DOI QR코드

DOI QR Code

하이브리드형 제약 외삽실험 계획법

Hybrid Constrained Extrapolation Experimental Design

  • 투고 : 20110800
  • 심사 : 20111000
  • 발행 : 2012.01.30

초록

실험영역을 벗어난 점에서의 분산을 최소화 하는 외삽-최적의 문제는 모형의 불확실성을 내재하고 있다. 즉, 실험영역을 벗어나게 되면 모형의 불확실성이 높아지므로 모형의 타당성 여부를 진단하여야 한다. 본 연구에서는 외삽(extrapolation) 문제하에서 기본적인 3가지 실험, 즉 제약 실험과 최대최소 실험, 그리고 복합실험 등을 융합한 새로운 제약조건형의 실험 2가지를 제안하였다. 그리고 실험영역의 바깥에 위치한 점의 위치가 실험영역을 얼마나 벗어나느냐에 따른 실험결과에 대한 영향력도 고려하여 보았다. 문제의 특성상 유전 알고리즘을 이용하여 해를 구하였다.

In setting an experimental design for the prediction outside the experimental region (extrapolation design), it is natural for the experimenter to be very careful about the validity of the model for the design because the experimenter is not certain whether the model can be extended beyond the design region or not. In this paper, a hybrid constrained type approach was adopted in dealing model uncertainty as well as the prediction error using the three basic principles available in literature, maxi-min, constrained, and compound design. Furthermore, the effect of the distance of the extrapolation design point from the design region is investigated. A search algorithm was used because the classical exchange algorithm was found to be complex due to the characteristic of the problem.

키워드

참고문헌

  1. 강명욱, 김영일 (2002). Multiple constrained optimal experimental design, The Korean Communications in Statistics, 9, 619-627. https://doi.org/10.5351/CKSS.2002.9.3.619
  2. 김영일, 임용빈 (2007). Hybrid approach multiple objective experimental design, The Korean Communications in Statistics, 9, 619-627.
  3. Atwood, C. L. (1969). Optimal and efficient designs of experiments, The Annals of Mathematical Statistics, 40, 1570-1602. https://doi.org/10.1214/aoms/1177697374
  4. Box, G. E. P. and Draper, N. R. (1975). Robust design, Biometrika, 62, 347-352. https://doi.org/10.1093/biomet/62.2.347
  5. Cook, R. D. and Fedorov, V. V. (1995). Constrained optimization of experimental design with discussion, Statistics, 26, 129-178. https://doi.org/10.1080/02331889508802474
  6. Cook, R. D. and Nachtsheim, C. J. (1982). Model-robust, linear optimal designs, Technometrics, 24, 49-54. https://doi.org/10.2307/1267577
  7. Cook, R. D. and Wong, W. K. (1994). On the equivalence between constrained and compound optimal designs, Journal of the American Statistical Association, 89, 687-692. https://doi.org/10.2307/2290872
  8. Dette, H. and Huang, M. (2000). Convex optimal designs for compound polynomial extrapolation, Annals of the Institute of Statistical Mathematics, 52, 557-573. https://doi.org/10.1023/A:1004185822838
  9. Dette, H. and Wong, W. K. (1996). Robust optimal extrapolation designs, Biometrika, 83, 667-680. https://doi.org/10.1093/biomet/83.3.667
  10. Fedorov, V. V. (1972). Theory of Optimal Experiments, Academic Press, Inc, New York.
  11. Hoel, P. G. and Levine, A. (1964). Optimal spacing and weighting in polynomial prediction, Annals of Statistics, 35, 1553-1560. https://doi.org/10.1214/aoms/1177700379
  12. Huang, M. L. and Studden, W. J. (1988). Model robust extrapolation designs, Journal of Statistical Planning and Inference, 18, 1-24. https://doi.org/10.1016/0378-3758(88)90012-2
  13. Huang, Y. C. andWong,W. K. (1998). Multiple-objective designs, Journal of Biopharmaceutical Statistics, 8, 635-643. https://doi.org/10.1080/10543409808835265
  14. Imhof, L. and Wong, W. K. (2000). A graphical method for finding maximin designs, Biometrics, 56, 113-117 https://doi.org/10.1111/j.0006-341X.2000.00113.x
  15. Kiefer, J. and Wolfowitz, J. (1960). The equivalence of two extremum problems, Canadian Journal of Mathematics, 12, 363-366. https://doi.org/10.4153/CJM-1960-030-4
  16. Kiefer, J. and Wolfowitz, J. (1964). Optimum extrapolation and interpolation designs, Annals of the Institute of Statistical Mathematics, 16, 79-108. https://doi.org/10.1007/BF02868564
  17. Kussmaul, K. (1969). Protection against assuming the wrong degree in polynomial regression, Technometrics, 11, 677-682. https://doi.org/10.2307/1266891
  18. Lauter, E. (1974). Experimental planning in a class of models, Mathematishe Operationsforshung und Statistik, 5, 673-708.
  19. Lee, C. M. S. (1987). Constrained optimal designs for regression models, Communications in Statistics, Part A-theory and Methods, 16, 765-783. https://doi.org/10.1080/03610928708829401
  20. Park, Y. J., Montgomery, D. C., Folwer, J. W. and Borror, C. M. (2005). Cost- constrained G-efficient response surface designs for cuboidal regions, Quality and Reliability Engineering International, 22, 121-139. https://doi.org/10.1002/qre.690
  21. Stigler, S. M. (1971). Optimal experimental design for polynomial regression, Journal of the American Statistical Association, 66, 311-318. https://doi.org/10.2307/2283928
  22. Studden, W. J. (1971). Optimal designs for multivariate polynomial extrapolation, The Annals of Mathematical Statistics, 42, 828-832. https://doi.org/10.1214/aoms/1177693442
  23. Wong,W. K. (1992). A unified approach to the construction of minimax designs, Biometrika, 79, 611-619. https://doi.org/10.1093/biomet/79.3.611
  24. Wong, W. K. (1995). A graphical approach for the construction of constrained D and L-optimal designs using efficiency plots, Journal of Statistical Computation and Simulation, 53, 143-152. https://doi.org/10.1080/00949659508811702
  25. Yum, J. K. and Nam, K. S. (2000). A study on D-optimal design using the genetic algorithm, The Korean Communications in Statistics, 7, 357-366.

피인용 문헌

  1. Robust Extrapolation Design Criteria under the Uncertainty of Model and Error Structure vol.28, pp.3, 2015, https://doi.org/10.5351/KJAS.2015.28.3.561
  2. Some Criteria for Optimal Experimental Design at Multiple Extrapolation Points vol.27, pp.5, 2014, https://doi.org/10.5351/KJAS.2014.27.5.693