Sigmoid Data Fitting by Least Squares Adjustment of Second and Third Divided Differences

  • Ioannis C. DemetriouEmail author
Part of the Springer Optimization and Its Applications book series (SOIA, volume 100)


We consider the performance of two data smoothing methods that provide sigmoid fits by adjustment of divided differences on some test problems. Thus we investigate the accuracy and the efficiency of the methods for smoothing a variety of data points, our conclusions being drawn from numerical results. The first method is a least squares data smoothing calculation subject to nonnegative third divided differences. The second method is a non-linear least squares data smoothing calculation subject to one sign change in the second divided differences. Both methods employ structured quadratic programming calculations, which take into account the form of the constraints and make efficient use of the banded matrices that occur in the subproblems during the iterations of the quadratic programming calculations. The total work of each method, in practice, is of quadratic complexity with respect to the number of data. Our results expose some weaknesses of the methods. Therefore they may be helpful to the development of new algorithms that are particularly suitable for sigmoid data fitting calculations. Our results expose also some strengths of the methods, which they may be useful to particular scientific analyses, e.g. sigmoid phenomena, and to strategic management practices, i.e. economic substitution.


Quadratic Programming Marginal Rate Quadratic Programming Problem Active Constraint Machine Accuracy 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



This work was partially supported by the University of Athens under Research Grant 11105.


  1. 1.
    Bejan, A., Lorente, S.: The constructal law origin of the logistics S curve. J. Appl. Phys. 110, 1–4 (2011)CrossRefGoogle Scholar
  2. 2.
    Bengisu, M., Nekhili, R.: Forecasting emerging technologies with the aid of science and technology databases. Technol. Forecast. Soc. Change 73, 835–844 (2006)CrossRefGoogle Scholar
  3. 3.
    Cullinan, M.P.: Data smoothing using non-negative divided differences and l2 approximation. IMA J. Numer. Anal. 10, 583–608 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Demetriou, I.C.: Algorithm 742: L2CXFT, a Fortran subroutine for least squares data fitting with non-negative second divided differences. ACM Trans. Math. Softw. 21(1), 98–110 (1995)CrossRefzbMATHGoogle Scholar
  5. 5.
    Demetriou, I.C.: Least squares convex-concave data smoothing. Comput. Optim. Appl. 29, 197–217 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Demetriou, I.C.: L2CXCV: a FORTRAN 77 package for least squares convex/concave data smoothing. Comput. Phys. Commun. 174, 643–668 (2006)CrossRefGoogle Scholar
  7. 7.
    Demetriou, I.C.: Applications of the discrete least squares 3-convex fit to sigmoid data. In: Ao, S.I., Gelman, L., Hukins, D.W.L., Hunter, A., Korsunsky, A.M. (eds) Lecture Notes in Engineering and Computer Science: Proceedings of the World Congress on Engineering 2012, WCE 2012, 4–6 July 2012 London, UK, pp. 285–290 (2012)Google Scholar
  8. 8.
    Demetriou, I.C.: Least squares data fitting subject to decreasing marginal returns. In: Yang, G.-C., Ao, S.I., Gelman, L. (eds.) IAENG Transactions on Engineering Technologies. Special Volume of the World Congress on Engineering 2012. Lecture Notes in Electrical Engineering, vol. 229, pp. 105–120. Springer, Berlin (2013)Google Scholar
  9. 9.
    Demetriou, I.C., Powell, M.J.D.: Least squares smoothing of univariate data to achieve piecewise monotonicity. IMA J. Numer. Anal. 11, 411–432 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Demetriou, I.C., Powell, M.J.D.: The minimum sums of squares change to univariate data that gives convexity. IMA J. Numer. Anal. 11, 433–448 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Dierckx, P.: Curve and Surface Fitting with Splines. Clarendon Press, Oxford (1995)zbMATHGoogle Scholar
  12. 12.
    Fisher, J.C., Pry, R.H.: A simple substitution model of technological change. Technol. Forecast. Soc. Change 2, 75–88 (1971)CrossRefGoogle Scholar
  13. 13.
    Fletcher, R.: Practical Methods of Optimization. Wiley, Chichester (2003)Google Scholar
  14. 14.
    Fubrycky, W.J., Thuesen, G.J., Verna, D.: Economic Decision Analysis, 3rd edn. Prentice Hall, Upper Saddle River (1998)Google Scholar
  15. 15.
    Goldfarb, D., Idnani, G.: A numerically stable dual method for solving strictly convex quadratic programs. Math. Program. 27, 1–33 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Gonzalez, R.C., Wintz, P.: Digital Image Processing, 2nd edn. Addison Wesley, Reading (1987)Google Scholar
  17. 17.
    Lindley, D.V.: Making Decisions, 2nd edn. Wiley, London (1985)Google Scholar
  18. 18.
    McKenna, C.J., Rees, R.: Economics: A Mathematical Introduction. Oxford University Press, New York (1996)Google Scholar
  19. 19.
    Meade, N., Islam, T.: Forecasting with growth curves: an empirical comparison. Int. J. Forecast. 11, 199–215 (1991)CrossRefGoogle Scholar
  20. 20.
    Medawar, P.B.: The laws of biological growth. Nature 148, 772–774 (1941)CrossRefGoogle Scholar
  21. 21.
    Modis, T.: Predictions - Society’s Telltale Signature Reveals the Past and Forecasts the Future. Simon and Schuster, New York (1992)Google Scholar
  22. 22.
    Modis, T.: Technological substitutions in the computer industry. Technol. Forecast. Soc. Change 43, 157–167 (1993)CrossRefGoogle Scholar
  23. 23.
    Morrison, J.S.: Life-cycle approach to new product forecasting. J. Bus. Forecast. Methods Syst. 14, 3–5 (1995)Google Scholar
  24. 24.
    Morrison, J.S.: How to use diffusion models in new product forecasting. J. Bus. Forecast. Methods Syst. 15, 6–9 (1996)Google Scholar
  25. 25.
    Porter, M.E.: Competitive Advantage, Creating and Sustaining Superior Performance. The Free Press/Collier Macmillan Publishers, London (1985)Google Scholar
  26. 26.
    Powell, M.J.D.: Approximation Theory and Methods. Cambridge University Press, Cambridge (1981)zbMATHGoogle Scholar
  27. 27.
    West, J.B.: Respiratory Physiology - The Essentials, 3rd edn. Williams and Wilkins, Baltimore (1985)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department of Economics, Division of Mathematics and InformaticsUniversity of AthensAthensGreece

Personalised recommendations