Comparison of beta,Johnson’s SB,Weibull and truncated Weibull functions for modeling the diameter distribution of forest stands in Catalonia (north-east of Spain)

The purpose of this study was to compare the beta, Johnson SB, Weibull and truncated Weibull functions in describing the diameter distributions of forest stands in Catalonia. The data consisted of permanent sample plots from the Spanish National Forest Inventory in Catalonia. The empirical data represent left-truncated distributions, as the smallest diameter measured in the field was 7.5 cm. A total of 1,242 plots were used to fit the functions and analyze their performance. The distribution functions were fitted to the diameter distributions of the number of stems (DD_N) and stand basal area (DD_G). The performance of the candidate functions was compared by means of their bias and RMSE for different diameter sums measuring the difference between the empirical and fitted distributions. The leftmost part (from 0 to 7.5 cm) of the non-truncated functions was ignored in this analysis. Sensitivity analyses were conducted to check whether the results depended on the number of trees measured in the stand, or the main species of the stand. The truncated Weibull function for the diameter distribution of stand basal area appeared to be in all cases the most accurate and consistent function. Generally, functions describing the distribution of stand basal area performed better than functions that described the distribution of the number of trees. Of the basal area distributions, beta and Johnson’s SB were the second best and nearly equally good with each other. The order of precision of the tested functions was: truncated Weibull for DD_G, truncated Weibull for DD_N, Johnson’s SB for DD_G, beta for DD_G, beta for DD_N and Weibull for DD_G, Weibull for DD_N, and Johnson’s SB for DD_N.     

Modelling the diameter distribution of Pinus sylvestris, Pinus nigra and Pinus halepensis forest stands in Catalonia using the truncated Weibull function

Parameter prediction models for the diameter distribution ofPinus sylvestris L., Pinus nigra Arn. and Pinus halepensis Mill.in Catalonia were developed using the truncated Weibull functionas the theoretical distribution. The parameter models allowone to use individual-tree models in the simulation of standdevelopment when only stand-level data are collected in forestinventories. Parameter models for the diameter distributionof stand basal area were developed. The data consisted of permanentsample plots from the Spanish National Forest Inventory in Catalonia.A total of 1780 empirical distributions of P. sylvestris, 1204distributions of P. nigra and 1535 distributions of P. halepensiswere used as modelling data. The empirical data represent left-truncateddistributions, as the smallest diameter measured in the fieldwas 7.5 cm. Two different approaches, namely, regression (two-stepmethod) and optimization approach (one-step method), were usedto find the coefficients of the parameter models. In the two-stepmodelling method, the Weibull parameters were first estimatedseparately for every empirical distribution by maximizing thelog-likelihood function of the Weibull density function. Inthe second-step, regression analysis was used to find the relationshipbetween Weibull parameters and stand basal area, number of treesper hectare and elevation of the site. The one-step method usedoptimization to find such coefficients for the parameter models,which minimized the mean of the squared differences betweenempirical and predicted cumulative tree frequencies in the wholemodelling data. The one-step optimization method performed betterthan the two-step regression method for all tree species. Theparameter prediction models developed in this study enable theprediction of the diameter distribution of P. sylvestris, P.nigra and P. halepensis in Catalonia from limited stand information.     

A model for predicting log yield from stand characteristics

A method for forecasting the log‐class distribution resulting from harvesting forest stands is investigated. The diameter distribution of a stand is approximated by a two‐parameter Weibull density function, the parameters of which are recovered from stand level characteristics. Theoretical bucking is performed for actual and simulated distributions and the resulting log yields are compared. An application to 40 stands indicates that an accurate prognosis can be expected, given a good approximation of the diameter distribution. For cases with multimodal diameter distributions or many small trees, the prognosis is unreliable.     

Characterising the diameter distribution of Sal plantations by comparing normal,lognormal and Weibull distributions at Tilagarh Eco-park,Bangladesh

For many years foresters have been using statistical probability density functions to describe and characterise stand structure. Predicting the current and future yields of a stand is essential for successful stand and timber management. Implicit prediction of current yield is accomplished by using diameter distribution methods. All diameter distribution yield systems predict the number of trees per unit area by diameter class. In this study, the normal, lognormal and the three-parameter Weibull probability density function were compared to characterise the diameter distributions of Sal (Shorea robusta) plantations grown at Tilagarh Eco-park, Bangladesh. Data from 70 plots, established in three plantations, were used for this study. The Weibull parameters were estimated by the maximum likelihood and moments estimator methods. A one-sample Kolmogorov–Smirnov test was used for the goodness of fit for all models. The Kolmogorov–Smirnov test results showed that both lognormal and Weibull distributions were suitable to characterise the diameter distributions of Sal plantations in the study area and may be applicable for other Sal forests in Bangladesh.     

