Modeling Pinus tree taper data using mixed-effects models

Abstract

Taper models are widely used to estimate log assortments and, consequently, forest yield from inventory data. However, for Pinus taeda, few studies have employed mixed-effects taper models that explicitly account for the hierarchical structure of forestry data and heterogeneity of variances. This study addresses this gap by developing and evaluating mixed-effects taper models based on modified versions of Kozak’s (1969) equation. The models incorporate random effects at the farm/forest region, stand, and tree levels and allow for different variance structures, enabling them to capture the heterogeneity commonly observed in P. taeda stands. Diagnostic procedures using least confounded residuals were applied to assess model adequacy. Compared with traditional fixed-effects taper models, the selected mixed-effects model achieved superior performance, including reduced bias, improved fit across stem sections, and better predictive accuracy. Additionally, in Appendices, we provide a tutorial outlining the computational procedures in R software for statistical modeling of data related to this species within the mixed-effects model framework.

Keywords

Aper modeling / Pinus taeda L. / Mixed-effects models / Least confounded residuals / Sustainable forest practices

References

[1] Adams JW (2005) Green weight, volume and taper equations for Virginia pine (Pinus virginiana) in the piedmont region of North Carolina. Master thesis, Graduate Faculty of North Carolina State University, Raleigh

[2] Alcarde R. Modelos lineares mistos em dados longitudinais com o uso do pacote ASReml-R, 2012, Tese de Doutorado, Universidade de São Paulo, São Paulo(in Portuguese)

[3] Alvares CA, Stape JL, Sentelhas PC, de Moraes Gonçalves JL, Sparovek G. Köppen’s climate classification map for Brazil. Metz, 2014, 22(6): 711-728

[4] Batista JLF (2014) Biometria florestal segundo o axioma da verossimilhança com aplicações em mensuração florestal. Universidade de São Paulo-USP, Piracicaba, pp 391. (in Portuguese)

[5] Behre EC. Preliminary notes on studies of tree form. J for, 1923, 21(5): 507-511

[6] Boczniewicz D, Mason EG, Morgenroth JA (2022) Developing fully compatible taper and volume equations for all stem components of Eucalyptus globoidea Blakely trees in New Zealand. N Z J For Sci 52. https://doi.org/10.33494/nzjfs522022x180x

[7] Calegario N, Daniels RF, Maestri R, Neiva R. Desenvolvimento de um modelo linear de efeito misto na estimativa do crescimento e produção de povoamentos clonais de Eucalyptus. Cerne, 2004, 10(1): 67-86(in Portuguese)

[8] Cao QV, Wang J. Calibrating fixed-and mixed-effects taper equations. For Ecol Manage, 2011, 262(4): 671-673

[9] Cui DY, Liang SL, Wang DD, Liu Z. A 1 km global dataset of historical (1979–2013) and future (2020–2100) Köppen-Geiger climate classification and bioclimatic variables. Earth Syst Sci Data, 2021, 13(11): 5087-5114

[10] da Silva BG, Demétrio CGB, Sermarini RA, Molenberghs G, Verbeke G, Behling A, Marques E, Accioly Y, Figura MA. Height-diameter models: a comprehensive review with new insights on relationships to generalized linear models and differential equations. Int Forest Rev, 2024, 26(4): 398-419

[11] Dantas D, Rodrigues Pinto LO, de Castro Nunes Santos Terra M, Calegario N, de Romarco Oliveira ML. Reduction of sampling intensity in forest inventories to estimate the total height of Eucalyptus trees. Bosque (Valdivia), 2020, 41(3): 353-364

[12] de Andra VCL. Novos modelos de taper do tipo expoente-Forma para descrever o perfil do fuste de árvores. Pesq Flor Bras, 2014, 34(80): 271-283

[13] de Oliveira XM, Scolforo HF, McTague JP, Dobner M, Scolforo JRS. A generalized mixed-effects height-diameter model for intensively managed Pinus taeda stands in Southern Brazil. Cerne, 2024, 30 e–103317

[14] Demaerschalk JP. Taper equations can be converted to volume equations and point sampling factors. For Chron, 1971, 47(6): 352-354

