Comparing Johnson’s SBB, Weibull and Logit-Logistic bivariate distributions for modeling tree diameters and heights using copulas
Abstract
Aim of study: In this study we compare the accuracy of three bivariate distributions: Johnson’s SBB, Weibull-2P and LL-2P functions for characterizing the joint distribution of tree diameters and heights.
Area of study: North-West of Spain.
Material and methods: Diameter and height measurements of 128 plots of pure and even-aged Tasmanian blue gum (Eucalyptus globulus Labill.) stands located in the North-west of Spain were considered in the present study. The SBB bivariate distribution was obtained from SB marginal distributions using a Normal Copula based on a four-parameter logistic transformation. The Plackett Copula was used to obtain the bivariate models from the Weibull and Logit-logistic univariate marginal distributions. The negative logarithm of the maximum likelihood function was used to compare the results and the Wilcoxon signed-rank test was used to compare the related samples of these logarithms calculated for each sample plot and each distribution.
Main results: The best results were obtained by using the Plackett copula and the best marginal distribution was the Logit-logistic.
Research highlights: The copulas used in this study have shown a good performance for modeling the joint distribution of tree diameters and heights. They could be easily extended for modelling multivariate distributions involving other tree variables, such as tree volume or biomass.
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References
Castedo-Dorado F, Ruiz-González AD, Álvarez-González JG, 2001. Modelización de la relación altura-diámetro para Pinus pinaster Ait. en Galicia mediante la función de densidad bivariante SBB. Invest. Agrar Sist Recur For 10(1): 111-125.
Hafley WL, Schreuder HT, 1977. Statistical distributions for fitting diameter and height data in even-aged stands. Can J For Res 7(3): 481-487. http://dx.doi.org/10.1139/x77-062
Johnson NL, 1949. Bivariate distributions based on simple translation systems. Biometrika 36: 297-304. http://dx.doi.org/10.1093/biomet/36.3-4.297
Knoebel BR, Burkhart HE, 1991. A bivariate distribution approach to modelling forest diameter distributions at two points of time. Biometrics 47: 241-253. http://dx.doi.org/10.2307/2532509
Li F, Zhang L, Davis CJ, 2002. Modeling the joint distribution of tree diameters and heights by bivariate generalized Beta distribution. For Sci 48(1): 47-58.
Mardia KV, 1970. Families of bivariate distributions. Griffin, London, UK. 231 pp.
Mønness E, 2015. The bivariate power-normal distribution and the bivariate Johnson system bounded distribution in forestry, including height curves. Can J For Res 45(3): 307-313. http://dx.doi.org/10.1139/cjfr-2014-0333
Plackett RL, 1965. A class of bivariate distributions. J Am Stat Assoc 60: 516-522. http://dx.doi.org/10.1080/01621459.1965.10480807
R Core Team, 2014. R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. URL http://www.R-project.org/.
Rennolls K, Wang M, 2005. A new parameterization of Johnson's SB distribution with application to fitting forest tree diameter data. Can J For Res 35(3): 575-579. http://dx.doi.org/10.1139/x05-006
Rupsys P, Petrauskas E, 2010. The Bivariate Gompertz Diffusion Model for Tree Diameter and Height Distribution. For Sci 56(3): 271-280.
Schreuder HT, Hafley WL, 1977. A useful bivariate distribution for describing stand structure of tree heights and diameters. Biometrics 33: 471-478. http://dx.doi.org/10.2307/2529361
Sklar A, 1973. Random variables, joint distribution functions and copulas. Kybernetika 9: 449-460.
Staudhammer CL, LeMay VM, 2001. Introduction and evaluation of possible indices of stand structural diversity. Can J For Res 31: 1105-1115. http://dx.doi.org/10.1139/x01-033
Tewari VP, Gadow Kv, 1999. Modelling the relationship between tree diameters and heights using SBB distribution. For Ecol Manage 119: 171-176.
Wang M, Rennolls K, 2007. Bivariate Distribution Modeling with Tree Diameter and Height Data. For Sci 53(1): 16-24.
Wang M, Rennolls K, Tang S, 2008. Bivariate Distribution Modeling of Tree Diameters and Heights: Dependency Modeling Using Copulas. For Sci 54(3): 284-293.
Zucchini W, Schmidt M., Gadow Kv, 2001. A model for the diameter-height distribution in an uneven-aged beech forest and a method to assess the fit of such models. Silva Fenn 35(2): 169-183. http://dx.doi.org/10.14214/sf.594
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