【摘要】：響應(yīng)曲面方法(RSM)是現(xiàn)代統(tǒng)計與數(shù)學的集成方法,被廣泛應(yīng)用于響應(yīng)變量與自變量之間函數(shù)關(guān)系研究中。通過響應(yīng)曲面設(shè)計、建模和優(yōu)化完成整個響應(yīng)曲面過程。在響應(yīng)曲面建模中,通常要求隨機誤差服從正態(tài)分布相互獨立且具有相同的方差;輸入變量相互獨立,沒有測量誤差。在實際生產(chǎn)過程中,這些假設(shè)條件只能近似地被滿足,并且自變量中很可能會出現(xiàn)測量誤差,響應(yīng)數(shù)據(jù)中也可能存在異常點情況。針對上述兩個問題,本文主要研究在輸入變量存在測量誤差時,響應(yīng)曲面設(shè)計的穩(wěn)健性以及在異常點存在時,響應(yīng)曲面模型的穩(wěn)健性。 本文介紹了響應(yīng)曲面方法的一些基本理論和傳統(tǒng)的模型估計方法、響應(yīng)曲面設(shè)計的評價準則、穩(wěn)健估計方法和誤差變量模型。在此基礎(chǔ)了比較分析了不同的響應(yīng)曲面設(shè)計和不同估計方法的穩(wěn)健性,具體從以下幾個方面進行研究。 首先,探討了不同的響應(yīng)曲面設(shè)計對響應(yīng)數(shù)據(jù)中異常點的穩(wěn)健性問題。主要包括中心復合設(shè)計——中心復合序貫設(shè)計、中心復合有界設(shè)計和Box-Behnken設(shè)計。基于這些設(shè)計的比較分析結(jié)果表明,中心點位置異常點極大地增加了模型的波動性,降低了模型的精度,軸線點和立方體點位置異常點影響模型二次項或交叉項的估計,增加了模型優(yōu)化和預(yù)測的難度。 其次,提出了兩種響應(yīng)曲面設(shè)計預(yù)測方差的評價指標,研究分析了誤差變量模型的單位預(yù)測方差和傳統(tǒng)的三因子響應(yīng)曲面設(shè)計預(yù)測性能。仿真分析表明,對于測量誤差的方差比較小時,中心復合表面設(shè)計的穩(wěn)健性是最優(yōu)的;當方差較大時,Box-Behnken設(shè)計的預(yù)測性更好一些。 再次,對存在測量誤差的二階球形響應(yīng)曲面設(shè)計優(yōu)化準則和預(yù)測方差性質(zhì)進行比較和評估。為了直觀展示不同設(shè)計對測量誤差的穩(wěn)健性,我們提出單位預(yù)測方差最大值與最小值圖和設(shè)計空間比率圖,計算了不同設(shè)計的G-效率值。得出了不同因子水平下,穩(wěn)健的響應(yīng)曲面設(shè)計方案。 最后,基于響應(yīng)數(shù)據(jù)中的異常點,探究了二階響應(yīng)曲面模型的穩(wěn)健估計方法,研究分析了異常點存在或非正態(tài)時不同估計方法的穩(wěn)健性。通過仿真和實例得出比較結(jié)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.
【學位授予單位】：天津大學
【學位級別】：博士
【學位授予年份】：2011
【分類號】：O211.3;C81

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