臺灣博碩士論文加值系統

本論文的研究主題為探討如何對長期追蹤資料 (longitudinal data) 之觀測對象分群的問題 。本文以函數型資料 (functional data) 的觀點來看待長期追蹤資料 , 亦即將資料視為一組互相獨立的時間函數或曲線在間斷時間點的觀測值 , 也因此文中所發展出的方法可處理資料具有密集的觀測時間點或是每個觀測對象之觀測時間不相同時的分群問題 。 此外 , 由於資料中同一個觀測對象在不同時間所得到的測量值存在著相關性 , 所以本文試圖根據資料隨時間變化的平均趨勢與在相異時間點之間的共變異結構對資料分群 。在分群曲線之平均函數 (mean function) 與共變異數函數 (covariance function) 相異的假設下 , 本文根據函數型主成份分析 (FPCA , functional principal components analysis) 建立分群曲線的基本模式 , 並依此模式提出一套以疊代重分群為主的分群演算法 , 簡稱為 rFPCA (reallocation based on FPCA) 。 rFPCA 分群演算法可分為初步分群與疊代重分群兩個步驟 , 其最終目的是想有效區分資料中平均函數與共變異數函數相異的曲線 。 在初步分群方面 , 主要是利用有限維度的函數型主成份分數 (functional principal component scores) 之分佈來探查資料在平均結構上的初步分群 。而在重分群方面 , 則是利用各群資料各別對等待被重分群的曲線做配適 , 然後以各群所得的配適曲線與此待重分群之觀測曲線間的最小 L2 距離來選擇此曲線的新群別 。除此之外 , 文中也從理論方面來探討 rFPCA 分群演算法的原理與特性 , 並經由模擬研究來驗證理論上的結果 。 結果發現 , 本文所提出的分群演算法在分群曲線之平均函數或共變異數函數相異的情況下 , 大部份都可達到預期的分群目的 。 尤其是當相異群組的共變異數函數有顯著差異時, 疊代重分群步驟更能大幅改善初步分群的結果 。 另一方面 , 本文亦提出一套兩階段的拔靴法檢定 (two-stage bootstrap test) , 以做為在實際應用上可判斷資料是否需要執行重分群步驟的工具 。 在本文的最後 ,將以成長曲線與隨時間量測的基因表現量等兩組資料的分群分析來說明所提出之方法的實際應用與可行性 。

論文摘要
誌謝辭
第一章 緒論
第二章 文獻回顧
2.1 多變量資料分群方法之文獻回顧
2.2 函數型資料分群方法之文獻回顧
2.2.1 以基底函數之係數分群
2.2.2 以機率模式為主的分群方法
2.2.3 以曲線型態為分群依據之分群法
2.2.4 其他無母數分群法
第三章 以函數型主成份分析所建構之分群方法
3.1 分群曲線的基本模式
3.2 估計方法
3.3 rFPCA 分群演算法
3.4 理論結果與推論
第四章 模擬研究
4.1 分群相似度指標
4.2模擬設計
4.3模擬結果
第五章 兩階段之拔靴法檢定
5.1檢定問題
5.2檢定步驟
5.3模擬驗證
第六章 實例分析
6.1身高曲線
6.2基因表現量
第七章 結論
附錄 A 模擬研究結果列表
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