×

Locating infinite discontinuities in computer experiments. (English) Zbl 1494.68216

Summary: Identification of input configurations so as to meet a prespecified output target under a limited experimental budget has been an important task for computer experiments. Such a task often involves the development of response models and design of experimental trials that rely on the models exhibiting continuity and differentiability properties. Motivated by two canonical examples in systems and manufacturing engineering, we propose a strategy for locating the boundary of the response surface in computer experiments, wherein on one side the response is finite, whereas on the other side it is infinite, leveraging ideas from active learning and quasi-Monte Carlo methods. The strategy is illustrated on an example from computer networks engineering and one from precision manufacturing and shown to allocate experimental trials in a fairly effective manner. We conclude by discussing extensions of the proposed strategy to characterize other types of output discontinuity or nondifferentiability in high-cost experiments, including jump discontinuities in the target output response or pathological structures such as kinks and cusps.

MSC:

68T05 Learning and adaptive systems in artificial intelligence
62K05 Optimal statistical designs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] M. Armony and N. Bambos, Queueing dynamics and maximal throughput scheduling in switched processing systems, Queueing Syst., 44 (2003), pp. 209-252, https://doi.org/10.1023/A:1024714024248. · Zbl 1035.90017
[2] J. Beck and S. Guillas, Sequential design with mutual information for computer experiments (MICE): Emulation of a tsunami model, SIAM/ASA J. Uncertain. Quantif., 4 (2016), pp. 739-766, https://doi.org/10.1137/140989613. · Zbl 1349.62364
[3] B. E. Boser, I. M. Guyon, and V. N. Vapnik, A training algorithm for optimal margin classifiers, in Proceedings of the 5th Annual ACM Workshop on COLT, 1992, pp. 144-152, https://doi.org/10.1145/130385.130401.
[4] M. Bramson, Stability of queueing networks, Probab. Surv., 5 (2008), pp. 169-345, https://doi.org/10.1214/08-PS137. · Zbl 1189.60005
[5] C. C. Chang and C. J. Lin, LIBSVM: A library for support vector machines, ACM Trans. Intell. Syst. Tech., 2 (2011), pp. 27:1-27:27. Software available at http://www.csie.ntu.edu.tw/ cjlin/libsvm.
[6] R. B. Chen, Y. W. Hsu, Y. Hung, and W. C. Wang, Discrete particle swarm optimization for constructing uniform design on irregular regions, Comput. Statist. Data Anal., 72 (2014), pp. 282-297, https://doi.org/10.1016/j.csda.2013.10.015. · Zbl 1506.62040
[7] R. B. Chen, Y. C. Hung, W. C. Wang, and S. W. Yen, Contour estimation via two fidelity computer simulators under limited resources, Comput. Statist., 28 (2013), pp. 1813-1834, https://doi.org/10.1007/s00180-012-0380-7. · Zbl 1306.65043
[8] W. J. Chen, Instability threshold and stability boundaries of rotor-bearing systems, J. Engrg. Gas Turbines Power, 118 (1996), pp. 115-121, https://doi.org/10.1115/1.2816526.
[9] S. C. Chuang and Y. C. Hung, Uniform design over general input domains with applications to target region estimation in computer experiments, Comput. Statist. Data Anal., 54 (2010), pp. 219-232, https://doi.org/10.1016/j.csda.2009.08.008. · Zbl 1284.62474
[10] C. Cortes and V. N. Vapnik, Support vector networks, Mach. Learn., 20 (1995), pp. 273-297, https://doi.org/10.1023/A:1022627411411. · Zbl 0831.68098
[11] H. Dette and A. Pepelyshev, Generalized Latin hypercube design for computer experiments, Technometrics, 52 (2010), pp. 421-429, https://doi.org/10.1198/TECH.2010.09157.
[12] K. T. Fang and D. K. J. Lin, Uniform experimental designs and their applications in industry, in Handbook of Statist., vol. 22, Elsevier, Amsterdam, 2003, pp. 131-170, https://doi.org/10.1016/S0169-7161(03)22006-X.
