On Differential Growth Equation to Stochastic Growth Model Using Hyperbolic Sine Function in Height/Diameter Modeling of Pines

Oyamakin Samuel Oluwafemi; Chukwu Angela Unna

doi:doi:10.11648/j.pamj.20140305.12

| Peer-Reviewed

On Differential Growth Equation to Stochastic Growth Model Using Hyperbolic Sine Function in Height/Diameter Modeling of Pines

Oyamakin Samuel Oluwafemi, Chukwu Angela Unna

Published in Pure and Applied Mathematics Journal (Volume 3, Issue 5)

Received: 15 September 2014 Accepted: 30 September 2014 Published: 10 October 2014

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

Richrads growth equation being a generalized logistic growth equation was improved upon by introducing an allometric parameter using the hyperbolic sine function. The integral solution to this was called hyperbolic Richards growth model having transformed the solution from deterministic to a stochastic growth model. Its ability in model prediction was compared with the classical Richards growth model an approach which mimicked the natural variability of heights/diameter increment with respect to age and therefore provides a more realistic height/diameter predictions using the coefficient of determination (R2), Mean Absolute Error (MAE) and Mean Square Error (MSE) results. The Kolmogorov Smirnov test and Shapiro-Wilk test was also used to test the behavior of the error term for possible violations. The mean function of top height/Dbh over age using the two models under study predicted closely the observed values of top height/Dbh in the hyperbolic Richards nonlinear growth models better than the classical Richards growth model.

Published in	Pure and Applied Mathematics Journal (Volume 3, Issue 5)
DOI	10.11648/j.pamj.20140305.12
Page(s)	99-104
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Height, Dbh, Forest, Pinus Caribaea, Hyperbolic, Richards, Stochastic

References

[1]	Bertalanffy, L. von. 1957. Quantitative laws in metabolism and growth. Quantitative Rev. Biology 32: 218-231.
[2]	Brisbin IL: Growth curve analyses and their applications to the conservation and captive management of crocodilians. Proceedings of the Ninth Working Meeting of the Crocodile Specialist Group. SSCHUSN, Gland, Switzerland 1989.
[3]	Burkhart, H.E., and Strub, M.R. 1974. A model for simulation or planted loblolly pine stands. In growth models for tree and stand simulation. Edited by J. Fries. Royal College of Forestry, Stockholm, Sweden. Pp. 128-135. Model of forest growth. J. Ecol. 60: 849–873.
[4]	Curtis, R.O. (1967). Height-diameter and height-diameter-age equations for second-growth Douglas-fir. For. Sci. 13: 365-375.
[5]	Draper, N.R. and H. Smith, 1981. Applied Regression Analysis. John Wiley and Sons, New York
[6]	Fekedulegn, D. 1996. Theoretical nonlinear mathematical models in forest growth and yield modeling. Thesis, Dept. of Crop Science, Horticulture and Forestry, University College Dublin, Ireland. 200p.
[7]	EK, A.R. ,E.T. Birdsall, and R.J.Spear (1984). As simple model for estimating total and merchantable tree heights. USDA. For. Serv. Res. Note NC-309.5p
[8]	Kansal AR, Torquato S, Harsh GR, et al.: Simulated brain tumor growth dynamics using a three dimentional cellular automaton. J Theor Biol 2000, 203:367-82.Khamis, A. and Z. Ismail, 2004. Comparative study on nonlinear growth curve to tobacco leaf growth data. J. Agro, 3: 147 – 153.
[9]	Kingland S: The refractory model: The logistic curve and history of population ecology. Quart Rev Biol 1982, 57: 29-51.
[10]	Marusic M, Bajzer Z, Vuk-Pavlovic S, et al.: Tumor growth in vivo and as multicellular spheroids compared by mathematical models. Bull Math Biol 1994, 56:617-31.
[11]	Marquardt, D.W., 1963. An algorithm for least squares estimation of nonlinear parameters. Journal of the society for Industrial and Applied Mathematics, 11: 431 – 441.
[12]	Myers, R.H. 1996. Classical and modern regression with applications. Duxubury Press, Boston. 359p.
[13]	Nelder, J.A. 1961. The fitting of a generalization of the logistic curve. Biometrics 17: 89-110.
[14]	Olea N, Villalobos M, Nunez MI, et al.: Evaluation of the growth rate of MCF-7 breast cancer multicellular spheroids using three mathematical models. Cell Prolif 1994, 27:213-23.
[15]	Oyamakin S. O., Chukwu A. U., (2014): On the Hyperbolic Exponential Growth Model in Height/Diameter Growth of PINES (Pinus caribaea), International Journal of Statistics and Applications, Vol. 4 No. 2, 2014, pp. 96-101. doi: 10.5923/j.statistics.20140402.03.
[16]	Oyamakin, S. O.; Chukwu, U. A.; and Bamiduro, T. A. (2013) "On Comparison of Exponential and Hyperbolic Exponential Growth Models in Height/Diameter Increment of PINES (Pinus caribaea)," Journal of Modern Applied Statistical Methods: Vol. 12: Iss. 2, Article 24. Pp 381 – 404.
[17]	Philip, M.S., 1994. Measuring trees and forests. 2nd Edition CAB International, Wallingford, UK
[18]	Ratkowskay, D.A., 1983. Nonlinear Regression modeling. Marcel Dekker, New York. 276p
[19]	Seber, G. A. F. and C. J. Wild, 1989. Nonlinear Regression. John Wiley and Sons: NY
[20]	Vanclay, J.K. 1994. Modeling forest growth and yield. CAB International, Wallingford, UK. 380p.
[21]	Zeide B (1993): Analysis of growth equations. Forest Sci 1993, 39:594-616.

