Modelling diameter distributions of two-cohort forest stands with various proportions of dominant species: a two-component mixture model approach

Abstract

In recent years finite-mixture models have been employed to approximate and model empirical diameter at breast height (DBH) distributions. We used two-component mixtures of either the Weibull distribution or the gamma distribution for describing the DBH distributions of mixed-species, two-cohort forest stands, to analyse the relationships between the DBH components, age cohorts and dominant species, and to assess the significance of differences between the mixture distributions and the kernel density estimates. The data consisted of plots from the Świętokrzyski National Park (Central Poland) and areas close to and including the North Carolina section of the Great Smoky Mountains National Park (USA; southern Appalachians). The fit of the mixture Weibull model to empirical DBH distributions had a precision similar to that of the mixture gamma model, slightly less accurate estimate was obtained with the kernel density estimator. Generally, in the two-cohort, two-storied, multi-species stands in the southern Appalachians, the two-component DBH structure was associated with age cohort and dominant species. The 1st DBH component of the mixture model was associated with the 1st dominant species sp1 occurred in young age cohort (e.g., sweetgum, eastern hemlock); and to a lesser degree, the 2nd DBH component was associated with the 2nd dominant species sp2 occurred in old age cohort (e.g., loblolly pine, red maple). In two-cohort, partly multilayered, stands in the Świętokrzyski National Park, the DBH structure was usually associated with only age cohorts (two dominant species often occurred in both young and old age cohorts). When empirical DBH distributions representing stands of complex structure are approximated using mixture models, the convergence of the estimation process is often significantly dependent on the starting strategies. Depending on the number of DBHs measured, three methods for choosing the initial values are recommended: min.k/max.k, 0.5/1.5/mean, and multistart. For large samples (number of DBHs measured ≥ 80) the multistage method is proposed--for the two-component mixture Weibull or gamma model select initial values using the min.k/max.k (for k=1,5,10) and 0.5/1.5/mean methods, run the numerical procedure for each method, and when no two solutions are the same, apply the multistart method also.

Keywords: Gamma distribution; Initial values; Kernel density estimator; Parameter estimation; Two-component mixture model; Weibull distribution.