Abstract
The quality of data used for data-driven modeling affects the model performance significantly. Thus, design of experiments (DoE) is an important part during model development. The design space is constrained in many applications. In this work, the constrained case is investigated. An Latin hypercube based approach is applied and analyzed for strongly constrained design spaces. Contrary to commonly used optimization techniques, an incremental procedure is proposed. In every step, new data are added to the design. Each new point is selected by a distance-based criterion. The performance of the created designs is evaluated by the quality of the trained models. For different constraints, artificial data sets are created with a function generator. The performance of local model networks and Gaussian process regression models trained with those designs is evaluated and compared to models trained on data sets based on Sobol’ sequences.
Zusammenfassung
Statistische Versuchsplanung (Design of Experiments) hat einen bedeutenden Einfluss auf die Qualität von datengetriebenen Modellen. Ein häufig vorkommender Sonderfall ist ein beschränkter Eingangsraum. Im Rahmen dieses Beitrages soll die Erstellung von Versuchsplänen für diesen Fall untersucht werden. Ein Ansatz, basierend auf Latin Hypercubes (LH), wird für stark eingeschränkte Eingangsräume untersucht. Im Gegensatz zu Optimierungsverfahren für unbeschränkte Versuchsräume wird der Versuchsplan in einem inkrementellen Verfahren aufgebaut. In jedem Schritt wird der neue Versuchspunkt anhand einer distanzbasierten Metrik ausgewählt. Anhand künstlicher Datensätze, erzeugt durch einen Funktionsgenerator, wird die Methode bewertet. Die Modellqualität wird über den Testfehler ermittelt. Als Vergleich dient der Testfehler von Modellen, welche mit Datensätzen, basierend auf Sobol’ Sequenzen, trainiert wurden. Zur Modellierung der Testprozesse werden Lokale Modellnetze und Gauß’sche Prozessmodelle verwendet.
About the authors
Fabian Schneider graduated with a Master of Science degree from University Siegen in 2020. He has joined the working group Automatic Control – Mechatronics of Prof. Nelles as a research assistant. His research topics focus on meta modeling for optimization tasks and design of experiments.
Ralph J. Hellmig is apl. Professor at the University of Siegen in the Department of Mechanical Engineering and institute of Material Science. He received his doctor’s degree in 2000 at the Technical University of Clausthal. His key research topics are material based questions in the field of joining technology and corrosion.
Oliver Nelles is Professor at the University of Siegen in the Department of Mechanical Engineering and chair of Automatic Control –Mechatronics. He received his doctor’s degree in 1999 at the Technical University of Darmstadt. His key research topics are nonlinear system identification, design of experiments, metamodeling and local model networks.
-
Research ethics: Not applicable.
-
Author contributions: The authors have accepted responsibility for the entire content of this manuscript and approved its submission.
-
Competing interests: All other authors state no conflict of interest.
-
Research funding: Not applicable.
-
Data availability: Not applicable.
References
[1] L. Pronzato and W. G. Müller, “Design of computer experiments: space filling and beyond,” Stat. Comput., vol. 22, no. 3, pp. 681–701, 2011, https://doi.org/10.1007/s11222-011-9242-3.Search in Google Scholar
[2] T. J. Santner, B. J. Williams, and W. I. Notz, The Design and Analysis of Computer Experiments, vol. 2, New York, Springer, 2018.10.1007/978-1-4939-8847-1Search in Google Scholar
[3] F. Viana, “Things you wanted to know about the Latin hypercube design and were afraid to ask,” in Proceedings of the 10th World Congress on Structural and Multidisciplinary Optimization, Orlando, FL, USA, 19–24 May 2013, 2013.Search in Google Scholar
[4] F. Schneider, M. Schüssler, R. J. Hellmig, and O. Nelles, “Constrained design of experiments for data-driven models,” in Proceedings – 32. Workshop Computational Intelligence: Berlin, 1.–2. Dezember 2022, Karlsruhe, Germany, KIT Scientific Publishing, 2022.Search in Google Scholar
[5] M. Vořechovský and J. Eliáš, “Modification of the maximin and ϕp (phi) criteria to achieve statistically uniform distribution of sampling points,” Technometrics, vol. 62, no. 3, pp. 371–386, 2020, https://doi.org/10.1080/00401706.2019.1639550.Search in Google Scholar
[6] G. Damblin, M. Couplet, and B. Iooss, “Numerical studies of space-filling designs: optimization of Latin Hypercube samples and subprojection properties,” J. Simulat., vol. 7, no. 4, pp. 276–289, 2013, https://doi.org/10.1057/jos.2013.16.Search in Google Scholar
[7] M. D. Morris and T. J. Mitchell, “Exploratory designs for computational experiments,” J. Stat. Plann. Inference, vol. 43, no. 3, pp. 381–402, 1995, https://doi.org/10.1016/0378-3758(94)00035-t.Search in Google Scholar
[8] R. A. Fisher, “The arrangement of field experiments,” J. Minist. Agric., vol. 33, pp. 503–513, 1926.Search in Google Scholar