Cite This Article

Plain Text BibTeX RIS

APA Style

Oyamakin Samuel Oluwafemi, Chukwu Angela Unna. (2014). On Differential Growth Equation to Stochastic Growth Model Using Hyperbolic Sine Function in Height/Diameter Modeling of Pines. Pure and Applied Mathematics Journal, 3(5), 99-104. https://doi.org/10.11648/j.pamj.20140305.12

Copy | Download

ACS Style

Oyamakin Samuel Oluwafemi; Chukwu Angela Unna. On Differential Growth Equation to Stochastic Growth Model Using Hyperbolic Sine Function in Height/Diameter Modeling of Pines. Pure Appl. Math. J. 2014, 3(5), 99-104. doi: 10.11648/j.pamj.20140305.12

Copy | Download

AMA Style

Oyamakin Samuel Oluwafemi, Chukwu Angela Unna. On Differential Growth Equation to Stochastic Growth Model Using Hyperbolic Sine Function in Height/Diameter Modeling of Pines. Pure Appl Math J. 2014;3(5):99-104. doi: 10.11648/j.pamj.20140305.12

Copy | Download

@article{10.11648/j.pamj.20140305.12,
  author = {Oyamakin Samuel Oluwafemi and Chukwu Angela Unna},
  title = {On Differential Growth Equation to Stochastic Growth Model Using Hyperbolic Sine Function in Height/Diameter Modeling of Pines},
  journal = {Pure and Applied Mathematics Journal},
  volume = {3},
  number = {5},
  pages = {99-104},
  doi = {10.11648/j.pamj.20140305.12},
  url = {https://doi.org/10.11648/j.pamj.20140305.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20140305.12},
  abstract = {Richrads growth equation being a generalized logistic growth equation was improved upon by introducing an allometric parameter using the hyperbolic sine function. The integral solution to this was called hyperbolic Richards growth model having transformed the solution from deterministic to a stochastic growth model. Its ability in model prediction was compared with the classical Richards growth model an approach which mimicked the natural variability of heights/diameter increment with respect to age and therefore provides a more realistic height/diameter predictions using the coefficient of determination (R2), Mean Absolute Error (MAE) and Mean Square Error (MSE) results. The Kolmogorov Smirnov test and Shapiro-Wilk test was also used to test the behavior of the error term for possible violations. The mean function of top height/Dbh over age using the two models under study predicted closely the observed values of top height/Dbh in the hyperbolic Richards nonlinear growth models better than the classical Richards growth model.},
 year = {2014}
}

Copy | Download

TY  - JOUR
T1  - On Differential Growth Equation to Stochastic Growth Model Using Hyperbolic Sine Function in Height/Diameter Modeling of Pines
AU  - Oyamakin Samuel Oluwafemi
AU  - Chukwu Angela Unna
Y1  - 2014/10/10
PY  - 2014
N1  - https://doi.org/10.11648/j.pamj.20140305.12
DO  - 10.11648/j.pamj.20140305.12
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 99
EP  - 104
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20140305.12
AB  - Richrads growth equation being a generalized logistic growth equation was improved upon by introducing an allometric parameter using the hyperbolic sine function. The integral solution to this was called hyperbolic Richards growth model having transformed the solution from deterministic to a stochastic growth model. Its ability in model prediction was compared with the classical Richards growth model an approach which mimicked the natural variability of heights/diameter increment with respect to age and therefore provides a more realistic height/diameter predictions using the coefficient of determination (R2), Mean Absolute Error (MAE) and Mean Square Error (MSE) results. The Kolmogorov Smirnov test and Shapiro-Wilk test was also used to test the behavior of the error term for possible violations. The mean function of top height/Dbh over age using the two models under study predicted closely the observed values of top height/Dbh in the hyperbolic Richards nonlinear growth models better than the classical Richards growth model.
VL  - 3
IS  - 5
ER  -

