响应曲面方法中试验设计与模型估计的比较研究
发布时间：2018-12-21 21:06
【摘要】：响应曲面方法(RSM)是现代统计与数学的集成方法,被广泛应用于响应变量与自变量之间函数关系研究中。通过响应曲面设计、建模和优化完成整个响应曲面过程。在响应曲面建模中,通常要求随机误差服从正态分布相互独立且具有相同的方差;输入变量相互独立,没有测量误差。在实际生产过程中,这些假设条件只能近似地被满足,并且自变量中很可能会出现测量误差,响应数据中也可能存在异常点情况。针对上述两个问题,本文主要研究在输入变量存在测量误差时,响应曲面设计的稳健性以及在异常点存在时,响应曲面模型的稳健性。 本文介绍了响应曲面方法的一些基本理论和传统的模型估计方法、响应曲面设计的评价准则、稳健估计方法和误差变量模型。在此基础了比较分析了不同的响应曲面设计和不同估计方法的稳健性,具体从以下几个方面进行研究。 首先,探讨了不同的响应曲面设计对响应数据中异常点的稳健性问题。主要包括中心复合设计——中心复合序贯设计、中心复合有界设计和Box-Behnken设计。基于这些设计的比较分析结果表明,中心点位置异常点极大地增加了模型的波动性,降低了模型的精度,轴线点和立方体点位置异常点影响模型二次项或交叉项的估计,增加了模型优化和预测的难度。 其次,提出了两种响应曲面设计预测方差的评价指标,研究分析了误差变量模型的单位预测方差和传统的三因子响应曲面设计预测性能。仿真分析表明,对于测量误差的方差比较小时,中心复合表面设计的稳健性是最优的;当方差较大时,Box-Behnken设计的预测性更好一些。 再次,对存在测量误差的二阶球形响应曲面设计优化准则和预测方差性质进行比较和评估。为了直观展示不同设计对测量误差的稳健性,我们提出单位预测方差最大值与最小值图和设计空间比率图,计算了不同设计的G-效率值。得出了不同因子水平下,稳健的响应曲面设计方案。 最后,基于响应数据中的异常点,探究了二阶响应曲面模型的稳健估计方法,研究分析了异常点存在或非正态时不同估计方法的稳健性。通过仿真和实例得出比较结果。
[Abstract]:Response surface method (RSM) is an integrated method of modern statistics and mathematics, which is widely used in the study of the functional relationship between response variables and independent variables. The whole process of response surface is completed by designing, modeling and optimizing the response surface. In response surface modeling, random errors are usually required to be independent from normal distribution and have the same variance, and input variables are independent without measurement errors. In the actual production process, these assumptions can only be approximately satisfied, and there may be measurement errors in the independent variables, and there may be outliers in the response data. In view of the above two problems, the robustness of response surface design and the robustness of response surface model when there are measurement errors in input variables are studied in this paper. This paper introduces some basic theories of response surface method and traditional model estimation method, evaluation criterion of response surface design, robust estimation method and error variable model. Based on this, the robustness of different response surface design and different estimation methods are compared and analyzed. Firstly, the robustness of different response surface designs to outliers in response data is discussed. It mainly includes central composite design, central composite sequential design, center composite bounded design and Box-Behnken design. The results of comparative analysis based on these designs show that the center point outliers greatly increase the volatility of the model and reduce the accuracy of the model. The location anomaly points of the axis and cube points affect the estimation of the quadratic or cross terms of the model. It increases the difficulty of model optimization and prediction. Secondly, two evaluation indexes of prediction variance of response surface design are proposed, and the unit prediction variance of error variable model and the prediction performance of traditional three-factor response surface design are studied and analyzed. Simulation results show that the robustness of the central composite surface design is optimal when the variance ratio of the measurement error is small, and the predictability of the Box-Behnken design is better when the variance is large. Thirdly, the optimization criteria and the properties of predictive variance for the design of second-order spherical response surfaces with measurement errors are compared and evaluated. In order to show the robustness of different designs to the measurement error, we propose the maximum and minimum values of unit predictive variance and the design space ratio diagram, and calculate the G- efficiency values of different designs. A robust response surface design scheme with different factor levels is obtained. Finally, based on the outliers in the response data, the robust estimation methods of the second-order response surface model are discussed, and the robustness of the different estimation methods for the existence or non-normal state of the outliers is analyzed. The comparison results are obtained by simulation and examples.
【学位授予单位】：天津大学
【学位级别】：博士
【学位授予年份】：2011
【分类号】：O211.3;C81
本文编号：2389463
[Abstract]:Response surface method (RSM) is an integrated method of modern statistics and mathematics, which is widely used in the study of the functional relationship between response variables and independent variables. The whole process of response surface is completed by designing, modeling and optimizing the response surface. In response surface modeling, random errors are usually required to be independent from normal distribution and have the same variance, and input variables are independent without measurement errors. In the actual production process, these assumptions can only be approximately satisfied, and there may be measurement errors in the independent variables, and there may be outliers in the response data. In view of the above two problems, the robustness of response surface design and the robustness of response surface model when there are measurement errors in input variables are studied in this paper. This paper introduces some basic theories of response surface method and traditional model estimation method, evaluation criterion of response surface design, robust estimation method and error variable model. Based on this, the robustness of different response surface design and different estimation methods are compared and analyzed. Firstly, the robustness of different response surface designs to outliers in response data is discussed. It mainly includes central composite design, central composite sequential design, center composite bounded design and Box-Behnken design. The results of comparative analysis based on these designs show that the center point outliers greatly increase the volatility of the model and reduce the accuracy of the model. The location anomaly points of the axis and cube points affect the estimation of the quadratic or cross terms of the model. It increases the difficulty of model optimization and prediction. Secondly, two evaluation indexes of prediction variance of response surface design are proposed, and the unit prediction variance of error variable model and the prediction performance of traditional three-factor response surface design are studied and analyzed. Simulation results show that the robustness of the central composite surface design is optimal when the variance ratio of the measurement error is small, and the predictability of the Box-Behnken design is better when the variance is large. Thirdly, the optimization criteria and the properties of predictive variance for the design of second-order spherical response surfaces with measurement errors are compared and evaluated. In order to show the robustness of different designs to the measurement error, we propose the maximum and minimum values of unit predictive variance and the design space ratio diagram, and calculate the G- efficiency values of different designs. A robust response surface design scheme with different factor levels is obtained. Finally, based on the outliers in the response data, the robust estimation methods of the second-order response surface model are discussed, and the robustness of the different estimation methods for the existence or non-normal state of the outliers is analyzed. The comparison results are obtained by simulation and examples.
【学位授予单位】：天津大学
【学位级别】：博士
【学位授予年份】：2011
【分类号】：O211.3;C81
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