幂函数型水位流量关系回归方法研究

为提高幂函数型水位流量关系的拟合精度，对幂函数回归的传统方法进行了剖析，导出了采用乘积随机误差时，幂函数型因变量的数学期望表达式和幂函数型非线性回归与其线性化回归的残差平方和的关系式。导出的公式表明，因变量的估计值并非是其数学期望的估计值；传统方法所求幂函数的回归系数不满足该因变量的残差平方和为最小。结合幂函数型水位流量关系回归计算的实例，选用5种计算方法进行分析比较。结果表明，高斯-牛顿法、化非线性为线性的加权回归法均显著优于传统方法；高斯-牛顿法优于自适应加速遗传算法，与拉格朗日乘子法相同。

Study on regression methods for power function type stage-discharge relation

In order to improve the fitting accuracy of stage-discharge relation,the traditional method of power function type regression is analyzed.The formula of mathematical expectation of dependent variable of power function type and the relation of residual sum of squares of linear regression and nonlinear regression are deduced when using the product random error.The analysis result indicates that the expected value of the dependent variable is not the expected value of mathematical expectation,the calculated regression coefficient of power function using traditional method does not make residual sum of squares of the dependent variable to be minimum.Combining the example of regression for power function type stage-discharge relation,five regression methods are selected and compared.The results show that the Gaussian-Newton and the nonlinear-to-linear weighted regression methods are notably better than the traditional method;the Gaussian-Newton method is better than the adaptive accelerative genetic algorithm,and same with the Lagrange multiplier method.