在因子設計下，可以利用數據來決定最有解釋力的編碼函數，這稱之為本質編碼。Huang (2020) 提出了如何在全因子設計下，定義及估計每個因子的本質編碼之方法。但是在因子數或因子水準數較高的情況下，可能因成本考量，無法實施全因子設計。本文探討在一個可估的部分因子設計下，如何估計本質編碼的問題。在部分因子設計下，Huang (2020) 的方法會在反應變數矩陣中產生無值的問題。在本文中，我們提出補值方法，透過在可估設計下建構的線性模型，利用最小平方法的不偏性質，對無值的反應變數進行補值，再利用 Huang (2020) 所提出的做法，對此補完值的反應變數矩陣求解本質編碼。我們並且在真實模型為 v 階模型的假設下，提出簡化此做法之計算量的方法。並利用模擬數據，在可估設計與不可估設計下，分別比較本論文在部分因子設計下的做法與 Huang (2020) 在全因子設計下的做法，所估得的本質編碼之差異。

For a factorial design, essence codings estimated by data can explain maximum variation of response in a sequential order. Under full factorial design, Huang (2020) proposed methods to define and estimate essence codings for each factors. However, when the numbers of factors or factor levels are large, a full factorial design would be less attractive due to cost considerations. In this work, we study the estimation problem of essence codings under a fractional factorial design (FFD). For an FFD, the estimation method in Huang (2020) would suffer from no-value problem in the matrix formed by the response data according to the full factorial structure. In this work, we propose an imputation method, which utilizes the unbiased property of the least square estimator under an estimable design to resolve the no-value problem. The method in Huang (2020) can be applied after the imputation to estimate essence codings. We also propose a method to reduce computation load under the assumption that the true model is a vth-order model. Through a simulated data, we compare the distrinction between the estimated essence codings obtained by Huang (2020) under a full factorial and our method under an estimable FFD and an unestimable FFD.

目錄
1 緒論--------------------------------------1
2 文獻回顧----------------------------------4
3 研究結果----------------------------------6
3.1 本質編碼之估計方法-----------------------6
3.2 簡化計算--------------------------------9
4 數據分析----------------------------------12
4.1 真實模型介紹----------------------------12
4.2 全因子設計下估計之本質編碼----------------14
4.3 部分因子設計下估計之本質編碼--------------16
4.4 在不可估設計下估計本質編碼的差異----------17
5 結論與討論--------------------------------20
參考文獻------------------------------------21

1. Huang, J.-M. (2020). Identifying essence codings and effects for multiple factors under full factorial designs, Master thesis, National Tsing Hua University, Hsinchu, Taiwan.