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Monotonic decrease of upper limit estimated with Gompertz model for data described using logistic model

  • Daisuke Satoh
  • Ryutaro Matsumura
Original Paper Area 3
  • 22 Downloads

Abstract

The Gompertz curve and logistic curve models are often used to forecast upper limits (saturation points). To accurately estimate an upper limit, an appropriate model selection as well as accurate parameter estimation is required. We mathematically analyze how an upper limit estimated with an inappropriate model changes as the data size increases, i.e., time elapses, when the Gompertz curve model is selected for data described on the exact solution of the logistic curve model. We prove that an estimated upper limit is strictly monotonically decreasing as the historical data size increases and that the upper limit estimated with the inappropriate model converges to the upper limit estimated with the appropriate model as the data size approaches infinity. These results can help in selecting an appropriate model.

Keywords

Gompertz curve model Logistic curve model Model-selection Applied discrete systems Discrete equation Exact solution 

Mathematics Subject Classification

39A60 62J05 91B62 

Notes

Acknowledgements

We thank anonymous reviewers for their careful reading and helpful comments.

References

  1. 1.
    Aggrey, S.: Comparison of three nonlinear and spline regression models for describing chicken growth curves. Poult. Sci. 81(12), 1782–1788 (2002)CrossRefGoogle Scholar
  2. 2.
    Bass, F.M.: A new product growth for model consumer durables. Manag. Sci. 15(5), 215–227 (1969).  https://doi.org/10.1287/mnsc.15.5.215 CrossRefzbMATHGoogle Scholar
  3. 3.
    Bemmaor, A.C.: Modeling the Diffusion of New Durable Goods: Word-of-Mouth Effect Versus Consumer Heterogeneity, pp. 201–223. Kluwer, Boston (1994)Google Scholar
  4. 4.
    Chu, W.L., Wu, F.S., Kao, K.S., Yen, D.C.: Diffusion of mobile telephony: an empirical study in Taiwan. Telecommun. Policy 33(9), 506–520 (2009)CrossRefGoogle Scholar
  5. 5.
    Franses, P.H.: A method to select between Gompertz and logistic trend curves. Technol. Forecast. Soc. Change 46(1), 45–49 (1994)CrossRefGoogle Scholar
  6. 6.
    Gregg, J., Hossel, C., Richardson, J.: Mathematical Trend Curves, An Aid to Forecasting. ICI Monograph 1. Oliver and Boyd, Edinburgh (1964)Google Scholar
  7. 7.
    Guidolin, M., Guseo, R.: Technological change in the U.S. music industry: within-product, cross-product and churn effects between competing blockbusters. Technol. Forecast. Soc. Change 99(1), 35–46 (2015)CrossRefGoogle Scholar
  8. 8.
    Gupta, R., Jain, K.: Diffusion of mobile telephony in India: an empirical study. Technol. Forecast. Soc. Change 79(4), 709–715 (2012)CrossRefGoogle Scholar
  9. 9.
    Guseo, R., Guidolin, M.: Modelling a dynamic market potential: a class of automata networks for diffusion of innovations. Technol. Forecast. Soc. Change 76(6), 806–820 (2009)CrossRefGoogle Scholar
  10. 10.
    Guseo, R., Mortarino, C.: Modeling competition between two pharmaceutical drugs using innovation diffusion models. Ann. Appl. Stat. 9(4), 2073–2089 (2015)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Hirota, R.: Nonlinear partial difference equations. V. nonlinear equations reducible to linear equations. J. Phys. Soc. Jpn. 46(1), 312–319 (1979)CrossRefGoogle Scholar
  12. 12.
    Hirota, R.: Lecture on discrete equations. Saiensusha, Tokyo (2000) (in Japanese) Google Scholar
  13. 13.
    Hirota, R., Takahashi, D.: Discrete and Ultradiscrete Systems. Kyoritsushuppan, Tokyo (2003) (in Japanese) Google Scholar
  14. 14.
    Karmeshu, Goswami, D.: Stochastic evolution of innovation diffusion in heterogeneous groups: study of life cycle patterns. IMA J. Manag. Math. 12(2), 107–126 (2001).  https://doi.org/10.1093/imaman/12.2.107 CrossRefzbMATHGoogle Scholar
  15. 15.
    Knízetová, H., Hyánek, J., Kníze, B., Roubícek, J.: Analysis of growth curve of fowl I. chickens. Br. Poult. Sci. 32(5), 1027–1038 (1991)CrossRefGoogle Scholar
