Abstract
Let measurements of a real function of one variable be given. If the function is convex but convexity has been lost due to errors of measurement, then we make the least sum of squares change to the data so that the second divided differences of the smoothed values are nonnegative. The underlying calculation is a quadratic programming algorithm and the piecewise linear interpolant to the solution components is a convex curve. Problems of this structure arise in various contexts in research and applications in science, engineering and social sciences. The sensitivity of the solution is investigated when the data are slightly altered. The sensitivity question arises in the utilization of the method. First some theory is presented and then an illustrative example shows the effect of specific as well as random changes of the data to the solution. As an application to real data, an experiment on the sensitivity of the convex estimate to the Gini coefficient in the USA for the time period 1947–1996 is presented. The measurements of the Gini coefficient are considered uncertain, with a uniform probability distribution over a certain interval. Some consequences of this uncertainty are investigated with the aid of a simulation technique.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Boot, J.C.G.: Quadratic Programming Algorithms - Anomalies - Applications. North-Holland, Amsterdam (1964)
de Boor, C.: A Practical Guide to Splines, Revised Edition. Applied Mathematical Sciences, vol. 27. Springer, New York (2001)
Demetriou, I.C.: Algorithm 742: L2CXFT, a Fortran 77 subroutine for least squares data fitting with non-negative second divided differences. ACM Trans. Math. Softw. 21(1), 98–110 (1995)
Demetriou, I.C.: Signs of divided differences yield least squares data fitting with constrained monotonicity or convexity. J. Comput. Appl. Math. 146, 179–211 (2002)
Demetriou, I.C., Powell, M.J.D.: The minimum sum of squares change to univariate data that gives convexity. IMA J. Numer. Anal. 11, 433–448 (1991)
Demetriou, I.C., Powell, M.J.D.: Least squares fitting to univariate data subject to restrictions on the signs of the second differences. In: Buhmann, M.D., Iserles, A. (eds.) Approximation Theory and Optimization. Tributes to M.J.D. Powell, pp. 109–132. Cambridge University Press, Cambridge (1997)
Fiacco, A.V., McCormick, G.P.: Nonlinear Programming: Sequential Unconstrained Minimization Techniques. Wiley, London (1968)
Fletcher, R.: Practical Methods of Optimization, 2nd edn. Wiley, Chichester (2003)
Georgiadou, S.A., Demetriou, I.C.: A computational method for the Karush-Kuhn-Tucker test of convexity of univariate observations and certain economic applications. IAENG Int. J. Appl. Math. 38(1), 44–53 (2008)
Goldfarb, D., Idnani, A.: A numerically stable dual method for solving strictly convex quadratic programs. Math. Program. 27, 1–33 (1983)
Golub, G., van Loan, C.F.: Matrix Computations, 2nd edn. The John Hopkins University Press, Baltimore (1989)
Groeneboom, P., Jongbloed, G., Wellner, A.J.: Estimation of a convex function: characterizations and asymptotic theory. Ann. Stat. 29, 1653–1698 (2001)
Hanson, D.L., Pledger, G.: Consistency in concave regression. Ann. Stat. 6(4), 1038–1050 (1976)
Hildreth, C.: Point estimates of ordinates of concave functions. J. Am. Stat. Assoc. 49, 598–619 (1954)
Lawson, C.L., Hanson, R.J.: Solving Least Squares Problems. SIAM, Philadelphia (1995)
Lindley, D.V.: Making Decisions, 2nd edn. Wiley, London (1985)
Marchiando, J.F., Kopanski, J.J.: Regression procedure for determining the dopant profile in semiconductors from scanning capacitance microscopy data. J. Appl. Phys. 92, 5798–5809 (2002)
Acknowledgements
This work was partially supported by the University of Athens under Research Grant 11105. The author feels grateful to the referees for valuable comments and suggestions that improved the chapter.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Demetriou, I.C. (2015). On the Sensitivity of Least Squares Data Fitting by Nonnegative Second Divided Differences. In: Migdalas, A., Karakitsiou, A. (eds) Optimization, Control, and Applications in the Information Age. Springer Proceedings in Mathematics & Statistics, vol 130. Springer, Cham. https://doi.org/10.1007/978-3-319-18567-5_5
Download citation
DOI: https://doi.org/10.1007/978-3-319-18567-5_5
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-18566-8
Online ISBN: 978-3-319-18567-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)