[15] Demaerschalk JP. Converting volume equations to compatible taper equations. For Sci, 1972, 18(3): 241-245

[16] Demaerschalk JP. Integrated systems for the estimation of tree taper and volume. Can J for Res, 1973, 3(1): 90-94

[17] Demaerschalk JP, Kozak A. The whole-bole system: a conditioned dual-equation system for precise prediction of tree profiles. Can J for Res, 1977, 7(3): 488-497

[18] Doyog ND, Lee YJ, Lee S. Stem taper equation analysis for Larix kaempferi species in the Central Region of South Korea. J Sustain for, 2017, 36(8): 747-763

[19] Gregoire TG, Schabenberger O. A non-linear mixed-effects model to predict cumulative bole volume of standing trees. J Appl Stat, 1996, 23(2–3): 257-272

[20] Hilden-Minton JA (1995) Multilevel diagnostics for mixed and hierarchical linear models. PhD thesis, University of California, Los Angeles

[21] Höjer A (1903) Bihang till fr. Lovén: Om vära barrskogar. Department of Agriculture, Forest Service, Stockholm, Sweden.

[22] Hradetzky J (1976) Analyse und interpretation statistischer abhängigkeiten:(biometrische beiträge zuaktuellen forschungsprojekten). (in German)

[23] Huang S, Price D, Titus SJ. Development of ecoregion-based height–diameter models for white spruce in boreal forests. For Ecol Manage, 2000, 129(1–3): 125-141

[24] IBÁ R (2023) Indústria Brasileira de Árvores – IBÁ. IBÁ, São Paulo, pp 91. (in Portuguese)

[25] IBÁ R (2024) Indústria Brasileira de Árvores – IBÁ. IBÁ, São Paulo, pp 99. (in Portuguese)

[26] Ko C, Kang JT, Mo Son Y, Kim DG. Estimating stem volume using stem taper equation for Quercus mongolica in South Korea. For Sci Technol, 2019, 15(2): 58-62

[27] Kozak A. A variable-exponent taper equation. Can J for Res, 1988, 18(11): 1363-1368

[28] Kozak A. My last words on taper equations. For Chron, 2004, 80(4): 507-515

[29] Kozak A, Smith JHG. Critical analysis of multivariate techniques for estimating tree taper suggests that simpler methods are best. For Chron, 1966, 42(4): 458-463

[30] Kozak A, Munro DD, Smith JHG. Taper functions and their application in forest inventory. For Chron, 1969, 45(4): 278-283

[31] Lanssanova R, Machado S, Orso G, Pelissari A, Filho AF, Silva F. Calibration of a mixed-effect stem taper model for Tectona grandis. J Trop for Sci, 2020, 32(4): 341-348

[32] Lappi J (1986) Mixed linear models for analyzing and predicting stem form variation of Scots pine

[33] Lee WK, Seo JH, Son YM, Lee KH, von Gadow K. Modeling stem profiles for Pinus densiflora in Korea. For Ecol Manag, 2003, 172(1): 69-77

[34] Li RX, Weiskittel A, Dick AR, Kershaw JAJr, Seymour RS. Regional stem taper equations for eleven conifer species in the Acadian Region of North America: development and assessment. North J Appl for, 2012, 29(1): 5-14

[35] Littell RC, Milliken GA, Stroup WW, Wolfinger RD. SAS system for mixed models, 1996, Cary, SAS Institute

[36] Liu CJ. Log volume estimation with spline approximation. For Sci, 1980, 26(3): 361-369

[37] López-Martínez JO, Vargas-Larreta B, Aguirre-Calderón OA, Aguirre-Calderón CG, Macario-Mendoza PA, Martínez-Salvador M, Álvarez-González JG. Compatible taper-volume systems for major tropical species in Mexico. Forestry (Lond), 2020, 93(1): 56-74

[38] Lynch TB, Zhao DH, Harges W, McTague JP. Deriving compatible taper functions from volume ratio equations based on upper-stem height. Can J for Res, 2017, 47(10): 1424-1431

