Research Express@NCKU - Articles Digest
Research Express@NCKU Volume 28 Issue 10 - April 10, 2015 [ http://research.ncku.edu.tw/re/articles/e/20150410/2.html ]
Discrete Particle Swarm Optimization for Constructing
Uniform Design on Irregular Regions
Ray-Bing Chen
1,*, Yen-Wen Shu
2, Ying Hung
3and Weichung Wang
41 Department of Statistics, National Cheng Kung University, Tainan 701, Taiwan 2 Department of Mathematics, National Taiwan University, Taipei 10617, Taiwan 3 Department of Statistics and Biostatistics, Rutgers University, NJ 08854, USA
4 Institute of Applied Mathematical Sciences, National Taiwan University, Taipei 10617, Taiwan rbchen@stat.ncku.edu.tw
Computational Statistics and Data Analysis, 72, 282-297.
E
fficient experimental design plays an important role in the study of the scientific problem. Since the experiments may have high-dimensional inputs and be costly, instead of running the experiments over a dense grid of input configurations, uniform design (Fang, 1980) is one of the most widely used approaches, because uniform design possesses a desirable space-filling property in which the design points are placed evenly over the experimental region.In previous literatures, uniform designs are developed mainly for regular experimental regions, such as rectangular or hypercubic regions. However, this regular region assumption is often violated (Ranjan et al., 2008; Hung et al., 2010; Hung, 2011). Recently, Chuang and Hung (2010) proposed central composite discrepancy (CCD) as a new uniformity criterion for irregular regions. The idea of CCD is to divide the K-dimensional experimental region, D, into several sub-regions and then compute both the ratio of the number of design points in each sub-region to the total number of design points and the ratio of the sub-region volume to the overall experimental region volume. In a uniform design, we would expect these two ratios to be close. Thus we would use this CCD criterion as our design criterion.
In this article, the discrete version of the CCD is used, and the candidate design points are limited to a set of grid points in the experimental region. Thus the objective can be formulated as solving an optimization problem,
minP∈Z(n,qK) CCD(P),
where Z(n, qK) is the set of all possible discrete designs with n design points. Here we focus on how to efficiently generate the CCD-based uniform design, because the computational challenges in optimal CCD design search and uniformity evaluations remain unsolved, especially for larger dimensionality and run sizes.
Particle swarm optimization (PSO) algorithm is a stochastic population-based heuristic that inherits the efficiency and capability of PSO for solving high-dimensional optimization problems with multiple optima. Here we propose a discrete version PSO (DPSO) for the CCD uniform design search problem. The complexity analysis of the discrete CCD computation is also given. The analysis result suggests that the computational cost increases rapidly. Parallel computing techniques based on the latest graphic processing unit (GPU) are thus applied to significantly accelerate the CCD function evaluations. The proposed DPSO algorithm is implemented and numerical results are compared with existing methods for two and three dimensional irregular experimental regions. Results on higher dimension regions are also provided to demonstrate the capability of the proposed algorithms. Finally the DPSO
Research Express@NCKU - Articles Digest
algorithm is used to solve a data center thermal management problem (Hung et al., 2010; Hung, 2011). This example shows how the proposed algorithms can easily be extended to handle disconnected experimental regions with additional constraints.
Reference:
1. Chuang, S.C., Hung, Y.C., 2010. Uniform design over general input domains with applications to target region estimation in computer experiments. Computational Statistics & Data Analysis 54, 219–232 2. Fang, K.-T., 1980. The uniform design: application of number-theoretic methods in experimental design.
Acta Mathematical Application Sinica 3, 363–372.
3. Hung, Y., 2011. Adaptive probability-based latin hypercube designs. Journal of the American Statistical Association 106 (493), 213–219.
4. Hung, Y., Amemiya, Y., Wu, C.F.J., 2010. Probability-based latin hypercube designs for slid-rectangular regions. Biometrika 97 (4), 961–968.
5. Ranjan, P., Bingham, D., Michailidis, G., 2008. Sequential experiment design for contour estimation from complex computer codes. Technometrics 50 (4), 527–541.
