部分因子設計是經過精心安排過的一種實驗，其目的在以最精簡的因子組合，而能獲得最多的資訊。由於實驗的次數有限，而無法估計實驗中所有可能因子的影響。基於因子稀疏假設，這些設計的重要目的之一是用來篩選有影響的因子。在本研究中，我們提出依據 Wang et al. (2012) 使用的影響分數 (I-score) 來進行變數的篩選。根據模型是否可能具有交互作用，調整變數篩選的方法，再以 Chen et al. (2013) 分量式吉布斯抽樣提升準確度。我們將使用文獻中之案例，進行模擬實驗驗證此篩選程序的有效性。最後，我們也與其他變數選篩選方法做比較，藉此希望找出更有效之變數篩選方法

Fractional factorial design is a well organized design aiming at obtaining as much
information as possible although with fewer factor combinations. As it is not possible
to estimate all effects in the experiment due to the limited size of experiments, the main
purpose of these types of designs is to screen out influential factors under the factor sparsity
assumption. In this study, we propose to screen the factors based on the influential score,
proposed by Wang et al. (2012). One of the benefits of the new screening procedure is
to be able to identify if there are interaction effects influencing the experimental results.
Then after screening out the important factors and interaction effects with fewer factors,
we may use the componentwise Gibbs sampler methodology again to improve the accuracy
of obtaining the exact set of significant factor effects. We will examine the effectiveness of
these screening procedure with simulations using design set ups in in several examples in
the literature. Finally, we compare our newly proposed screening methods with others to
examine the performances of screening factor effects

論文審定書 i
誌謝 ii
摘要 iii
Abstract iv
1 前言 1
2 研究方法 2
2.1 影響分數 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.2 差分 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.3 逐步迴歸 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.4 套索迴歸 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.5 選模準則 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.6 分量式吉布斯抽樣 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 分析流程及方法測試 5
3.1 分析流程 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.2 流程測試 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.2.1 模型只有主效應 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2.2 模型含有主效應及交互作用 . . . . . . . . . . . . . . . . . . . . . . . 12
4 案例分析 15
5 模擬模型 19
5.1 模擬模型只有主效應 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5.2 模擬模型含有主效應及交互作用 . . . . . . . . . . . . . . . . . . . . . . . . . 21
6 變數篩選方法比較 24
6.1 模型模擬之比較 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
6.2 實際案例之比較 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
v
7 結論 26
參考文獻 27
A 附錄 29
附錄 29
A.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
A.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
A.3 Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
A.4 Example 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
A.5 Example 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
B 附錄 33
附錄 33

[1] Beattie, S.D., Fong, D.K.H., Lin, D.K.J. (2002). A two-stage Bayesian model selection
strategy for supersaturated designs. Technometrics, 44, 55-63.
[2] Box, G.E.P., Meyer, R.D. (1986). An analysis for unreplicated fractional factorials.
Technometrics, 4, 489-495.
[3] Chen, R.B., Chu, C.H., Lai, T.H., Wu, Y.N. (2011). Stochastic matching pursuit for
Bayesian variable selection. Statistics and Computing, 21, 247-259.
[4] Chen, R.B., Weng, J.Z., Chu, C.H. (2013). Screening procedure for supersaturated
designs using a Bayesian variable selection method. Quality and Reliability Engineering
International, 29, 89-101.
[5] Chipman, H.,Hamada, H., Wu, C.F.J. (1997). A Bayesian variable selection approach
for analyzing designed experiments with complex aliasing. Technometrics, 39, 372-381.
[6] Georgiou, S.D. (2008). Modelling by supersaturated designs. Compuational Statistic
& Data Analysis, 53, 528-435.
[7] Hurvich, C.M., Tsai, C.L. (1989). Regression and time series model selection in small
samples. Biometrika, 76, 297-307.
[8] Li, R., Lin, D.K.J. (2003). Analysis method for supersaturated design: some comparisons.
Journal of Data Science, 1, 249-260.
[9] Lin, D.K.J. (1993). A New Class of Supersaturated Design. Technometrics, 35, 28-31.
[10] Marley, C.J.,Woods, D.C. (2010). A comparison of design and model selection methods
for supersarurated experiments. Compuational Statistic & Data Analysis, 54, 3158-
3167.
[11] Montgomery, D.C. (2013). Design and Analysis of Experiments, 8
th Edition, Wiley,
New York.
[12] Phoa, F.K.H, Pan, Y.H, Xu, H. (2009). Analysis of supersaturated designs via the
Dantzig selector. Journal of Statistical Planning and Inference, 139, 2362-2372.
[13] Rais, F., Kamoun, A., Chaavouni, M., Claeys-Bruno M., Phan-Ten-Luu R., Sergent
M. (2009). Supersaturated design for screening factors influencing the preparation
of sulfated of olive pomace oil fatty acids. Chemometrics and Intelligent Laboratory
Systems, 99, 71-79.
[14] Wang, H., Lo, S.H., Zheng, T. & Hu, I. (2012). Interaction-based feature selection and
classification for high-dimensional biological data. Bioinformatics, 28, 2834-2842.
[15] Westfall, P.H., Young, S.S., Lin, D.K.J., (1998). Forward selection error control in the
analysis of supersaturated designs. Statistica Sinica, 8, 101-117.
[16] Williams, K.R. (1968). Designed Experiments. Rubber Age, 100, 65-71.
[17] Wu, C.F.J., Hamada, M. (2000). Experiments: Planning, analysis and parameter design
optimization, Wiley, New York.
[18] Zhang, Q.Z., Zhang, R.C., Liu, M.Q., (2007). A method for screening active effects in
supersaturated designs. Journal of Statistical Planning and Inference, 137, 235–248.