在因子實驗中，因子的效應編碼函數常為實驗者事先給定，但其函數形式可能並非是最能解釋反應變數的形式。在本文中，我們將已知的效應編碼函數，改以未知的本質編碼$
hi_{i,v}(x)$'s取代，並考慮線性模型$y_{mj} = \sum\limits_{\textbf{v}} w_{
hi, \textbf{v}}
hi_{\textbf{v}}(\textbf{x}_i)+{\varepsilon}_{mj}$，其中$w_{
hi, \textbf{v}}$'s稱為本質效應。此模型中$
hi_{\textbf{v}}(\textbf{x})$和$w_{
hi, \textbf{v}}$皆為未知，都需利用資料來估計，故原本由實驗者主觀決定的編碼，可由數據來客觀決定。但因$
hi_{\textbf{v}}(\textbf{x})$和$w_{
hi, \textbf{v}}$皆為未知，故此模型是無法辨識的。對此問題，本文使用全因子設計的性質，將模型針對某一因子$x_i$改寫成多變量線性模型，再套用Liao（2018）的做法引入準則，使模型變成可辨識。我們亦套用Liao（2018）的估計準則，來對每個因子個別估計本質編碼。在估得本質編碼後，我們提出決定重要本質編碼個數的方法。我們並使用一組電腦實驗的全因子設計資料為例，估計本質編碼並分析本質效應。另外，我們也提出一個本質編碼的新應用，利用本質編碼分析解構複雜的無母數模型。

In the conventional factorial design, the effect codings of factors are determined by the experimenters. They are treated as known functions in the design and analysis of experiments. There is, however, no guarantee that the chosen codings have the best functional forms for explaining the response. In this work, we propose to replace known effect codings by unknown essence codings $
hi_{i,v}(x)$'s, and consider the linear model $y_{mj} = \sum\limits_{\textbf{v}} w_{
hi, \textbf{v}}
hi_{\textbf{v}}(\textbf{x}_i)+{\varepsilon}_{mj}$, where $w_{
hi, \textbf{v}}$'s are called essence effects. In this model, both $
hi_{\textbf{v}}(\textbf{x})$ and $w_{
hi, \textbf{v}}$ are unknown, and must be estimated using data. Through estimating essence codings, the determination of codings that originally depends on the subjective judgment of the experimenters can be now objectively implemented using data. Because $
hi_{\textbf{v}}(\textbf{x})$ and $w_{
hi, \textbf{v}}$ are unknown, this model is unidentifiable. To resolve this problem, we use a property of full factorial designs to write this model into a multivariate linear model for a certain factor, and apply the criteria in Liao (2018) to make it identifiable. We apply the estimation criteria in Liao (2018) to individually estimate the essence codings for every factors. We also propose a method to decide the number of important essence codings. Our method of estimating essence codings and analyzing essence effects for multiple factors is demonstrated on a dataset of a computer experiment using a full factorial design. We also propose a new application of essence codings: decomposing and explaining complex non-parametric models such as tree models and neural networks.

目錄
1 緒論 1
2 文獻回顧 5
3 研究方法 8
3.1 本質編碼之估計方法 10
3.2 本質效應參數之估計 14
3.3 本質編碼的解釋比例 15
3.4 重要本質編碼個數的挑選 16
3.5 單一因子之討論 18
4 數據分析 20
4.1 實驗模型與模擬資料介紹 20
4.2 本質編碼與效應之分析結果 22
5 本質編碼之新應用：解構無母數模型 27
5.1 解構迴歸樹模型 28
5.1.1 迴歸樹模型配適結果 28
5.1.2 本質編碼之估計 29
5.1.3 本質效應之估計與分析 31
5.1.4 利用本質編碼對原始數據配適模型 34
5.2 解構類神經網路模型 35
5.2.1 類神經網路模型配適結果 35
5.2.2 本質編碼之估計 37
5.2.3 本質效應之估計與分析 40
5.2.4 利用本質編碼對原始數據配適模型 43
6 結論與討論 45
參考文獻 46

[1] Dua, D. and Graff, C. (2019). UCI Machine Learning Repository [http://archive.ics.uci.edu/ml]. Irvine, CA: University of California, School of Information and Computer Science.
[2] Eldred, M. and Burkardt, J. (2009). Comparison of non-intrusive polynomial chaos and stochastic collocation methods for uncertainty quantification. In the 47th AIAA Aerospace Sciences Meeting Including the New Horizons Forum
and Aerospace Exposition, Orlando, Florida.
[3] Harrison Jr, D. and Rubinfeld, D. L. (1978). Hedonic housing prices and the demand for clean air, Journal of Environmental Economics and Management, Volume 5, pp. 81-102.
[4] Liao, Y.-F. (2018). Identifying essence codings and effects in functional linear models with homogeneous and independent errors, Master thesis, National Tsing Hua University, Hsinchu, Taiwan.
[5] Peng, P.-R. (2018). Identifying essence codings and effects in functional linear models with heterogeneous and correlated errors, Master thesis, National Tsing Hua University, Hsinchu, Taiwan.
[6] Saltelli, A., Ratto, M., Andres, T., Campolongo, F., Cariboni, J., Gatelli, D., Saisana, M., and Tarantola, S. (2008). Global Sensitivity Analysis. The Primer. John Wiley and Sons, Ltd.