European Journal of Forest Research

, Volume 131, Issue 5, pp 1313–1326

Testing copula regression against benchmark models for point and interval estimation of tree wood volume in beech stands

  • Francesco Serinaldi
  • Salvatore Grimaldi
  • Mohammad Abdolhosseini
  • Piermaria Corona
  • Dora Cimini
Original Paper


This study compares copula regression, recently introduced in the forest biometric literature, with four benchmark regression models for computing wood volume V in forest stands given the values of diameter at breast height D and total height H, and suggests a set of statistical techniques for the accurate assessment of model performance. Two regression models deduced from the trivariate copula-based distribution of VD, and H are tested against the classical Spurr’s model and Schumacher-Hall’s model based on allometric and geometric concepts, and two regression models that rely on Box-Cox transformed variables and are in a middle ground, in terms of model complexity, between copula-based regression and classical models. The accuracy of the point estimates of V is assessed by a suitable set of performance criteria and the nonparametric sign test, whereas the associated uncertainty is evaluated by comparing empirical and nominal coverage probabilities of the prediction intervals. Focusing on point estimates, the Schumacher-Hall’s model outperforms the other models in terms of several performance criteria. The sign test points out that the differences among the models that involve D and H as separate covariates are not definitely significant, whereas these models outperform the models with a single covariate. As far as the interval estimates are of concern, the four benchmark models provide comparable interval estimates. The copula-based model with parametric marginals is definitely outperformed by its competitors according to all criteria, whereas the copula-based model with nonparametric marginals provides quite accurate point estimates but biased interval estimates of V.


Fagus sylvatica Weighted regression Box-Cox transformation Copula regression Normal quantile transformation Uncertainty 


