在此論文中,我們提出兩個新方法用以估計非平穩共變異函數與預測缺失值，
第一種方法適用於重覆觀測資料，第二種則適用在單次觀測資料上。
第一種方法將空間訊號表示為數個基底函數與數個平穩過程的線性組合，
假設空間訊號在各個基底上的分量與各個平穩過程間均為獨立，利用迴歸分析處理此模式的參數估計問題，
並採取Tibshirani(1996)所提出的選取變數之方法來挑選適合的基底函數與平穩過程，
至於預測方法則採用最佳線性不偏預測法(best linear unbiased prediction)。

第二種方法是在小波(wavelet)域上建構非平穩空間模式，這個模式需要將資料的空間位置映在規則格點上。
空間過程經由小波轉換後，假設其變異數的對數(log variance)服從條件自迴歸模型
(conditional autoregressive model) (Besag, 1974),統計推論則採用貝氏分析法(Bayesian inference)。
這個方法允許有缺失值，因此對於非規則格點的資料，只要將其對應到夠細的格子點上，亦可適用。

在特定轉換下，兩種模式的共變異矩陣為對角或近似對角矩陣，
因此在處理龐大的空間資料時，能迅速有效地運算。透過模擬，
發現所提之兩種模式對於平穩或非平穩過程都有很好的近似結果，應用在實際資料上也有不錯的預測表現。

In this thesis, we develop two approaches for approximating nonstationary spatial covariance function
and predicting missing data. The first approach is proposed for handling spatial data with
multiple replications and the second is for spatial data with single replication. The first method
represents a spatial process as a linear combination of some local basis
functions with uncorrelated random coefficients plus some independent stationary processes.
The covariance function estimation problem is
formulated as a regression problem and a constrained least squared method proposed by Tibshirani
(1996) is applied for selecting appropriate basis functions and stationary processes, and
estimating parameters. The best linear unbiased prediction is then used for prediction.

The second approach is constructed on the wavelet domain. It can be applied to lattice data
with single measurement. In wavelet domain, the log variances of the
transformed data are assumed to follow the conditional autoregressive model (Besag, 1974).
Bayesian inference is implemented for estimation and prediction.
Since missing data are allowed in this approach, it can be adopted to irregularly spaced data by
mapping irregular data on a finer grid.

Both approaches are computationally efficient for handling large data sets since the
covariance matrix of either method is diagonal or nearly diagonal under a certain transformation.
Simulation experiments show that the proposed methods approximate both stationary and
nonstationary dependence structures very well. Both methods also perform well in
prediction for real applications.

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