J. For. Sci., 2012, 58(6):245-252 | DOI: 10.17221/66/2011-JFS

Modelling of tree diameter growth using growth functions parameterised by least squares and Bayesian methods

R. Sedmák, L. Scheer
Department of Forest Management and Geodesy, Faculty of Forestry, Technical University in Zvolen, Zvolen, Slovak Republic

The purpose of this paper is to present a new growth and yield function (denoted as KM-function), which was empirically derived from the cumulative density function of the Kumaraswamy probability distribution. KM-function is theoretically well disposed for the prediction of future growth; however, the function also has other theoretical features that make it useful also for retrospective estimation of the past growth frequently used in biological analyses of growth in the initial life stages. In order to demonstrate the practical applicability of the KM-function for growth reconstruction, an investigation of the accuracy of five-year retrospective projections of the real tree diameters obtained by stem analyses of 35 beech trees was done. Bias and accuracy of the new function were compared with bias and accuracy of some well-known growth functions on the same database. Compared functions were parameterised in two ways: by the method of nonlinear least squares and Bayesian methods. Empirical validation of the KM-function confirmed its good theoretical properties when it was used for retrospective estimation of the tree diameter growth. The valuable knowledge of this research is also a finding that the incorporation of a great deal of a priori known facts about the growth of trees and stands in natural conditions of Slovakia into Bayesian parameter estimation led to a decrease in the bias and magnitude of reconstruction errors.

Keywords: growth function; nonlinear least squares; Bayes; diameter; estimation

Published: June 30, 2012  Show citation

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Sedmák R, Scheer L. Modelling of tree diameter growth using growth functions parameterised by least squares and Bayesian methods. J. For. Sci. 2012;58(6):245-252. doi: 10.17221/66/2011-JFS.
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    References

    1. Bock R.D., Du Toit S.H.C. (2003): Parameter estimation in the context of nonlinear longitudinal growth models. In: Hauspie R.C., Cameron N., Molinari L. (eds): Methods in Human Growth Research. Cambridge Studies in Biological and Evolutionary Anthropology No. 39. Cambridge, Cambridge University Press: 198-220.
    2. Carlin B.P., Louis T.A. (2000): Bayes and Empirical Bayes Methods for Data Analysis. Texts in Statistical Science. Chapman & Hall/CRC, Boca Raton-London-New YorkWashington, DC: 419. Go to original source...
    3. D'Agostini G. (2003): Bayesian inference in processing experimental data principles and basic applications. Reports on Progress in Physics, 66: 1-40. Go to original source...
    4. Ek A.R., Monserud R.A. (1979): Performance and comparison of stand growth models based on individual tree diameter. Canadian Journal of Forest Research, 9: 231-244. Go to original source...
    5. Fabrika M. (2005): Rastový simulátor SIBYLA (koncepcia, konštrukcia, softvérové riešenie). [Growth simulator SIBYLA (conception, construction, software solution)]. [Habilitation Thesis.] Zvolen, Technical University in Zvolen: 187.
    6. Fitzburgh H.A. (1976): Analysis of growth curves and strategies for altering their shape. Journal of Animal Science, 42: 1036-1051. Go to original source... Go to PubMed...
    7. Halaj J. (1957): Matematicko-štatistický prieskum hrúbkovej štruktúry slovenských porastov. [Mathematical and statistical research of diameter structures of Slovak stands.] Lesnícky časopis, 3: 39-74.
    8. Halaj J., Grék J., Pánek F., Petráš R., Řehák J. (1987): Rastové tabuľky hlavných drevín ČSSR. [Growth and Yield Tables of Main Tree Species in Czechoslovakia.] Bratislava, Príroda: 362.
    9. Halaj J., Petráš R. (1998): Rastové tabuľky hlavných drevín pre priemerné prírodné pomery Slovenska. [Growth and Yield Tables of Main Tree Species in Slovakia.] Bratislava, Slovak Academic Press: 325.
    10. Karkach A.S. (2006). Trajectories and models of individual growth. Demographic Research, 15: 347-400. Go to original source...
    11. Korf V. (1939): Příspevěk k matematické formulaci vzrůstového zákona lesních porostů. [Contribution to mathematical definition of the law of stand volume growth.] Lesnická práce, 18: 339-379.
    12. Kumaraswamy P. (1980): A generalized probability density function for double-bounded random processes. Journal of Hydrology, 46: 79-88. Go to original source...
    13. Lunn D.J., Thomas A., Best N., Spiegelhalter D. (2000): WinBUGS - a Bayesian modelling framework: concepts, structure, and extensibility. Statistics and Computing, 10: 325-337. Go to original source...
    14. Pagan J. (1992): Forestry Dendrology. Zvolen, Technical university in Zvolen: 347.
    15. Pretzsch H., Biber P., Ďurský J. (2002): The single treebased stand simulator SILVA: construction, application and evaluation. Forest Ecology and Management, 162: 3-21. Go to original source...
    16. Pretzsch H. (2009): Forest Dynamics, Growth and Yield. Springer-Verlag, Berlin Heidelberg: 664. Go to original source...
    17. Richards F.J. (1959): A flexible growth function for empirical use. Journal of Experimental Botany, 10: 290-300. Go to original source...
    18. Seber G.A.F., Wild C.J. (2003): Nonlinear Regression. Wiley Series in Probability and Statistics. New Jersey, John Wiley and Sons, Inc.: 768. Go to original source...
    19. Sedmák R. (2009): Modelovanie rastu bukových stromov a porastov. [Growth and Yield Modelling of Beech Trees and Stands.] [Ph.D. Thesis.] Zvolen, Technical University in Zvolen: 181.
    20. StatSoft, Inc. (2005). Electronic Statistics Textbook. Tulsa. Available at http://www.statsoft.com/textbook/
    21. Šmelko Š. (1982): Biometrické zákonitosti rastu a prírastku lesných stromov a porastov. [Biometrical Laws of Growth and Yield of Forest Trees and Stands.] Bratislava, VEDA: 183.
    22. Šmelko Š., Wenk G., Antanaitis V. (1992): Rast, štruktúra a produkcia lesa. [Forest Growth, Structure and Yield.] Bratislava, Príroda: 342.
    23. Tsoularis A., Wallace J. (2002): Analysis of logistic growth models. Mathematical Biosciences, 179: 21-25. Go to original source... Go to PubMed...
    24. Van Laar A., Akça A. (2007): Forest Mensuration. 2nd Ed. Dordrecht, Springer: 383. Go to original source...
    25. Vanclay J.K. (1994): Modelling Forest Growth and Yield. (Application to mixed tropical forests). Wallingford, CAB International: 312.
    26. Weibull W. (1951): A statistical distribution of wide applicability. Journal of Applied Mechanics, 18: 293-297. Go to original source...
    27. Yin X., Goudriaan J., Lantinga E.A., Vos J., Spiertz H.J. (2003): A flexible sigmoid function of determinate growth. Annals of Botany, 91: 361-371. Go to original source... Go to PubMed...
    28. Xiao Y. (2005): An equation of biological windows and its application. Ecological Modelling, 181: 521-533. Go to original source...
    29. Zeide B. (1993): Analysis of growth equations. Forest Science, 39: 594-616. Go to original source...
    30. Zhang L. (1997): Cross-validation of non-linear growth functions for modelling tree height-diameter relationships. Annals of Botany, 79: 251-257. Go to original source...

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