[13] K. T. Fang, D. K. J. Lin, P. Winker, and Y. Zhang, Uniform design: Theory and applications, Technometrics, 42 (2000), pp. 237-248, https://doi.org/10.2307/1271079. · Zbl 0996.62073
[14] K. T. Fang and Y. Wang, Number-Theoretic Methods in Statistics, Chapman and Hall, London, 1994. · Zbl 0925.65263
[15] Y. Fang and K. A. Loparo, Stochastic stability of jump linear systems, IEEE Trans. Automat. Control, 47 (2002), pp. 1204-1208, https://doi.org/10.1109/TAC.2002.800674. · Zbl 1364.93844
[16] A. I. J. Forrester, A. Sóbester, and A. J. Keane, Multi-fidelity optimization via surrogate modelling, in Proc. A, 463 (2007), pp. 3251-3269, https://doi.org/10.1098/rspa.2007.1900. · Zbl 1142.90489
[17] Y. Freund, Boosting a weak learning algorithm by majority, Inform. and Comput., 121 (1995), pp. 256-285, https://doi.org/10.1006/inco.1995.1136. · Zbl 0833.68109
[18] Y. Freund and R. E. Schapire, A decision-theoretic generalization of on-line learning and an application to boosting, J. Comput. System Sci., 55 (1997), pp. 119-139, https://doi.org/10.1006/jcss.1997.1504. · Zbl 0880.68103
[19] R. B. Gramacy and H. H. Lee, Bayesian treed Gaussian process models with an application to computer modeling, J. Amer. Statist. Assoc., 103 (2008), pp. 1119-1130, https://doi.org/10.1198/016214508000000689. · Zbl 1205.62218
[20] M. Gu and L. Wang, Scaled Gaussian stochastic process for computer model calibration and prediction, SIAM/ASA J. Uncertain. Quantif., 6 (2018), pp. 1555-1583. · Zbl 1409.62185
[21] F. J. Hickernell, A generalized discrepancy and quadrature error bound, Math. Comp., 67 (1998), pp. 299-322, https://doi.org/10.1090/S0025-5718-98-00894-1. · Zbl 0889.41025
[22] C. M. Huang, Y. J. Lee, D. K. J. Lin, and S. Y. Huang, Model selection for support vector machines via uniform design, Comput. Statist. Data Anal., 52 (2007), pp. 335-346, https://doi.org/10.1016/j.csda.2007.02.013. · Zbl 1452.62073
[23] Y. C. Hung and C. C. Chang, Dynamic scheduling for switched processing systems with substantial service-mode switching times, Queueing Syst., 60 (2008), pp. 87-109, https://doi.org/10.1007/s11134-008-9088-3. · Zbl 1166.60057
[24] Y. C. Hung and G. Michailidis, Stability and control of acyclic stochastic processing networks with shared resources, IEEE Trans. Automat. Control, 57 (2012), pp. 489-494, https://doi.org/10.1109/TAC.2011.2164012. · Zbl 1369.90044
[25] G. Imbens and T. Lemieux, Regression discontinuity designs: A guide to practice, J. Econometrics, 142 (2008), pp. 615-635, https://doi.org/10.1016/j.jeconom.2007.05.001. · Zbl 1418.62475
[26] M. E. Johnson, L. M. Moore, and D. Ylvisaker, Minimax and maximin distance designs, J. Statist. Planning Inference, 26 (1990), pp. 131-148, https://doi.org/10.1016/0378-3758(90)90122-B.
[27] V. R. Joseph, Limit kriging, Technometrics, 48 (2006), pp. 458-466, https://doi.org/10.1198/004017006000000011.
[28] V. R. Joseph, Y. Hung, and A. Sudjianto, Blind kriging: A new method for developing metamodels, J. Mech. Des., 130 (2008), 0311021 https://doi.org/10.1115/1.2829873.
[29] A. Karatzoglou, D. Meyer, and K. Hornik, Support vector machines in R, J. Statist. Softw., 15 (2006), pp. 1-28, https://doi.org/10.18637/jss.v015.i09.
[30] C. G. Kaufman, D. Bingham, S. Habib, K. Heitmann, and J. A. Frieman, Efficient emulators of computer experiments using compactly supported correlation functions, with an application to cosmology, Ann. Appl. Stat., 5 (2011), pp. 2470-2492, https://doi.org/10.1214/11-AOAS489. · Zbl 1234.62166
[31] S. S. Keerthi and C. J. Lin, Asymptotic behaviors of support vector machines with Gaussian kernel, Neural Comput., 15 (2003), pp. 1667-1689, https://doi.org/10.1162/089976603321891855. · Zbl 1086.68569
[32] T. A. Khawli, U. Eppelt, and W. Schulz, Advanced metamodeling techniques applied to multidimensional applications with piecewise response, in Proceedings of the International worshop on Machine Learning, Optimization, and Big Data, Springer International Publishing, Switzerland, 2015, pp. 93-104, https://doi.org/10.1007/978-3-319-27926-8_9.
[33] J. Kremer, K. S. Pedersen, and C. Igel, Active learning with support vector machines, Wiley Interdiscip. Rev. Data Mining Knowledge Discovery, 4 (2014), pp. 313-326, https://doi.org/10.1002/widm.1132.
[34] P. R. Kumar and S. P. Meyn, Stability of queueing networks and scheduling policies, IEEE Trans. Automat. Control, 40 (1995), pp. 251-260, https://doi.org/10.1109/9.341782. · Zbl 0834.90059
[35] D. Lee and T. Lemieux, Regression discontinuity designs in economics, J. Econom. Literature, 48 (2010), pp. 281-355, https://doi.org/10.1257/jel.48.2.281.
[36] D. Lewis and J. Catlett, Heterogeneous uncertainty sampling for supervised learning, in Proceedings of the Eleventh International Conference on Machine Learning, 1994, pp. 148-156, https://doi.org/10.1016/B978-1-55860-335-6.50026-X.
[37] C. H. Li, C. T. Lin, and B. C. Kuo, An automatic method for selecting the parameter of the RBF kernel function to support vector machines, in Proceedings of the IEEE International Conference on Geoscience and Remote Sensing Symposium, 2010, pp. 836-839, https://doi.org/10.1109/IGARSS.2010.5649251.
