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A robust unscented transformation for uncertain moments. (English) Zbl 1456.65180
In numerous problems of statistics and stochastic filtering, one is often interested in calculating the posterior expectation of a continuous random variable that undergoes a nonlinear transform.
The authors propose a robust version of the unscented transform (UT) for one-dimensional random variables. The principle behind UT is to approximate the continuous distribution by the discrete distribution by equating the first \(m\) moments of these distributions. UT is a deterministic sampling technique and has less computational burden than the Monte Carlo integration method.
In the paper, it is proposed to use the Chebyshev center of the semialgebraic set defined by the possible choices of sigma points and their weights as a robust UT. As the moments are not usually exactly known in practical situations, it is assumed that they lie in some intervals. Two approaches for generating robust sigma points are proposed. The moment matching equations of UT are reformulated as a system of polynomial equations with polynomial inequalities. As this system can have more than one solution, it is possible to choose a set of sigma points which minimizes a given cost function by formulating the problem as a polynomial optimization problem solved by using Lasserre’s hierarchy of semidefinite programming relaxations.
The main contribution of the paper is the introduction of the concept of UT robustness in the sense of exploiting the upper and lower bounds for moments. Robustness is achieved by matching precisely known high order moments. Lasserre’s hierarchy of relaxations is applied to polynomial equations with polynomial inequalities.
60E10 Characteristic functions; other transforms
90C23 Polynomial optimization
90C17 Robustness in mathematical programming
Full Text: DOI
[1] Geddes, K. O.; Czapor, S. R.; Labahn, G., Algorithms for Computer Algebra, (1992), Kluwer Academic Publishers · Zbl 0805.68072
[2] Papoulis, A.; Pillai, S. U., Probability, Random Variables, and Stochastic Processes, (2002), McGraw-Hill: McGraw-Hill New York
[3] Julier, S. J.; Uhlmann, J. K.; Durrant-Whyte, H. F., A new approach for filtering nonlinear systems, Proceedings of the 1995 American Control Conference, 3, 1628-1632, (1995)
[4] de Menezes, L. R.A. X.; Soares, A. J.M.; Silva, F. C.; Terada, M. A.B.; Correia, D., A new procedure for assessing the sensitivity of antennas using the unscented transform, IEEE Trans. Antennas Propag., 58, 3, 988-993, (2010)
[5] Carneiro, M. L.; de Carvalho, P. H.P.; Deltimple, N.; Brito, L.d. C.; de Menezes, L. R.A. X.; Kerherve, E.; de Araujo, S. G.; Rocira, A. S., Doherty amplifier optimization using robust genetic algorithm and unscented transform, Proceedings of the Ninth IEEE International Conference on New Circuits and Systems Conference, 77-80, (2011), IEEE
[6] Menegaz, H. M.T.; Ishihara, J. Y.; Borges, G. A.; Vargas, A. N., A systematization of the unscented Kalman filter theory, IEEE Trans. Autom. Control, 60, 10, 2583-2598, (2015) · Zbl 1360.93705
[7] Krishnamoorthy, K., Handbook of Statistical Distributions with Applications, (2010), CRC Press · Zbl 1341.62013
[8] Murty, K. G.; Kabadi, S. N., Some NP-complete problems in quadratic and nonlinear programming, Math. Program., 39, 2, 117-129, (1987) · Zbl 0637.90078
[9] Lasserre, J. B., Global optimization with polynomials and the problem of moments, SIAM J. Optim., 11, 3, 796-817, (2001) · Zbl 1010.90061
[10] Lasserre, J. B., Moments, Positive Polynomials and Their Applications. Moments, Positive Polynomials and Their Applications, Optimization Series, (2009), Imperial College Press: Imperial College Press Singapore
[11] Zandt, J. R.V., A more robust unscented transform, Proceedings of the International Symposium on Optical Science and Technology, 371-380, (2001), International Society for Optics and Photonics
[12] Mehrotra, S.; Papp, D., Generating moment matching scenarios using optimization techniques, SIAM J. Optim., 23, 2, 963-999, (2013) · Zbl 1273.90137
[13] Dette, H.; Studden, W. J., The Theory of Canonical Moments with Applications in Statistics, Probability, and Analysis, Vol. 338, (1997), John Wiley & Sons · Zbl 0886.62002
