Nonlinear versus linearised model on stand density model fitting and stand density index calculation: analysis of coefficients estimation via simulation

Abstract

The stand density index, one of the most important metrics for managing site occupancy, is generally calculated from empirical data by means of a coefficient derived from the “self-thinning rule” or stand density model. I undertook an exploratory analysis of model fitting based on simulated data. I discuss the use of the logarithmic transformation (i.e., linearisation) of the relationship between the total number of trees per hectare (N) and the quadratic mean diameter of the stand (QMD). I compare the classic method used by Reineke (J Agric Res 46:627–638, 1933), i.e., linear OLS model fitting after logarithmic transformation of data, with the “pure” power-law model, which represents the native mathematical structure of this relationship. I evaluated the results according to the correlation between N and QMD. Linear OLS and nonlinear fitting agreed in the estimation of coefficients only for highly correlated (between − 1 and − 0.85) or poorly correlated (> − 0.39) datasets. At average correlation values (i.e., between − 0.75 and − 0.4), it is probable that for practical applications, the differences were relevant, especially concerning the key coefficient for Reineke’s stand density index calculation. This introduced a non-negligible bias in SDI calculation. The linearised log–log model always estimated a lower slope term than did the exponent of the nonlinear function except at the extremes of the correlation range. While the logarithmic transformation is mathematically correct and always equivalent to a nonlinear model in case of prediction of the dependent variable, the difference detected in my studies between the two methods (i.e., coefficient estimation) was directly related to the correlation between N and QMD in each simulated/disturbed dataset. In general, given the power law as the “natural” structure of the N versus QMD relationship, the nonlinear model is preferred, with a logarithmic transformation used only in the case of violation of parametric assumptions (e.g. data distributed non-normally).

Keywords

Ordinary least squares / Power law / Reineke function / Silviculture / Ecological mathematics / Forest mathematics

References

[1] Anfodillo T, Carrer M, Simini F, Popa I, BanavarJR MaritanA. An allometry-based approach for understanding forest structure, predicting tree-size distribution and assessing the degree of disturbance. Proc Biol Sci, 2013, 280: 20122375.

[2] Baskerville GL. Use of logarithmic regression in the estimation of plant biomass. Can J For, 1972, 2: 49-53.

[3] Bi H, Wan G, Turvey ND. Estimating the self-thinning boundary line as a density-dependent stochastic biomass frontier. Ecology, 2000, 81: 1477-1483.

[4] Cabon A, Mouillot F, Lempereur M, Ourcival J, Simioni G, Limousin J. Thinning increases tree growth by delaying drought-induced growth cessation in a Mediterranean evergreen oak coppice. For Ecol Manag, 2018, 409: 333-342.

[5] Castagneri D, Vacchiano G, Lingua E, Motta R. Analysis of intraspecific competition in two subalpine Norway spruce (Picea abies (L.) Karst.) stands in Paneveggio (Trento, Italy). For Ecol Manag, 2008, 255: 651-659.

[6] Corona P. Forestry research to support the transition towards a bio-based economy. Ann Silvic Res, 2015, 38: 37-38.

[7] Ducey MJ, Woodall CW, Bravo-Oviedo A. Climate and species functional traits influence maximum live tree stocking in the Lake States, USA. For Ecol Manag, 2017, 386: 51-61.

[8] Fabbio G, Cantiani P, Ferretti F, Di Salvatore U, Bertini G, Becagli C, Chiavetta U, Marchi M, Salvati L. Sustainable land management, adaptive silviculture, and new forest challenges: evidence from a latitudinal gradient in Italy. Sustainability, 2018, 10: 2520.

[9] Fassnacht FE, Hartig F, Latifi H, Bergerd C, Hernández J, Corvalán P, Koch B. Importance of sample size, data type and prediction method for remote sensing-based estimations of aboveground forest biomass. Remote Sens Environ, 2014, 154: 102-114.

[10] Ferrara C, Marchi M, Fares S, Salvati L. Sampling strategies for high quality time-series of climatic variables in forest resource assessment. iForest - Biogeosci For, 2017, 10: 739-745.

[11] Fonseca T, Duarte J. A silvicultural stand density model to control understory in maritime pine stands. iForest - Biogeosci For, 2017, 10: 829-836.

[12] Ge F, Zeng W, Ma W, Meng J. Does the slope of the self-thinning line remain a constant value across different site qualities? An implication for plantation density management. Forests, 2017, 8: 355.

[13] Liira J, Sepp T, Kohv K. The ecology of tree regeneration in mature and old forests: combined knowledge for sustainable forest management. J For Res, 2011, 16: 184-193.

[14] Marchi M, Chiavetta U, Cantiani P. Assessing the mechanical stability of trees in artificial plantations of Pinus nigra J. F. Arnold using the LWN tool under different site indexes. Ann Silvic Res, 2017, 41: 48-53.

[15] Marchi M, Ferrara C, Bertini G, Fares S, Salvati L. A sampling design strategy to reduce survey costs in forest monitoring. Ecol Indic, 2017, 81: 182-191.

