函數型迴歸模式是一個能將非傳統類型的變數放入模型進而估計的模型，在倖
存資料中，Kong et al. (2014) 提出函數型線性Cox 迴歸模式，其利用函數型主
成分分析加上Cox 比例風險模式，以此分析大腦影像與阿茲海默症之間的關
係。本論文推廣更多處理這類型資料的方法，且將模式替換成更廣義的半母數
轉換模型，可使研究者更自由的進行分析，以找出函數型或影像型變數與病人
的存活時間之間的關係，因此在這篇文章中，將綜合上述的方法，提出函數型
半母數轉換模式，其又可細分為四個方法，最後利用數值模擬來比較彼此之間
的結果，並將此方法我們將運用到阿茲海默症的資料中，以驗證這些方法實際
應用的成效。

The functional regression model is a model that can put non-traditional types of variables in and make further estimations. In survival data, Kong et al. (2014) proposed a Functional Linear Cox Regression Model (FLCRM). They combine both Functional Principal Component Analysis (FPCA) and Cox proportional hazards model to analyze relations between brain images and Alzheimer’s disease. In this thesis, more method is presented to deal with this type of survival data; furthermore, the Cox proportional hazards model in FLCRM is replaced with a more general semiparametric transformation model, which makes finding relations between functional or image types of data and patients’ survival time more convenient. Thus, a functional semiparametric transformation model is proposed, which can be further subdivided into four methods, and in the end comparisons will be made between these four methods through numerical simulation, then validate these methods outcome by applying actual Alzheimer’s disease neuroimaging initiative data through above model.

第一章緒論1
第二章文獻回顧3
2.1 函數型主成分分析. . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1.1 主成分分析. . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1.2 函數型主成分分析. . . . . . . . . . . . . . . . . . . . . . 5
2.2 樣條基底. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 哈爾小波轉換. . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.4 右設限倖存資料. . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.5 半母數轉換模式. . . . . . . . . . . . . . . . . . . . . . . . . . . 12
第三章研究方法16
3.1 函數型主成分分析法. . . . . . . . . . . . . . . . . . . . . . . . 18
3.2 函數型主成分及樣條基底分析法. . . . . . . . . . . . . . . . . . 19
3.3 哈爾小波轉換分析法. . . . . . . . . . . . . . . . . . . . . . . . 20
3.4 函數型主成分及哈爾小波轉換分析法. . . . . . . . . . . . . . . 22
第四章數值模擬24
4.1 模擬設定. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.2 模擬結果. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
第五章實例分析28
5.1 資料介紹. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5.2 分析結果. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
第六章結論33
參考文獻34
附錄36
A 函數型主成分分析法之模擬結果. . . . . . . . . . . . . . . . . . . 37
B 函數型主成分及樣條基底分析法之模擬結果. . . . . . . . . . . . . 43
C 哈爾小波轉換分析法之模擬結果. . . . . . . . . . . . . . . . . . . 45
D 函數型主成分及哈爾小波轉換分析法之模擬結果. . . . . . . . . . 46

Cattell, R. B. (1966). The scree test for the number of factors. Multivariate Behavioral Research, 1(2):245–276.
Chen, Y.-H. and Zucker, D. M. (2009). Case-cohort analysis with semiparametric transformation models. Journal of Statistical Planning and Inference, 139(10):3706–3717.
Cox, D. R. (1992). Regression models and life-tables. In Breakthroughs in Statistics, pages 527–541. Springer.
Goldsmith, J., Bobb, J., Crainiceanu, C. M., Caffo, B., and Reich, D. (2011). Penalized functional regression. Journal of Computational and Graphical Statistics, 20(4):830–851.
Kong, D., Ibrahim, J., Lee, E., and Zhu, H. (2014). Flcrm: Functional linear cox regression models.
Reiss, P. T., Huo, L., Zhao, Y., Kelly, C., and Ogden, R. T. (2015). Wavelet domain regression and predictive inference in psychiatric neuroimaging. The Annals of Applied Statistics, 9(2):1076.
Ruppert, D., Wand, M. P., and Carroll, R. J. (2003). Semiparametric Regression. Number 12. Cambridge university press.
Vidakovic, B. and Mueller, P. (1994). Wavelets for kids. Instituto de Estadística, Universidad de Duke.
Wang, X., Nan, B., Zhu, J., and Koeppe, R. (2014). Regularized 3d functional regression for brain image data via haar wavelets. The Annals of Applied Statistics, 8(2):1045.
Yang, G., Yu, Y., Li, R., and Buu, A. (2016). Feature screening in ultrahigh dimensional cox’s model. Statistica Sinica, 26:881.
Zhang, H. H. and Lu, W. (2007). Adaptive lasso for cox’s proportional hazards model. Biometrika, 94(3):691–703.