精準化識別函數數據顯著差異區段之研究

Abstract

在函數數據的雙樣本檢定中，我們除了判定樣本的差異與否，還特別關注差異的所在位置。在目前的文獻中，學者們針對這類型的問題 (區間檢定)，提供了不少方法，並且多數要求在執行方法前，對資料定義域均勻地切割，而切割的精細度決定了資料標定差異區間的準確度。然而，當差異區間越小，便須越精細的切割，否則容易檢測出過大的區間，並增加方法的錯誤拒絕率。此外，即便選擇一個相對更密集的切割，也得面臨計算時間上的需求，或是控制錯誤拒絕率對大樣本的要求。因此這篇論文我們針對區間檢定的前置步驟，分割，進行改良，以增強檢測的準確性。同時將其中一種現存的均值檢定法，從相同共變異數均值檢定推廣至相異共變異數均值檢定。最後將改良後的區間均值檢定法，應用於兩組資料 (氣象、車流量) 以展示改良前後與推廣前後的差異性。

In two-sample tests for functional data, it is essential not only to determine overall differences but also to identify the specific locations of these differences. This presents what is known as an interval testing problem. Most existing methods require a pre-defined partition whose resolution may affect detection accuracy. However, a higher resolution necessitating a denser partition could lead to excessive computational time. Besides, although a higher resolution generally increases the detection precision, it also requires a larger sample size to control the same level of false discovery rates. To address these issues, we have introduced a novel adjustment strategy that involves constructing a suitable partition as a preliminary step before implementing interval testing methods. This approach enhances the detection precision across various resolution scenarios. The concept is also extended to tests involving unequal sample covariances. Our simulation study indicates that the proposed approach yields higher statistical power in identifying significant intervals under small sample situations. We apply the approach to weather and traffic data, examining temperature differences at different locations and traffic flow differences on weekends, respectively.