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T-optimal designs for multi-factor polynomial regression models via a semidefinite relaxation method. (English) Zbl 1430.62176
Summary: We consider T-optimal experiment design problems for discriminating multi-factor polynomial regression models where the design space is defined by polynomial inequalities and the regression parameters are constrained to given convex sets. Our proposed optimality criterion is formulated as a convex optimization problem with a moment cone constraint. When the regression models have one factor, an exact semidefinite representation of the moment cone constraint can be applied to obtain an equivalent semidefinite program. When there are two or more factors in the models, we apply a moment relaxation technique and approximate the moment cone constraint by a hierarchy of semidefinite-representable outer approximations. When the relaxation hierarchy converges, an optimal discrimination design can be recovered from the optimal moment matrix, and its optimality can be additionally confirmed by an equivalence theorem. The methodology is illustrated with several examples.
62K05 Optimal statistical designs
62J02 General nonlinear regression
90C22 Semidefinite programming
Full Text: DOI
[1] ApS, M: The MOSEK optimization toolbox for MATLAB manual. Version 8.1. http://docs.mosek.com/8.1/toolbox/index.html. (2017)
[2] Atkinson, AC, Optimum experimental designs for choosing between competitive and non competitive models of enzyme inhibition, Commun. Stat. Theory Methods, 41, 2283-2296, (2012) · Zbl 1271.62168
[3] Atkinson, A.; Cox, DR, Planning experiments for discriminating between models, J. R. Stat. Soc. Ser. B (Methodol.), 36, 321-348, (1974) · Zbl 0291.62095
[4] Atkinson, AC; Fedorov, V., The design of experiments for discriminating between two rival models, Biometrika, 62, 57-70, (1975) · Zbl 0308.62071
[5] Atkinson, AC; Fedorov, VV, The design of experiments for discriminating between several rival models, Biometrika, 62, 289-303, (1975) · Zbl 0321.62085
[6] Bisschop, J: AIMMS optimization modeling. Lulu.com (2006)
[7] Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004) · Zbl 1058.90049
[8] Carlos Monteiro Ponce de Leon, A.: Optimum experimental design for model discrimination and generalized linear models. Ph.D. thesis, London School of Economics and Political Science (United Kingdom) (1993)
[9] De Castro, Y., Gamboa, F., Henrion, D., Hess, R., Lasserre, J.B.: D-optimal design for multivariate polynomial regression via the christoffel function and semidefinite relaxations. (2017) arXiv preprint arXiv:170301777
[10] Leon, AP; Atkinson, AC, Optimum experimental design for discriminating between two rival models in the presence of prior information, Biometrika, 78, 601-608, (1991) · Zbl 0741.62073
[11] Dette, H., Discrimination designs for polynomial regression on compact intervals, Ann. Stat., 22, 890-903, (1994) · Zbl 0806.62059
[12] Dette, H.; Melas, VB, Optimal designs for estimating individual coefficients in fourier regression models, Ann. Stat., 31, 1669-1692, (2003) · Zbl 1046.62076
[13] Dette, H.; Titoff, S., Optimal discrimination designs, Ann. Stat., 37, 2056-2082, (2009) · Zbl 1168.62066
[14] Dette, H.; Melas, VB; Shpilev, P., T-optimal designs for discrimination between two polynomial models, Ann. Stat., 40, 188-205, (2012) · Zbl 1246.62176
[15] Dette, H., Guchenko, R., Melas, V., Wong, W.K.: Optimal discrimination designs for semi-parametric models. Biometrika 105(1), 185-197 (2018) · Zbl 07072400
[16] Diamond, S.; Boyd, S., Cvxpy: a python-embedded modeling language for convex optimization, J. Mach. Learn. Res., 17, 2909-2913, (2016) · Zbl 1360.90008
[17] Duarte, BP; Wong, WK; Atkinson, AC, A semi-infinite programming based algorithm for determining t-optimum designs for model discrimination, J. Multivar. Anal., 135, 11-24, (2015) · Zbl 1314.62164
[18] Duarte, BP; Wong, WK; Dette, H., Adaptive grid semidefinite programming for finding optimal designs, Stat. Comput., 28, 441-460, (2018) · Zbl 06844376
[19] Fedorov, VV, The design of experiments in the multiresponse case, Theory Probab. Appl., 16, 323-332, (1971) · Zbl 0255.62060
