本論文提出多層解析混合自迴歸樹狀空間模型、及其預測空間資料的快速
演算法。此一新方法根源於自迴歸樹狀模型，但單一自迴歸樹狀模型的缺點是其預測值會有塊狀叢聚現象。本文提出的混合自迴歸樹狀空間模型，具有平穩的空間共變異結構，因此其預測值不會出現塊狀現象。根據該模型可導證出一快速演算法用於處理大量空間資料，當資料中有遺漏值時，並不會影響其計算速度。該模型的另一項優點是可以 EM 演算法迭代求得其參數的最大概似估計量。此外，在實際問題中，選擇適合的模型對於預測舉足輕重，但相關文獻卻付之闕如。本文將討論如何以加權最小平方判則進行模型篩選。

In this article, we propose an autoregressive tree-structured mixture model and develop a computationally efficient algorithm for spatial prediction. The algorithm allows us to handle a huge dataset, even when there are missing observations. The
proposed mixture model has a stationary covariance structure and is free from blocky artifacts in prediction, which may be produced by a single autoregressive tree-structured model. We shall also show how to obtain the maximum likelihood estimators of the model parameters using an EM algorithm, and develop a model-selection criterion, which has not been addressed in the past literature.

Contents
1 Introduction 1
2 Autoregressive Tree-structured Models for Spatial Data 3
3 Autoregressive Tree-structured Mixture Models 6
3.1 The Model . . . . . . . . . . . . . . . . . . . . . . . 6
3.2 A Fast Spatial Prediction Algorithm . . . . . . . . . . 9
3.3 Parameter Estimation . . . . . . . . . . . . . . . . . . 12
3.4 Model Selection . . . . . . . . . . . . . . . . . . . . 14
4 An illustrated Example 15
5 Discussion 21
Appendix A 23
Appendix B 27
References 30

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