Abstract
Design of Experiments (DoE) is essential for data-driven models. It is applied to create uniformly distributed designs for the model’s input space. For good uniformity, the point density of the generated design shall be kept roughly constant, i.e., too close points and too big data holes shall be avoided. The generation of space-filling designs, even for the unconstrained design space, is challenging, especially with increasing dimensionality and design size. These difficulties increase in the presence of constraints that occurs in many engineering applications. Equality constraints are one group, i.e., mixture or volume constraints. Specialized approaches are required for such constraints. A point-distance-based optimization is not suitable. The computational effort for large designs is too high, and for higher-dimensional designs, too many points are placed close to the boundary of the design space. The proposed approach, instead, is based on the projection of uniformly distributed designs onto the constraint. Thus, this work presents the idea of using maximin Latin Hypercubes (LH) with their good space-filling properties and projecting them onto the constraint. The resulting design might be non-uniformly distributed so that large data holes could occur. To fill these holes, this projection of an LH onto the constraint is repeated for the remaining inputs by going over all inputs. This method is analyzed in a first two-dimensional case study. In a second case study, it is applied to a nonlinear constraint. A design has to be placed on a spherical surface. The results are compared to an existing state-of-the-art method.
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References
Pronzato, L., Müller, W.G.: Design of computer experiments: space filling and beyond. Stat. Comput. 22(3), 681–701 (2011)
Viana, F.: Things you wanted to know about the Latin hypercube design and were afraid to ask. In: Proceedings of the10th World Congress on Structural and Multidisciplinary Optimization, Orlando, FL, USA, 19–24 May 2013 (2013)
Sobol’, I.: On the distribution of points in a cube and the approximate evaluation of integrals. USSR Comput. Math. Math. Phys. 7(4), 86–112 (1967)
Santner, T.J., Williams, B.J., Notz, W.I.: The Design and Analysis of Computer Experiments. Springer, New York (2018). https://doi.org/10.1007/978-1-4939-8847-1
Schneider, F., Schüssler, M., Hellmig, R.J., Nelles, O.: Constrained design of experiments for data-driven models. In: Proceedings - 32. Workshop Computational Intelligence, Berlin, 1–2 December 2022 (2022)
Petelet, M., Iooss, B., Asserin, O., Loredo, A.: Latin hypercube sampling with inequality constraints. AStA Adv. Stat. Anal. 94(4), 325–339 (2010)
Khan, S., Gunpinar, E.: An extended Latin hypercube sampling approach for cad model generation. Anadolu Univ. J. Sci. Technol.: Appl. Sci. Eng. 18, 301–314 (2017)
Kayacier, A., Yüksel, F., Karaman, S.: Simplex lattice mixture design approach on physicochemical and sensory properties of wheat chips enriched with different legume flours: an optimization study based on sensory properties. LWT Food Sci. Technol. 58(2), 639–648 (2014)
Scheffé, H.: The simplex-centroid design for experiments with mixtures. J. Roy. Stat. Soc.: Ser. B (Methodol.) 25(2), 235–251 (2018)
Snee, R.D., Marquardt, D.W.: Extreme vertices designs for linear mixture models. Technometrics 16(3), 399–408 (1974)
Hao, Z., Liu, Z., Feng, B.: Application of uniform design for mixture experiments in multi-objective optimization. In: 2014 IEEE International Conference on Progress in Informatics and Computing, pp. 350–354 (2014)
Borkowski, J.J., Piepel, G.F.: Uniform designs for highly constrained mixture experiments. J. Qual. Technol. 41(1), 35–47 (2009)
Zhao, H., Li, G., Li, J.: Uniform test on the mixture simplex region. Symmetry 14(7), 1371 (2022)
Stumpf, J., Naumann, T., Vogt, M.E., Duddeck, F., Zimmermann, M.: On the treatment of equality constraints in mechanical systems design subject to uncertainty. In: Balancing Innovation and operation. The Design Society (2020)
Li, H., Castillo, E.D.: Optimal design of experiments on Riemannian manifolds. J. Am. Stat. Assoc. 1–12 (2022)
Morris, M.D., Mitchell, T.J.: Exploratory designs for computational experiments. J. Stat. Plann. Inference 43(3), 381–402 (1995)
Ebert, T., Fischer, T., Belz, J., Heinz, T.O., Kampmann, G., Nelles, O.: Extended deterministic local search algorithm for maximin Latin hypercube designs. In: 2015 IEEE Symposium Series on Computational Intelligence, pp. 375–382 (2015)
Bates, S., Sienz, J., Toropov, V.: Formulation of the optimal Latin hypercube design of experiments using a permutation genetic algorithm (2004)
Belz, J.: Fighting the curse of dimensionality with local model networks. Ph.D. thesis, Universität Siegen (2018)
Murray-Smith, R., Johansen, T.: Local learning in local model networks. In: 1995 Fourth International Conference on Artificial Neural Networks, pp. 40–46 (1995)
Nelles, O., Isermann, R.: Basis function networks for interpolation of local linear models. In: IEEE Conference on Decision and Control (CDC), pp. 470–475 (1996)
Javier, C.S.: Selecting the slack variable in mixture experiment. Ingeniería Invest. Tecnol. 16(4), 613–623 (2015)
Ning, J., Fang, K.T., Zhou, Y.: Uniform design for experiments with mixtures. Commun. Stat.-Theory Methods 40, 1734–1742 (2011)
Ma, C., Fang, K.T.: A new approach to construction of nearly uniform designs. Int. J. Mater. Prod. Technol. 20, 115–126 (2004)
Ning, J.H., Zhou, Y.D., Fang, K.T.: Discrepancy for uniform design of experiments with mixtures. J. Stat. Plann. Inference 141(4), 1487–1496 (2011)
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Schneider, F., Hellmig, R.J., Nelles, O. (2023). Uniform Design of Experiments for Equality Constraints. In: Quaresma, P., Camacho, D., Yin, H., Gonçalves, T., Julian, V., Tallón-Ballesteros, A.J. (eds) Intelligent Data Engineering and Automated Learning – IDEAL 2023. IDEAL 2023. Lecture Notes in Computer Science, vol 14404. Springer, Cham. https://doi.org/10.1007/978-3-031-48232-8_29
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