A practical measure for determining if diameter (D) and height (H) should be combined into D2H in allometric biomass models

Abstract

Tree diameter at breast height (D) and tree height (H) are often used as predictors of individual tree biomass. Because D and H are correlated, the combined variable D2H is frequently used in regression models instead of two separate independent variables, to avoid collinearity related issues. The justification for D2H is that aboveground biomass is proportional to the volume of a cylinder of diameter, D, and height, H. However, the D2H predictor constrains the model to produce parameter estimates for D and H that have a fixed ratio, in this case, 2.0. In this paper we investigate the degree to which the D2H predictor reduces prediction accuracy relative to D and H separately and propose a practical measure, Q-ratio, to guide the decision as to whether D and H should or should not be combined into D2H. Using five training biomass datasets and two fitting approaches, weighted nonlinear regression and linear regression following logarithmic transformations, we showed that the D2H predictor becomes less efficient in predicting aboveground biomass as the Q-ratio deviates from 2.0. Because of the model constraint, the D2H-based model performed less well than the separate variable model by as much as 12 per cent with regard to mean absolute percentage residual and as much as 18 per cent with regard to sum of squares of log accuracy ratios. For the analysed datasets, we observed a wide variation in Q-ratios, ranging from 2.5 to 5.1, and a large decrease in efficiency for the combined variable model. Therefore, we recommend using the Q-ratio as a measure to guide the decision as to whether D and H may be combined further into D2H without the adverse effects of loss in biomass prediction accuracy.

Allometric Biomass Models for European Beech and Silver Fir: Testing Approaches to Minimize the Demand for Site-Specific Biomass Observations

“…In the case of silver fir, the parameters showed quite a different pattern compared to European beech and other species. The scaling exponent of D (i.e., 1.33), surprisingly, was smaller than the scaling exponent of H (i.e., 1.45), showing a Q-ratio of 0.92 [54]. That means, in the case of silver fir, a 1% increase in D (while H constant) produced a 1.33% increase in AGB, whereas a 1% increase in H (under constant D) produced a 1.45% increase in AGB.…”

“…For models based on D and H, the scaling exponent of D was approximately 2.0, whereas the scaling exponent of H was smaller than 1.0 for most species [35]. Consequently, it has also been shown that the ratio between the parameter estimate of D and parameter estimate of H (i.e., the Q-ratio) is frequently larger than 2.0, usually between 3 and 4 [54]. In the case of silver fir, the parameters showed quite a different pattern compared to European beech and other species.…”

Allometric Models for Estimating Tree Biomass of Dryland Secondary Forest in East Halmahera

“…Parameter-parameter model regresi (a, b, c) ditentukan melalui analisis regresi non-linier menggunakan metode Generalized Non-linear Least Square (GNLS) yang efektif untuk mereduksi heteroskedastisitas ragam sisaan model (Pinheiro et al, 2020). Analisis regresi non-linier dilakukan menggunakan paket program nlme (Pinheiro et al, 2020) dalam software R versi 3.6.3 (R Core Team, 2020), karena paket program tersebut menerapkan metode GNLS untuk menghitung parameter model alometrik biomassa (Dutcă et al, 2019;Huy et al, 2016;Tiryana et al, 2011).…”

The Effects of Combining the Variables in Allometric Biomass Models on Biomass Estimates over Large Forest Areas: A European Beech Case Study

“…However, combining the variables D and H into D 2 H is not that harmless. Dutcă et al [18] showed that combining the variables into D 2 H can affect the accuracy of biomass prediction under certain conditions. They proposed a practical measure, the Q-ratio:…”

“…where β11 and β12 are the parameter estimates from Equation (1), to guide the decision as to whether D and H may be combined into D 2 H without the adverse effects of loss in biomass prediction accuracy. The authors showed that there is no preference in using a logtransformation or a weighted nonlinear approach to fit the allometric model (Equation ( 1)) in order to calculate the Q-ratio (Equation (3)), since the differences in estimated Q-ratio were minor [18]. The combined variable model (Equation ( 2)) is the equivalent of the separate variable model (Equation ( 1)), when Q = 2.0, and, therefore, the combined variable model can be safely used instead of separate variable model when Q = 2.0.…”