[39] Machado SA, Filho AF (2014) Dendrometria–Segunda edição. Unicentro, Guarapuava, pp 316. (in Portuguese)

[40] Madi JS. Modelagem e otimização no sortimento de árvores de Tectona grandis L, 2024, Tese de Doutorado, Universidade Federal do Paraná, Curitiba(in Portuguese)

[41] McTague JP, Weiskittel A. Evolution, history, and use of stem taper equations: a review of their development, application, and implementation. Can J for Res, 2021, 51(2): 210-235

[42] Miguel EP (2009) Avaliação biométrica e prognose da produção de Eucalyptus urophylla (S.T. Blake) na região norte do estado de Goiás. Dissertação de Mestrado, Universidade Federal do Paraná, Curitiba. (in Portuguese)

[43] Monteiro BC, de Abreu JC, de Souza RLF, dos Santos BC, de Oliveira IR, de Lima RB. Uso de modelos mixtos para estimativa do volume de árvores individuais em tipologias florestais no estado do Amapá. Biota Amazonia, 2021, 11(2): 7-10

[44] Morrone S, Green PC. Regional differences in stem form between southern and northern red spruce (Picea rubens Sarg.) populations. Forestry, 2024, 97(5): 771-784

[45] Muhairwe CK. Taper equations for Eucalyptus pilularis and Eucalyptus grandis for the north coast in New South Wales, Australia. For Ecol Manage, 1999, 113(2–3): 251-269

[46] Newnham RM. Variable-form taper functions for four Alberta tree species. Can J for Res, 1992, 22(2): 210-223

[47] Nicoletti MF, de Pádua Chaves e Carvalho S, do Amaral Machado S, Costa VJ, Silva CA, Topanotti LR. Bivariate and generalized models for taper stem representation and assortments production of loblolly pine (Pinus taeda L.). J Environ Manage, 2020, 270 110865

[48] Nunes MH, Görgens EB. Artificial intelligence procedures for tree taper estimation within a complex vegetation mosaic in Brazil. PLoS ONE, 2016, 11(5 e0154738

[49] Ormerod DW. A simple bole model. For Chron, 1973, 493): 136-138

[50] Özçelik R, Diamantopoulou MJ, Brooks JR, Wiant HVJr. Estimating tree bole volume using artificial neural network models for four species in Turkey. J Environ Manage, 2010, 91(3): 742-753

[51] Özçelik R, Brooks JR, Jiang LC. Modeling stem profile of Lebanon cedar, Brutian pine, and Cilicica fir in Southern Turkey using nonlinear mixed-effects models. Eur J for Res, 2011, 130(4): 613-621

[52] Parresol BR. Assessing tree and stand biomass: a review with examples and critical comparisons. For Sci, 1999, 45(4): 573-593

[53] Pélico Netto S, Brena DA (1997) Inventário florestal. Curitiba, p 316. (in Portuguese)

[54] Pinheiro JC, Bates DM. Mixed-effects models in sand S-PLUS. Springer, New York, 2000

[55] Pinheiro JC, Bates DM, DebRoy S, Sarkar D, R Core Team (2021) nlme: Linear and Nonlinear Mixed Effects Models. R package version 3.1–153. https://CRAN.R-project.org/package=nlme

[56] Prodan M (1965) Holzmesslehre. Technical report, Sauerländer

[57] Quiñonez-Barraza G, Zhao DH, De los Santos-Posadas HM. Compatible taper and stem volume equations for five pine species in mixed-species forests in Mexico. For Sci, 2019, 65(5): 602-613

[58] R Core Team (2021) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria

[59] Resende RT, Borém A, Leite HG (2023) Pinus: do plantio à colheita. Oficina de Textos. (in Portuguese)

[60] Sahin A. Analyzing regression models and multi-layer artificial neural network models for estimating taper and tree volume in Crimean pine forests. Iforest, 2024, 17(1): 36-44

[61] Sakici OE, Ozdemir G. Stem taper estimations with artificial neural networks for mixed oriental beech and kazdaği fir stands in karabük region, Turkey. Cerne, 2018, 24(4): 439-451

