Derivation of Inflection Points of Nonlinear Regression Curves - Implications to Statistics
In this paper, we derive inflection points for the commonly known growth curves, namely, generalized logistic, Richards, Von Bertalanffy, Brody, logistic, Gompertz, generalized Weibull, Weibull, Monomolecular and Mitscherlich functions. The functions often represent the mean part of non-linear regression models in Statistics. Inflection point of a growth curve is the point on the curve at which the rate of growth gets maximum value and it represents an important physical interpretation in the respective application area. Not only the model parameters but also the inflection point of a growth curve is of high statistical interests.
Ayele Taye Goshu,
Purnachandra Rao Koya,
Derivation of Inflection Points of Nonlinear Regression Curves - Implications to Statistics, American Journal of Theoretical and Applied Statistics.
Vol. 2, No. 6,
2013, pp. 268-272.
Paine, C.E.T., Marthews, T.R., Vogt, D.R., Purves D., Rees M., Hector, A., Turnbull, L.A., "How to fit nonlinear plant growth models and calculate growth rates: an update for ecologists", Methods in Ecology and Evolution, Vol. 3(2012), 245–256 doi: 10.1111/j.2041-210X.2011.00155.x
Ayele T.,"Simulation study of the commonly used mathematical growth models", J. Ethio. Stat. Assoc., Vol. 17(2008), 44-53.
Edwards, Jr., C.H. and Penney, D.E., "Calculus with analytic geometry", Printice Hall International, New Jersey, 1994.
Fischer, S., "The role of macroeconomic factors in growth", Journal of Monetary Economics, Vol. 32(1993): 485 – 512.
Bawa, S. and Abdullahi, I.S., "Threshold effect of inflation on economic growth in Nigeria", CBN Journal of Applied Statistics, Vol. 3(2011), No.1, 43-63.
Ahmad, N.; Khan, M.G.M.; Rafi, L.S., "Analysis of an inflection s-shaped software reliability model considering log-logistic testing - effort and imperfect debugging", International Journal of Computer Science and Network Security, Vol. 11(2011), No. 1, 161-171.
Ahmad, N., Khan, M.G.M., Rofi, L.S., "A study of testing-effort dependent inflection s-shaped software reliability growth models with imperfect debugging", International Journal of Quality and Reliability Management, Vol. 27(2010), No. 1, 89-110.
Ahmad, N.; Khan, M.G.M. Rafi, L.S., "Inflection s-shaped software reliability growth models with testing-effort functions", VI International Symposium on Optimization and Statistics (ISOS-2008), Aligarh Muslim University, Aligarh, India, December 29-31, 2008.
http://epp.eurostat.ec.europa.eu/statistics_explained/index.php/Population_projections (accessed on August 2013).
http://www.td.com/document/PDF/economics/special/RecentMortgageRateHikesInCanada.pdf (accessed on August 2013).
Eberhardt, L.L. and Breiwick, J.M.,"Models for population growth curves", International Scholarly Research Network, ISRN Ecology (2012), doi:10.5402/2012/815016.
Ersoy, İ.E., Mendeş, M., Keskin, S., "Estimation of parameters of linear and nonlinear growth curve models at early growth stage in California Turkeys". Arch. Geflügelk. Vol. 71(2007), No.4, 175–180.
France, J., Dijkstra, J., MDhanoa, M.S., "Growth functions and their application in animal science", Ann. Zootechn Vol. 45(1996), 165-174.
Brown, J.E., Fitzhugh, Jr., H.A., Cartwright, T.C., "A comparison of nonlinear models for describing weight-age relationship in cattle", J. Animal Sci. Vol. 42(1976), 810-818.
Brody, S., "Bioenergetics and growth", Rheinhold Pub. Corp. N.Y, 1945.
Winsor, C.P., "The Gompertz curve as a growth curve", Proc. National Academy of Science, Vol. 18(1932), No.1.
Robertson, T.B., "On the normal rate of growth of an individual and its biochemical significance", Arch Entwicklungsmech Org Vol. 25(1906), 581-614.
Lei, Y.C. and Zhang, S.Y., "Features and partial derivatives of Bertalanffy-Richards growth model in forestry", Nonlinear Analysis: Modelling and Control, Vol. 9(2004), No. 1, 65-73.
Zeide, B., "Analysis of growth equations", Forest Science, Vol. 39(1993), No. 3, 594-616.
Fekedulegn, D., Mac Siurtain, M.P., Colbert, J.J., "Parameter estimation of nonlinear growth models in forestry", Silva Fennica Vol. 33(1999), No. 4, 327-336.
Koya, P.R., Goshu, A.T., "Solutions of rate-state equation describing biological growths", American Journal of Mathematics and Statistics, Vol. 3(2013), No. 6, 305-311, http://dx.doi.org/ 10.5923/j.ajms.20130306.02.
Koya, P.R., Goshu, A.T., "Generalized mathematical model for biological growths", Open Journal of Modelling and Simulation, Vol. 1(2013), 42-53 http://dx.doi.org/10.4236/ojmsi. 2013.14008.
http://en.wikipedia.org/w/index.php?oldid=472125857 (accessed on December 2012).
Richards, F.J., "A flexible growth function for empirical use", J. Exp. Bot. Vol. 10(1959), 290-300.
Bertalanffy, von L., "Quantitative laws in metabolism and growth", Quart. Rev. Biol. Vol. 3(1957), No. 2, 218.
Nelder, J.A., "The fitting of a generalization of the logistic curve", Biometrics Vol. 17(1961), 89-110.
Rawlings, J.O., Pantula, S.G., Dickey, D.A., "Applied regression analysis: a research tool". Second Edition, 1998.
Mombiela, R.A. and Nelson, L.A., "Relationships among some biological and empirical fertilizer response models and use of the power family of transformations to identify an appropriate model", Agronomy Journal, Vol. 73(1981), 353-356.