研究探討具空間相關誤差的反應曲面模型下之最少點數之 D 型最適設計。空間相關誤差透過共變異數函數
C(d)=exp(-rd) 依據兩觀測點的距離來描述它們之間的相關性。在一維度的設計空間下，具空間相關誤差的
多項式模型之最少點數 D 型最適設計，必須包含兩個端點，且會對稱於設計區間的中心點。對於線性和
二次迴歸模型我們給出了真實解。至於三階或三階以上的模型，我們給出數值解。在二維設計空間方面，
最少點數 D 型最適設計具備位移、旋轉以及翻轉不變性。此外，數值結果顯示一個在實驗區域為一個圓上的
正三角形設計會是一個最少點數 D 型最適設計。

In this work minimally supported D-optimal designs for response surface models with spatially
correlated errors are studied. The spatially correlated errors describe the correlation between two
measurements depending on their distance d through the covariance function C(d)=exp(-rd). In one
dimensional design
space, the minimally supported D-optimal designs for polynomial models with spatially correlated errors
include two end points and are symmetric to the center of the design region. Exact solutions for simple
linear and quadratic regression models are presented. For models with third or higher order, numerical
solutions are given. While in two dimensional design space, the minimally supported D-optimal designs
are invariant under translation、rotation and reflection. Numerical results show that a regular triangle
on the experimental region of a circle is a minimally supported D-optimal design for the first-order
response surface model.

誌謝i
摘要ii
Abstract iii
1 Introduction 1
2 Minimally Supported D-optimal Designs 2
2.1 D-optimal Designs in one dimensional design space . . . . . . . . . 2
2.2 D-optimal Designs in two dimensional design space . . . . . . . . . 9
2.2.1 Simulated Annealing . . . . . . . . . . . . . . . . . . . . . . 11
3 Conclusion 17
References 18
Appendix 19

1. Atkinson, A. C. and Donev, A. N. (1992). Optimum Experimental Designs,
Oxford University Press, Oxford.
2. Fedorov, V. V. (1972). Theory of optimal experiments, Academic Press, New
York.
3. Muller, W. G. (2007). Collecting Spatial Data: Optimum Designs of Experiments
for Random Fields, 3rd ed. Heidelberg: Springer-Verlag.