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Keywords:
correlation; D-efficiency; D-optimal chemical balance weighing design; Hadamard matrix; simulated annealing algorithm; tabu search
correlation; D-efficiency; D-optimal chemical balance weighing design; Hadamard matrix; simulated annealing algorithm; tabu search
Summary:
In this paper we consider D-optimal and highly D-efficient chemical balance weighing designs. The errors are assumed to be equally non-negatively correlated and to have equal variances. Some necessary and sufficient conditions under which a design is D*-optimal design (regular D-optimal design) are proved. It is also shown that in many cases D*-optimal design does not exist. In many of those cases the designs constructed by Masaro and Wong (2008) and some new designs are shown to be highly D-efficient. Theoretical results are accompanied by numerical search, suggesting D-optimality of designs under consideration.
References:
[1] Angelis, L., Bora-Senta, E., Moyssiadis, C.: Optimal exact experimental designs with correlated errors through a simulated annealing algorithm. Comput. Statist. Data Anal. 37 (2001), 275-296. DOI 10.1016/s0167-9473(01)00011-1 | MR 1862514 | Zbl 0990.62061
[2] Banerjee, K. S.: Weighing Designs for Chemistry, Medicine, Economics, Operations Research, Statistics. Marcel Dekker Inc., New York 1975. MR 0458751 | Zbl 0334.62030
[3] Bora-Senta, E., Moyssiadis, C.: An algorithm for finding exact D- and A-optimal designs with $n$ observations and $k$ two-level factors in the presence of autocorrelated errors. J. Combinat. Math. Combinat. Comput. 30 (1999), 149-170. MR 1705339 | Zbl 0937.62074
[4] Bulutoglu, D. A., Ryan, K. J.: D-optimal and near D-optimal $2^k$ fractional factorial designs of resolution $V$. J. Statist. Plann. Inference 139 (2009), 16-22. DOI 10.1016/j.jspi.2008.05.012 | MR 2460547 | Zbl 1284.62473
[5] Ceranka, B., Graczyk, M.: Optimal chemical balance weighing designs for $v+1$ objects. Kybernetika 39 (2003), 333-340. MR 1995737 | Zbl 1248.62128
[6] Ceranka, B., Graczyk, M.: Robustness optimal spring balance weighing designs for estimation total weight. Kybernetika 47 (2011), 902-908. MR 2907850 | Zbl 1274.62492
[7] Ceranka, B., Graczyk, M., Katulska, K.: A-optimal chemical balance weighing design with nonhomogeneity of variances of errors. Statist. Probab. Lett. 76 (2006), 653-665. DOI 10.1016/j.spl.2005.09.012 | MR 2234783 | Zbl 1090.62074
[8] Ceranka, B., Graczyk, M., Katulska, K.: On certain A-optimal chemical balance weighing design. Comput. Statist. Data Analysis 51 (2007), 5821-5827. DOI 10.1016/j.csda.2006.10.021 | MR 2407680
[9] Cheng, C. S.: Optimal biased weighing designs and two-level main-effect plans. J. Statist. Theory Practice 8 (2014), 83-99. DOI 10.1080/15598608.2014.840520 | MR 3196641
[10] Domijan, K.: tabuSearch: R based tabu search algorithm. R package version 1.1. \url{ http://CRAN.R-project.org/package=tabuSearch} (2012)
[11] Ehlich, H.: Determinantenabschätzungen für binäre Matrizen. Math. Zeitschrift 83 (1964), 123-132. DOI 10.1007/bf01111249 | MR 0160792 | Zbl 0115.24704
[12] Ehlich, H.: Determinantenabschätzungen für binäre Matrizen mit $n\equiv 3 \mathrm{mod} 4$. Math. Zeitschrift 84 (1964), 438-447. DOI 10.1007/bf01109911 | MR 0168573
[13] Galil, Z., Kiefer, J.: D-optimum weighing designs. Ann. Statist. 8 (1980), 1293-1306. DOI 10.1214/aos/1176345202 | MR 0594646 | Zbl 0598.62087
[14] Graczyk, M.: A-optimal biased spring balance weighing design. Kybernetika 47 (2011), 893-901. MR 2907849 | Zbl 1274.62495
[15] Graczyk, M.: Some applications of weighing designs. Biometr. Lett. 50 (2013), 15-26. DOI 10.2478/bile-2013-0014