  1. Akaike H (1974) A new look at the statistical model identification. IEEE Trans Autom Control 19(6):716–723CrossRefGoogle Scholar
  2. Baskerville GL (1972) Use of logarithmic regression in the estimation of plant biomass. Can J For Res 2:49–53CrossRefGoogle Scholar
  3. Box GEP, Cox DR (1964) An analysis of transformations. J R Stat Soc Ser B 26(2):211–252Google Scholar
  4. Christakos G (2011) Integrative problem-solving in a time of decadence. Springer, London, UKCrossRefGoogle Scholar
  5. Cienciala E, Èerný M, Apltauer J, Exnerová Z (2005) Biomass functions applicable to European beech. J For Sci 51(4):147–154Google Scholar
  6. Cleveland WS, Devlin SJ (1988) Locally-weighted regression: an approach to regression analysis by local fitting. J Am Stat Assoc 83(403):596–610CrossRefGoogle Scholar
  7. Cunia T (1979a) On tree biomass tables and regression: some statistical comments. In: Freyer WE (ed) Forest resource inventories, workshop proceedings, vol 2. Colorado State University, Colorado, pp 629–642Google Scholar
  8. Cunia T (1979b) On sampling trees for biomass tables construction: some statistical comments. In: Freyer WE (ed) Forest resource inventories, workshop proceedings, vol 2. Colorado State University, Colorado, pp 643–664Google Scholar
  9. Dawson CW, Abrahart RJ, See LM (2007) Hydrotest: a web-based toolbox of evaluation metrics for the standardised assessment of hydrological forecasts. Environ Modell Softw 22:1034–1052CrossRefGoogle Scholar
  10. Eamus D, Burrows W, McGuinness K, Australian Greenhouse Office (2000) Review of allometric relationships for estimating woody biomass for Queensland, the Northern Territory and Western Australia. Australian Greenhouse Office, Canberra,
  11. Embrechts P (2009) Copulas: a personal view. J Risk Insur 76(3):639–650CrossRefGoogle Scholar
  12. Fox J (2006) car: companion to applied regression.,, R package version 1.2-1
  13. Furnival GM (1961) An index for comparing equations used in constructing volume tables. For Sci 7:337–341Google Scholar
  14. Genest C, Favre AC (2007) Everything you always wanted to know about copula modeling but were afraid to ask. J Hydrol Eng 12(4):347–368CrossRefGoogle Scholar
  15. Genest C, Rémillard B (2008) Validity of the parametric bootstrap for goodness-of-fit testing in semiparametric models. Ann de l’Institut Henri Poincaré-Probabilités et Statistiques 44(6):1096–1127. doi:10.1214/07-AIHP148 CrossRefGoogle Scholar
  16. Genest C, Gendron M, Bourdeau-Brien M (2009) The advent of copulas in finance. Eur J Finance 15(7–8):609–618. doi:10.1080/13518470802604457 Google Scholar
  17. Hosking JRM (1990) L-moments: analysis and estimation of distributions using linear combinations of order statistics. J R Stat Soc Ser B 52:105–124Google Scholar
  18. Hosking JRM (1994) The four-parameter kappa distribution. IBM J Res Dev 38:251–258. doi:10.1147/rd.383.0251 CrossRefGoogle Scholar
  19. Hosking JRM (2009) L-moments., R package, version 1.5
  20. Hutson AD (2002) A semi-parametric quantile function estimator for use in bootstrap estimation procedures. Stat Comput 12:331–338CrossRefGoogle Scholar
  21. Hyndman RJ, Koehler AB (2006) Another look at measures of forecast accuracy. Int J Forecast 22:679–688CrossRefGoogle Scholar
  22. Jachner S, van den Boogaart KG, Petzoldt T (2007) Statistical methods for the qualitative assessment of dynamic models with time delay. J Stat Softw 22(8):1–30. doi:10.1007/s00703-006-0199-2 Google Scholar
  23. Joe H (2006) Discussion of copulas: tales and facts, by Thomas Mikosch. Extremes 9:37–41. doi:10.1007/s10687-006-0019-6 CrossRefGoogle Scholar
  24. Keith H, Barrett D, Keenan R, Australian Greenhouse Office (2000) Review of allometric relationships for estimating woody biomass for New South Wales, the Australian Capital Territory, Victoria, Tasmania and South Australia. Australian Greenhouse Office, Canberra,
  25. Kelly KS, Krzysztofowicz R (1997) A bivariate meta-gaussian density for use in hydrology. Stoch Hydrol Hydraul 11:17–31CrossRefGoogle Scholar
  26. Kershaw JA Jr, Richards EW, McCarter JB, Oborn S (2010) Original paper: spatially correlated forest stand structures: a simulation approach using copulas. Comput Electron Agric 74:120–128. doi:10.1016/j.compag.2010.07.005 CrossRefGoogle Scholar
  27. Kitanidis PK, Bras RL (1980) Real-time forecasting with a conceptual hydrologic model: 2. Application and results. Water Resour Res 16(6):1034–1044CrossRefGoogle Scholar
  28. Kojadinovic I, Yan J (2011) A goodness-of-fit test for multivariate multiparameter copulas based on multiplier central limit theorems. Stat Comput 21:17–30. doi:10.1007/s11222-009-9142-y CrossRefGoogle Scholar
  29. Kottegoda NT, Rosso R (2008) Applied statistics for civil and environmental engineers. 2nd edn. Wiley, New YorkGoogle Scholar
  30. Kvålseth TO (1985) Cautionary note about R2. Am Stat 39(4):279–285Google Scholar
  31. Leggett RW, Williams LR (1981) A reliability index for models. Ecol Model 13:303–312CrossRefGoogle Scholar
  32. Lehmann EL (1975) Nonparametrics, statistical methods based on ranks. McGraw-Hill, San FranciscoGoogle Scholar
  33. Mac Berthouex P, Brown LC (2002) Statistics for environmental engineers. 2nd edn. Lewis Publishers/CRC Press, Boca RatonGoogle Scholar
  34. Mercier G, Bouchemakh L, Smara Y (2007) The use of multidimensional copulas to describe amplitude distribution of polarimetric SAR data. In: Geoscience and remote sensing symposium, 2007. IGARSS 2007. IEEE International, pp 2236–2239 doi:10.1109/IGARSS.2007.4423284
  35. Meyer HA (1934) Die rechnerischen grundlagen der kontrollmethode. Beiheft zu den Zeitschriften der Forstvereins 13:122Google Scholar
  36. Khan NI, Farugue O (2010) Allometric relationships for predicting the stem volume in a dalbergia sissoo roxb. plantation in Bangladesh. iForest 3:153–158, doi:10.3832/ifor0554-003,
