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Modelling joint distribution of tree diameter and height using Frank and Plackett copulas

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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 (−ΛΛ). 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.

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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 Frank copula

Burr XII-2P :

Bivariate Burr XII distribution from Plackett copula

LL-2F :

Bivariate Logit-Logistic distribution from Frank copula

LL-2P :

Bivariate Logit-Logistic distribution from Plackett copula

LogL-2F :

Bivariate Log-Logistic distribution from Frank copula

LogL-2P :

Bivariate Log-Logistic distribution from Plackett copula

Gweibull-2F :

Bivariate generalized Weibull distribution from Frank copula

Gweibull-2P :

Bivariate generalized Weibull distribution from Plackett copula

Weibull-2F :

Bivariate Weibull distribution from Frank copula

Weibull-2P :

Bivariate Weibull distribution from Plackett copula

Kum-2F :

Bivariate Kumaraswamy distribution from Frank copula

Kum-2P :

Bivariate Kumaraswamy distribution from Plackett copula

−ΛΛ:

Negative loglikelihood value

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Authors and Affiliations

  1. Department of Social and Environmental Forestry, University of Ibadan, Ibadan, Nigeria

    Friday Nwabueze Ogana & Johnson Sunday Ajose Osho

  2. Department of Innovation, Föra Forest Technologies, Lugo, Spain

    Jose Javier Gorgoso-Varela

Authors
  1. Friday Nwabueze Ogana
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  2. Jose Javier Gorgoso-Varela
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  3. Johnson Sunday Ajose Osho
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Corresponding author

Correspondence to Friday Nwabueze Ogana.

Additional information

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” (AGL2008-02259).

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Corresponding editor: Tao Xu.

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Ogana, F.N., Gorgoso-Varela, J.J. & Osho, J.S.A. Modelling joint distribution of tree diameter and height using Frank and Plackett copulas. J. For. Res. 31, 1681–1690 (2020). https://doi.org/10.1007/s11676-018-0869-1

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  • DOI: https://doi.org/10.1007/s11676-018-0869-1

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