本論文主要研究經驗正交函數(empirical orthogonal functions)對非穩定空間相關函數的估計問題，我們假設資料在空間中不同的時間點重複蒐集，但採樣的地點可能是稀少、不規則分佈的。針對這樣的問題目前有兩種處理方法，一種方法是將空間隨機過程局限在一組給定函數所展開的空間中，另一種方法則先在密集格點上內插資料，再利用主成分分析法獲得經驗正交函數的估計。有別於這兩種方法，本論文提出兩個半母數空間模型，結合平滑樣條函數(smoothing splines)或迴歸樣條函數(regression splines)，以最大懲罰概似函數法，直接得到經驗正交函數的估計。我們以期望條件最大演算法(expectation conditional maximization algorithm)，同時獲得所有模型參數的估計值。模擬結果顯示我們所提出的方法無論對於平穩或是非平穩的空間隨機過程，其空間相關函數估計或空間預測準確度皆有良好的表現。此外我們也將所發展的方法應用於分析美國科羅拉多州的雨量資料，並進一步將空間模型擴展至時空模型，用以分析同時具有時間及空間相關性的資料。This thesis considers nonstationary spatial covariance estimation using empirical orthogonal functions (EOFs) under the consideration that data may be observed only at some sparse, irregularly spaced locations with repeated measurements. Instead of obtaining EOFs by principal component analysis based on a class of pre-specified basis functions or a pre-smoothing step with data imputed on a regular grid, two semiparametric approaches are advocated for EOF estimation, which are based on smoothing splines and regression splines using penalized likelihood. An expectation-conditional-maximization algorithm is proposed to obtain the penalized maximum likelihood estimates of the mean and the covariance parameters simultaneously. Some simulation results show that the proposed methods perform well in both spatial prediction and covariance function estimation, regardless of whether the underlying spatial process is stationary or nonstationary. In addition, the methods are applied to analyze a precipitation dataset in Colorado. Some further extension to spatio-temporal models is also provided.