[62] Salekin S, Catalán CH, Boczniewicz D, Phiri D, Morgenroth J, Meason DF, Mason EG. Global tree taper modelling: a review of applications, methods, functions, and their parameters. Forests, 2021, 12(7): 913

[63] Sánchez-Banda MB, De los Santos-Posadas HM, Valdez-Lazalde JR, Roldán-Cortés MA (2022) Heartwood functions and compatible taper and merchantable volume systems for Acacia mangium Willd. in, Tabasco, Mexico. Agrociencia. https://doi.org/10.47163/agrociencia.v56i1.2704

[64] Sanquetta CR, de Oliveira Piva LR, Sanquetta MNI, Maas GCB, Dalla Corte AP. Mercado de celulose no Brasil e em Cinco grandes países. BIOFIX Sci J, 2020, 52189

[65] Schöepfer W (1966) Automatisierung des Massen-, Sorten- und Wertberechnung stehender Waldbestände. WurttForstl, Berlin. (in German)

[66] Scolforo HF, McTague JP, Burkhart H, Roise J, Carneiro RL, Stape JL. Generalized stem taper and tree volume equations applied to Eucalyptus of varying genetics in Brazil. Can J for Res, 2019, 495447-462

[67] Singer JM, Rocha FM, Nobre JS (2024) Sub-rotinas computacionais. http://www.ime.usp.br/~jmsinger/lmmdiagnostics.zip [Accessed on 19 July 2025]

[68] Stefanello FR, Péllico Netto S, Behling A, Pelissari AL, Orso GA (2021) Preussner functions for volume estimation of Pinus taeda L. In: Southern Brazil. N Z J For Sci. 51: https://doi.org/10.33494/nzjfs512021x66x

[69] Stepka TF, Junior GZ, Lisboa GS, Cerqueira CL, Pesck VA, Roik M. Modelos volumétricos e funções de afilamento para Pinus taeda L. na região dos Campos Gerais, Paraná, Brasil. Espacios, 2017, 38(21): 26-34(in Portuguese)

[70] Tasissa G, Burkhart HE. An application of mixed effects analysis to modeling thinning effects on stem profile of loblolly pine. For Ecol Manag, 1998, 103(1): 87-101

[71] Thomas CE, Parresol BR. Simple, flexible, trigonometric taper equations. Can J for Res, 1991, 21(7): 1132-1137

[72] Trincado G, Burkhart HE. A generalized approach for modeling and localizing stem profile curves. For Sci, 2006, 52(6): 670-682

[73] Valentine HT, Gregoire TG. A switching model of bole taper. Can J for Res, 2001, 31(8): 1400-1409

[74] West BT, Welch KB, Galecki AT. Linear mixed models: a practical guide using statistical software, 2014, Boca Raton, CRC Press339

[75] Wilms F, Berendt F, Bronisz K, Bashutska U, Fotelli M, Radoglou K, Spyroglou G. Applying taper function models for black locust plantations in Greek post-mining areas. Sci Rep, 2024, 14 13557

[76] Zapata-Cuartas M, Bullock BP, Montes CR. A taper equation for loblolly pine using penalized spline regression. For Sci, 2021, 67(1): 1-13

[77] Zhang SS, Sun JJ, Duan AG, Zhang JG. Variable-exponent taper equation based on multilevel nonlinear mixed effect for Chinese fir in China. Forests, 2021, 12(2 126

[78] Zhang ZP, He P, Wang JJ, Xin SD, Jiang LC. Evaluation of taper measurement schemes for modeling stem profiles: a case study of two conifer species. Can J for Res, 2025, 55: 1-14

[79] Zhao DH, Kane M. New variable-top merchantable volume and weight equations derived directly from cumulative relative profiles for loblolly pine. For Sci, 2017, 633): 261-269

[80] Zhao DH, Lynch TB, Westfall J, Coulston J, Kane M, Adams DE. Compatibility, development, and estimation of taper and volume equation systems. For Sci, 2019, 65(1): 1-13

RIGHTS & PERMISSIONS

Northeast Forestry University