[9] O. Nelles, Nonlinear System Identification from Classical Approaches to Neural Networks, Fuzzy Models, and Gaussian Processes, 2nd ed., Cham, Springer International Publishing; Imprint: Springer, 2020, Nelles2020.10.1007/978-3-030-47439-3Search in Google Scholar
[10] X. Yue, Y. Wen, J. H. Hunt, and J. Shi, “Active learning for Gaussian process considering uncertainties with application to shape control of composite fuselage,” IEEE Trans. Autom. Sci. Eng., vol. 18, no. 1, pp. 36–46, 2021, https://doi.org/10.1109/tase.2020.2990401.Search in Google Scholar
[11] J. Belz, “Fighting the curse of dimensionality with local model networks,” Ph.D. thesis, Universität Siegen, 2018.Search in Google Scholar
[12] T. J. Peter and O. Nelles, “Fast and simple dataset selection for machine learning,” at-Automatisierungstechnik, vol. 67, no. 10, pp. 833–842, 2019, https://doi.org/10.1515/auto-2019-0010.Search in Google Scholar
[13] D. W. Scott, Multivariate Density Estimation, New York, Wiley, 1992.10.1002/9780470316849Search in Google Scholar
[14] J. L. Bentley, “Multidimensional binary search trees used for associative searching,” Commun. ACM, vol. 18, no. 9, pp. 509–517, 1975, https://doi.org/10.1145/361002.361007.Search in Google Scholar
[15] H. Song, K. K. Choi, I. Lee, L. Zhao, and D. Lamb, “Sampling-based RBDO using probabilistic sensitivity analysis and virtual support vector machine,” in Volume 3: 38th Design Automation Conference, Parts A and B. International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, American Society of Mechanical Engineers, 2012, pp. 1213–1225.10.1115/DETC2012-70715Search in Google Scholar
[16] I. M. Sobol’, “On the distribution of points in a cube and the approximate evaluation of integrals,” USSR Comput. Math. Math. Phys., vol. 7, no. 4, pp. 86–112, 1967, https://doi.org/10.1016/0041-5553(67)90144-9.Search in Google Scholar
[17] S. Joe and F. Y. Kuo, “Constructing Sobol sequences with better two-dimensional projections,” SIAM J. Sci. Comput., vol. 30, pp. 2635–2654, 2008, https://doi.org/10.1137/070709359.Search in Google Scholar
[18] T. Ebert, T. Fischer, J. Belz, T. O. Heinz, G. Kampmann, and O. Nelles, “Extended deterministic local search algorithm for maximin Latin hypercube designs,” in 2015 IEEE Symposium Series on Computational Intelligence, New York, IEEE, 2015, pp. 375–382.10.1109/SSCI.2015.63Search in Google Scholar
[19] F. A. C. Viana, G. Venter, and V. Balabanov, “An algorithm for fast optimal Latin hypercube design of experiments,” Int. J. Numer. Methods Eng., vol. 82, no. 2, pp. 135–156, 2010, https://doi.org/10.1002/nme.2750.Search in Google Scholar
[20] A. Grosso, A. R. M. J. U. Jamali, and M. Locatelli, “Finding maximin Latin hypercube designs by Iterated Local Search heuristics,” Eur. J. Oper. Res., vol. 197, no. 2, pp. 541–547, 2009, https://doi.org/10.1016/j.ejor.2008.07.028.Search in Google Scholar
[21] T. Voigt, M. Kohlhase, and O. Nelles, “Incremental Latin hypercube additive design for LOLIMOT,” in 2020 25th IEEE International Conference on Emerging Technologies and Factory Automation (ETFA), vol. 1, IEEE, 2020, pp. 1602–1609.10.1109/ETFA46521.2020.9212173Search in Google Scholar
[22] V. V. Garg and R. H. Stogner, “Hierarchical Latin hypercube sampling,” J. Am. Stat. Assoc., vol. 112, no. 518, pp. 673–682, 2017, https://doi.org/10.1080/01621459.2016.1158717.Search in Google Scholar
[23] Y.-L. Chen, W. Wei, C.-N. J. Liu, and L. He, “Incremental Latin hypercube sampling for lifetime stochastic behavioral modeling of analog circuits,” in The 20th Asia and South Pacific Design Automation Conference, New York, IEEE, 2015, pp. 556–561.Search in Google Scholar
[24] J. Belz and O. Nelles, “Proposal for a function generator and extrapolation analysis,” in 2015 International Symposium on Innovations in Intelligent Systems and Applications (INISTA), 2015, pp. 1–6.10.1109/INISTA.2015.7276762Search in Google Scholar
[25] C. E. Rasmussen and C. K. I. Williams, Gaussian Processes for Machine Learning (Adaptive Computation and Machine Learning), Cambridge, The MIT Press, 2005.10.7551/mitpress/3206.001.0001Search in Google Scholar
[26] R. Murray-Smith and T. A. Johansen, “Local learning in local model networks,” in 1995 Fourth International Conference on Artificial Neural Networks, 1995, pp. 40–46.10.1049/cp:19950526Search in Google Scholar
[27] O. Nelles, “Axes-oblique partitioning strategies for local model networks,” in IEEE International Symposium on Intelligent Control, 2006, pp. 2378–2383.10.1109/ISIC.2006.285656Search in Google Scholar
[28] O. Nelles and R. Isermann, “Basis function networks for interpolation of local linear models,” in IEEE Conference on Decision and Control (CDC), 1996, pp. 470–475.Search in Google Scholar
[29] S. Ernst, “Hinging hyperplane trees for approximation and identification,” in Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171), vol. 2, 1998, pp. 1266–1271.Search in Google Scholar
[30] L. Piegl and W. Tiller, The NURBS Book, Berlin, Heidelberg, Springer, 1995.10.1007/978-3-642-97385-7Search in Google Scholar
© 2023 Walter de Gruyter GmbH, Berlin/Boston