertainty assessment through a precipitation dependent hydrologic uncertainty processor:An application to a small catchment in southern Italy[J].Journal of Hydrology,2010,386(1):38-54.
[17]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.DONG Leihua,XIONG Lihua,WAN Min.Uncertainty analysis of hydrological modeling using the Bayesian model averaging method[J].Journal of Hydraulic Engineering,2011,42(9):1065-1074.(in Chinese)
[18]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):114-188.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)
[19]Box G E P,Cox D R.An analysis of transformations[J].Journal of the Royal Statistical Society,1964,26(2):211-252.
[20]刘章君,郭生练,李天元,等.贝叶斯概率洪水预报模型及其比较应用研究[J].水利学报,2014,45(9):1019-1028.LIU Zhangjun,GUO Shenglian,LI Tianyuan,et al.Comparative study of Bayesian probabilistic flood forecasting models[J].Journal of Hydraulic Engineering,2014,45(9):1019-1028.(in Chinese)
[21]陈璐,卢韦韦,周建中,等.水文预报不确定性对水库防洪调度的影响分析[J].水利学报,2016,47(1):77-84.CHEN Lu,LU Weiwei,ZHOU Jianzhong,et al.Effect of streamflow forecast uncertainty on reservoir operation[J].Journal of Hydraulic Engineering,2016,47(1):77-84.(in Chinese)
[22]李明亮,杨大文,陈劲松.基于采样贝叶斯方法的洪水概率预报研究[J].水力发电学报,2011,30(3):27-33.LI Mingliang,YANG Dawen,CHEN Jinsong.Probabilistic flood forecasting by a sampling-based Bayesian model[J].Journal of Hydroelectric Engineering,2011,30(3):27-33.(in Chinese)
[23]Chiang Y M,Chang L C,Chang F J.Comparison of staticfeedforward and dynamic-feedback neural networks for rainfall-runoff modeling[J].Journal of Hydrology,2004,290(3/4):297-311.
[24]Santhosh D,Srinivas V V.Bivariate frequency analysis of floods using a diffusion based kernel density estimator[J].Water Resources Research,2013,49(12):8328-8343.
[25]Chen X,Tung Y K.Investigation of polynomial normal transform[J].Structural Safety,2003,25(4):423-445.
[26]Hamasaki T,Kim S Y.Box and Cox power-transformation to confined and censored non-normal responses in regression[J].Computational Statistics&Data Analysis,2007,51(8):3788-3799.
[27]徐炜,张驰,彭勇,等.基于降雨预报信息的梯级水电站不确定优化调度研究I:聚合分解降维[J].水利学报,2013,44(8):924-933.XU Wei,ZHANG Chi,PENG Yong,et al.Stochastic optimization operation for cascade hydropower reservoirs by using precipitation forecasts I.Using aggregation decomposition methodology to reduce high computational costs[J].Journal of Hydraulic Engineering,2013,44(8):924-933.(in Chinese)
[28]韩义超,徐炜,张弛,等.径流贝叶斯概率预报在水库发电优化调度中的应用[J].水利水电科技进展,2014,34(4):39-45.HAN Yichao,XU Wei,ZHANG Chi,et al.Building optimization model for reservoir hydropower generation using Bayesian forecasting system and stochastic dynamic programming[J].Advances in Science and Technology of Water Resources,2014,34(4):39-45.(in Chinese)
[29]Li L,Xu C Y,Xia J,et al.Uncertainty estimates by Bayesian method with likelihood of AR(1)plus Normal model and AR(1)plus Multi-Normal model in different time-scales hydrological models[J].Journal of Hydrology,2011,406(1):54-65.