一種基於函數型資料主成分分析的曲線對齊方式

Abstract

函數型資料分析的是一組曲線資料，通常定義域為一段時間範圍。常見的如某一個地區人口在成長期的身高紀錄表或是氣候統計資料。函數型資料主要特色曲線間常有共同趨勢，而且個別曲線反應共同趨勢時也有時間和強度上的差異。本文研究主要是使用Kneip 和 Ramsay提出，結合對齊程序和主成分分析的想法作為模型架構，來分析函數型資料的特性。首先在對齊過程中，使用時間轉換函數(warping function)，解決觀測資料上時間的差異；並使用主成分分析方法，幫助研究者探討資料的主要特性。基於函數型資料被預期的共同趨勢性，我們可以利用此一特色作為各種類型資料分類上的依據。此外本研究會對幾種選取主成分個數的方法，進行綜合討論與比較。

In this thesis, a procedure combining curve alignment and functional principal component analysis is studied. The procedure is proposed by Kneip and Ramsay .In functional principal component analysis, if the data curves are roughly linear combinations of k basis curves, then the data curves are expected to be explained well by principle component curves. The goal of this study is to examine whether this property still holds when curves need to be aligned. It is found that, if the aligned data curves can be approximated well by k basis curves, then applying Kneip and Ramsay`s procedure to the unaligned curves gives k principal components that can explain the aligned curves well. Several approaches for selecting the number of principal components are proposed and compared.