有關空間建模的方法，在現今的研究與發展中，有許多可以延伸和探討的面向，本研究即著手於透過解釋變數來瞭解造成空間相關結構改變的原因，並透過統計檢定方法確認解釋變數效應與空間相關結構的關係是否顯著，以及瞭解其關聯性的屬性為何者。
本研究涵蓋兩個主題，其一是在Fixed rank kriging模型（Cressie and Johannesson, 2008）的架構下，將解釋變數的影響以參數化方式建構在模型的空間相關函數中，並以迴歸式將解釋變數鏈結在 Cholesky 矩陣分解式中，用以建構非平穩空間隨機效應模型：FRK-x, 此模型是 Fixed rank kriging 模型的推廣。其二是透過統計檢定，檢驗空間資料或其空間相關結構與解釋變數之間是否存在關聯性。本論文提供線性與非線性兩種典型相關檢定方法，線性方法不僅可以推論兩筆資料之間的線性相關，亦可用於找尋影響空間相關函數的重要線性組合變量，非線性檢定則可進一步檢驗解釋變數與空間相關函數之非線性關係。

This thesis concerns about the the non-stationary spatial covariance structure affected by the covariates. For example, the spatial covariance function for ozone concentrations might be affected by the wind conditions as well as the temperature around the spatial domain of interest. In this thesis, two tasks are achieved. First, adopted the Fixed rank kriging (FRK) model, a new non-stationary spatial model, called FRK-x model, is proposed which incorporates the covariate information into the spatial covariance function through the matrix Cholesky decomposition. Second, two statistical tests are proposed to check possible connections between the spatial covariance and the covariates. These tests are derived based on the canonical correlation analysis (CCA) and the corresponding likelihood ratio test. In particular, linear CCA and kernel CCA are both considered. Through the simulations, both tests have good ability to detect the connections between spatial correlation structure and the covariates. The linear CCA also provides a linear combination of covariates as an effective summary for FRK-x modeling.

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