A growth model for Pinus radiata D. Don stands in north-western Spain

A dynamic whole-stand growth model for radiata pine (Pinus radiata D. Don) stands in north-western Spain is presented. In this model, the initial stand conditions at any point in time are defined by three state variables (number of trees per hectare, stand basal area and dominant height), and are used to estimate total or merchantable stand volume for a given projection age. The model uses three transition functions derived with the generalized algebraic difference approach (GADA) to project the corresponding stand state variables at any particular time. These equations were fitted using the base-age-invariant dummy variables method. In addition, the model incorporates a function for predicting initial stand basal area, which can be used to establish the starting point for the simulation. Once the state variables are known for a specific moment, a distribution function is used to estimate the number of trees in each diameter class by recovering the parameters of the Weibull function, using the moments of first and second order of the distribution. By using a generalized height-diameter function to estimate the height of the average tree in each diameter class, combined with a taper function that uses the above predicted diameter and height, it is then possible to estimate total or merchantable stand volume.     

抚育间伐对栓皮栎人工林直径和树高分布的影响

【目的】以河南登封林场栓皮栎人工林为研究对象,研究抚育间伐对林分不同生长阶段林木株数、林木直径分布和树高分布的影响,为制定科学合理的抚育经营措施奠定理论和技术基础。【方法】在株数强度为31.55%的间伐林分和条件基本一致的未间伐林分内,分别设置1个1 hm2样地,间伐作业2 a后获取每木检尺数据,分别利用Normal分布、Gamma分布、Lognormal分布、Weibull分布、Logistic分布函数对间伐样地和对照样地栓皮栎人工林林木个体直径分布、树高分布进行拟合,并应用Kolmogorov-Smirnov(KS)拟合优度检验,选择拟合效果最好的分布模型。【结果】1)间伐和对照样地幼树、小树和大树林木株数比例分别是9:642:582和229:978:585。相比对照样地,间伐样地林木小树和大树生长阶段株数比重较高。2)间伐样地林木直径分布范围主要集中在8~18 cm,高于对照样地的4~16 cm。KS拟合优度检验结果表明,间伐样地林木直径分布采用Weibull分布拟合效果最好,Normal和Logistic分布次之;对照样地林木直径分布采用Weibull分布拟合效果最好,Normal和Gamma分布次之。3)间伐样地和对照样地林木树高级分别为8~16 m和4~14 m,但KS拟合优度检验结果表明,5种分布函数对间伐样地和对照样地树高分布拟合优度排序一致,说明抚育间伐对林分树高分布的影响不明显。【结论】抚育间伐调整了栓皮栎人工林林分的直径结构,林分直径分布向较大径阶方向偏移,使林分直径分布结构更趋于对称分布,但对树高分布影响不明显。     

Diameter distributions in Picea ables described by the Weibull model

The three‐parameter Weibull function met specified statistical standards for goodness of fit as a model for the diameter distribution of moderately thinned Norway spruce stands in Denmark. Weibull distributions estimated by percentile estimators fit the majority of 522 observed diameter distributions (material made available by the Danish Forest Experiment Station) at the 10% level of significance in the Chi‐square and Kolmogoroff‐Smirnow tests. The Weibull distribution was less suited to describe the diameter distribution in unthinned stands and heavily thinned stands. Weibull parameter predictions were developed and used in connection with a Danish yield table for Norway spruce on clay‐rich coastal soils. The use of the Weibull distribution to predict stem frequencies in different diameter classes before and after a specified thinning of stands is demonstrated.     

Predicting and calibrating diameter distributions of Eucalyptus grandis (Hill) Maiden plantations in Zimbabwe

Diameter distribution models for even-aged Eucalyptus grandis plantations in Zimbabwe were developed using the two-parameter Weibull function. The analysis was based on data from Correlated Curve Trend (CCT) experiments replicated on four different sites. Parameters of the Weibull distribution were predicted using stand characteristics as regressors. Two sets of parameter models were estimated: a set with and one without stand basal area as a predictor. Stand variables such as dominant height, age, site index and number of stems were used in both sets. The models were further calibrated to result in a given number of stems and stand basal area simultaneously. The usability of constructed models was tested both in prediction of yield in a stand inventory situation and in simulation of growth in connection with different growth models. The results indicated that models not including stand basal area produce considerably less precise stand volume estimates compared to models including also stand basal area. Calibration improved the accuracy of diameter distribution models. In growth simulation diameter distribution models can be connected both to single tree growth models and to stand projection models. The usability of calibration in growth simulation depends on the accuracy of the prediction of stand characteristics.     