[14] Hiriart-Urruty, J. B.; Lemaréchal, C., Fundamentals of convex analysis, Grundlehren Text Editions, (2001), Springer: Springer Germany · Zbl 0998.49001
[15] Radhakrishnan, R.; Yadav, A.; Date, P.; Bhaumik, S., A new method for generating sigma points and weights for nonlinear filtering, IEEE Control Syst. Lett., 2, 3, 519-524, (2018)
[16] Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P., Numerical Recipes: The Art of Scientific Computing, (2007), Cambridge University Press: Cambridge University Press New York · Zbl 1132.65001
[17] Julier, S. J.; Uhlmann, J. K., Unscented filtering and nonlinear estimation, Proc. IEEE, 92, 3, 401-422, (2004)
[18] Stoer, J.; Bulirsch, R., Introduction to Numerical Analysis, Texts in Applied Mathematics, (2002), Springer: Springer New York · Zbl 1004.65001
[19] Caballero-Aguila, R.; Hermoso-Carazo, A.; Linares-Pérez, J., Extended and unscented filtering algorithms in nonlinear fractional order systems with uncertain observations, Appl. Math. Sci., 6, 30, 1471-1486, (2012) · Zbl 1253.62067
[20] Boyd, S.; Vandenberghe, L., Convex Optimization, (2004), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1058.90049
[21] Li, Z.; He, S.; Zhang, S., Approximation methods for polynomial optimization: models, algorithms, and applications, Springer Briefs in Optimization, (2012), Springer: Springer New York
[22] Markowitz, H., Portfolio selection, J. Finance, 7, 1, 77-91, (1952)
[23] Jondeau, E.; Rockinger, M., Optimal portfolio allocation under higher moments, Eur. Financ. Manag., 12, 1, 29-55, (2006)
[24] P.M. Kleniati, P. Parpas, B. Rustem, Partitioning procedure for polynomial optimization: application to portfolio decisions with higher order moments, 2009, Technical Report WPS-023, COMISEF Working Papers Series · Zbl 1226.90077
[25] Roberts, A. P.; Newmann, M. M., Polynomial optimization of stochastic feedback control for stable plants, IMA J. Math. Control Inf., 5, 3, 243-257, (1988) · Zbl 0648.93077
[26] Henrion, D.; Lasserre, J. B., Solving nonconvex optimization problems, Control Syst. IEEE, 24, 3, 72-83, (2004)
[27] Mariere, B.; Luo, Z. Q.; Davidson, T. N., Blind constant modulus equalization via convex optimization, signal processing, IEEE Trans., 51, 3, 805-818, (2003) · Zbl 1369.94396
[28] Qi, L.; Teo, K. L., Multivariate polynomial minimization and its application in signal processing, J. Glob. Optim., 26, 4, 419-433, (2003) · Zbl 1023.90064
[29] Dahl, G.; Leinaas, J. M.; Myrheim, J.; Ovrum, E., A tensor product matrix approximation problem in quantum physics, Linear Algebra Appl., 420, 2, 711-725, (2007) · Zbl 1118.15027
[30] Soare, S.; Yoon, J. W.; Cazacu, O., On the use of homogeneous polynomials to develop anisotropic yield functions with applications to sheet forming, Int. J. Plast., 24, 6, 915-944, (2008) · Zbl 1394.74024
[31] Henrion, D.; Lasserre, J. B.; Löfberg, J., GloptiPoly 3: moments, optimization and semidefinite programming, Optim.Methods Softw., 24, 4-5, 761-779, (2009) · Zbl 1178.90277
[32] Waki, H.; Kim, S.; Kojima, M.; Muramatsu, M.; Sugimoto, H., Algorithm 883: SparsePop—a sparse semidefinite programming relaxation of polynomial optimization problems, ACM Trans. Math. Softw., 35, 2, 15, (2008)
[33] Wittek, P., Algorithm 950: Ncpol2sdpa-sparse semidefinite programming relaxations for polynomial optimization problems of noncommuting variables, ACM Trans. Math. Softw. (TOMS), 41, 3, 21, (2015) · Zbl 1347.65110
[34] Cerone, V.; Lasserre, J. B.; Piga, D.; Regruto, D., A unified framework for solving a general class of conditional and robust set-membership estimation problems, IEEE Trans. Autom. Control, 59, 11, 2897-2909, (2014) · Zbl 1360.93330
[35] Indyk, P., Algorithmic applications of low-distortion geometric embeddings, Proceedings of 42nd IEEE Symposium on Foundations of Computer Science, 10-33, (2001)
[36] A.A. Ahmadi, A. Majumdar, DSOS and SDSOS optimization: more tractable alternatives to sum of squares and semidefinite optimization, arXiv preprint, 2018, arXiv:1706.02586. To appear in SIAM Journal on Applied Algebraic Geometry, 2018.
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