[16] Harman, R., Bachratá, A., Filová, L.: Construction of efficient experimental designs under multiple resource constraints. Appl. Stochast. Models in Business and Industry 32 (2015), 1, 3-17. DOI 10.1002/asmb.2117 | MR 3460885
[17] Jacroux, M., Wong, C.S., Masaro, J.C.: On the optimality of chemical balance weighing designs. J. Statist. Planning Inference 8 (1983), 231-240. DOI 10.1016/0378-3758(83)90041-1 | MR 0720154 | Zbl 0531.62072
[18] Jenkins, G. M., Chanmugam, J.: The estimation of slope when the errors are autocorrelated. J. Royal Statist. Soc., Ser. B (Statistical Methodology) 24 (1962), 199-214. MR 0138154 | Zbl 0116.11401
[19] Jung, J. S., Yum, B. J.: Construction of exact D-optimal designs by tabu search. Comput. Statist. Data Analysis 21 (1996), 181-191. DOI 10.1016/0167-9473(95)00014-3 | MR 1394535 | Zbl 0900.62403
[20] Katulska, K., Smaga, Ł.: D-optimal chemical balance weighing designs with $n\equiv 0 (\text{mod} 4)$ and $3$ objects. Comm. Statist. - Theory and Methods 41 (2012), 2445-2455. DOI 10.1080/03610926.2011.608587 | MR 3003795 | Zbl 1271.62175
[21] Katulska, K., Smaga, Ł.: D-optimal chemical balance weighing designs with autoregressive errors. Metrika 76 (2013), 393-407. DOI 10.1007/s00184-012-0394-8 | MR 3041462
[22] Katulska, K., Smaga, Ł.: A note on D-optimal chemical balance weighing designs and their applications. Colloquium Biometricum 43 (2013), 37-45.
[23] Katulska, K., Smaga, Ł.: On highly D-efficient designs with non-negatively correlated observations. REVSTAT - Statist. J. (accepted).
[24] Li, C. H., Yang, S. Y.: On a conjecture in D-optimal designs with $n\equiv 0 (\mathrm{mod} 4)$. Linear Algebra Appl. 400 (2005), 279-290. DOI 10.1016/j.laa.2004.11.020 | MR 2132491
[25] Masaro, J., Wong, C. S.: D-optimal designs for correlated random vectors. J. Statist. Planning Inference 138 (2008), 4093-4106. DOI 10.1016/j.jspi.2008.03.012 | MR 2455990 | Zbl 1147.62062
[26] Neubauer, M. G., Pace, R. G.: D-optimal $(0,1)$-weighing designs for eight objects. Linear Algebra Appl. 432 (2010), 2634-2657. DOI 10.1016/j.laa.2009.12.007 | MR 2608182 | Zbl 1185.62134
[27] Pukelsheim, F.: Optimal Design of Experiments. John Wiley and Sons Inc., New York 1993. MR 1211416 | Zbl 1101.62063
[28] Team, R Core: R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. URL http://www.R-project.org/ (2015).
[29] Smaga, Ł.: D-optimal Chemical Balance Weighing Designs with Various Forms of the Covariance Matrix of Random Errors. Ph.D. Thesis, Adam Mickiewicz University, 2013 (in polish).
[30] Smaga, Ł.: Necessary and sufficient conditions in the problem of D-optimal weighing designs with autocorrelated errors. Statist. Probab. Lett. 92 (2014), 12-16. DOI 10.1016/j.spl.2014.04.027 | MR 3230466
[31] Smaga, Ł.: Uniquely E-optimal designs with $n\equiv 2 (\mathrm{mod} 4)$ correlated observations. Linear Algebra Appl. 473 (2015), 297-315. DOI 10.1016/j.laa.2014.08.022 | MR 3338337
[32] Wojtas, M.: On Hadamard's inequality for the determinants of order non-divisible by $4$. Colloquium Mathematicum 12 (1964), 73-83. DOI 10.4064/cm-12-1-73-83 | MR 0168574 | Zbl 0126.02604
[33] Yang, C. H.: On designs of maximal $(+1,-1)$-matrices of order $n\equiv 2 (\text{mod} 4)$. Math. Computat. 22 (1968), 174-180. DOI 10.1090/s0025-5718-1968-0225476-4 | MR 0225476 | Zbl 0167.01703
[34] Yeh, H. G., Huang, M. N. Lo: On exact D-optimal designs with $2$ two-level factors and $n$ autocorrelated observations. Metrika 61 (2005), 261-275. DOI 10.1007/s001840400336 | MR 2230375 | Zbl 1079.62078
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