兩個具時空效應之隨機場的獨立性檢定

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曾逸群(Yih-Chyurn Tseng) 查詢紙本館藏

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數學系

論文名稱

兩個具時空效應之隨機場的獨立性檢定
(Testing Independence Between Two Spatial-Temporal Random Fields)

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摘要(中)

隨著人類在軟硬體上的進步，影像資料的解析度越來越高，細節越來越豐富。在資料分析時卻也面臨變數維度大量增加隨之而來的檢定力下降與計算複雜度大增的問題。目前主流的做法便是將資料維度降低後再做下一步的分析。本文旨在保持資料固有的結構下對具有時空效應的成對圖像資料進行獨立性檢定。在本文中我們提出兩階段獨立性檢定法，分開處理高維度空間影像資料的維度縮減問題，以及處理成對低維度時間序列之獨立性檢定。在階段一中我們將原始資料投影至主要的多分辨率薄板樣條 (MRTS) 基底函數所展開的空間中，在保留原始資料的固有相關結構的同時能夠有效地降低資料維度;而在階段二中我們將降維後的資料以成對低維度時間序列之獨立性檢定加以分析與推論。我們藉由模擬資料試驗分別在基於虛無假設成立時以及在對立假設成立時分別比較各種獨立性檢定方法的表現，並將這些方法應用在海平面溫度與降雨量之遙相關分析中。

摘要(英)

With the advancements in hardware and software, the resolution of image data has been continuously increasing, leading to richer details. However, data analysis encounters challenges such as decreased statistical power and increased computational complexity due to the substantial increase in variable dimensions. The most popular approaches are to reduce the data dimensionality before conducting further analysis. This thesis aims to test independence between paired image data with spatiotemporal dependence while preserving the inherent spatial and temporal structures. A two-stage independence testing method is proposed, combining dimension reduction for high dimensional spatial image data and an independent test for paired low-dimensional time series. In the first stage, we project the original data onto a space spanned by the primary MultiResolution Thin-plate Splines (MRTS) basis functions. This projection effectively reduces the data dimensionality while preserving the inherent spatial correlation structure of the original data. In the second stage, we analyze and infer the independence of the dimension-reduced data using an independence test for paired low-dimensional time series. We compare the performance of these independence testing methods through simulation experiments under the null hypothesis and alternative hypothesis, respectively. Additionally, we apply these methods to the teleconnection analysis of sea surface temperature and rainfall datasets.

關鍵字(中)

★ 維度縮減
★ 高斯隨機場
★ 獨立性檢定
★ 多分辨率薄板樣條基底函數
★ 時空資料
★ 遙相關

關鍵字(英)

★ Dimension Reduction
★ Guassian Random Field
★ Independence Test
★ Multiresolution Thin-plate Spline Basis Function
★ Spatial-Temporal Data
★ Teleconnection

論文目次

一、緒論 1
二、方法介紹 5
2.1 利用 MRTS 基底函數之維度縮減 ................................... 5
2.2 獨立性檢定 ............................................................... 8
2.3 MRTS-CCA 對成對時空資料之獨立性檢定法及比較. . . . . . . . . . . 14
三、統計模擬
3.1 實驗一:獨立資料之型一誤差控制研究 ........................... 17
3.2 實驗二:兩種非獨立成對資料之檢定力研究 ..................... 21
3.2.1 實驗二 (a):隨機誤差具相關性............................. 21
3.2.2 實驗二 (b):資料具有跨時間相關性 ...................... 22
四、實際資料分析 25
五、結論 27
參考文獻 29
附錄 A 2.2 節所使用之 VARMA(1,0) 模型 33

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指導教授

黃世豪(Shih-Hao Huang)

審核日期

2023-8-16

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