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Two−component mixtures of either the Weibull distribution or the gamma distribution and the kernel density estimator were used for describing the diameter at breast height (dbh) empirical distributions of two−cohort stands. The data consisted of study 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). Kernel density estimators belong to a class of nonparametric density estimators. Nonparametric estimators have no fixed structure and depend upon all the data points to reach an estimate. In this study the Weibull and the gamma mixture distributions were the most versatile models. The results also support the conclusion that there are only minor differences between the parametric models and the kernel density estimates.
The goals of this study are (1) to analyse the accuracy of the approximation of empirical distributions of diameter at breast height (dbh) using two−component mixtures of either the Weibull distribution or the gamma distribution in two−cohort stands, and (2) to discuss the procedure of choosing goodness−of−fit tests. The study plots were located in the Świętokrzyski National Park (central Poland) and in the Southern Appalachian Mountains (eastern USA). The results of the goodness−of−fit tests (chi−squared, Kolmogorov−Smirnov, Cramér−von Mises, and Anderson−Darling), normalised bias and normalised root mean square error, indicate that dbh empirical distributions of two−cohort stands are compatible with the mixture models investigated. The chi−squared test and the generalization of the Anderson−Darling test to discrete distributions should be used to assess whether empirical dbh data are consistent with a hypothesized null distribution.
Study assessed the usefulness of various methods for choosing the initial values for the numerical procedures for estimating the parameters of mixture distributions and analysed variety of mixture models to approximate empirical diameter at breast height (dbh) distributions. Two−component mixtures of either the Weibull distribution or the gamma distribution were employed. The study plots, representing two−cohort stands, were located in the Świętokrzyski National Park (central Poland) and in the Southern Appalachian Mountains (eastern USA). A new strategy using three methods for choosing initial values (min.k/max.k for k=1, 5, 10; 0,5/1,5/mean; wp) for maximizing the likelihood during parameter estimation for mixture models for small and large plots is proposed.
The objectives of the study were (1) to determine the models of diameter at breast height (dbh) distributions in two− and multi−storied mixed stands with fir Abies alba Mill. and beech Fagus sylvatica L. as well as (2) to assess the usefulness of single Weibull distribution and two−component mixture of Weibull distribution to approximation of empirical dbh distributions for distinguished models of dbh structures. In the Świętokrzyskie Mountains, 21 sample plots ranging in size from 0.2 to 0.4 ha were established. To identify the models of dbh distributions, in two− and multi−storied stands with similar empirical dbh distributions, the hierarchical cluster analysis (HCA) with the Jaccard's measure and the Ward's minimum variance agglomeration method were used. To approximate dbh distributions, the single Weibull distribution and the two−component mixture of Weibull distribution were employed. In two− and multi−storied mixed stands with fir and beech, with the mean age between 50 and 70 at the dbh, four models of dbh distributions were determined (fig. 1). Two of them were decreasing, strongly asymmetric (OS and OJ models; fig. 2) and the other two were increasing in the initial phase and decreasing in the final, having two maximums (DM1 and DM2 models; fig. 3). In the stands with the complex structure mixed distribution should be used to approximate empirical data. The analysis revealed high suitability and versatility of Weibull distribution both as single form and two−component mixture.
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