空間迴歸模型中若隨機效應項與解釋變數間存在共線性的關係稱之為空間混淆(spatial confounding)，此時迴歸係數估計量會有嚴重的偏誤並將導致較不精確的空間預測。此議題在空間統計中已受到關注，但是如何有效的修正迴歸係數的偏誤至今仍然未有明確且完善的方法。此研究計畫將使用一個半母數的方法去估計空間隨機效應項，進而提出修正迴歸係數估計量的準則，此做法不須事先給定空間型態資料的共變異結構，因此使用上更具彈性，同時因為結合固定秩克利金(Fixed rank kriging)的技巧，使得此方法可以處理高維度的資料分析而不需處理高維度反矩陣的計算問題。所提方法中需要決定一組基底函數，且基底函數的個數預期將影響隨機效應項的解析度與迴歸係數的估計，因此我們分別從參數估計與空間預測的觀點出發，提出兩個選取基底函數個數的準則。在完成參數估計與空間預測的問題之後，我們進一步嘗試提出一個模型變數的選取準則，使得空間迴歸模型的配適更為完善。本研究計畫將探討相關的統計理論並設計完整的模擬實驗驗證所提方法的有效性，同時也將藉由分析帶有空間混淆效應的實際資料說明所提方法的可行性。 ;Spatial regression with spatial confounding is an important issue in statistical modeling, because it will lead to biased estimators of regression coefficients and inaccurate spatial predictors. This issue has received much attention in spatial statistics, but foundational questions of how to modify the biases of coefficient estimators in the presence of spatial confounding have not been adequately addressed under the frequentist framework. In this proposal, we attempt to propose a semiparametric method to estimate regression coefficients based on a fixed rank kriging technique. The idea does not require specifying a parametric covariance structure and hence is more flexible in modeling spatial covariance functions. In our proposal, a class of basis functions exacted from thin-plate splines is used, where the number of basis functions is expected to impact the resolution of the spatial random process and the estimation of regression coefficients. We will develop two aspects to select the number of basis functions which are designed toward two different inferences when the main goals lie respectively in the estimation of regression coefficients and spatial prediction. As a result, two estimators of regression coefficients and the consequent spatial predictors will be established. The proposed methodology can be applied to stationary or nonstationary spatial processes and it also can be applied to massive datasets without handling the computational issue of high-dimensional inverse matrices. Further, a variable selection criterion under the presence of spatial confounding will be discussed as well. Statistical inferences associated with the proposed methodology will be justified in theories and via simulation studies. Finally, a real data example will be analyzed for illustration.
