本文利用 Poisson ANOVA 模型描述了多站點多機台生產製程流程配置與最終產品品質的關聯，並藉此探討製程中最佳的機台組合問題。最終目標是希望能找到較好的生產路徑，進而改善產品良率。延續李少芃 (2017) 的論文，透過在參數估計中引入正規化 (regularization) 的參數限制式，採用不同的懲罰項藉以辨識重要的站點並估計其機台效應值。也特別在標準化 (standardized) 與非標準化 (Simon and Tibshirani, 2012) 的參數限制條件之間做探討並比較其參數估計值表現的差異。驗證結果顯示對於不平衡數據，標準化的估計方法表現較優良。在本文的第二部分則考量到模型配適與機台效應估計的線上更新，藉由線上蒐集數據與分析來達到模型配適。對於實際的生產線，能隨時監控機台的效應並即時做調整，可更有效解決實務上的需求，且本文也另外提出了一種運用在線上更新中的參數正規化演算法。最後並藉由數據模擬以及真實資料，來驗證我們所提出的方法。

本文內容中引用了各種不同的 Lasso 演算法，在求算估計值等問題的同時，也考量到是否存在理論解析解。透過參數的各種轉換以及嘗試不同變因的組合，推導出更適用於正交設計矩陣的最佳估計解。藉由數值上的模擬以及真實資料，驗證出本論文所提出的參數估計法，進而改善在兩階段估計法 (李少芃, 2017) 中，參數估計上的表現。最後，透過各種估計法的比較，可了解到對於不同資料結構，資料的平衡與否以及是否執行正規化，在 BIC 法則的選模下， standardization 更能提高配適力，進而了解其中的變因。

This thesis uses the Poisson regression model to describe the relationship between the manufacturing process and the quality of the products in a multi-stage-multi-tool production. The goal is to find a better production path leading to a satisfactory production quality. Following Lee’s thesis (2017), a penalized likelihood approach is adopted to identify important stages and estimate the tool effects, via incorporating parameter regularizations in the estimation. In particular, standardized and unstandardized types of regularization (Simon and Tibshirani, 2012) are both considered and their inference performance is compared. It turns out that the penalized estimations with standardized regularization perform better for unbalance data. The second part of this thesis considers the online updating of the model fitting and parameter estimation. For real production lines, it is practically useful if the tool effects are monitored online and adjusted for the production management. This can be achieved by adaptively fitting the model according to online data collection and analysis. This thesis suggests an updating algorithm taking into account parameter regularization. The proposed methodologies are demonstrated via simulation and a real application.

1. 李少芃. (2017). 計數型模型下的類別變數選取與水準合併, 清華大學統計所碩士論文。

2. Boyd, S., Parikh, N., Chu, E., Peleato, B. and Eckstein, J. (2011). Distributed optimization and statistical learning via the alternating direction method of multipliers. \textit{Foundations and Trends in Machine Learning}, \textbf{3}(1), 1–122.

3. Breheny, P. (2015). The group exponential lasso for bi-level variable selection. \textit{Biometrics}, \textbf{71}, 731-740.

4. Breheny, P. and Huang, J. (2009). Penalized methods for bi-level variable selection. \textit{Statistics and Its Interface}, \textbf{2} 369-380.

5. Fan, J. and Li, R. (2001). Variable selection via nonconcave penalized likelihood and its oracle properties. \textit{Journal of the American Statistical Association}, \textbf{96}, 1348-1360.

6. Simon, N. and Tibshirani, R. (2012). Standardization and the group Lasso penalty. \textit{Statistica Sinica}, \textbf{22}(3), 983–1001.

7. Tibshirani, R., Saunders, M., Rosset, S., Zhu, J. and Knight, K. (2005). Sparsity and smoothness via the fused Lasso. \textit{Journal of the Royal Statistical Society: Series B}, \textbf{67}(1), 91–108.

8. Yuan, M. and Lin, Y. (2006). Model selection and estimation in regression with grouped variables. \textit{Journal of the Royal Statistical Society: Series B}, \textbf{68}(1), 49–67.