本論文探討兩成對函數型資料之相關性，提出成對函數型資料的函數型相關係數估計方法，借鏡Huang (2018)所提出的局部組內變異量(SSW)估計，計算成對函數型資料{X_{i}(t), Y_{i}(t)} i = 1,..., N$各自之局部組內變異量，並計算局部組間共變量(Cov_{X(t),Y(t)})，藉由局部組內變異量及局部組間共變量計算得函數型相關係數(FCor_{X(t),Y(t)})。將提出之方法與既有估計方法比較Zhou et al. (2018)，透過MIBias、MIVar、MISE三個準則衡量在不同模擬設定下比較兩方法之優劣，隨後探討函數型相關係數於特定格點(grid points)、帶寬(bandwidth)下之分佈。最後將本論文所提出之方法應用至臺灣臺北市古亭空氣品質量測站所測得之臭氧濃度數據及氮氧化物數據上，並嘗試了解臭氧濃度及氮氧化物濃度間相關性變化。


關鍵字：局部多項式迴歸，函數型資料，函數型相關係數

We investigate a new approach to estimate functional correlation between two paired functional data. This approach is inspired by an estimator of within-group variation for functional data introduced in Huang (2018). We adopt this estimator for measuring local variance within each group of functional data, propose an estimate of local covariance between paired functional data, and form an estimator of functional correlation. We conduct simulation studies to compare the proposed method to Zhou et al. (2018) using three criteria, MIBias, MIVar and MISE. In addition, we explore the empirical distribution of the proposed functional correlation estimate. Finally, the proposed method is applied to air pollution data collected by Executive Yuan in Taiwan, to estimate the functional correlation between two chemical compounds, concentrations of ozone and nitrogen oxide in Taipei.

ch1 緒論 ...1
ch2 背景介紹 ...2
ch2-1 函數型資料 ...2
ch2-2 成對函數型資料 ...3
ch2-3 局部多項式迴歸 ...3
ch2-4 函數型資料的相關係數估計 ...4
ch2-5 階層變異數分析 ...6
ch2-6 Chi-squared type approximation ...7
ch3 函數型相關係數估計 ...9
ch4 函數型相關係數之模擬 ...11
ch4-1 模擬一 ...13
ch4-1-1 模擬一小結 ...14
ch4-2 模擬二 ...16
ch4-2-1 模擬二小節 ...16
ch4-3 函數型相關係數之分佈討論 ...19
ch4-4 研究結果比較 ...20
ch5 實際資料分析 ...21
ch5-1 背景介紹 ...21
ch5-1-1 相關文獻回顧 ...21
ch5-1-2 大氣中臭氧及氮氧化物之光化反應 ...22
ch5-2 資料分析 ...23
ch5-2-1 資料探索 ...24
ch5-2-2 分析結果 ...26
ch5-2-3 結果討論 ...26
ch6 結論與後續研究 ...28
ch7 附錄 ...28

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