在水位流量关系曲线拟合中,非线性最小二乘法(Non-linear Least Square,NLS)是广泛运用的方法。但NLS中采用的对数变换有时不能起到稳定方差的作用,且没有考虑异方差会导致水位流量关系中的参数和流量估计值不可靠的问题。为了克服这个局限性,采用Box-Cox变换进行改进,并利用极大似然估计法(Maximum Likelihood Estimation,MLE)对变换后的模型进行参数估计。结果表明:相比于NLS中的对数变换模型,基于Box-Cox变换模型能更好地得到稳定方差,更倾向产生正态分布。并且发现对数变换模型是Box-Cox变换模型的一个特例,因此Box-Cox变换模型能够更合理地推断水位流量关系,在实际应用中的适用范围更广。
Abstract
The nonlinear least squares (NLS) is widely used in stage-discharge curve fitting. The logarithmic transformation used by NLS, however, sometimes cannot play the role of stabilizing the variance with no consideration of the heteroskedasticity, which leads to unreliable parameters and flow rates estimated in stage-discharge relation. To overcome this limitation, the Box-Cox transformation was adopted and the maximum likelihood estimation method (MLE) was used to estimate the parameters of the transformed model. Results show that compared with the logarithmic transformation model in NLS, the stability variance based on the Box-Cox transformation model can be better obtained, more favorable for normal distribution. It is also found that the logarithmic transformation model is a special case of the Box-Cox transformation model. Therefore, the Box-Cox transformation model could more reasonably infer the stage-discharge relation and has a wider application range in practical applications.
关键词
水位流量关系 /
非线性最小二乘法 /
Box-Cox变换 /
极大似然估计法 /
对数变换
Key words
stage-discharge relation /
non-linear least square /
Box-Cox transform /
maximum likelihood estimation /
logarithmic transform
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基金
国家自然科学基金项目(11731012,11571052);湖南省自然科学基金项目(2017JJ2271,2017JJ2274,2018JJ2417)
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