Stand-level diameter distribution yield model for black spruce plantations

The objectives of this study were to develop and demonstrate a stand-level diameter distribution yield model and associated algorithm for black spruce (Picea mariana (Mill.) B.S.P) plantations. Employing a parameter prediction approach within the context of a stand density management diagram (SDMD), model development consisted of four sequential steps: (1) obtaining maximum likelihood estimates for the location, scale and shape parameters of the Weibull probability density function (PDF) for 296 empirical diameter frequency distributions; (2) developing and evaluating parameter prediction equations in which the parameter estimates of the Weibull PDF were expressed as functions of stand-level variables employing stepwise regression and seemingly unrelated regression techniques; (3) explicitly incorporating the parameter prediction equations into the SDMD modelling framework; and (4) developing an associated PC-based algorithm and demonstrating its utility in density management decision-making. The results indicated that the parameter prediction equations described 74.4, 87.1 and 66.8% of the variation in location, scale and shape parameter estimates, respectively. Incorporating the parameter prediction equations into the structure of the SDMD enabled the prediction of the temporal dynamics of the diameter frequency distribution by density management regime, site quality and region. An algorithmic version of the model is provided as a decision-support aid in which forest managers are able to simultaneously contrast multiple density management regimes in terms productivity, product value and optimal site occupancy.     

林分材种出材量表编制理论和方法的研究

本文论述了直径分布在林分结构和林分材种出材量表中的重要意义。介绍了使用Weibull直径函数研制林分材种出材量表的技术和方法。根据辽宁省人工落叶松林分资料,系统地介绍了利用一致性削度方程和Weibull直径函数编制林分材种出材量表的技术和方法。最后,对林分材种出材量表的一些技术和方法问题进行了讨论。     

Fitting irregular diameter distributions of forest stands by Weibull, modified Weibull, and mixture Weibull models

Irregular diameter frequency distributions of forest stands include multimodal structure of mixed-species stands, highly skewed and highly irregular shapes of uneven-aged stands, and rotated sigmoid form of old-growth stands. In this study, a traditional two-parameter Weibull model, a modified two-parameter Weibull model, and a finite mixture of two-parameter Weibull models were used to fit four artificial example plots. The model fitting and comparison results indicate that the mixture Weibull model is more flexible to fit various irregular diameter distributions, while the traditional Weibull model fails in every case to adequately describe these frequency distributions. The modified Weibull model is a good choice for fitting the “rotated-sigmoid” diameter distribution of an uneven-aged old-growth stand. However, it may not be sufficient when a diameter frequency distribution is multimodal or highly irregular in shape.     

金沟岭林场云冷杉过伐林林分直径结构的研究

以吉林省汪清林业局金沟岭林场云冷杉过伐林112块固定样地1987—2003年间4次调查数据为依据,分析其株数径阶分布特点,并用Weibull分布和负指数分布描述其直径结构。研究表明:该林分直径结构为异龄林结构;Weibull分布函数和负指数分布函数均能表达云冷杉过伐林直径结构,Weibull分布函数拟合的效果比负指数分布好。     

Comparing normal,lognormal and Weibull distributions for fitting diameter data from Akashmoni plantations in the north-eastern region of Bangladesh

Statistical probability density functions are widely used to model tree diameter distributions and to describe stand structure. The objective of this study was to compare the performance of normal, logarithmic-normal and three-parameter Weibull distributions for fitting diameter data from Akashmoni (Acacia auriculiformis A. Cunn. ex Benth.) plantations grown in the north-eastern region of Bangladesh. Data from 96 plots, established in 24 plantations in north-eastern Bangladesh and ranging in age from 1 to 6 years, were used for this study. The parameters of the Weibull distribution were calculated using maximum likelihood estimation (MLE) and moment estimation (ME) methods. The goodness of fit of normal, lognormal, Weibull MLE and Weibull ME were tested using one-sample Kolmogorov-Smirnov (KS) tests. The KS test results showed that both lognormal and Weibull distributions were equally effective for describing the diameter distributions of these Akashmoni plantations grown in the north-eastern region of Bangladesh.     

