Baybeian Theory For Estimating P-Ⅲ Distribution Parameters And Uncertainty

The hydrological frequency analysis is to derive hydrologic design values and to provide the basis for water resources and hydropower engineering designing plan. There are a lot of traditional hydrologic frequency analysis parameters estimation methods, such as moment method, optimal fitting curve method, maximum likelihood method and probability weighted moment method, these methods have been widely applied in engineering practice, but these methods have some shortcomings.With the emergence of Markov Chain Monte Carlo (MCMC) sampling methods, Bayesian theory was paid attention again, and is widely used in hydrologic frequency analysis. The paper summarizes the bayesian theory’s application in the hydrologic frequency analysis, comparing the methods used at home and abroad of sampling from the posterior distribution parameters synthetically, at the same time, introduce some sampling methods in other fields into the hydrologic frequency analysis, such as simulated annealing method, simulation method, particle filter method, important sampling method, etc. In addition, the paper apply two nonparametric methods, Bootstrap method and Bayes Bootstrap method,to sample from the measured sample creatively, and constitute a large number of sample group to statistical analysis the parameters. Finally,the author take precipitation samples of meteorological station at Weihe basin fit annual precipitation data by P-Ⅲ distribution, according the bayesian formula to infer the posterior stations parameters, use the Metropolis-Hastings method (MH), Metropolis method (RWM), the Adaptive Metropolis method (AM), of Rejection method (DR), DRAM algorithm(DRAM), Data expansion algorithm (DA), OBMC method, stochastic gradient MCMC method (SGLD), simulated annealing method (SA), simulated fire back (ST), parallel holds back (PT), the Equi-Energy algorithm (EE), the Bootstrap method(B), Bayes theorem the Bootstrap method (BB), importance sampling method (IS) and the particle filter sampling methods(PF) to sample from the posterior distribution parameters, and compare the convergence、the sampling speed and the efficiency index etc. synthetically of various sampling method. After compared all the sampling methods, employ optimize the proper sampling method to estimate distribution parameters and hydrologic design values then analyse the uncertainty. In this paper, the main research contents and conclusions are as follows:(1) P-Ⅲ distribution posterior distribution parameter. Apply the bayesian formula to calculate the P-Ⅲ distribution posterior distribution parameter, then sample from the posterior distribution parameters through the MCMC sampling methods,and statistical the generated sample, extract the posteriori parameter information.(2) The comparison and analysis among the performance of various sampling method. On the base of correspondence with the prior distribution of sampling methods, the sampling proposal distribution and the number of sampling, with the sampling histogram, convergence, sampling time and sampling efficiency as the index of contrast.The results show that in addition to the DA algorithm, the sampling histogram of other sampling algorithms is balanced, and explore the posterior distribution parameters state space well. In addition to the EE and ST method, the rest of the sampling methods have good convergence and basically stable when sampling to a certain number, its sample sequence converges to the posterior distribution parameters. The fast sampling speed of these sampling speed are AM, DRAM, MH, PT and DR.(3) The contrast of bayesian method and moment method, probability weighted moment method. Compare all sampling method based on bayesian theory with the traditional moment method and probability weighted moment methods, take the sum of squared residuals of frequency and theoretical frequency as a comparison standard. The results show that compared with the moment method and probability weighted moment method, there are smaller the sum of squared residuals by AM, DRAM, DR., MH, ST and PT,which illustrate that the bayesian theory can further improve the calculation accuracy of design storm.(4) The uncertainty analysis of P-Ⅲ distribution parameters and hydrological design value. We measure the uncertainty with confidence interval, using the Bootstrap method to calculate 90% confidence interval of the distribution parameters and hydrological design under a given frequency value, select AM, DRAM and BB these three methods as test objects,which explore the state space of posterior distribution parameters well, then calculate all the distribution parameters and 90% confidence interval of the hydrologic design values under 0.1%,1% and 10% these three frequency. The results show that for the same station, it doesn’t appear to be much different from each other.(5)The best methods are AM and DRAM for annual precipitation data of Weihe basin.