Modelling joint distribution of tree diameter and height using Frank and Plackett copulas
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ORIGINAL PAPER
Modelling joint distribution of tree diameter and height using Frank and Plackett copulas Friday Nwabueze Ogana1 • Jose Javier Gorgoso-Varela2 • Johnson Sunday Ajose Osho1
Received: 12 September 2018 / Accepted: 14 October 2018 Ó Northeast Forestry University 2018
Abstract Bivariate distribution models are veritable tools for improving forest stand volume estimations. Their accuracy depends on the method of construction. To-date, most bivariate distributions in forestry have been constructed either with normal or Plackett copulas. In this study, the accuracy of the Frank copula for constructing bivariate distributions was assessed. The effectiveness of Frank and Plackett copulas were evaluated on seven distribution models using data from temperate and tropical forests. The bivariate distributions include: Burr III, Burr XII, Logit-Logistic, Log-Logistic, generalized Weibull,
Weibull and Kumaraswamy. Maximum likelihood was used to fit the models to the joint distribution of diameter and height data of Pinus pinaster (184 plots), Pinus radiata (96 plots), Eucalyptus camaldulensis (85 plots) and Gmelina arborea (60 plots). Models were evaluated based on negative log-likelihood (-KK). The result show that Frank-based models were more suitable in describing the joint distribution of diameter and height than most of their Plackett-based counterparts. The bivariate Burr III distributions had the overall best performance. The Frank copula is therefore recommended for the construction of more useful bivariate distributions in forestry.
Project funding: This work was supported by the Government of Spain, Department of Economy, Industry and Competitiveness under the Torres Quevedo Contract PTQ-16-08445. The study also was financially supported by the Gobierno del Principado de Asturias through the project entitled ‘‘Estudio del crecimiento y produccion de Pinus pinaster Ait. en Asturias’’ (CN-07-094); by the Ministerio de Ciencia e Innovacio through the project entitled ‘‘Influencia de los tratamientos selvicolas de claras en la produccion, estabilidad mecanica y riesgo de incendios forestales en masas de Pinus radiata D. Don y Pinus pinaster Ait. en el Noroeste de Espana’’ (AGL200802259).
Keywords Bivariate distributions Frank copula Plackett copula Diameter Height
The online version is available at http://www.springerlink.com Corresponding editor: Tao Xu.
Electronic supplementary material The online version of this article (https://doi.org/10.1007/s11676-018-0869-1) contains supplementary material, which is available to authorized users. & Friday Nwabueze Ogana [email protected] 1
Department of Social and Environmental Forestry, University of Ibadan, Ibadan, Nigeria
2
Department of Innovation, Fo¨ra Forest Technologies, Lugo, Spain
Abbreviations CDF Cumulative distribution function PDF Probability density function Burr III-2F Bivariate Burr III distribution from Frank copula Burr III-2P Bivariate Burr III distribution from Plackett copula Burr XII-2F Bivariate Burr XII distribution from
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