反應曲面法中最佳平均均方誤差設計與最陡上升搜尋法之研究

本論文的第一個主題是探討最佳設計問題，在這個階段中，平均均方誤差(AMSE) 值被用來當作執行測量的準則。在AMSE的領域中，有三種估計反應曲面的方法分別為最小平方估計量(LSE)、最小偏估計量(MBE)與廣義估計量(GE)。在 MBE中，我們針對Karson, Manson and Hader (1969)在反應曲面文獻中探討的單變數設計問題提出一個有用的數學性質，由這個數學性質可發現最佳設計僅與實驗總點數和中心點數有關。在GE中，我們針對多因子設計問題提出幾個有用的數學性質。此外，我們也提出GE ( Kupper and Meydrech (1973, 1974)) 與LSE在AMSE準則下的比較結果，針對這些個別的比較問題可發現LSE下的AMSE模式將依設計點與實際的曲率而定。但是，GE下的AMSE模式將依賴原始模式中參數、參數的權重、設計點位置和曲率有關。為了公平比較LSE與GE，首先給定一個曲率值，再給定一個存在假設模式中的參數與曲率的相對比值，即可發現當曲率呈現顯著時，GE相較於 LSE而言，可得到較小的AMSE值。論文的第二個主題是關於製程最佳化問題。在反應曲面法的領域中，對實驗最佳化而言，最陡上升法扮演一個重要的角色。然而，早期對於如何去停止最陡上升方向的搜尋在文獻中並沒有完備的分析與定義。通常，一個便於使用的停止規則是當出現1次、2次或3次下滑即停止實驗。另一方面，Myers 與 Khuri (1979)及del Castillo (1997)已提出兩個正式的停止規則。而在本論文中，我們提出一個新的停止規則是使用一個conjugate方向來修正最陡上升方向。在這個新停止規則中，我們不需要估計達到最佳化的步伐數，同時也可避免複雜的數學與統計計算。除此之外，在藉由模擬的個案研究中可發現，新停止規則在執行最佳反應值的搜尋上比經驗式及Myers與Khuri式的停止規則好。

關鍵字

最小平方估計量；最小偏估計量；廣義估計量；反應曲面法；平均均方誤差；最陡上升。

並列摘要

The first topic of this dissertation is concerned with optimum designs problem. In this stage, the average mean squared error (AMSE) value is used as the performance criterion. Within the realm of AMSE, three methods of estimating a response surface are standard least-square estimation (LSE), minimum bias estimation (MBE) and generalized estimation (GE). In MBE, we presents a supplementary, valuable property of the minimum bias estimation procedure, addressed in Karson, Manson and Hader (1969), within the context of a single variable design problem appearing in the response surface methodology (RSM) literature. It is discovered that this simple property only depends on the relationship between the total number of experimental runs and the number of center replicates. In GE, we present some useful mathematical properties of GE for the multi-factor design problem. Furthermore, we also present the comparison results of the generalized estimation approach (GE, Kupper and Meydrech (1973, 1974)) to the least-squares estimation (LSE) approach within the context of Box and Draper’s (1959, 1963) AMSE criterion. For these particular problems, it has been discovered that the AMSE of the LSE approach depends only on the design allocation and the actual scaled curvature. Nonetheless, the AMSE of the GE approach relies on the parameter weight, the design allocation, the actual scaled parameter existing in the initial model, and the actual scaled curvature. To make fair comparison possible, a fixed scaled curvature is first given for LSE and GE, and then a variety of relative curvature ratios are specified to define the scaled parameter already exiting in the assumed model. The improvement of the GE approach over the LSE approach becomes more prominent as the potential bias error is significantly present. The second topic of this dissertation is concerned with process optimization problem. The method of steepest ascent plays an important role for experimental optimization in the area of RSM. However, the details of how to stop a search in the steepest ascent direction are not completely, analytically defined in the literature. Usually, it is convenient to use the simpler stopping rule of stopping after 1, 2, and 3 drops in a row. On the other hand, there are two formal mathematical stopping rules have been proposed by Myers and Khuri (1979) and del Castillo (1997). A new stopping rule is proposed in this dissertation to suggest a conjugate direction to correct the direction of steepest ascent. In this new stopping rule, we do not require an initial estimate of the number of steps to reach the optimum and complex mathematic and statistic computation can be avoided. In addition, in our case study via simulations, the new stopping rules perform considerably better than the classical rules and Myers and Khuri’s rules.

並列關鍵字

Least squares estimation (LSE), minimum bias estimation (MBE), Generalized estimation (GE), Response surface methodology (RSM), Average mean squared error (AMSE), Steepest Ascent.

參考文獻

Box, G. E. P. and Draper, N. R., (1959). A Basis for the Selection of a Response Surface Design. Journal of the American Statistical Association, 54, pp. 622-654.

Box, G. E. P. and Draper, N. R., (1963). The Choice of a Second Order Rotatable Design. Biometrika, 50, pp. 335-352.

Box, G. E. P. and Wilson, K. B., (1951). On the Experimental Attainment of Optimum Conditions. Journal of the Royal Statistical Society, B13, 1-38.

David, H.A. and Arens, B.E., (1959). Optimal Spacing in Regression Analysis. Annals of Mathematical Statistics, 30, pp. 111-118.

Draper, N. R. and Lawrence W. E., (1965). Designs Which Minimize Model Inadequacies: Cuboidal Regions of Interest, Biometrika, 52, pp. 111-118.

國際替代計量

反應曲面法中最佳平均均方誤差設計與最陡上升搜尋法之研究

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