剖面資料在先進製程時常被用來監控產品的製程品質，張(2014) 改良了函數化迴歸模型以及函數化主成分分析，使模型能夠解釋剖面資料中兩個解釋變數在不同時間點的交互作用，該交互作用模型能夠解釋兩種機台參數在不同時間點下如何影響製程品質；本篇文章承襲了張(2014) 的作法，使用函數化迴歸模型以及函數化主成分分析，不同的地方是將時間當作新的變數，將資料中所有的變數投影到一維度的主曲線上，再利用主曲線的參數來代表不同解釋變數在相同時間點上的同步作用，我們利用同步作用模型來解釋多個因子對於反應變數的影響；因此，本論文的主要貢獻在於我們能夠利用此同步作用模型來監控多個機台參數在同一時間點發生問題如何影響產品的製程品質。
本篇文章會先介紹張(2014) 的函數化迴歸模型，以及主曲線的作法，再衍伸到同步作用模型的產生，最後利用該模型做實際資料分析並比較同步作用模型以及交互作用模型的分析結果。

Profile data are used to assess the quality of the product in many advance process control. Chang (2014) modified functional regression model and functional principal component to make functional regression model for the temporal action of two factors of profile data. This interaction model explains how the quality of produce is influenced by two tool parameters in different seconds. This thesis follows the method of Chang (2014) and use functional regression model and functional principal component analysis. The different is that we take the time as a factor and map the data of all factors to one dimension principal curve. Then we use the parameter of the principal curve to represent the influence of synchronized action from different factors at same time. We use the synchronized action model to explain how the response variable was influenced by many factors. Therefore, the main contribution of this thesis is we use this synchronized action model to assess the influence from more than two variables at the same time for the quality of the product.
This thesis will introduce the functional regression model and functional principal component analysis by Chang (2014) first, then illustrate how to build the synchronized action model. Next, we use the synchronized action model to analysis the data from Chang (2014) and illustrate the difference between the analysis result of the synchronized action model and the interaction model.

Chang (2014). Model building of multistage manufacturing process with profile data.
Chen, D., Müller, H.-G., et al. (2012). Nonlinear manifold representations for functional data. The Annals of Statistics, 40(1):1–29.
Chien, C.-F., Hsu, C.-Y., and Chen, P.-N. (2013). Semiconductor fault detection and classification for yield enhancement and manufacturing intelligence. Flexible Services and Manufacturing Journal, 25(3):367–388.
Cook, E. R. and Peters, K. (1981). The smoothing spline: a new approach to standardizing forest interior tree-ring width series for dendroclimatic studies. Tree-ring bulletin.
Creighton, J. and Ho, P. (2001). Introduction to chemical vapor deposition (cvd). Chemical vapor deposition, 2:1–22.
Einbeck, J., Evers, L., and Powell, B. (2010). Data compression and regression through local principal curves and surfaces. International journal of neural systems, 20(03):177–192.
Einbeck, J., Isaac, B., Evers, L., and Parente, A. (2012). Penalized regression on principal manifolds with application to combustion modelling.
Febrero-Bande, M. and Oviedo de la Fuente, M. (2012). Statistical computing in functional data analysis: the r package fda. usc. Journal of Statistical Software, 51(4):1–28.
Gerber, S., Tasdizen, T., and Whitaker, R. (2009). Dimensionality reduction and principal surfaces via kernel map manifolds. In Computer Vision, 2009 IEEE 12th International Conference on, pages 529–536. IEEE.
Hastie, T. and Stuetzle, W. (1989). Principal curves. Journal of the American Statistical Association, 84(406):502–516.
Yao, F. and Müller, H.-G. (2010). Functional quadratic regression. Biometrika, 97(1):49–64.
Yao, F., Müller, H.-G., Wang, J.-L., et al. (2005). Functional linear regression analysis for longitudinal data. The Annals of Statistics, 33(6):2873–2903.