Application Of Maximum Enteropy Principle On Hydrological Frequency Distribution Models

Hydrological frequency analysis is an important research topic in hydrological calculation. Through quantitative estimating possible long-term changes of hydrological phenomena in the future, it can provide a scientific basis for the planning, construction and operation management of Water Resources and Hydropower Engineering design. Flood frequency analysis includes curve selection and parameters estimation. In line-type selection, we must make specific frequency analysis for the corresponding region and then choose a proper curve for this region from a variety of distributions, because hydrological frequency distribution is greatly depend on different regions. In parameters estimation, the current common methods for parameter estimation are moments, maximum likelihood, probability-weighted moment method, weight function method and the fitting method, etc., but these traditional methods for parameter estimation have some limitations.Maximum entropy principle is a promising solution to estimate the parameters of hydrological frequency analysis, and its application include hydrological variables probability distribution derivation and parameter estimation. The hydrological frequency probability distribution derived from maximum entropy principle has minimum human error, and the results are objective, reasonable, of high accuracy and have good statistical performance, so it provides a new way for estimating hydrological frequency parameter. The application of maximum entropy principle for estimating hydrological frequency parameter in China usually choose Pearson typeⅢdistribution, and still lack systematic research.Based on the above background, this paper reviewed the recent hydrological frequency research literature using maximum entropy principle, and investigated the parameter estimation method based on maximum entropy principle, and then programmed a set of procedures to systematically analyze and apply hydrological distribution models. Through example applications, analyzed the performance of parameter estimation methods based on maximum entropy principle to provide design basis for water project planning and design for study area. The main result of this paper is as follows:(1)The status and progress of application of maximum entropy principle to study hydrological frequency in recent years was summed up and the application of the probability distribution function on hydrological frequency analysis and conventional parameter estimation methods(moment method and the maximum likelihood method) were introduced. The probability distribution function includes normal distribution type, extreme value distribution type, Gamma distribution type, Logistic and Pareto distribution type.(2)The author applied the maximum entropy method proposed by Professor Singh and developed a set of computer programs to estimate hydrological parameters based on the principle of maximum entropy. This paper detailed the derivation process of parameter estimation equation based on maximum entropy principle of the normal type (normal distribution, two parameter log-normal distribution, three-parameter log-normal distribution), extreme value distribution type (extreme value type I distribution, log-extreme value type I distribution, generalized extreme value distribution, Weibull distribution), gamma distribution type (exponential distribution, two parameter gamma distribution, Pearson typeⅢdistribution and log-Pearson typeⅢdistribution), Logistic distribution type (two-parameters log-Logistic distribution) and the Pareto distribution type (two-parameter Pareto distribution, two-parameter generalized Pareto distribution) and developed a set of computer programs to estimate hydrological frequency parameters based on maximum entropy principle.(3)The research employed the annual runoff series data of main stations in Northern Shaanxi, analyzed the hydrological frequency based on maximum entropy principle according to 14 type distribution curve for each station, calculated the parameters of each distribution based on maximum entropy principle, and plotted theoretical frequency curve of each station.The theoretical frequency distribution curve figures based on maximum entropy principle of each station showed that the fitting effect and flexible of normal distribution and exponential distribution is not so good for the reason that the numbers of parameters of these two distributions were rare and the form of the probability density distribution is simple; two-parameter generalized Pareto distribution usually only suited for Cs≥2, so the fitting effect was poor too; normal distribution, Pearson typeⅢ,and log-Pearson typeⅢdistribution fitted well with the annual runoff data of most stations.Using the goodness of fit evaluation criteria of OLS and AIC, the author compared the maximum entropy principle method with the moment method and the maximum likelihood method. Comparison of various methods and analysis showed that the parameters estimated by maximum entropy method and moment method and maximum likelihood estimation is the same for normal distribution and exponential distribution which had relatively less number of parameters and simpler form., for most other distributions, the maximum entropy method and the maximum likelihood method was closer, and better than the moment method; the distribution parameter of Pearson typeⅢfor Xinghe station can't be estimated with the maximum likelihood method, and then the maximum entropy method can be used., for the log-Pearson typeⅢdistribution, the distribution parameters of Ansai station, Xinghe station, and Suide station can't be estimated with the moment method and the maximum likelihood method, while maximum entropy method can be used here, so it was shown that the maximum entropy principle (method) had significant advantages.Through the statistical analysis of the minimum values of OLS and AIC, we obtained the optimal distribution model and the corresponding optimal parameters for annual runoff data of each station in Northern Shaanxi. The optimal distribution models under these two criteria were consistent. The optimal distribution models of each station are: log-extreme value type I distribution for Liujihe station; extreme value type I distribution for Jiaokouhe station; extreme value type I distribution for Jiaokouhe station; three-parameter lognormal distribution for Zhidan station; two-parameter lognormal distribution for Zhangcunyi station; two-parameter log-Logistic distribution for Ansai station; two-parameter lognormal distribution for Xinghe station; Pearson typeⅢfor Suide station. Considering the values of OLS and AIC for all stations comprehensively, it was suggested that lognormal distribution can be used as theoretical distribution model for series of annual runoff in Northern Shaanxi, and Pearson typeⅢdistribution model can also be referred to with specific circumstances of Northern Shaanxi to do further research.