本論文感興趣的議題是監控計數型資料之參數變化。若資料為連續型，可以利用線性迴歸模型估計參數，並透過Kalman filter 進行參數更新。本篇論文，利用泰勒展開式將卜瓦松模型(Poisson regression)的對數概似函數(log-likelihood function)近似成常態，並利用貝氏定理進行參數後驗分配的推論，發展了一套更新參數的non-linear filter ，可以系統性的進行參數更新，達到監控參數變化的目的。

此外，如何辨識重要因子也是模型配適的重要議題，若在廣義線性模型的估計方程式加入L1-norm懲罰項，常面臨到不可微分的情形，本篇論文進一步在更新參數的流程中，加入參數估計的正規化(regularization)限制條件，除了達到監控參數的目的外，也能夠辨識重要因子。

本論文的研究僅針對卜瓦松模型進行預測推論，並探討了加入Lasso、Group Lasso、Adaptive Lasso 及 Adaptive Group Lasso不同懲罰項所帶來的效益。但所提出的方法可以廣泛地適用於其他廣義線性模型(generalized linear model)，亦可搭配更一般化的參數限制條件。

This thesis proposes an on-line updating scheme for parameters in a dynamic Poisson regression model. In particular, the regularization is considered in the updating scheme to simultaneously identify important variables. The proposed method is a two-stage procedure: (1) re-estimate the parameters, and (2) select the important variables. In the first stage (re-estimate stage), the likelihood function of Poisson regression model and the prior are approximated as a normal density via Taylor expansion and then the resulting posterior for parameters remains normal-distributed and the corresponding mean and variance structure can be computed recursively. In the second stage, the posterior mean estimates are regularized after parameter updated to further remove some insignificant effects from the fitting. As a good result, the algorithm has simple closed forms and easy to implement like Kalman filter. For illustration, the proposed method is applied to failure count data from a multi-stage-multi-tool manufacturing process. The updating scheme works effectively on identifying good production paths during the production stage.

1 緒論 1
2 資料簡介與模型介紹 3
2.1 製程資料介紹. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 模型介紹. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3 參數推論及預測誤差7
3.1 t 後驗分配的推導. . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.1.1 牛頓法(Newton’s method) 更新演算法. . . . . . . . . . . . 10
3.1.2 貝氏更新（Bayesian updating）演算法. . . . . . . . . . . 11
3.2 正規化參數估計. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.3  維度改變時演算法的調整. . . . . . . . . . . . . . . . . . . . . . 13
4 調控參數的選取16
5 數值模擬17
5.1 模擬一：Algorithm 1 及Algorithm 2 的驗證與比較. . . . . . . . . 17
5.2 模擬二：加入懲罰項. . . . . . . . . . . . . . . . . . . . . . . . . . 20
5.3 模擬三：機台數量變動. . . . . . . . . . . . . . . . . . . . . . . . . 28
6 案例分析30
6.1 最佳製程路徑. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
6.2 製程參數更新與監控. . . . . . . . . . . . . . . . . . . . . . . . . . 32
7 結論36
參考文獻 37

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