[38] D. K. J. Lin, C. Sharpe, and P. Winker, Optimized u-type designs on flexible regions, Comput. Statist. Data Anal., 54 (2010), pp. 1505-1515, https://doi.org/10.1016/j.csda.2010.01.032. · Zbl 1284.62476
[39] B. McDonald, P. Ranjan, and H. Chipman, GPFIT: An R package for fitting a Gaussian process model to deterministic simulator outputs, J. Statist. Softw., 64 (2015), pp. 1-23, https://doi.org/10.18637/jss.v064.i12.
[40] M. D. Morris, T. J. Mitchell, and D. Ylvisaker, Bayesian design and analysis of computer experiments: use of derivatives in surface prediction, Technometrics, 35 (1993), pp. 243-255, https://doi.org/10.2307/1269517. · Zbl 0785.62025
[41] K. P. Murphy, Machine Learning: A Probabilistic Perspective, MIT Press, Cambridge, MA, 2012. · Zbl 1295.68003
[42] J. Platt, Probabilistic outputs for support vector machines and comparison to regularized likelihood methods, in Advances in Large margin Classifiers, A. J. Smola, P. L. Bartlett, B. Schölkopf, and D. Schuurmans, eds., MIT Press, Cambridge, 2000, pp. 61-74.
[43] M. T. Pratola, O. Harari, D. Bingham, and G. E. Flowers, Design and analysis of experiments on nonconvex regions, Technometrics, 59 (2017), pp. 36-47, https://doi.org/10.1080/00401706.2015.1115674.
[44] P. Ranjan, D. Bingham, and G. Michailidis, Sequential experimental design for contour estimation from complex computer codes, Technometrics, 50 (2008), pp. 527-541, https://doi.org/10.1198/004017008000000541.
[45] J. Sacks, S. B. Schiller, and W. J. Welch, Design for computer experiments, Technometrics, 31 (1989), pp. 41-47, https://doi.org/10.2307/1270363.
[46] J. Sacks, W. J. Welch, T. J. Mitchell, and H. P. Wynn, Design and analysis of computer experiments, Statist. Sci., 4 (1989), pp. 409-423, https://doi.org/10.1214/ss/1177012420. · Zbl 0955.62619
[47] T. J. Santner, B. J. Williams, and W. I. Notz, The Design and Analysis of Computer Experiments, Springer Verlag, New York, 2003. · Zbl 1041.62068
[48] B. Schölkopf, Support Vector Learning, Oldenbourg Verlag, Munich, 1997. · Zbl 0915.68137
[49] J. Shawe-Taylor and N. Cristianini, Kernel Methods for Pattern Analysis, Cambridge University Press, New York, 2004, https://doi.org/10.1017/CBO9780511809682. · Zbl 0994.68074
[50] M. A. L. Thathachar and P. Viswanath, On the stability of fuzzy systems, IEEE Trans. Fuzzy Syst., 5 (1997), pp. 145-151, https://doi.org/10.1109/91.554461.
[51] D. Thistlewaite and D. Campbell, Regression-discontinuity analysis: An alternative to the ex post facto experiment, J. Educ. Phychol., 51 (1960), pp. 309-317, https://doi.org/10.1037/h0044319.
[52] S. Tong and E. Chang, Support vector machine active learning for image retrieval, in Proceedings of the International Conference on Multimedia (MM), 2001, pp. 107-118, https://doi.org/10.1145/500141.500159.
[53] S. Tong and D. Koller, Support vector machine active learning with applications to text classification, J. Mach. Learn. Res., 2 (2001), pp. 45-66, https://doi.org/10.1162/153244302760185243. · Zbl 1009.68131
[54] V. N. Vapnik, Statistical Learning Theory, Wiley, New York, 1998. · Zbl 0935.62007
[55] D. Wang, X. Zhang, M. Fan, and X. Ye, Hierarchical mixing linear support vector machines for nonlinear classification, Pattern Recogn., 59 (2016), pp. 255-267, https://doi.org/10.1016/j.patcog.2016.02.018. · Zbl 1414.68083
[56] W. Wang, Z. Xu, W. Lu, and X. Zhang, Determination of the spread parameter in the Gaussian kernel for classification and regression, Neurocomputing, 55 (2003), pp. 643-663, https://doi.org/10.1016/S0925-2312(02)00632-X.
[57] J. Wieland, R. Pasupathy, and B. W. Schmeiser, Queueing-network stability: Simulation-based checking, in Proceedings of the Winter Simulation Conference, 2003, pp. 520-527, https://doi.org/10.1109/WSC.2003.1261464.
[58] Z. Xu, M. Dai, and D. Meng, Fast and efficient strategies for model selection of Gaussian support vector machine, IEEE Trans. Syst., Man Cybernet., 39 (2009), pp. 1292-1307, https://doi.org/10.1109/TSMCB.2009.2015672.
[59] X. Yang, Q. Song, and Y. Wang, A weighted support vector machine for data classification, J. Pattern Recogn. Artif. Intell., 21 (2007), pp. 961-976, https://doi.org/10.1142/S0218001407005703.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.