iForest - Biogeosciences and Forestry


Modeling extreme values for height distributions in Pinus pinaster, Pinus radiata and Eucalyptus globulus stands in northwestern Spain

J Javier Gorgoso-Varela (1)   , J Daniel García-Villabrille (2), Alberto Rojo-Alboreca (2)

iForest - Biogeosciences and Forestry, Volume 9, Issue 1, Pages 23-29 (2015)
doi: https://doi.org/10.3832/ifor1447-008
Published: Jul 25, 2015 - Copyright © 2015 SISEF

Research Articles

Methods of estimating extreme height values can be used in forest modeling to improve fits to the marginal distribution of heights in the following bivariate diameter-height models: the SBB Johnson’s distribution, the bivariate beta (GDB-2) distribution, the bivariate Logit-Logistic (LL-2) distribution and the power-normal (PN) distribution. Some applications to LiDAR derived data are also possible, e.g., for error calibration. Practical applications in forest management may also be considered, e.g., for pruning. In probability theory and statistics, the generalized extreme value (GEV) distribution, also known as the Fisher-Tippett distribution, is a family of continuous probability distributions that combine the Gumbel, Fréchet and Weibull distributions. This study compared the three distributions for fitting extreme values of tree heights (maximum and minimum heights), which were measured in 185 permanent research plots in Pinus pinaster Ait. stands, 97 research plots in Pinus radiata D. Don stands, and 128 research plots in Eucalyptus globulus Labill. Most of the eucalyptus stands were measured three times giving a total of 304 measurements. All plots are located in northwestern Spain. The Bias, Mean Absolute Error (MAE) and Mean Square Error (MSE) of the mean relative frequency of trees were used to evaluate the goodness-of-fit of the different functions, as well as the Kolmogorov-Smirnov statistic Dn. The Gumbel and the Weibull cumulative distribution functions (CDFs) proved suitable for describing extreme values of height distributions of the above-mentioned tree species in northwestern Spain. The Fréchet distribution was only used to model maximum values and yielded the poorest results in all cases.


Gumbel, Fréchet, Weibull, Minimum Height, Maximum Height

Authors’ address

J Javier Gorgoso-Varela
Departamento de Biología de Organismos y Sistemas, Universidad de Oviedo, Escuela Politécnica de Mieres, c/ Gonzalo Gutiérrez Quirós s/n, E-33600 Mieres, Asturias (Spain)
J Daniel García-Villabrille
Alberto Rojo-Alboreca
Unidade de Xestión Forestal Sostible (UXFS), Departamento de Enxeñaría Agroforestal, Universidade de Santiago de Compostela, Escola Politécnica Superior, Campus Universitario s/n, E-27002 Lugo, Galicia (Spain)

Corresponding author

J Javier Gorgoso-Varela


Gorgoso-Varela JJ, García-Villabrille JD, Rojo-Alboreca A (2015). Modeling extreme values for height distributions in Pinus pinaster, Pinus radiata and Eucalyptus globulus stands in northwestern Spain. iForest 9: 23-29. - doi: 10.3832/ifor1447-008

Academic Editor

Emanuele Lingua

Paper history

Received: Sep 18, 2014
Accepted: Mar 21, 2015

First online: Jul 25, 2015
Publication Date: Feb 21, 2016
Publication Time: 4.20 months

Breakdown by View Type

(Waiting for server response...)

Article Usage

Total Article Views: 22819
(from publication date up to now)

Breakdown by View Type
HTML Page Views: 17280
Abstract Page Views: 860
PDF Downloads: 3481
Citation/Reference Downloads: 20
XML Downloads: 1178

Web Metrics
Days since publication: 3252
Overall contacts: 22819
Avg. contacts per week: 49.12

Article Citations

Article citations are based on data periodically collected from the Clarivate Web of Science web site
(last update: Nov 2020)

Total number of cites (since 2016): 3
Average cites per year: 0.60


Publication Metrics

by Dimensions ©

Articles citing this article

List of the papers citing this article based on CrossRef Cited-by.

