空間迴歸分析中，如何將解釋變數有效地加入共變異結構中以獲取更精確的空間預測是本文感興趣的議題。本文提供一種矩陣分解的方法，將解釋變數引入共變異結構中並能保證共變異矩陣為非負定矩陣，然後透過最大概似估計法估計模型參數。此外，本文將參數估計的結果結合自我迴歸模型(Autoregressive model)用以推測下一個時間點的參數值，並依此建立對應的研究區域之空間預測曲面。透過數值模擬實驗得知，本文所提的方法較一般常用的泛克利金迴歸法(universal kriging)有較佳的預測表現。同時本文亦透過分析臺灣PM2.5濃度的資料說明所提方法的實用性。

In spatial regression analysis, how to incorporate possible covariates into the covariance structure and obtain a more accurate spatial prediction both are interesting issues. In this thesis, a method of matrix decomposition is proposed which not only can incorporate possible covariates into the covariance structure but also guarantees the resulting covariance matrix to be positive semidefinite. Then, model parameters are estimated by the maximum likelihood method. Based on the estimation results of model parameters and the autoregressive model, model parameter values and the corresponding predicted surface for the next time point can be obtained. Numerical results show that the proposed method is superior to the commonly used universal kriging method in terms of spatial prediction. Also, a real data example regarding the fine particulate matter concentration in Taiwan is analyzed for illustration.

Reference
Akaike, H.(1973). Information theory and the maximum likelihood principle. In: B.N.Petrov and F. Csaki(eds.), Second International Symposium on Information Theory,
Akademiai Kiado, Budapest.

Chiu, T.Y.M., Leonard, T., Tsui, K. W. (1996). The matrix-logarithm covariance model. J. Am. Statist. Assoc, 98, 198-210.

Chen, S. (2013). Applied Time−Series Econometrics for Macroeconomics and Finance,2nd Edition. Tung Hua Book Co., Ltd.

Cheng, Y., He, K., Du, Z., Zheng, M., Feng, K., Ma, Y. (2015). Humidity plays an
important role in the PM2.5 pollution in Beijing. Environmental Pollution, 197,68-75.

Cressie, N. (1993). Statistics for Spatial Data (revised edition), Wiley: New York.
Ejigu, A., Wencheko, E., Moraga, P., Giorgi, E. (2020). Geostatistical methods for modelling non-stationary patterns in disease risk. Spatial Statistics, 35, 100397.

McCullafh, P. and Nelder, J. A. (1989). Generalized Linear Models, 2nd ed. London: Chapman and Hall.
Mat´ern, B. (2013). Spatial Variation. Springer Science and Business Media.

Newton, H. J.(1988). TIMESLAB: A Time Series Analysis Laboratory. Pacific Grove,
CA: Wadsworth and Brooks/Cole.

Pourahmadi, M. (1999). Joint mean-covariance models with applications to longitudinal data: Unconstrained parameterization. Biometrika, 86, 677-690.32

Pinheiro, J. D. and Bates, D. M. (1996). Unconstrained parameterizations for variance-covariance matrices. Statist. Comp, 6, 289-96.

Schabenberger, O. and Gotway, C.A. (2005). Statistical Methods for Spatial Data Analysis.
Chapman and Hall/CRC Press.