臺灣博碩士論文加值系統

函數型線性迴歸(Functional linear regression)是一種分析長期資料(longtudinal data)並預測其未來趨勢的方法。在長期資料分析中，每個樣本函數的觀測時間點可能不規則且稀少，並帶有測量誤差，而且相同樣本函數在不同時間的觀測值也存在著相關性，本篇論文的方法便適用於這樣的情況。

本論文旨在改良函數對函數之主成份迴歸模型，原先作法是將自變函數
X(•) 與應變函數 Y(•) 各自作Karhunen-Loéve 分解，取前幾個主要的特徵函數及其對應的特徵值求出主成份分數，藉此將函數對函數線性迴歸問題轉成主成份對主成份的多變量迴歸問題。然而只取前幾個主要主成份的作法可能造成有用的信息卻被併入誤差項處理的情況。

本論文將主成份迴歸模型改良成用共變異函數求得迴歸解，
希望藉此解決上述問題，並將迴歸模型從一個應變函數(functional response) 對一個自變函數，擴展到一個應變函數對多個自變函數。另外，本模型亦適用於多維度的長期資料，且在資料的觀測時間點規律且密集的情況下亦有良好的表現。

論文的最後，將針對不同情況的模擬資料，比較此模型和其他函數-函數型迴歸方法的預測結果，來說明其可行性。

Functional linear regression is an important tool to analyze longitudinal data. In longitudinal data analysis, the observations are made on irregular time points (or locations) with measurement error. Moreover, observations of the same subject are correlated. Our method is suitable for the mentioned situations.

Our method aims at improving function-on-function linear regression. Traditionally, the regression coefficients for function-on-function linear regression models are estimated by the first few principal components of both the predictor and response functions. However, some useful information might be treated as error term and thus be discarded if we just adapt the first few important principal components. Consequently, we resolve this issue by estimating the coefficients directly from the covariance functions of predictor and response functions. The proposed estimation approach can be used in multiple and multidimensional function-on-function linear regression models as well.

1 緒論(P.1)

2 文獻回顧(P.3)

2.1 局部多項式迴歸(P.3)
2.1.1 簡介(P.3)
2.1.2 模型及估計(P.4)
2.2 多維度函數型主成份分析(P.6)
2.2.1 簡介(P.6)
2.2.2 模型與假設(P.7)
2.2.3 估計方法(P.8)
2.3 函數對函數之線性迴歸(P.13)

3 研究方法(P.14)

3.1 簡介(P.14)
3.2 模型(P.16)
3.3 推導與估計(P.17)
3.3.1 Karhunen-Loéve 分解(P.17)
3.3.2 符號定義(P.18)
3.3.3 推導及估計(P.21)
3.4 預測(P.23)

4 模擬與比較(P.24)

4.1 模擬設定I (P.25)
4.2 模擬設定II(P.30)

5 結論與討論(P.41)

附錄A: 多變量線性迴歸(P.42)

A.1 模型(P.42)
A.2 估計(P.43)

附錄B: 多變量函數型主成份迴歸(P.44)

B.1 模型(P.44)
B.2 估計(P.45)

參考文獻(P.50)

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2005.

[2] Fang Yao, Hans-Georg Müller, and Jane-Ling Wang. Functional data analysis for sparse longitudinal data. Journal of the American Statistical Association,
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[9] Andrada E. Ivanescu, Ana-Maria Staicu, Fabian Scheipl, and Sonja Greven. Penalized function-on-function regression. Computational Statistics, 30(2):539–568, June 2015. ISSN 0943-4062. doi: 10.1007/s00180-014-0548-4. URL http:
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[10] Jeff Goldsmith, Jennifer Bobb, Ciprian M. Crainiceanu, Brian Caffo, and Daniel Reich. Penalized functional regression. Journal of Computational and Graphical Statistics, 20(4):830–851, 2011. ISSN 10618600. URL http://www.jstor.org/
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[11] Fabian Scheipl and Sonja Greven. Identifiability in penalized functionon-function regression models. Electronic Journal of Statistics, 10(1):495–526, 2016. doi: 10.1214/16-EJS1123. URL http://dx.doi.org/10.1214/16-EJS1123.

[12] Guozhong He, Hans-Georg Müller, and Jane-Ling Wang. Extending correlation and regression from multivariate to functional data*. Asymptotics in Statistics and Probability (M.L. Puri, ed.), (197-210), 2000.