响应曲面建模的稳健M-回归方法

摘要/Abstract

摘要： 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，〖JP2〗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。

关键词: 响应曲面, 稳健M-回归, 中心复合设计

Abstract: Response surface methodology is a powerful tool for product/process improvement and optimization. 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 observations, the constant variance assumption does not hold in practice. To dampen the effect of such observation random errors on the least square regression model, robust regression techniques are employed. Consider that the outlier which may occur in different experimental region and based on central composite design, 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 comparison, 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 industry 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).

Key words: response surface methodology, robust Mestimators, central composite design

方俊涛1，何桢2，宋琳曦2，张阳2. 响应曲面建模的稳健M-回归方法[J]. 工业工程, 2012, 15(3): 98-103.

Fang Juntao1, He Zhen2, Song Linxi2, Zhang Yang2. The robust M-estimators in Response Surface Modeling[J]. Industrial Engineering Journal , 2012, 15(3): 98-103.

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