响应曲面建模的稳健M-回归方法

The robust M-estimators in Response Surface Modeling

响应曲面方法是生产过程改进和优化的一种非常有效的方法。在传统的响应曲面模型的建立过程中,通常假定随机误差服从正态分布且相互独立具有相同的方差。但是实际生产中随机误差的方差并不是完全相同,观测值中会存在异常点,这就需要稳健的估计方法来抑制异常点对模型估计的影响。为了降低异常点对响应曲面模型最优值的影响,针对响应曲面方法中的中心复合设计,充分考虑到不同实验设计位置上可能出现异常点的情况,对稳健M-回归方法:Huber-估计、Tukey-估计和Welsch-估计进行了理论比较研究。研究结果表明Welsch和Tukey-估计能有效改善异常点对响应曲面模型最优值的影响,消弱异常点对中心复合设计的干扰。通过一个来自化工方面的案例,计算了中心复合设计不同位置存在异常点与不存在异常点时,响应曲面模型的最优值,对比分析得出当异常点与响应均值的偏离程度较大时(10倍标准差),稳健M-估计尤其是Welsch和Tukey-估计显著提高响应曲面建模的稳健性。

Response surface methodology is a powerful tool for product/process improvement and optimiza- tion. In response surface modeling, the random errors are assumed to be normally distributed independent random variables with constant variance. However, due to the fact that outliers are inevitable in the obser- vations, the constant variance assumption does not hold in practice. To dampen the effect of such observa- tion random errors on the least square regression model, robust regression techniques are employed. Con- sider that the outlier which may occur in different experimental region and based on central composite de- sign, performance analysis of reducing the influence of outliers for the M-estimators of robust regression is made. It includes three estimators： Huber-estimator, Tukey-estimator, and Welsch-estimator. By compar- ison, it shows that Welsch and Tukey-estimators are better than Huber-estimator in reducing the effect of outliers among response surface optimization and in response surface design. An example from chemical in- dustry is used to calculate the optimal value of response surface model based on different experiment region of central composite design with and without outlier. In other words, the robust M-estimators, especially Welsch and Tukey-estimators, significantly improve the robustness of response surface modeling in large magnitude outliers （10 standard deviation）.