[20] Fedorov, V.: Theory of Optimal Experiments. Elsevier, New York (1972)
[21] Fedorov, V.V., Hackl, P.: Model-Oriented Design of Experiments, vol. 125. Springer, Berlin (2012) · Zbl 0878.62052
[22] Fedorov, V.V., Leonov, S.L.: Optimal Design for Nonlinear Response Models. CRC Press, Boca Raton (2013) · Zbl 1373.62001
[23] Fedorov, VV; Malyutov, MB, Optimal designs in regression problems, Math Operationsforsch Statist, 3, 281-308, (1972) · Zbl 0258.62044
[24] Gahinet, P., Nemirovskii, A., Laub, A.J., Chilali, M.: The lmi control toolbox. Decision and Control, 1994. In: Proceedings of the 33rd IEEE Conference on Decision and Control, vol. 3, pp. 2038-2041 (1994)
[25] Goh, J.; Sim, M., Robust optimization made easy with rome, Oper. Res., 59, 973-985, (2011) · Zbl 1235.90107
[26] Grant, M., Boyd, S., Ye, Y.: Cvx: Matlab software for disciplined convex programming (2008)
[27] Henrion, D., Lasserre, J.B.: Detecting global optimality and extracting solutions in gloptipoly 312, 293-310 (2005) · Zbl 1119.93301
[28] Henrion, D.; Lasserre, JB; Löfberg, J., Gloptipoly 3: moments, optimization and semidefinite programming, Optim. Methods Softw., 24, 761-779, (2009) · Zbl 1178.90277
[29] Hess, R: Some approximation schemes in polynomial optimization. Ph.D. thesis, Université de Toulouse, Université Toulouse III-Paul Sabatier (2017)
[30] Karlin, S., Studden, W.: Tchebycheff systems: with applications in analysis and statistics, Interscience, New York, vol. 15. Interscience Publishers (1966) · Zbl 0153.38902
[31] Lasserre, JB, Global optimization with polynomials and the problem of moments, SIAM J. Optim., 11, 796-817, (2001) · Zbl 1010.90061
[32] Lasserre, J.B.: Moments, Positive Polynomials and Their Applications, vol. 1. World Scientific, Singapore (2009)
[33] Lasserre, J.B.: An Introduction to Polynomial and Semi-algebraic Optimization, vol. 52. Cambridge University Press, Cambridge (2015) · Zbl 1320.90003
[34] Lofberg, J: Yalmip: A toolbox for modeling and optimization in matlab. In: 2004 IEEE International Conference on Robotics and Automation, pp. 284-289 (2004)
[35] López-Fidalgo, J.; Tommasi, C.; Trandafir, P., An optimal experimental design criterion for discriminating between non-normal models, J. R. Stat. Soc. Ser. B (Stat. Methodol.), 69, 231-242, (2007) · Zbl 1123.62056
[36] Nie, J., The \({{\cal{A}}}\)-truncated k-moment problem, Found. Comput. Math., 14, 1243-1276, (2014) · Zbl 1331.65172
[37] Rockafellar, R.: Convex Analysis. Princeton University Press, Princeton (1970) · Zbl 0193.18401
[38] Scheiderer, C., Spectrahedral shadows, SIAM J. Appl. Algebra Geom., 2, 26-44, (2018) · Zbl 1391.90462
[39] Shohat, J.A., Tamarkin, J.D.: The Problem of Moments, vol. 1. American Mathematical Society, Providence (1943) · Zbl 0063.06973
[40] Sturm, JF, Using sedumi 1.02, a matlab toolbox for optimization over symmetric cones, Optim. Methods Softw., 11, 625-653, (1999) · Zbl 0973.90526
[41] Toh, KC; Todd, MJ; Tütüncü, RH, Sdpt3—a matlab software package for semidefinite programming, version 1.3, Optim. Methods Softw., 11, 545-581, (1999) · Zbl 0997.90060
[42] Uciński, D.: Optimal Measurement Methods for Distributed Parameter System Identification. CRC Press, Boca Raton (2004) · Zbl 1155.93003
[43] Uciński, D.; Bogacka, B., T-optimum designs for discrimination between two multiresponse dynamic models, J. R. Stat. Soc. Ser. B (Stat. Methodol.), 67, 3-18, (2005) · Zbl 1060.62084
[44] Waterhouse, T.; Eccleston, J.; Duffull, S., Optimal design criteria for discrimination and estimation in nonlinear models, J. Biopharm. Stat., 19, 386-402, (2009)
[45] Wiens, DP, Robust discrimination designs, J. R. Stat. Soc. Ser. B (Stat. Methodol.), 71, 805-829, (2009) · Zbl 1248.62132
[46] Wong, WK; Chen, RB; Huang, CC; Wang, W., A modified particle swarm optimization technique for finding optimal designs for mixture models, PLoS ONE, 10, e0124720, (2015)
[47] Wynn, HP, The sequential generation of d-optimum experimental designs, Ann. Math. Stat., 41, 1655-1664, (1970) · Zbl 0224.62038
[48] Yang, M.; Biedermann, S.; Tang, E., On optimal designs for nonlinear models: a general and efficient algorithm, J. Am. Stat. Assoc., 108, 1411-1420, (2013) · Zbl 1283.62161
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