Two-Component Mixture Models for Diameter Distributions in Mixed-Species, Two-Age Cohort Stands

Abstract

Abstract The objectives of this study were to investigate the suitability of two-component Weibull and gamma mixtures to model the dbh distribution of a mixed-species, two-age cohort stands and for age cohort determination and to compare several methods to choose initial parameter values for maximum likelihood estimation of mixture models. Investigations were carried out in near-natural, fir (Abies alba Mill.)–beech (Fagus sylvatica L.), two-age cohort stands in the Świętokrzyski National Park (Central Poland), where unusually high mortality of fir followed by its recovery and revitalization has been observed. The age cohort YG1 is composed of trees from 60 to ∼150 years of breast height age, and the age cohort YG2 is composed of trees less than 60 years of breast height age. The empirical distributions for the stands in this study were equally well fit by both the mixture Weibull and gamma models. It has been assumed that the estimated values, the weights (fractions), the means, and the standard deviations of two-component mixture models, are the predicted values of dbh statistics of age cohorts. The mean absolute relative errors used to evaluate this assumption were least for age cohort YG2 (from 14.8 to 29.6%) and largest for age cohort YG1 (from 17.7 to 45.0%). The dbh component 1 of mixture models can be identified in the stands investigated with age cohort YG2 and to a lesser degree the dbh component 2 with age cohort YG1. The multistart method for choosing initial values for the numerical procedure (a combination of the expectation-maximization algorithm with the Newton-type method) was best but also the most labor-intensive. The optimal way to estimate parameters in two-component mixtures with the Weibull or the gamma distributions is to apply min/max and 0.5/1.5/mean methods and, additionally, but only if necessary, a multistart method.