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

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 Świętokrzyski 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.