Bailey RL, Dell TR (1973)
Quantifying diameter distributions with the Weibull function. Forest Science 19: 97-104.
Online | Gscholar
Castedo-Dorado F, Ruiz-Gonzalez AD, Alvarez-González JG (2001)
Modelización de la relación altura-diámetro para Pinus pinaster Ait. en Galicia mediante la función de densidad bivariante SBB [Modeling the height-diameter relationship for Pinus pinaster Ait. in Galicia using the bivariate SBB function]. Investigación Agraria: Sistemas y Recursos Forestales 10 (1): 111-125. [in Spanish]
Online | Gscholar
Chen Q, Baldocchi D, Gong P, Kelly M (2006)
Isolating individual trees in a savanna woodland using small footprint lidar data. Photogrammetric Engineering and Remote Sensing 72: 923-932.
CrossRef | Gscholar
Coles S (2001)
An introduction to statistical modeling of extreme values. Springer-Verlag, London, UK, pp. 209.
CrossRef | Gscholar
Coops NC, Wulder MA, Culvenor DS, St-Onge B (2004)
Comparison of forest attributes extracted from fine spatial resolution multispectral and lidar data. Canadian Journal of Remote Sensing 30: 855-866.
CrossRef | Gscholar
Del Río M (1999)
Régimen de claras y modelo de producción para Pinus sylvestris L. en los sistemas Central e Ibérico [A thinning program and yield model for Pinus sylvestris L. in Spanish Central and Iberian Ranges]. PhD Thesis, Serie Forestal 2, INIA, Madrid, Spain, pp. 257. [in Spanish]
Fisher RA, Tippett LHC (1928)
Limiting forms of the frequency distribution of the largest or smallest member of a sample. Proceedings of the Cambridge Philosophical Society 24: 190-190.
CrossRef | Gscholar
Fonseca TF, Marques CP, Parresol BR (2009)
Describing maritime pine diameter distributions with Johnson’s SB distribution using a new all-parameter recovery approach. Forest Science 55(4): 367-373.
Online | Gscholar
Fréchet M (1927)
Sur la loi de probabilité de l’écart maximum [On the probabilistic law of maximum deviance]. Annales de la Sociéte Polonaise de Mathematique 6: 93. [in French]
Gerald CF, Wheatley PO (1989)
Applied numerical analysis (4th edn). Addison-Wesley Publishing Co, Reading, MS, USA, pp. 597.
Gorgoso JJ, Rojo A, Cámara-Obregón A, Diéguez-Aranda U (2012)
A comparison of estimation methods for fitting Weibull, Johnson’s SB and beta functions to Pinus pinaster, Pinus radiata and Pinus sylvestris stands in northwest Spain. Forest Systems 21 (3): 446-459.
CrossRef | Gscholar
Gorgoso-Varela JJ, Rojo-Alboreca A (2014)
Use of Gumbel and Weibull functions to model extreme values of diameter distributions in forest stands. Annals of Forest Science 71: 741-750.
CrossRef | Gscholar
Gumbel EJ (1954)
Statistical theory of extreme values and some practical applications. Applied Mathematics Series 33, US Department of Commerce, National Bureau of Standards, Washington, DC, USA, pp. 51.
Hall SA, Burke IC, Box DO, Kaufmann MR, Stoker JM (2005)
Estimating stand structure using discrete-return LiDAR: an example from low density, fire prone ponderosa pine forests. Forest Ecology and Management 208 (1-3): 189-209.
CrossRef | Gscholar
Holmgren J, Persson A (2004)
Identifying species of individual trees using airborne laser scanner. Remote Sensing of Environment 90 (4): 415-423.
CrossRef | Gscholar
Holmgren J, Nilsson M, Olsson H (2003)
Estimation of tree height and stem volume on plots using airborne laser scanning. Forest Science 49: 419-428.
Online | Gscholar
Johnson NL (1949)
Bivariate distributions based on simple translation systems. Biometrika 36: 297-304.
CrossRef | Gscholar
Knoebel BR, Burkhart HE (1991)
A bivariate distribution approach to modeling forest diameter distributions at two points in time. Biometrics 47: 241-253.
CrossRef | Gscholar
Li F, Zhang L, Davis CJ (2002)
Modeling the joint distribution of tree diameters and heights by bivariate generalized Beta distribution. Forest Science 48 (1): 47-58.
Online | Gscholar
Lim KS, Treitz PM (2004)
Estimation of aboveground forest biomass from airborne discrete return laser scanner data using canopy-based quantile estimators. Scandinavian Journal of Forest Research 19: 558-570.
CrossRef | Gscholar
Liu C, Zhang SY, Lei Y, Newton PF, Zhang L (2004)
Evaluation of three methods for predicting diameter distributions of black spruce (Picea mariana) plantations in central Canada. Canadian Journal of Forest Research 34: 2424-2432.
CrossRef | Gscholar
Maltamo M, Puumalainen J, Päivinen R (1995)
Comparison of beta and Weibull functions for modelling basal area diameter distribution in stands of Pinus sylvestris and Picea abies. Scandinavian Journal of Forest Reseach 10: 284-295.
CrossRef | Gscholar
MMAMRM (2011)
Cuarto Inventario Forestal Nacional [Fourth National Forest Inventory]. Ministerio de Medio Ambiente y Medio Rural y Marino, Galicia, Spain, pp. 52. [in Spanish]
Mønness E (2011)
The power-normal distribution: application to forest stands. Canadian Journal of Forest Research 41: 707-714.
