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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.
In Poland, the majority of silver fir stands is characterised by a single−layer stand structure. The current silvicultural activities aim at obtaining multilayer stands, which requires a clearer definition of the structure, and more precisely, a better knowledge of the pattern of diameters at breast height (DBH) distribution. The aim of the research was to develop a pattern of DBH distribution of trees in a multilayer silver fir stand in the Świętokrzyskie Mountains (central Poland) using the BDq method, through the mathematical determination of function parameters, including the productive capacity of the habitat. The application of this pattern will allow the selection of the management methods aimed to obtain and then maintain model stands with a different layer structure. The research was based on the empirical material collected in 56 stands that were characterized by both a single− and a multi−layered structure. They grew in coniferous forest habitats, mostly in the upland mixed coniferous forest, upland mixed deciduous forest and upland deciduous forest or mountain forest. One big (up to 1 ha) or 3−5 smaller (0.04 ha) sample plots were established in each stand in which diameters at breast height of all trees and heights of usually 25 trees, selected from the entire range of diameters, were measured. These were used to determine the height growth curve equation coefficients and then the height of each tree. For each stand the basal area was calculated and a graph for DBH distribution was prepared. Site index was established using the original empirical equation. In order to develop a model for multilayer fir stands, individual parameters of the equation 5, such as the basal area (B), the target diameter at breast height (D) and the coefficient q were to be determined using the BDq method. Taking into consideration only stands with one−tailed DBH, individual parameters of the function were found to be associated with site index. Empirical equations were developed for the determination of B and D, and, based on the DBH distribution graphs, coefficient q associated with site index was also calculated (tab.). The developed DBH distribution pattern is a mathematical description of the target fir stand. When the value of stand parameters approaches those of the model stand, it should be managed using the shelterwood cutting system, which results from the spatial variation in the stand height structure, and consequently from the need for different silvicultural treatments in different stand fragments.
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.
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Optymalizacja rozkladu piersnic w lesie przerebowym

75%
Sylwan
|
2005
|
tom 149
|
nr 02
12-24
The study on the structure of Norway spruce old−growth stands of original provenience in upper mountain zone was carried out in a nature reserve ‘Śnieżnik Kłodzki', located in the Lądek Zdrój Forest District (SW Poland). The fieldwork was conducted in three spruce stands located at the altitude of 1,215 and 1,235 m a.s.l. The breast height diameters (dbh) of living trees were characterized. The aim of this study is to characterize the structure of dbh of trees in spruce stands of subalpine forest. Implementation of the objective will be associated with the characteristics of empirical distributions dbh and an indication of theoretical distributions of continuous random variable best approximating the structure of dbh. The comparison of empirical dbh distributions with 36 theoretical distributions were carried out, but goodness−of−fit tests proven statistically significant compatibility with seventeen of them: beta (4−parameter), Cauchy, exponential power, folded normal, gamma (3−parameter), generalized logistic, Laplace, logistic, loglogistic, loglogistic (3−parameter), lognormal (3−parameter), noncentral chi−square, normal, smallest extreme value, triangular, Weibull and Weibull (3−parameter). The dbh distributions in upper mountain spruce stands are the best fitted with logistic distribution, and subsequently loglogistic (3−parameter), generalized logistic and Weibull (3−parameter) distributions. That four distributions can be used in individual tree growth models while generating structure dbh of trees in the stand. The knowledge of dbh structure in protected stands could be helpful in shaping spruce stand structure in planned silvicultural treatments, and shows the need for silvicultural treatments in upper mountain spruce stands.
Although modelling of the diameter at the breast height (DBH) distributions has long history, theoretical discrete distributions have not so far been used for this purpose. In this study we use measurements covering 25 years (six inventories) without silvicultural influence on 14 even−aged Scots pine (Pinus sylvestris L.) stands to develop the DBH distribution models. Analysed stands are located in Murowana Goślina Forest Experimental Station (W Poland). The objective of the study was to elaborate the most simplified model that applies stand variables easy to assess and uncomplicated theoretical distribution. We employed two−parameter Gamma Poisson (GP) distribution and compared it to Weibull (W) and Sb Johnson (SbJ) ones. When maximum likelihood estimation (MLE) was used, GP gave similar results to W and both were slightly worse than SbJ. We found that both DBH standard deviation and stand density have substantial impact for model bias when above distributions used. Stepwise regression analysis was used for obtaining linear equations for parameter prediction of GP distribution. Then, GP model was simplified by removing standard deviation of DBH (SDD) from equation for overdispersion parameter. Results showed slight increase in mean of error values (i.e. modified Reynolds e index and root mean square error) for simplified model (SGP) compared with those for model including SDD, but differences in means were insignificant. Minimal and mean DBH represent enough variability of diameter distribution to obtain appropriate model based on Gamma Poisson distribution. Error was only 5% greater from four−parameter SbJ (MLE) distribution with similar range: 8.2−28.8% against 5.1−25.5% for SGP and SbJ respectively. The presented model can be used in many branches of forestry for more accurate calculation of stand level variables, when additional allometric equations employed, for instance assimilation apparatus volume or below− and above− ground biomass.
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