Abstract
The fundamental problem in the orthogonal design theory is the design isomorphism, which involves two classes of methods in the statistical literature. One is to identify the isomorphic designs by costly computation, another is only to detect the non-isomorphic designs as a feasible alternative. In this paper we explore the design structure to propose the degree of isomorphism, as a novel criterion showing the similarity between orthogonal designs. A column-wise framework is proposed to accommodate different issues of the design isomorphism, including the detection of non-isomorphism, identification of isomorphism and determination of subclasses for symmetric orthogonal designs. Our framework shows surprisingly high efficiency, where the average time of identifying the isomorphism between two designs in selected classes is all down to about one second. By applying the hierarchical clustering on the average linkage, a novel classification is also presented for non-isomorphic orthogonal designs in a combinatorial view.
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References
Chen JH (1992) Some results on \( 2^{n-k} \) fractional factorial designs and search for minimum aberration designs. Ann Stat 20(4):2124–2141
Chen JH, Lin DKJ (1991) On the identity relationships of \( 2^{k-p} \) designs. J Stat Plan Inference 28(1):95–98
Chen JH, Sun DX, Wu CFJ (1993) A catalogue of two-level and three-level fractional factorial designs with small runs. Int Stat Rev/Revue Internationale de Statistique 61:131–145
Clark JB, Dean AM (2001) Equivalence of fractional factorial designs. Stat Sin 11:537–547
Deng LY, Tang BX (2002) Design selection and classification for hadamard matrices using generalized minimum aberration criteria. Technometrics 44(2):173–184
Draper NR, Mitchell TJ (1967) The construction of saturated \( 2_{R}^{k-p} \) designs. Ann Math Stat pp 1110–1126
Draper NR, Mitchell TJ (1968) Construction of the set of 256-run designs of resolution \( \ge \) 5 and the set of even 512-run designs of resolution \( \ge \) 6 with special reference to the unique saturated designs. Ann Math Stat 39:246–255
Draper NR, Mitchell TJ (1970) Construction of a set of 512-run designs of resolution \( \ge \) 5 and a set of even 1024-run designs of resolution \( \ge \) 6. Ann Math Stat 41(3):876–887
Elsawah AM, Fang KT, Ke X (2021) New recommended designs for screening either qualitative or quantitative factors. Stat Pap 62(1):267–307
Evangelaras H, Koukouvinos C, Lappas E (2006) An efficient algorithm for the identification of isomorphic orthogonal arrays. J Discret Math Sci Cryptogr 9(1):125–132
Evangelaras H, Koukouvinos C, Lappas E (2007) 18-run nonisomorphic three level orthogonal arrays. Metrika 66(1):31–37
Fang KT, Zhang AJ (2004) Minimum aberration majorization in non-isomorphic saturated designs. J Stat Plan Inference 126(1):337–346
Johnson RA, Wichern DW (2002) Applied multivariate statistical analysis, vol 5. Prentice Hall, Upper Saddle River
Katsaounis TI, Dean AM (2008) A survey and evaluation of methods for determination of combinatorial equivalence of factorial designs. J Stat Plan Inference 138(1):245–258
Katsaounis TI, Dean AM, Jones B (2013) On equivalence of fractional factorial designs based on singular value decomposition. J Stat Plan Inference 143(11):1950–1953
Ke X, Fang KT, Elsawah AM et al (2020) New non-isomorphic detection methods for orthogonal designs. Commun Stat Simul Comput. https://doi.org/10.1080/03610918.2020.1844895
Lai JF, Weng LC, Peng XL et al (2021) Construction of symmetric orthogonal designs with deep Q-network and orthogonal complementary design. Comput Stat Data Anal 171:107448
Lam C, Tonchev VD (1996) Classification of affine resolvable 2-(27, 9, 4) designs. J Stat Plan Inference 56(2):187–202
Leon JS (1979) An algorithm for computing the automorphism group of a hadamard matrix. J Comb Theory A 27(3):289–306
Lin CD, Sitter RR (2008) An isomorphism check for two-level fractional factorial designs. J Stat Plan Inference 138(4):1085–1101
Lin CY, Cheng SW (2012) Isomorphism examination based on the count vector. Stat Sin 22:1253–1272
Liu Y, Yang JF, Liu MQ (2011) Isomorphism check in fractional factorial designs via letter interaction pattern matrix. J Stat Plan Inference 141(9):3055–3062
Ma CX, Fang KT, Lin DKJ (2001) On the isomorphism of fractional factorial designs. J Complex 17(1):86–97
Schoen ED, Eendebak PT, Nguyen MV (2010) Complete enumeration of pure-level and mixed-level orthogonal arrays. J Comb Des 18(2):123–140
Sun DX, Li W, Ye KQ (2002) An algorithm for sequentially constructing nonisomorphic orthogonal designs and its applications. Tech. rep., Department of Applied Mathematics and Statistics, State University of New York at Stony Brook, SUNYSB-AMS-01-13
Xu H, Cheng SW, Wu CJ (2004) Optimal projective three-level designs for factor screening and interaction detection. Technometrics 46(3):280–292
Xu HQ, Deng LY (2005) Moment aberration projection for nonregular fractional factorial designs. Technometrics 47(2):121–131
Acknowledgements
The authors thank the Editor, Associate Editor and two referees for their constructive comments leading to significant improvement of this paper, and the UIC Statistics Bayes Cluster (USBC) for its high-performance computing service. This work was partially supported by the UIC Grants (Nos. R201810, R201912 and R202010) and the Zhuhai Premier Discipline Grant.
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Weng, LC., Fang, KT. & Elsawah, A.M. Degree of isomorphism: a novel criterion for identifying and classifying orthogonal designs. Stat Papers 64, 93–116 (2023). https://doi.org/10.1007/s00362-022-01310-2
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DOI: https://doi.org/10.1007/s00362-022-01310-2
Keywords
- Combinatorial classification
- Design isomorphism
- Degree of isomorphism
- Saturated orthogonal design
- Non-saturated orthogonal design