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On the quantification of aleatory and epistemic uncertainty using sliced-normal distributions. (English) Zbl 1428.93036
Summary: This paper proposes a means to characterize multivariate data. This characterization, given in terms of both probability distributions and data-enclosing sets, is instrumental in assessing and improving the robustness properties of system designs. To this end, we propose the ‘sliced-normal’ (SN) class of distributions. The versatility of SNs enables characterizing complex parameter dependencies with minimal modeling effort. A polynomial mapping which injects the physical space into a higher dimensional (so-called) feature space is first defined. Optimization-based strategies for the estimation of SNs from data in both physical and feature space are proposed. The non-convex formulations in physical space yield SNs having the best performance. However, the formulations in feature space either admit an analytical solution or yield a convex program thereby facilitating their application to high-dimensional datasets. The semi-algebraic form of the superlevel sets of a SN, form which a tight data-enclosing set can be readily identified, makes them amenable to rigorous worst-case based approaches to robustness analysis and robust design. Furthermore, we propose a chance-constrained optimization framework for identifying and eliminating the effects of outliers in the prescription of such a set. In addition, the distribution-free and non-asymptotic scenario theory framework is used to rigorously bound the probability of unseen data falling outside the identified data-enclosing set.

93B30 System identification
93C35 Multivariable systems, multidimensional control systems
62P30 Applications of statistics in engineering and industry; control charts
Cuba; nwSpGr; SDPT3; Sostools
Full Text: DOI
[1] Oberkampf, W.; Helton, J. C.; Joslyn, C. A.; Wojtkiewicz, S. F.; Ferson, S., Challenge problems: Uncertainty in system response given uncertain parameters, Reliab. Eng. Syst. Saf., 85, 11-19 (2004)
[2] Roy, C.; Oberkampf, W., A comprehensive framework for verification, validation, and uncertainty quantification in scientific computing, Comput. Methods Appl. Mech. Engrg., 200, 2131-2144 (2011) · Zbl 1230.76049
[3] Der-Kiureghian, A.; Ditlevsen, O., Aleatory or epistemic? Does it matter?, Struct. Saf., 31, 105-112 (2009)
[4] Li, J.; Qi, X.; Xiu, D., On upper and lower bounds for quantity of interest in problems subject to epistemic uncertainty, SIAM J. Sci. Comput., 36, 2, 364-376 (2014) · Zbl 1300.65046
[5] Nelsen, R. B., An Introduction to Copulas (2007), Springer Science & Business Media
[6] Embrechts, P.; Lindskog, F.; McNeil, A., Modelling Dependence with Copulas (2001), Rapport technique, Département de mathématiques, Institut Fédéral de Technologie de Zurich: Rapport technique, Département de mathématiques, Institut Fédéral de Technologie de Zurich Zurich
[7] Joe, H., Multivariate Models and Multivariate Dependence Concepts (1997), CRC Press · Zbl 0990.62517
[8] Kurowicka, D.; Cooke, R. M., Uncertainty Analysis with High Dimensional Dependence Modelling (2006), John Wiley & Sons · Zbl 1096.62073
[9] Segers, J., Copulas: An Introduction (2013), Columbia University: Columbia University New York
[10] Aas, K.; Czado, C.; Frigessi, A.; Bakken, H., Pair-copula constructions of multiple dependence, Insur.: Math. Econom., 44, 2, 182-198 (2009) · Zbl 1165.60009
[11] C. Czado, Model selection of vine copulas with applications, in: International Workshop on High-Dimensional Dependence and Copulas, Beijing, China, 2014.
[12] Haff, I. H., Parameter estimation for pair-copula constructions, Bernoulli, 19, 2, 462-491 (2013) · Zbl 1456.62033
[13] Schirmacher, D.; Schirmacher, E., Multivariate Dependence Modeling using Pair-CopulasTechnical Report (2008)
[14] Chesi, G., LMI techniques for optimization over polynomials in control: A survey, IEEE Trans. Automat. Control (2010) · Zbl 1368.93496
[15] Pauwels, E.; Lasserre, J., Sorting out typicality with the inverse moment matrix SOS polynomial, (Advances in Neural Information Processing Systems (2016))
[16] H. El-Samad, S. Prajna, A. Papachristodoulou, M. Khammash, J. Doyle, Model validation and robust stability analysis of the bacterial heat shock response using sostools, in: IEEE Conference on Decision and Control, 2003.
[17] Campi, M.; Garatti, S., The exact feasibility of randomized solutions of uncertain convex programs, SIAM J. Optim., 19, 3, 1211-1230 (2008) · Zbl 1180.90235
[18] Campi, M.; Garatti, S., A sampling-and-discarding approach to chance-constrained optimization: Feasibility and optimality, J. Optim. Theory Appl., 148, 1, 257-280 (2011) · Zbl 1211.90146
[19] Campi, M.; Garatti, S.; Ramponi, F., A general scenario theory for non-convex optimization and decision making, Trans. Autom. Control, 63, 12 (2018) · Zbl 1423.90196
[20] Hodge, V.; Austin, J., A survey of outlier detection methodologies, Artif. Intell. Rev., 22, 2 (2004) · Zbl 1101.68023
[21] Zimek, A.; Campello, R. J.; Sander, J., Ensembles for unsupervised outlier detection: Challenges and research questions a position paper, SIGKDD Explor. Newsl., 15, 1 (2014)
[22] Myung, I., Tutorial on maximum likelihood estimation, J. Math. Psychol. (2003) · Zbl 1023.62112
[23] Dwyer, P. S., Some applications of matrix derivatives in multivariate analysis, J. Amer. Statist. Assoc., 62, 318 (1967) · Zbl 0152.36303
[24] Colbert, B.; Crespo, L. G.; Giesy, D.; M., P., A sum of squares optimization approach to uncertainty quantification, Am. Control Conf. (2019)
[25] Toh, K.; Todd, M.; Tutuncu, R., SDPT3 - a Matlab software package for semidefinite programming, Optim. Methods Softw. (1999) · Zbl 0997.90060
[26] Campi, M.; Care, A., Wait-and-judge scenario optimization, Math. Program., 167, 1 (2018) · Zbl 1388.90090
[27] Care, A.; Garatti, S.; Campi, M., Scenario min-max optimization and the risk of empirical costs, SIAM J. Optim., 4 (2015) · Zbl 1327.90197
[28] Charnes, A.; Cooper, W. W.; Symonds, G. H., Cost horizons and certainty equivalents: An approach to stochastic programming of heating oil, J. Inst. Oper. Res. Manage. Sci., 4, 3 (1958)
[29] Miller, L.; Wagner, H., Chance constrained programming with joint constraints, Oper. Res., 13, 930-945 (1965) · Zbl 0132.40102
[30] Hahn, T., Cuba: A library for multidimensional numerical integration, Comput. Phys. Comm., 168, 2, 78-95 (2005) · Zbl 1196.65052
[31] Heiss, F.; Winschel, V., Likelihood approximation by numerical integration on sparse grids, J. Econometrics, 1 (2008) · Zbl 1418.62466
[32] Van Zandt, J. R., Efficient cubature rules (2018) · Zbl 1431.65021
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