Nearly Uniform Design Construction on Flexible Region
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Acta Mathemacae Applicatae Sinica, English Series The Editorial Office of AMAS & Springer-Verlag GmbH Germany 2020
Nearly Uniform Design Construction on Flexible Region Jian-hui NING1 , Wen-wen YIN1 , Li PENG2,3,† 1 School 2 School
of Mathematics and Statistics, Center China Normal University, Wuhan 430079, China of statistics, Renmin University of China, Beijing 100872, China
3 Department
of Statistics, General Administration of Customs of the People’s Republic of China, Beijing 100730,
China († E-mail: [email protected])
Abstract
In this paper, we deduced an iteration formula for the computation of central composite discrepancy.
By using the iteration formula, the computational complexity of uniform design construction in flexible region can be greatly reduced. And we also made a refinement to threshold accepting algorithm to accelerate the algorithm’s convergence rate. Examples show that the refined algorithm can converge to the lower discrepancy design more stably. Keywords
uniform design; flexible region; central composite discrepancy; threshold accepting algorithm
2000 MR Subject Classification
1
62K15; 62K10
Introduction
As a kind of space filling design, uniform design can be used for computer experimental design and industry design without model assumption. Since proposed by [8] and [20], the uniform design was popularly used, and there are a lots of publications studied its theory and application (see [7, 21, 22]). However, most of these studies are based on the standard settings, in which the shape of the target experimental region is hypercube C s = [0, 1]s . Most of research about uniform design, including the criteria for measuring the design’s uniformity and corresponding design construction approaches, are all based on the regular cuboidal shape. A comprehensive description on it can refer to [5]. However, in many practical cases, aforementioned standard setting might not provide a good approximation to the real experimental region, such as the 16component waste glass mixture problem introduced by [1] and data center thermal management problem introduced by [11, 12]. As mentioned by [14, 17], simply project or transform the uniform design from hypercube to the irregular target region may not maintain the uniformity well. As far as we know, there are no projection or transformation can promise the uniformity. It’s important to find some methods to construct the uniform or nearly uniform design on irregular shaped regions directly. There are some different irregular shape regions have been studied by statisticians, such as the spherical[7] and simplex regions[1, 12, 16, 17] . Another more flexible and complex region was proposed by [3]. They developed a flexible region approach, which can be viewed as a combination of spherical and cuboidal. The flexible region in dimension s can be defined as {(x1 , · · · , xs ) : |x1 |m + · · · + |xs |m ≤ 1}. By adjusting the parameter m, it is possible to obtain an infinite variety of intermediate symmetrical shapes. For the convenience of theore
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