  37. Montanari A, Brath A (2004) A stochastic approach for assessing the uncertainty of rainfall-runoff simulations. Water Resour Res 40:W01,106. doi:10.1029/2003WR002540
  38. Montanari A, Grossi G (2008) Estimating the uncertainty of hydrological forecasts: a statistical approach. Water Resour Res 44:W00B08. doi:10.1029/2008WR006897 CrossRefGoogle Scholar
  39. Mood AM, Graybill FA, Boes DC (1974) Introduction to the theory of statistics. 3rd edn. McGraw-Hill, New YorkGoogle Scholar
  40. Napolitano G, Serinaldi F, See L (2011) Impact of EMD decomposition and random initialisation of weights in ANN hindcasting of daily stream flow series: An empirical examination. J Hydrol 406(3–4):199–214. doi:10.1016/j.jhydrol.2011.06.015 CrossRefGoogle Scholar
  41. Nash J, Sutcliffe J (1970) River flow forecasting through conceptual models part I—a discussion of principles. J Hydrol 10:282–290CrossRefGoogle Scholar
  42. Nelsen RB (2006) An introduction to copulas. 2nd edn. Springer, New YorkGoogle Scholar
  43. Parresol BR (1999) Assessing tree and stand biomass: a review with examples and critical comparisons. For Sci 45:573–593Google Scholar
  44. Pinheiro JC, Bates DM (2000) Mixed-effects models in S and Splus. Springer, New York, NYCrossRefGoogle Scholar
  45. Pinheiro J, Bates D, DebRoy S, Sarkar D, the R Core team (2009) nlme: linear and nonlinear mixed effects models. R package version 3.1-93Google Scholar
  46. R Development Core Team (2009) R: A language and environment for statistical computing. R foundation for statistical computing, Vienna, Austria,, ISBN 3-900051-07-0
  47. Reusser DE, Blume T, Schaefli B, Zehe E (2009) Analysing the temporal dynamics of model performance for hydrological models. Hydrol Earth Syst Sci 13:999–1018CrossRefGoogle Scholar
  48. Robinson AP, Hamann JD (2011) Forest analytics with R: an introduction. 1st edn. Springer, New YorkCrossRefGoogle Scholar
  49. Rupšys P, Petrauskas E (2010) Development of q-exponential models for tree height, volume and stem profile. Int J Phys Sci 5(15):2369–2378Google Scholar
  50. Salvadori G, De Michele C (2007) On the use of copulas in hydrology: theory and practice. J Hydrol Eng 12(4):369–380CrossRefGoogle Scholar
  51. Salvadori G, De Michele C, Kottegoda NT, Rosso R (2007) Extremes in nature: an approach using copulas. Springer, New YorkGoogle Scholar
  52. Schaefli B, Gupta HV (2007) Do Nash values have value? Hydrol Process 21:2075–2080CrossRefGoogle Scholar
  53. Schumacher F, Hall FDS (1933) Logarithmic expression of timber tree volume. J Agric Res 47:719–734Google Scholar
  54. Serinaldi F (2009) Assessing the applicability of fractional order statistics for computing confidence intervals for extreme quantiles. J Hydrol 376(3–4):528–541Google Scholar
  55. Serinaldi F (2011) Analytical confidence intervals for index flow flow duration curves. Water Resour Res 47:W02,542. doi:10.1029/2010WR009408
  56. Sklar A (1959) Fonction de répartition à n dimensions et leurs marges. Publications de l’Institut de Statistique de L’Université de Paris 8:229–231Google Scholar
  57. Sprugel DG (1983) Correcting for bias in log-transformed allometric equations. Ecology 64:209–210CrossRefGoogle Scholar
  58. Spurr SH (1952) Forest inventory. Ronald Press Co., New York, NYGoogle Scholar
  59. Tabacchi G, Di Cosmo L, Gasparini P (2011) Aboveground tree volume and phytomass prediction equations for forest species in Italy. Eur J For Res 1–24. doi:10.1007/s10342-011-0481-9
  60. Ter–Mikaelian MT, Korzukhin MD (1997) Biomass equations for sixty-five North American tree species. For Ecol Manage 97(1):1–24. doi:10.1016/S0378-1127(97)00019-4 CrossRefGoogle Scholar
  61. Tsallis C (2004) What should a statistical mechanics satisfy to reflect nature? Phys D Nonlinear Phenomena 193:3–34CrossRefGoogle Scholar
  62. Villarini G, Serinaldi F, Krajewski WF (2008) Modeling radar-rainfall estimation uncertainties using parametric and non-parametric approaches. Adv Water Resour 31:1674–1686CrossRefGoogle Scholar
  63. Wang M, Upadhyay A, Zhang L (2010) Trivariate distribution modeling of tree diameter, height, and volume. For Sci 56(3):290–300Google Scholar
  64. West PW (2009) Tree and forest measurement. 2nd edn. Springer, BerlinCrossRefGoogle Scholar
  65. Williams MS, Schreuder HT (2000) Guidelines for choosing volume equations in the presence of measurement error in height. Can J For Res 30:306–310CrossRefGoogle Scholar
  66. Yan J (2007) Enjoy the joy of copulas: with a package copula. J Stat Softw 21(4):1–21Google Scholar
  67. Zianis D, Mencuccini M (2004) On simplifying allometric analyses of forest biomass. For Ecol Manage 187(2–3):311–332. doi:10.1016/j.foreco.2003.07.007 CrossRefGoogle Scholar
  68. Zianis D, Muukkonen P, Mäkipää R, Mencuccini M (2005) Biomass and stem volume equations for tree species in Europe. Silva Fennica Monographs 4:1–63Google Scholar
  69. Zianis D, Xanthopoulos G, Kalabokidis K, Kazakis G, Ghosn D, Roussou O (2011) Allometric equations for aboveground biomass estimation by size class for Pinus Brutia Ten. trees growing in North and South Aegean Islands, Greece. Eur J For Res 130(2):145–160CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Francesco Serinaldi
    • 1
    • 2
  • Salvatore Grimaldi
    • 3
    • 4
    • 5
  • Mohammad Abdolhosseini
    • 6
  • Piermaria Corona
    • 3
  • Dora Cimini
    • 3
  1. 1.School of Civil Engineering and GeosciencesNewcastle UniversityNewcastle Upon TyneUK
  2. 2.Willis Research NetworkLondonUK
  3. 3.Dipartimento DIBAFUniversità della TusciaViterboItaly
  4. 4.Honors Center of Italian Universities, H2CU, Sapienza Università di RomaRomaItaly
  5. 5.Department of Mechanical and Aerospace EngineeringPolytechnic Institute of New York University, Six MetroTech CenterBrooklynUSA
  6. 6.Department of Water EngineeringCollege of Agriculture, Isfahan University of TechnologyIsfahanIran

Personalised recommendations