臺灣博碩士論文加值系統

許多研究在多個時間點上對實驗參與者重複測量相同的變項，如此收集的數據稱為函數型資料。根據每人的多筆測量結果進行集群分析，在很多領域有重要的應用。本論文針對（一維）函數型資料的相異程度，提出新穎且容易執行的方法。此方法同時適用於規律時間點或不規律時間點上的紀錄。首先以平滑樣條插捕每位實驗參與者的函數曲線，決定每人之平滑參數。任兩參與者間的相異程度是透過共用兩人之平滑參數所估計的曲線計算，參與者不同則使用的一對平滑參數也隨之變動。此方法的直觀意義是平滑參數相當於平滑樣條中訊號-噪音比值的倒數，而兩條相似曲線上的觀測數據若以同一平滑參數估計背後的函數，可預期將得到相似的估計曲線。作法看似複雜，實際上並未增加計算量，因給定平滑參數時平滑樣條有特殊的快速演算法。此法透過平滑參數互換，將曲線估計的不確定性納入考量。且由於成對的相異程度較不受離群值的影響，它也可以用於找出離群值。文中將透過模擬與現有其他方法比較，及應用於美沙酮劑量變化的分群，驗證本法的有效性。本文亦敘述將此方法拓展至二維或三維空間之函數資料的困難，並討論可能的解決方案。

Many studies measure the same type of information longitudinally on the same subject at multiple time points, and clustering of such functional data has many important applications. We propose a novel and easy method to implement dissimilarity measure for functional data clustering based on smoothing splines and smoothing parameter commutation. This method handles data observed at regular or irregular time points in the same way. We measure the dissimilarity between subjects based on varying curve estimates with pairwise commutation of smoothing parameters. The intuition is that smoothing parameters of smoothing splines reflect the inverse of the signal-to-noise ratios and that when applying an identical smoothing parameter the smoothed curves for two similar subjects are expected to be close. Our method takes into account the estimation uncertainty using smoothing parameter commutation and is not strongly affected by outliers. It can also be used for outlier detection. The effectiveness of our proposal is shown by simulations comparing it to other dissimilarity measures and by a real application to methadone dosage maintenance levels. We also discuss the challenge and the potential solution to extend this method to functional data on two- or three- dimensional space.

Acknowledgements. . . . . i
中文摘要. . . . . ii
Abstract. . . . . iii
1 Introduction. . . . . 1
1.1 Motivation. . . . . 1
1.2 Outline. . . . . 3
2 Literature Review. . . . . 4
2.1 Cluster analysis for vector-valued data. . . . . 4
2.1.1 Hierarchical clustering. . . . . 4
2.1.2 Partition-based methods. . . . . 6
2.1.3 Distribution-based methods. . . . . 9
2.1.4 Density-based methods. . . . . 11
2.2 Clustering result comparison. . . . . 12
2.3 Outlier detection. . . . . 16
2.4 Types of functional data. . . . . 19
2.5 Smoothing techniques. . . . . 21
3 Method. . . . . 24
3.1 Challenges and strategies in functional data clustering. . . . . 24
3.2 Smoothing splines. . . . . 29
3.3 Smoothing Parameter Commutation dissimilarity. . . . . 31
4 Results. . . . . 33
4.1 Simulation study. . . . . 33
4.2 Real data application with outlier detection . . . . . 42
5 Discussion. . . . . 51
5.1 Future directions for 1D problems . . . . . . . . . . 51
5.2 2D or 3D extensions via spatial random-effects models. . . . . 52
6 Conclusion. . . . . 57
Reference. . . . . 58

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