  16. 16.
    Krishnan, T.V., Bass, F.M.: Impact of a late entrant on the diffusion of a new product/service. J. Mark. Res. 37(2), 269–278 (2000)CrossRefGoogle Scholar
  17. 17.
    Lechman, E.: ICT Diffusion in Developing Countries: Towards a New Concept of Technological Takeoff. Springer, Berlin (2015)CrossRefGoogle Scholar
  18. 18.
    Li, T.Y., Yorke, J.A.: Period three implies chaos. Am. Math. Mon. 82(10), 985–992 (1975)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Martino, J.P.: A review of selected recent advances in technological forecasting. Technol. Forecast. Soc. Change 70(8), 719–733 (2003)CrossRefGoogle Scholar
  20. 20.
    Meade, N.: The use of growth curves in forecasting market development—a review and appraisal. J. Forecast. 3(4), 429–451 (1984)CrossRefGoogle Scholar
  21. 21.
    Meade, N., Islam, T.: Forecasting with growth curves: an empirical comparison. Int. J. Forecast. 11(2), 199–215 (1995)CrossRefGoogle Scholar
  22. 22.
    Morisita, M.: The fitting of the logistic equation to the rate of increase of population density. Res. Popul. Ecol. 7(1), 52–55 (1965)CrossRefGoogle Scholar
  23. 23.
    Narinc, D., Karaman, E., Firat, M.Z., Aksoy, T.: Comparison of non-linear growth models to describe the growth in Japanese quail. J. Anim. Vet. Adv. 9(14), 1961–1966 (2010)CrossRefGoogle Scholar
  24. 24.
    Nguimkeu, P.: A simple selection test between the Gompertz and logistic growth models. Technol. Forecast. Soc. Change 88(1), 98–105 (2014)CrossRefGoogle Scholar
  25. 25.
    Richards, F.: A flexible growth model for empirical use. J. Exp. Bot. 10(2), 290–301 (1959).  https://doi.org/10.1093/jxb/10.2.290 CrossRefGoogle Scholar
  26. 26.
    Roush, W., Branton, S.: A comparison of fitting growth models with a genetic algorithm and nonlinear regression. Poult. Sci. 84(3), 494–502 (2005)CrossRefGoogle Scholar
  27. 27.
    Satoh, D.: A discrete Gompertz equation and a software reliability growth model. IEICE Trans. E83–D(7), 1508–1513 (2000)Google Scholar
  28. 28.
    Satoh, D.: A discrete Bass model and its parameter estimation. J. Oper. Res. Soc. Jpn. 44(1), 1–18 (2001)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Satoh, D., Uchida, M.: Computer warm model describing infection via e-mail. Bull. Jpn. Soc. Ind. Appl. Math. 20(3), 50–55 (2010) (in Japanese) Google Scholar
  30. 30.
    Satoh, D., Yamada, S.: Discrete equations and software reliability growth models. In: Proceedings of 12th International Symposium on Software Reliability Engineering, pp. 176–184 (2001)Google Scholar
  31. 31.
    Satoh, D., Yamada, S.: Parameter estimation of discrete logistic curve models for software reliability assessment. Jpn. J. Ind. Appl. Math. 19(1), 39–53 (2002)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Savin, S., Terwiesch, C.: Optimal product launch times in a duopoly: balancing life-cycle revenues with product cost. Oper. Res. 53(1), 26–47 (2005)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Yamada, S.: Software Reliability Modeling—Fundamentals and Applications. Springer, New York (2014)CrossRefGoogle Scholar
  34. 34.
    Yamada, S., Tamura, Y.: OSS Reliability Measurement and Assessment. Springer, New York (2016)CrossRefGoogle Scholar
  35. 35.
    Yamada, S., Inoue, S., Satoh, D.: Statistical data analysis modeling based on difference equations for software reliability assessment. Trans. Jpn. Soc. Ind. Appl. Math. 12(2), 155–168 (2002) (in Japanese) Google Scholar
  36. 36.
    Yamakawa, P., Rees, G.H., Salas, J.M., Alva, N.: The diffusion of mobile telephones: an empirical analysis for Peru. Telecommun. Policy 37(6–7), 594–606 (2013)CrossRefGoogle Scholar
  37. 37.
    Young, P., Ord, J.: Model selection and estimation for technological growth curves. Int. J. Forecast. 5(4), 501–513 (1989)CrossRefGoogle Scholar

Copyright information

© The JJIAM Publishing Committee and Springer Japan KK, part of Springer Nature 2018

Authors and Affiliations

  1. 1.NTTMusashino-shiJapan

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