CrossRef | Gscholar
Naesset E, Bjerknes KO (2001)
Estimating tree heights and number of stems in young forest stands using airborne laser scanner data. Remote Sensing of Environment 78 (3): 328-340.
CrossRef | Gscholar
Nanang DM (1998)
Suitability of the Normal, Log-normal and Weibull distributions for fitting diameter distributions of neem plantations in Northern Ghana. Forest Ecology and Management 103: 1-7.
CrossRef | Gscholar
Nanos N, Montero G (2002)
Spatial prediction of diameter distributions models. Forest Ecology and Management 161: 147-158.
CrossRef | Gscholar
Nilsson M (1996)
Estimation of tree heights and stand volume using an airborne LiDAR system. Remote Sensing of Environment 56 (1): 1-7.
CrossRef | Gscholar
Palahí M, Pukkala T, Blasco E, Trasobares A (2007)
Comparison of beta, Johnson’s SB, Weibull and truncated Weibull functions for modeling the diameter distribution of forest stands in Catalonia (north-east of Spain). European Journal of Forest Research 126: 563-571.
CrossRef | Gscholar
Parresol BR (2003)
Recovering parameters of Johnson’s SB distribution. Research Paper SRS-31, Southern Research Station, USDA Forest Service, Ashville, NC, USA, pp. 9.
Online | Gscholar
Persson K, Rydén J (2010)
Exponentiated Gumbel distribution for estimation of return levels of significant wave height. Journal of Environmental Statistics 1 (3): 1-12.
Online | Gscholar
Popescu SC, Zhao K (2008)
A voxel-based lidar method for estimating crown base height for deciduous and pine trees. Remote Sensing of Environment 112: 767-781.
CrossRef | Gscholar
Popescu SC, Wynne RH, Scrivani JA (2004)
Fusion of small-footprint lidar and multispectral data to estimate plot-level volume and biomass in deciduous and pine forests in Virginia, USA. Forest Science 50: 551-565.
Online | Gscholar
Rennolls K, Geary DN, Rollinson TJ (1985)
Characterizing diameter distributions by the use of the Weibull distribution. Forestry 58: 57-66.
CrossRef | Gscholar
Roberts SD, Dean TJ, Evans DL, McCombs JW, Harrington RL, Glass PA (2005)
Estimating individual tree leaf area in loblolly pine plantations using LiDAR-derived measurements of height and crown dimensions. Forest Ecology and Management 213 (1-3): 54-70.
CrossRef | Gscholar
SAS Institute Inc (2003)
SAS/STATTM user’s guide (Version 9.1). Cary, NS, USA, pp. 409.
Schmidt VM, Von Gadow K (1999)
Baumhöhenschätzung mit Hilfe der bivariaten Johnson’s SBB-Funktion [Individual tree high estimation by using the bivariate Johnson’s SBB function]. Forstw. Cbl. 118: 355-367. [in German]
CrossRef | Gscholar
Schreuder HT, Hafley WL (1977)
A useful bivariate distribution for describing stand structure of tree heights and diameters. Biometrics 33: 471-478.
CrossRef | Gscholar
Siipilehto J (2000)
A comparison of two parameter prediction methods for stand structure in Finland. Silva Fennica 34 (4): 331-349.
CrossRef | Gscholar
Stankova TV, Zlatanov TM (2010)
Modeling diameter distribution of Austrian black pine (Pinus nigra Arn. ) plantations: a comparison of the Weibull frequency distribution function and percentile-based projection methods. European Journal of Forest Research 129: 1169-1179.
CrossRef | Gscholar
Tewari VP, Von Gadow K (1997)
Fitting a bivariate distribution to diameter-height data of forest trees. Indian Forester 123: 815-820.
Online | Gscholar
Tewari VP, Von Gadow K (1999)
Modelling the relationship between tree diameters and heights using SBB distribution. Forest Ecology and Management 119: 171-176.
CrossRef | Gscholar
Wang M, Rennolls K (2005)
Tree diameter distribution modeling: introducing the logit-logistic distribution. Canadian Journal of Forest Research 35: 1305-1313.
CrossRef | Gscholar
Wang M, Rennolls K (2007)
Bivariate distribution modeling with tree diameter and height data. Forest Science 53 (1): 16-24.
Online | Gscholar
Watt P, Meredith A, Yang C, Watt MS (2013)
Development of regional models of Pinus radiata height from GIS spatial data supported with supplementary satellite imagery. New Zealand Journal of Forestry Science 43: 11.
Online | Gscholar
Weibull W (1951)
A statistical distribution function of wide applicability. Journal of Applied Mechanics 18 (3): 293-297.
Willemse WJ, Kaas R (2007)
Rational reconstruction of frailty-based mortality models by a generalisation of Gompertz’ law of mortality. Insurance: Mathematics and Economics 40 (3): 468-484.
CrossRef | Gscholar
Zhang L, Packard KC, Liu C (2003)
A comparison of estimation methods for fitting Weibull and Johnson’s SB distributions to mixed spruce-fir stands in northeastern North America. Canadian Journal of Forest Research 33: 1340-1347.
CrossRef | Gscholar
Zucchini W, Schmidt M, Von Gadow K (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 Fennica 35 (2): 169-183.
CrossRef | Gscholar

This website uses cookies to ensure you get the best experience on our website. More info