[16] Marchi M, Paletto A, Cantiani P, Bianchetto E, De Meo I. Comparing thinning system effects on ecosystem services provision in artificial black pine (Pinus nigra J. F. Arnold) forests. Forests, 2018, 9: 188.

[17] Mason B, Connolly T. Long-term development of experimental mixtures of Scots pine (Pinus sylvestris L.) and silver birch (Betula pendula Roth.) in northern Britain. Ann Silvic Res, 2016, 40: 11-18.

[18] Newman MC. Regression analysis of log-transformed data: statistical bias and its correction. Environ Toxicol Chem, 1993, 12: 1129-1133.

[19] Newton PF. Asymptotic size–density relationships within self-thinning black spruce and jack pine stand-types: parameter estimation and model reformulations. For Ecol Manag, 2006, 226: 49-59.

[20] O’Neill GA, Hamann A, Wang TL. Accounting for population variation improves estimates of the impact of climate change on species’ growth and distribution. J Appl Ecol, 2008, 45: 1040-1049.

[21] Packard GC. Is logarithmic transformation necessary in allometry?. Biol J Linn Soc, 2013, 109: 476-486.

[22] Packard GC. On the use of log-transformation versus nonlinear regression for analyzing biological power laws. Biol J Linn Soc, 2014, 113: 1167-1178.

[23] Pecchi M, Marchi M, Giannetti F, Bernetti I, Bindi M, Moriondo M, Maselli F, Fibbi L, Corona P, Travaglini D, Chirici G. Reviewing climatic traits for the main forest tree species in Italy. iForest - Biogeosci For, 2019, 12: 173-180.

[24] Pommerening A, Särkkä A. What mark variograms tell about spatial plant interactions. Ecol Modell, 2013, 251: 64-72.

[25] Poschenrieder W, Biber P, Pretzsch H. An inventory-based regeneration biomass model to initialize landscape scale simulation scenarios. Forests, 2018, 9: 212.

[26] Pourmajidian MR, Jalilvand H, Fallah A, Hosseini SA, Parsakhoo A, Vosoughian A, Rahmani A. Effect of shelterwood cutting method on forest regeneration and stand structure in a Hyrcanian forest ecosystem. J For Res, 2010, 21: 265-272.

[27] Pretzsch H, Biber P. A re-evaluation of Reineke’ s rule and stand density index. For Sci, 2005, 51: 304-320.

[28] Ray D, Petr M, Mullett M, Bathgate A, Marchi M, Beauchamp K. A simulation-based approach to assess forest policy options under biotic and abiotic climate change impacts: A case study on Scotland’s National Forest Estate. For Policy Econ, 2017

[29] Reineke LH. Perfecting a stand-density index for even-aged forests. J Agric Res, 1933, 46: 627-638.

[30] Rivoire M, Le Moguedec G. A generalized self-thinning relationship for multi-species and mixed-size forests. Ann For Sci, 2012, 69: 207-219.

[31] Shaw JD (2006) Reineke’s stand density index: where are we and where do we go from here? Proceedings: Society of American Foresters 2005 National Convention. October 19–23, 2005, Ft. Worth, TX [published on CD-ROM]: Society of American Foresters, Bethesda, p 13

[32] Smith RJ. Logarithmic transformation bias in allometry. Am J Phys Anthropol, 1993, 90: 215-228.

[33] Solomon DS, Zhang L. Maximum size-density relationships for mixed softwoods in the northeastern USA. For Ecol Manag, 2002, 155: 163-170.

[34] Spearman C. The proof and measurement of association between two things. By C. Spearman, 1904. Am J Psychol, 1987, 100: 441-471.

[35] Vacchiano G. Valutazione dello Stand density index in popolamenti di abete bianco (Abies alba Mill.). Ital For e Mont, 2005, 3: 269-286.

[36] Vospernik S, Sterba H. Do competition-density rule and self-thinning rule agree?. Ann For Sci, 2015, 72: 379-390.

[37] Wang Y, Titus SJ, LeMay VM. Relationships between tree slenderness coefficients and tree or stand characteristics for major species in boreal mixedwood forests. Can J For Res, 1998, 28: 1171-1183.

[38] Weiskittel A, Gould P, Temesgen H. Sources of variation in the self-thinning boundary line for three species with varying levels of shade tolerance. For Sci, 2009, 55: 84-93.

[39] Westoby M. The self-thinning rule. Adv Ecol Res, 1984, 14: 167-225.

[40] Yoda K, Kira T, Ogawa H, Hozumi K. Self-thinning in overcrowded pure stands under cultivated and natural conditions (intraspecific competition among higher plants XI). J Biol, 1963, 14: 107-129.

[41] Zhang L, Bi H, Gove JH, Heath LS. A comparison of alternative methods for estimating the self-thinning boundary line. Can J For Res, 2005, 35: 1507-1514.

[42] Zhang X, Zhang J, Duan A, Deng Y. A hierarchical Bayesian model to predict self-thinning line for Chinese fir in Southern China. PLoS ONE, 2015, 10: e0139788.