凸面區域之均勻設計及其在電腦實驗上之應用

Abstract

過去二十年來，均勻設計在所謂的電腦實驗領域已深受矚目。但是在執行電腦實驗時，我們往往會發現有許多“非矩形” 的輸入因子區域，這些區域並不適用於傳統的均勻設計方法。本研究計畫主要可分為兩部分，在第一部分我們首先介紹兩個啟發本研究計畫的複雜網路系統。接下來我們提出一個新的均勻性測量值，此測量值適用於任何“凸面形” 的輸入因子區域且有所謂“無坐標旋轉性” 的特質。為了可以實際應用此方法，我們也提出一個高效率的演算法來建構所謂的“近似均勻設計”，並利用數值結果証明所提演算法非常接近真正的均勻設計之解。利用此新的均勻設計，在第二部分我們提出一個方法來估計電腦實驗的“目標區域”。此方法是有步驟性的，其目的在於使用最少的實驗次數(或成本)並提供適合且具有彈性的模型來描述(或預測)對應目標區域的實驗輸出值。最後，我們將此方法應用在不同的例子上並証明當給定相同的實驗成本時，我們的方法在估計目標區域的準確性上勝過現有的所有方法。

The power of uniform design (UD) has received great attention in the area of computer experiments over the last two decades. However, when conducting a typical computer experiment, one finds many non-rectangular types of input domains on which traditional UD methods can not be adequately applied. The goal of this study is summarized as the following two parts. In the first part, we first introduce two examples from queueing systems that are used to motivate this research. We next construct a rotation-invariant UD solution by introducing a new measure of uniformity for any convex types of input domains. For practical implementation, we also develop an efficient algorithm to construct a so-called nearly uniform design (NUD) and show that it approximates very well the optimal UD solution. In the second part, we develop a methodology for estimating the target region of computer experiments by utilizing the proposed UD method. The methodology is sequential and aims to: (i) provide adaptive models that predict well the output measures related to the experimental target; and (ii) minimize the number of experimental trials. Finally, we illustrate the developed methodology on various examples and show that, given the same experimental budget, it outperforms other approaches in estimating the prespecified target region of computer experiments.