Modelling irregular and multimodal tree diameter distributions by finite mixture models: an approach to stand structure characterisation

Abstract

Tree structural diversity is assessed by modelling stand diameter at breast height (DBH) distribution. The aim of this study was to verify the suitability of a mixture of two-and three-component Weibull and gamma models for describing irregular and multimodal DBH distributions. Investigations were carried out in natural Abies alba Mill. and Fagus sylvatica L. stands, representing the growing-up stage, in the Ś więtokrzyski National Park (Central Poland) and in the Pieniny National Park (Southern Poland). Sample plots (0.25-0.5 ha in area) were selected from a database used for forest dynamics analysis. The parameters of the mixture model were estimated using maximum likelihood estimation (MLE) and a combination of an expectation-maximisation (EM) algorithm with a Newton-type (NT) method. The multistart method (MM) of choosing initial values was used. The DBH distributions in mixed Abies-Fagus forests of complex structure were accurately approximated by two-and three-component mixture models. The root mean square error (RMSE) values varied from 3.4515 to 7.6682 for the Weibull mixture and from 3.6457 to 7.1419 for the gamma mixture. The results indicated that the mixture Weibull and gamma models were, in general, equally suitable for modelling irregular and multimodal DBH distributions. Finite mixture models may be used to characterise tree structural diversity, especially in uneven-aged stands of complex structure.

Tree diameter structural diversity in Central European forests withAbies albaandFagus sylvatica: managed versus unmanaged forest stands

“…The normal, Weibull, gamma and negative exponential distribution as well as the finite mixture normal, Weibull and gamma models consisting of two components were employed. These distributions, especially models with Weibull and gamma functions, are very useful for fitting the empirical DBH data in Central European A. alba-F. sylvatica forests (Podlaski and Zasada 2008;Podlaski 2011a, b;Jaworski and Podlaski 2012). The normal, Weibull, gamma and negative exponential distributions have the probability density functions (PDFs) given by:…”

Forest structure as a predictor of tree species diversity in the North Carolina Piedmont

“…While there is not a distinct biological basis for this function, it derives its utility from its superior performance in fitting a wide variety of stand‐size distributions, despite having just two parameters (Rennolls et al. ; Jaworski & Podlaski ). For a distribution starting at zero, the Weibull probability distribution function is expressed as:where x is the size or age class, λ is the scale parameter, and k is the shape parameter such that k > 0 and λ > 0 (Weibull ).…”

“…We also adopt the Weibull function to describe stand size distribution for its flexible statistical properties and its long history in forestry applications (Weibull 1951;Leak 1964;Bailey & Dell 1973). While there is not a distinct biological basis for this function, it derives its utility from its superior performance in fitting a wide variety of stand-size distributions, despite having just two parameters (Rennolls et al 1985;Jaworski & Podlaski 2011). For a distribution starting at zero, the Weibull probability distribution function is expressed as:…”

Generating Virtual Even-Aged Silver Fir Stand Structure Based on the Measured Sample Plots

“…The deviations were primarily a result of the mathematical fitting of diameter distribution and the variability of tree heights within the same diameter class. Although the best possible fitting functions were used, they are still equations, and therefore they cannot completely stochastically describe all the variability in a forest stand [29]. In Figure 3 it can be seen that particular subpopulations are not continuously represented in all diameter classes, but appear intermittently in fitted distribution.…”

“…According to Jaworski and Podlaski [29] and Burkhart and Tomé [30], the diameter distribution with diameter classes of 5 cm is fitted by using one of the following functions: normal distribution [31], logarithmic normal distribution [32], Weibull's function [33] or gamma distribution. The fitted distribution which has the lowest values of parameters AIC and BIC is taken to be optimal, as according to Dziak et al [34].…”