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).
Highlights
The maximum degree of competition that forest populations of a species can sustain is described by the principle of self-thinning (Yoda et al 1963; Westoby 1984)
The aims of this study were to evaluate the performance of ordinary least squares (OLS) and power law (PWL) methods for estimating k in stand density index (SDI) calculations and to analyse the discrepancies between them
The corresponding quadratic mean diameter (QMD) for each of the 200 normally distributed N values was calculated with the following equation, derived from Eq (2) and assuming k = - 1.605 and b = 1.2Á105: sffiffiffi
Summary
The maximum degree of competition that forest populations of a species can sustain is described by the principle of self-thinning (Yoda et al 1963; Westoby 1984) This rule, a power function linearised by a logarithmic transformation, states that environmental resources can satisfy only a limited number of trees within a stand, and this number grows progressively smaller as tree age and size. After the boundary is exceeded, competition between individuals begins and, in the absence of disturbance, growth trends of dominated individuals progressively slow in favour of more competitive species. This process leads the population to a maximum number of plants of a given size that can coexist within a given unit area of land (Liira et al 2011; Vospernik and Sterba 2015)
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