Copy | Download

Author Information

Oyamakin Samuel Oluwafemi

Dept. of Statistics, University of Ibadan, Ibadan, Oyo State, Nigeria
Chukwu Angela Unna

Dept. of Statistics, University of Ibadan, Ibadan, Oyo State, Nigeria

Download PDF

Sections

Plain Text BibTeX RIS

APA Style

Oyamakin Samuel Oluwafemi, Chukwu Angela Unna. (2014). On Differential Growth Equation to Stochastic Growth Model Using Hyperbolic Sine Function in Height/Diameter Modeling of Pines. Pure and Applied Mathematics Journal, 3(5), 99-104. https://doi.org/10.11648/j.pamj.20140305.12

Copy | Download

ACS Style

Oyamakin Samuel Oluwafemi; Chukwu Angela Unna. On Differential Growth Equation to Stochastic Growth Model Using Hyperbolic Sine Function in Height/Diameter Modeling of Pines. Pure Appl. Math. J. 2014, 3(5), 99-104. doi: 10.11648/j.pamj.20140305.12

Copy | Download

AMA Style

Oyamakin Samuel Oluwafemi, Chukwu Angela Unna. On Differential Growth Equation to Stochastic Growth Model Using Hyperbolic Sine Function in Height/Diameter Modeling of Pines. Pure Appl Math J. 2014;3(5):99-104. doi: 10.11648/j.pamj.20140305.12

Copy | Download

@article{10.11648/j.pamj.20140305.12,
  author = {Oyamakin Samuel Oluwafemi and Chukwu Angela Unna},
  title = {On Differential Growth Equation to Stochastic Growth Model Using Hyperbolic Sine Function in Height/Diameter Modeling of Pines},
  journal = {Pure and Applied Mathematics Journal},
  volume = {3},
  number = {5},
  pages = {99-104},
  doi = {10.11648/j.pamj.20140305.12},
  url = {https://doi.org/10.11648/j.pamj.20140305.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20140305.12},
  abstract = {Richrads growth equation being a generalized logistic growth equation was improved upon by introducing an allometric parameter using the hyperbolic sine function. The integral solution to this was called hyperbolic Richards growth model having transformed the solution from deterministic to a stochastic growth model. Its ability in model prediction was compared with the classical Richards growth model an approach which mimicked the natural variability of heights/diameter increment with respect to age and therefore provides a more realistic height/diameter predictions using the coefficient of determination (R2), Mean Absolute Error (MAE) and Mean Square Error (MSE) results. The Kolmogorov Smirnov test and Shapiro-Wilk test was also used to test the behavior of the error term for possible violations. The mean function of top height/Dbh over age using the two models under study predicted closely the observed values of top height/Dbh in the hyperbolic Richards nonlinear growth models better than the classical Richards growth model.},
 year = {2014}
}

Copy | Download

TY  - JOUR
T1  - On Differential Growth Equation to Stochastic Growth Model Using Hyperbolic Sine Function in Height/Diameter Modeling of Pines
AU  - Oyamakin Samuel Oluwafemi
AU  - Chukwu Angela Unna
Y1  - 2014/10/10
PY  - 2014
N1  - https://doi.org/10.11648/j.pamj.20140305.12
DO  - 10.11648/j.pamj.20140305.12
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 99
EP  - 104
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20140305.12
AB  - Richrads growth equation being a generalized logistic growth equation was improved upon by introducing an allometric parameter using the hyperbolic sine function. The integral solution to this was called hyperbolic Richards growth model having transformed the solution from deterministic to a stochastic growth model. Its ability in model prediction was compared with the classical Richards growth model an approach which mimicked the natural variability of heights/diameter increment with respect to age and therefore provides a more realistic height/diameter predictions using the coefficient of determination (R2), Mean Absolute Error (MAE) and Mean Square Error (MSE) results. The Kolmogorov Smirnov test and Shapiro-Wilk test was also used to test the behavior of the error term for possible violations. The mean function of top height/Dbh over age using the two models under study predicted closely the observed values of top height/Dbh in the hyperbolic Richards nonlinear growth models better than the classical Richards growth model.
VL  - 3
IS  - 5
ER  -

Copy | Download