Abstract
We show how a recently developed multivariate data fitting technique enables to solve a variety of scientific computing problems in filtering, queueing, networks, metamodelling, computational finance, graphics, and more. We can capture linear as well as nonlinear phenomena because the method uses a generalized multivariate rational model. The technique is a refinement of the basic ideas developed in Salazar et al. (Numer Algorithms 45:375–388, 2007. https://doi.org/10.1007/s11075-007-9077-3) and interpolates interval data. Intervals allow to take the inherent data error in measurements and simulation into consideration, whilst guaranteeing an upper bound on the tolerated range of uncertainty. The latter is the main difference with a best approximation or least squares technique which does as well as it can, but without respecting an a priori imposed threshold on the approximation error. Compared to the best approximations, the interval interpolant is relatively easy to compute. In applications where industry standards need to be guaranteed, the interval interpolation technique may be a valuable alternative.
Similar content being viewed by others
References
Allouche, H., Cuyt, A.: On the structure of a table of multivariate rational interpolants. Constr. Approx. 8, 69–86 (1992). https://doi.org/10.1007/BF01208907
Allouche, H., Cuyt, A.: Unattainable points in multivariate rational interpolation. J. Approx. Theory 72, 159–173 (1993). https://doi.org/10.1006/jath.1993.1013
Berrut, J.P.: Rational functions for guaranteed and experimentally well-conditioned global interpolation. Comput. Math. Appl. 15, 1–16 (1988). https://doi.org/10.1016/0898-1221(88)90067-3
Boehm, B.: Existence of best rational tchebycheff approximations. Pac. J. Math. 15(1), 19–28 (1965)
Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)
Gilewicz, J., Magnus, A.: Valleys in \({C}\)-table. LNM 765, 135–149 (1979)
Goldman, A.J., Tucker, A.W.: Polyhedral convex cones. In: Kuhn, H.W., Tucker, A.W. (eds.) Linear Inequalities and Related Systems. Annals of Mathematics Studies, vol. 38, p. 1940. Princeton University Press, Princeton (1956)
Gorinevsky, D., Boyd, S.: Optimization-based design and implementation of multi-dimensional zero-phase IIR filters. IEEE Trans. Circuits Syst. I 53(2), 372–383 (2006). https://doi.org/10.1109/TCSI.2005.856048
Harris, D.: Design of 2-D rational digital filters. In: IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP ’81, pp. 696–699 (1981)
Ibrahimoglu, B.A., Cuyt, A.: Sharp bounds for Lebesgue constants of barycentric rational interpolation. Exp. Math. 25(3), 347–354 (2016). https://doi.org/10.1080/10586458.2015.1072862
Li, M.: Approximate inversion of the Black–Scholes formula using rational functions. Eur. J. Oper. Res. 185(2), 743–759 (2008). https://doi.org/10.1016/j.ejor.2006.12.028
Matusik, W., Pfister, H., Brand, M., McMillan, L.: A data-driven reflectance model. ACM Trans. Graph. 22(3), 759–769 (2003)
Reddy, H., Khoo, I.H., Rajan, P.: 2-D symmetry: theory and filter design applications. IEEE Circuits Syst. Mag. Third Quarter, 4–33 (2003)
Rusinkiewicz, S.M.: A new change of variables for efficient BRDF representation. In: In Eurographics Workshop on Rendering, pp. 11–22 (1998)
Salazar Celis, O., Cuyt, A., Van Deun, J.: Symbolic and interval rational interpolation: the problem of unattainable data. In: Simos, T., Psihoyios, G., Tsitouras, C. (eds.) International conference on numerical analysis and applied mathematics, AIP Conference Proceedings, vol. 1048, pp. 466–469 (2008). https://doi.org/10.1063/1.2990963
Salazar Celis, O., Cuyt, A., Verdonk, B.: Rational approximation of vertical segments. Numer. Alg. 45, 375–388 (2007). https://doi.org/10.1007/s11075-007-9077-3
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Tom Lyche.
Rights and permissions
About this article
Cite this article
Cuyt, A., Salazar Celis, O. Multivariate data fitting with error control. Bit Numer Math 59, 35–55 (2019). https://doi.org/10.1007/s10543-018-0721-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10543-018-0721-1