• 地球信息科学理论与方法 •

最小二乘法估算Pearson-Ⅲ型分布参数的改进

1. 1. 成都信息工程大学资源与环境学院,成都 610225
2. 四川省局气候中心,成都 610071
• 收稿日期:2015-11-25 修回日期:2016-02-20 出版日期:2016-09-27 发布日期:2016-09-27
• 通讯作者: 陈军 E-mail:494834920@qq.com
• 作者简介:

作者简介：袁典(1992-),男,湖北荆州人,硕士生,研究方向为3S集成与气象应用。E-mail: 837513724@qq.com

• 基金资助:
四川省教育厅项目(15ZB0184)四川省国土资源厅科学研究计划(KJ-2015-18)威海市科学技术发展计划项目“威海市暴雨次生灾害预报预警系统研究”(2014GNS014)

The Improvements in Estimating Parameters of Pearson-Ⅲ Distribution Based on the Least Squares Method

YUAN Dian1, CHEN Jun1,*(), QING Qingtao2, DENG Guowei2, LI Yuting1

1. 1. College of Resources and Environment, Chengdu University of Information Technology, Chengdu 610225, China
2. Climate Center of Sichuan Province, Chengdu 610771, China
• Received:2015-11-25 Revised:2016-02-20 Online:2016-09-27 Published:2016-09-27
• Contact: CHEN Jun E-mail:494834920@qq.com

Pearson-Ⅲ曲线分布在气象、水文和农业等领域有广泛的应用,其概率密度函数包含形状参数$(α)$、尺度参数$(β)$和起始值$(α0)$3个待估计参数。应用Pearson-Ⅲ分布来估算强度的关键在于这3个参数估算的精度。由于原有最小二乘法在估算参数时未考虑各参数的有效区间,参数$α0$可能小于0,并导致估算强度出现负值,从而使雨强、水速、水位等强度估算出现异常值。经理论推导证明,若Pearson-Ⅲ概率密度函数中的参数$α0$非负,则估算出的强度值不会出现负值。以该推导为基础,提出了一种改进的Pearson-Ⅲ分布三参数估算算法。首先根据最小二乘法估算迭代计算一系列参数组合;然后按离差平方和的大小排列各组参数;接着利用各组参数分别计算$α0$值,过滤使$α0$小于0的参数组合;最后在剩下的参数中选取离差平方和最小的一组作为最优参数。以估算暴雨风险值作算法测试实验,结果表明改进算法模拟得到的$α0$值始终大于等于0,估算出的暴雨风险值均在合理范围之内。对比改进算法与传统算法的参数分布拟合检验结果,发现改进算法能使更多的气象站点通过了置信度为0.05的分布拟合检验。因此,利用改进后的算法估算出的Pearson-Ⅲ概率密度函数参数更合理,且强度估算结果更准确,对编制城市暴雨强度公式有一定的参考价值。

Abstract:

Pearson-Ⅲ distribution is widely applied in various fields such as meteorology, hydrology, agriculture and so on. Pearson-Ⅲ probability density equation has three parameters to be estimated, including the shape parameter of alpha, the scale parameter of beta and the initial parameter of alpha-0. The precision of estimating the three parameters is one of the key focuses when using Pearson-Ⅲ distribution to forecast rainstorm intensity in actual applications. Without considering the effective range, the traditional method used in estimating Pearson-Ⅲ distribution's three parameters may cause alpha-0 to be less than zero, thus making the predicted rainstorm intensity to be negative. This is impossible to happen when predicting the real physical quantities such as the rainstorm intensity, flow velocity, water depth and so on. Strict theoretical derivation has proved in this paper that the results of rainstorm intensity estimation would be nonnegative as long as alpha-0 maintained nonnegative. Based on the above conclusion, an improved algorithm aiming at fixing the drawbacks of the traditional estimation algorithm was proposed in this paper. Firstly, a series of parameters are estimated by iterative calculations based on the Least Square Method. Secondly, the sum of the squared deviations was calculated for each parameter group and each group was sorted by its value for the sum of the squared deviations. On that basis, different alpha-0 values are calculated for each group respectively to exclude groups with negative alpha-0 values. Finally, one group of parameters which has the minimum sum of the squared deviations was selected as the optimal combination for the three parameters of Pearson-Ⅲ. In the end, this algorithm was verified by estimating the rainstorm intensity in Sichuan Province. The result shows that the proposed algorithm can assure the alpha-0 to always be greater than or equal to zero and the value of rainstorm intensity to always be in a reasonable range. Furthermore, it makes more weather stations pass the distribution fit test with a 95% confidence level. In short, the improved algorithm is more reasonable and accurate than the traditional algorithm, and it provides a reference to the preparation of city rainstorm intensity formula.