空間相關通常是複雜的，且非平穩的現象普遍存在，這使得傳統的平穩模型無法適切地描述實際問題的空間相依結構。為了解決這個問題，現今已發展許多關於非平穩空間模型建模的研究方法，其中空間變形法是主要方法之一，其藉由引入變形函數的手法轉換空間座標，將在原座標空間 (G) 之非平穩隨機過程等價表示為在轉換後座標空間 (D) 之平穩性和等向性的過程。然而，這個方法在變形函數的估計上極具挑戰，估計的空間變形函數需具備雙射性質，否則可能造成映射後的變形空間發生折疊的情況。本論文提出了一種基於仿射耦合 (AC) 函數的空間變形法。仿
射耦合函數具有一對一映射的特性，可以避免空間折疊的問題。與傳統變形法的相比，本論文採用最大概似估計同時估計仿射耦合函數和空間相關函數之參數，不需進行兩階段的多維度縮放估計 (MDS) 及樣條函數 (splines) 估計，從而簡化了建模過程。在模型選取上，本論文提出以 AIC 準則選取適當的 AC 函數。此外，本論文的方法可適用於單次觀測和重複觀測資料的兩種空間資料情境，表示其能更彈性地應用於多種實際情況。經由模擬實驗和實際資料應用，驗證了此論文提出之 AC空間變形法確實能有效的估計非平穩空間相關，提供良好的配適結果及空間預測，並避免空間折疊的問題。

Spatially correlated non-stationary phenomena are commonly observed and often exhibit complexity, which poses challenges for accurately estimating spatial dependence structures using traditional stationary models. To address this issue, numerous research methods have been developed for non-stationary spatial modeling, among which spatial deformation approaches are one of the main methods. These approaches transform non-stationary stochastic processes into stationary and isotropic processes using bijective deformation functions. However, a significant challenge in this method is the estimation of the deformation functions. If the estimated spatial deformation function lacks bijectivity, it may lead to spatial folding issues in the transformed space. In this paper, we propose a spatial deformation method based on affine coupling functions. Affine coupling functions possess one-to-one mapping properties, effectively avoiding spatial folding problems. Compared to traditional methods, our approach simultaneously estimates the parameters of both the affine coupling function and the spatial correlation function using maximum likelihood estimation (MLE), eliminating the need for multidimensional scaling and thin-plate spline estimation, thus simplifying the modeling process. Furthermore, our proposed method is applicable to both single and repeated observations, providing flexibility for various practical situations. Through simulation experiments and a real dataset, we evaluate the fitting performance of our method and demonstrate that it indeed avoids spatial folding issues.

目次
摘要 i
Abstract ii
表目次 v
圖目次 vi
第一章 緒論 1
第二章 高斯隨機過程之模型、估計與預測 4
2.1 平穩的 GP 模型 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 參數之最大概似估計 . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 空間預測 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
第三章 空間變形法之非平穩高斯過程建模 7
3.1 仿射耦合變形法 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2 多項式內部方程式 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.3 AC 模型參數估計及選模 . . . . . . . . . . . . . . . . . . . . . . . . . 11
第四章 正則化估計及參數選擇 14
第五章 模擬實驗 16
5.1 實驗 I：真實空間變形函數為仿射耦合 . . . . . . . . . . . . . . . . . 17
5.2 實驗 II：真實空間變形函數不為仿射耦合 . . . . . . . . . . . . . . . 22
5.3 實驗 III：未知的 covariance 模型 . . . . . . . . . . . . . . . . . . . . 25
5.4 正則化估計實驗 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5.5 計算 MLE 的時間比較 . . . . . . . . . . . . . . . . . . . . . . . . . . 31
第六章 實際應用 32
第七章 結論與未來展望 34
參考文獻 35
附錄 37

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