臺灣博碩士論文加值系統

我們想要建立一個模型，能使一裂區設計最佳化。許多準則方法只注重在最佳化變異數參數估計，而忽略偏誤的部分。例如：D 準則最佳設計、A 準則最佳設計與I 準則最佳設計，然而，因子設計通常是複雜的，在模型假設下容易發生錯誤，若有重要的作用項沒有放在模型裡，則會有較大的偏誤。所以，一個好的裂區設計，是要掌控好參數估計中偏誤與變異數。我們建立均方誤差(MSE) 與損失函數(Loss function)，藉由損失函數來判斷裂區設計優劣。此論文使用I 準則，利用預測變異數，建立好損失函數，來比較兩設計。

In this paper, we construct split-plot designs that are robust under model misspecification. Many criteria such as the D- criterion, A-criterion and I-criterion only
focus on minimizing the variance of the parameter estimation while ignoring the bias. However, the fitted model is often misspecified. If there exist significant effects that are not in the model, the bias will be large. Therefore, a good split plot design should control the variance and bias of the estimates. We establish the mean square error (MSE) and loss function. We use the loss function to determine good designs.

1. 緒論 1
2. 文獻回顧 3
2.1. D 最佳部分因子設計. . . . . . . . . . . . . . . . . 3
2.2. A 最佳部分因子設計. . . . . . . . . . . . . . . . . 5
2.3. D 最佳裂區設計. . . . . . . . . . . . . . . . . . . 6
2.4. A 最佳裂區設計. . . . . . . . . . . . . . . . . . . 9
2.5. I 最佳裂區設計. . . . . . . . . . . . . . . . . . 10
3. 研究方法與準則 12
3.1. I 準則與損失函數(Loss Function) . . . . . . . . . 12
3.2. 討論特殊情況下的I 準則損失函數. . . . . . . . . . . 14
4. 演算法(Algorithm) 15
5. 模擬(Simulation) 20
6. 結論 50
7. 證明(appendix) 51
8. 文獻參考 54
9. 程式碼 55

[ 1 ] Dennis K. J. and Lin. and Zhou,J.(2013). “D-optimal Minimax Fractional Factorial Designs”. Journal of the Royal Statistical Society. Series C, Applied Statistics 41, 2, 325-340.
[ 2 ] Goos,P. and Vandebroek,M.(2011).“Optimal Split-Plot Designs”. Journal of Quality Technology, 33, 4, 436-450.
[ 3 ] Jones, B. and Goos, P. (2007).“A Candidate-Set-Free Algorithm for Generation D-Optimal Split-Plot Designs”. Journal of the Royal Statistical Society. Series C, Applied Statistics 56, pp. 347-364.
[ 4 ] Jones, B. and Nachtsheim, C. J. (2009).“Split-Plot Design: What, Why, and How”. Journal of Quality Technology 41, pp. 340-361.
[ 5 ] Jones, B. and Nachtsheim, C. J. (2011).“Efficient Designs with Minimal Aliasing”. Technometrics 53, pp. 62-71.
[ 6 ] Jones, B. and Goos, P. (2012).“I-Optimal Versus D-Optimal Split-Plot Response Surface Designs”. Journal of Quality Technology 44, pp. 85-101.
[ 7 ] Lin, C. Y. (2017).“Robust Split-Plot Designs for Model Misspecification”. Journal of Quality Technology ,In press.
[ 8 ] Wilmut,M. and Zhou,J.(2011). .“D-optimal Minimax Design Criterion For Two-Level Fractional Factorial Designs”. Journal of the Royal Statistical Society. Series C, Applied Statistics 141, 1, 576-587.
[ 9 ] Yin,Y. and Zhou,J.(2015)“. Minimax Design Criterion For Fractional Factorial Designs”. Ann Inst Stat Math, 67, 673–685.