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研究生： 劉怡禎
Liu, Yi-Chen
論文名稱： 空間迴歸模型在模型錯誤假設下之係數估計及其在溫度增量的應用
Coefficients Estimation in Spatial Regression Models under Model Misspecification with Applications to Thermal Increment
指導教授： 徐南蓉
Hsu, Nan-Jung
口試委員: 黃信誠
Huang, Hsin-Cheng
陳春樹
Chen, Chun-Shu
學位類別： 碩士
Master
系所名稱： 理學院 - 統計學研究所
Institute of Statistics
論文出版年： 2021
畢業學年度： 109
語文別： 英文
論文頁數： 29
中文關鍵詞： 高斯過程 、模型錯誤設定 、局部線性迴歸模型 、空間自迴歸模型
外文關鍵詞： Gaussian process model, model misspecification, partial linear regression, spatial autoregressive model
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    • 本研究討論使用具有空間隨機效應的迴歸模型在模型錯誤設定(model misspecification)下估計迴歸係數的估計表現。所考慮的模型包含空間自迴歸模型(Spatial Autoregressive Model)、高斯過程迴歸模型(Gaussian Process Model with Linear Regression)，以及局部線性迴歸模型(Partial Linear Regression Model)。
    在模型具有不同空間效應下，推導出變數的迴歸係數及估計誤差的一般式與估計係數變異數。此外，本研究根據 Chang (2020) 論文中 PCBA 資料做模擬設計，討論了不同模型在模型錯誤設定下的估計誤差表現。
    在此研究中，發現在真實模型為空間效應較弱的空間自迴歸模型情況下，以高斯過程迴歸模型作為擬合模型估計迴歸係數時，表現相當穩健。相反地，若在真實模型為具有較強空間效應的高斯過程迴歸模型情況，以空間自迴歸模型作為擬合模型估計迴歸係數時，則會具有顯著的估計誤差。

    This thesis concerns the estimation properties on the regression coefficient estimates in regression models involving spatial random effects under model misspecification.
    The models considered include spatial autoregressive models (SAR), regression models with Gaussian spatial processes, partial linear regressions.
    Under different specifications for spatial effects, the estimation bias and variance for the regression coefficients on the covariates are derived theoretically. Moreover, the empirical performance on the estimation mean squared errors is investigated in a simulation study designed according to the PCBA data studied in Chang (2020).
    This work found that in a SAR model with weak spatial influences,
    this work found that the regression coefficient estimates are fairly robust against the misspecified GP modeling.
    On the contrary, in a regression GP model with a wide range of spatial dependence,
    the regression coefficient estimates have non-negligible efficiency loss when the fitted model is misspecified as a SAR model.

    摘要 i Abstract ii List of Figures vi List of Tables vii 1 Introduction 1 1.1 Motivating Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Literature Review and Thesis Structure . . . . . . . . . . . . . . . . . . . . . . . 2 2 Introduction to PCBA Data 4 2.1 Data Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Definition of Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3 Various Models for PCBA Data 6 3.1 Spatial Autoregressive Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.2 Gaussian Process Model with Linear Regression . . . . . . . . . . . . . . . . . . 7 3.3 Partial Linear Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4 Estimation Bias and Variance 10 5 Simulation 13 5.1 Data Generation Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 5.2 Methods To Be Compared . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 5.2.1 Performance Measures on Parameter Estimation . . . . . . . . . . . . . . 15 5.2.2 Performance Measures on Prediction . . . . . . . . . . . . . . . . . . . . 19 6 Application 22 7 Conclusion 24 References 25 Appendix : Results of Simulation Experiment with p = 6 26

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