空間統計廣泛應用於地質、大氣、水文、生態等相關領域的資料分析。而研究區域的空間預測和解釋變數的選擇都是空間統計學中的重要研究議題。為了瞭解固定效應項，研究者需要收集足夠的解釋變數並選擇適當的解釋變數來做後續的建模與空間預測。在實際資料分析中，解釋變數收集不易且常有收集不完全的情況，且收集多個解釋變數通常需花費大量的人力與成本。本篇論文使用一個以抽樣位置所決定的基底函數集合來取代解釋變數的角色，省去收集解釋變數的人力與成本，同時也可以估計固定效應項的趨勢，進而得到準確的空間預測曲面。從模擬實驗結果可知，我們所提的方法有不錯的預測表現；最後，本篇論文也應用所提的方法去分析孟加拉地下水的數據，並得到砷汙染在孟加拉地區的汙染濃度預測曲面，說明我們所提方法的實用性與有效性。

Spatial statistics is widely used in geology, atmosphere, hydrology, ecology and other related fields. In spatial statistics, spatial prediction and selection of appropriate covariates both are important issues. To understand the fixed effect clearly, researchers generally need to collect enough covariates and select a suitable subset of covariates based on some selection criteria. Then, the corresponding spatial predicted surface can be obtained. In practice, collecting covariates is difficult and it is often incomplete. In this thesis, a class of basis functions only determined by the sampling locations is applied to replace the possible covariates. In other words, we do not need any covariates and it saves the manpower and the cost of collecting covariates. Combining a selection criterion to select the number of basis functions, the trend of fixed effect can be estimated and the consequent spatial predicted surface also can be obtained. From simulation studies, we can see that our proposed method has better performance than the conventional methods under various situations. Finally, we illustrate the utility of the proposed method by a real data application concerning the groundwater data in Bangladesh.

Altman, N. (2000), “Krige, Smooth, Both or Neither?” (with discussion), Aus tralian & New Zealand Journal of Statistics, 42, 441-461.
Akaike, H. (1973), “Information Theory and an Extension of the Maximum Likelihood Principle,” in Proceedings of the Second International Symposium on Information Theory, eds. Petro, B. and Csa ́ki, F., Budapest: Akade ́miai Kiado ́, pp. 267–281.
Bookstein, F. L. (1989), “Principal Warps: Thin Plate Splines and the Decomposition of Deformations,” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 11, No. 6, 567–585.
Bradley, J. R., Cressie, N., and Shi, T. (2012), “Local Spatial-Predictor Selection,” Proceedings of the Joint Statistical Meetings, Alexandria, VA. Section on Statistics and the Environment: 3098 - 3110.
Bradley, J. R., Cressie, N., and Shi, T. (2011), “Selection of Rank and Basis Functions in the Spatial Random Effects Model,” in Proceedings of the Joint Statistical Meetings, 3393–3406. Alexandria, VA: American Statistical Association.
Burnham, K. P. and Anderson, D. R. (2002), Model Selection and Multimodel Inference: A Practical Informationtheoretic Approach (2nd ed.), New York: Springer.
Chen, C. S. and Chen, C. S. (2018), “A Composite Spatial Predictor via Local Criteria under a Misspecified Model,” Stoch Environ Res Risk Assess, 32, 341–355.
Chen, C. S. and Huang, H. C. (2011), “Geostatistical Model Averaging Based on Conditional Information Criteria,” Environmental and Ecological Statistics, 19, 23–35.
Craven, P. and Wahba, G. (1979), “Smoothing Noisy Data with Spline Functions: Estimating the Correct Degree of Smoothing by the Method of Generalized Cross-Validation,” Numerische Mathematik, 31, 377–403.
Cressie, N. (1993), Statistics for Spatial Data, Revised edition: Wiley, New York.
Cressie, N. and Lahiri, S. N. (1996), “Asymptotics for REML Estimation of Spatial Covariance Parameters,” Journal of Statistical Planningand Inference, 50, 327–341.
Cressie, N., and Johannesson, G. (2008), “Fixed RankKriging for very Large Spatial Data Sets,” Journal of the Royal Statistical Society, Series B, 70, 209–226.
Duchon, J. (1976), “Splines Minimizing Rotation Invariant Semi-norms in Sobolev Spaces,” Constructive theory of functions of several variables, Springer-Verlag, Berlin, pp 85–100.
Furrer, R., Genton, M. G. and Nychka, D. (2006), “Covariance Tapering for Interpolation of Large Spatial Datasets,” J Comput Graph Stat, 15, 502–523.
Green, P. J. and Silverman, B. W. (1993), Nonparametric Regression and Generalized LinearModels: A Roughness Penalty Approach, BocaRaton, FL: CRC Press.
Huang, H. C. and Chen, C. S. (2007), “Optimal Geostatistical Model Selection,” Journal of the American Statistical Association, 102, 1009–1024.
Hurvich, C. M. and Tsai, C. L. (1989), “Regression and Time Series Model Selection in Small Samples,” Biometrika, 76, 297–307.
Hutchinson, M. F. and de Hoog, F. R. (1985), “Smoothing Noisy Data with Spline Functions,” Numerische Mathematik, 47, 99–106.
Laslett, G. M. (1994), “Kriging and Splines: An Empirical Comparison of Their Predictive Performance in Some Applications,” Journal of the American Statistical Association, 89, 391-409.
Kaufman, C. G., Schervish, M. J. and Nychka, D. W. (2008), “Covariance Tapering for Likelihood-Based Estimation in Large Spatial Data Sets,” J Am Stat Assoc, 103, 1545–1555.
Kinniburgh, D. and Smedley, P. (2001), “Arsenic Contamination of Groundwater in Bangladesh,” Technical report, British Geological Survey.
Mate ́rn, B. (1986), Spatial Variation (2nd ed.), New York: Springer.
McGilchrist, C. A. (1989), “Bias of ML and REML Estimators in Regression Models with ARMA Errors,” Journal of Statistical Computation and Simu lation, 32, 127-136.
Micchelli, C. A. (1986), “Interpolation of Scattered Data: DistanceMatrices and Conditionally Positive Definite Functions,” Constructive Approximation, 2, 11–22.
Patterson, H. D and Thompson, R. (1971), “Recovery of Inter-Block Information when Block Sizes Are Unequal,” Biometrika, 58:545–554.
Roy, P. K. and Hossain, S. S. (2014), “Predicting Arsenic Concentration in Groundwater of Bangladesh Using Bayesian Geostatistical Model,” Environmental and Ecological Statistics, 21, 583–597.
Schabenberger, O. and Gotway, C. A. (2005), Statistical Methods for Spatial Data Analysis, Boca Raton, FL: Chapman & Hall/CRC.
Schwarz, G. (1978), “Estimating the Dimension of a Model,” The Annals of Statistics, 6,461-464.
Tzeng, S. and Huang, H. C. (2018), “Resolution Adaptive Fixed Rank Kriging,” Technometrics, 60, 198-208.
Vaida, F. and Blanchard, S. (2005), “Conditional Akaike Information for Mixed-Effects Models,” Biometrika, 92, 351 – 370.
Wahba, G. (1990), Spline Models for Observational Data, Philadelphia: Society for Industrial and Applied Mathematics.
Wahba, G., and Wendelberger, J. (1980), “Some New Mathematical Methods for Variational Objective Analysis using Splines and Cross Validation,” MonthlyWeather Review, 108, 1122–1143.
Wang, Y (1998), “Smoothing Spline Models with Correlated Random Errors,” Journal of the American Statistical Association, 93, 341-348.