Modeling frequency distributions of tree height,diameter and crown area by six probability functions for open forests of <Emphasis Type="Italic">Quercus persica</Emphasis> in Iran

The prediction of the distribution of quantitative variables in a forest stand is of great interest to forest managers, for the evaluation of forest resources and scheduling of future silvicultural treatments. The aim of this research was to model the distribution of quantitative variables for Quercus persica in open forests in Iran. To investigate the probability distribution of trees in natural stands, 642 trees were selected for measurement using a systematic random sampling method. Selected trees were measured and data were analyzed. Gamma, beta, normal,lognormal, exponential and Weibull probability distributions were fitted to the height distribution of trees. Variables of distribution functions were estimated using the maximum likelihood estimation method. Actual probability and probability which derived from functions was compared using Kolmogorov–Smirnov and Anderson–Darling tests. Beta, Weibull and Weibull probability distributions explained the distributions of tree height, DBH and crown area.     

油松、侧柏人工林树冠率的Weibull分布及动态预测

应用Weibull函数来拟合油松(Pinus tabulaeformis)、侧柏(Platycladus orientalis)人工林树冠率的分布,研究发现林分平均树冠率与相对林分密度指数之间呈极显著线性关系,并随林分相对密度指数增加而降低;Weibull函数参数b与油松、侧柏人工林林分平均树冠率呈极显著线性关系,参数c与侧柏林分平均树冠率呈二次方关系,在油松林分中则为一常数值;通过对未参加建模的油松、侧柏样地进行动态预测,发现Weibull函数对林分树冠率分布预测值与实测值之间没有显著性差异。因此,该方法可以应用于预测油松和侧柏人工林树冠率的动态分布。     

辽东山区天然次生林林分直径分布模型研究

以峰度、偏度和变动系数为指标研究了辽东山区天然次生林直径株数分布特征。运用Johnson’s SB分布、Gamma分布、3参数和2参数Weibull分布等4种概率密度函数研究了辽东山区天然次生林直径株数分布规律,同时利用Kolmogorov-Smirnov(KS)检验法来检验4种概率分布的拟合效果,结果表明:林分直径的株数分布以Johnson’s SB分布拟合效果最好,Weibull分布(包括2参数和3参数)仅仅适合中度区和弱度区,但不是最好。所有的分布模型均通过了优势度检验,说明均适用于直径株数分布的拟合。     

Estimation of stem volume using laser scanning-based canopy height metrics

The aim of this study was to test different stem volume predictorsthat are capable of utilizing laser scanning-based canopy heightmetrics as independent variables. The three laser scanning-basedmethods compared were (1) a direct prediction model for thestem volume at plot level, (2) a volume prediction system basedon the modelled percentiles of the basal area diameter distribution,and (3) a parameter prediction method used to determinate Weibull-basedbasal area diameter distributions for the plot-level stem volumeprediction. The predicted volumes were also compared with field-measuredvolumes obtained with the Finnish conventional inventory bycompartments. The best results were obtained with the firstmethod, i.e. the model that predicts plot-level stem volumesdirectly, which is logical. Furthermore, the simulated reductionof point density of laser data had no effect on the accuracyof stem volume predictions. The percentile-based modelling ofdiameter distributions was applied, in particular, to the determinationof non-homogenous stand structure; using this method, it iseven possible to fit multimodal distributions. In terms of theaccuracy of the predicted plot-level stem volumes, the volumeprediction method based on modelled percentiles of basal areadiameter distributions was the second best, whereas the volumeprediction method based on the parameter prediction of the Weibull-basedbasal area diameter distributions resulted in slightly worseresults. However, the accuracies of the three laser-based volumeprediction methods tested were superior to the published resultsof spectral value-based remote sensing studies implemented usingdata collected from Finland. Furthermore, the accuracy of plot-levelstem volume estimates calculated from field assessments wasconsiderably weaker than the accuracy of the three volume predictionmethods that utilized measures obtained with laser scanning.     

Comparison of beta and weibull functions for modelling basal area diameter distribution in stands of pinus sylvestris and picea abies

The purpose of this study was to compare beta and Weibull distributions in describing basal area diameter distributions in stands dominated by Scots pine and Norway spruce. The material of the study consisted of 535 stands located in eastern Finland. Parameters for both two‐ and three‐parameter approaches of the Weibull distribution were estimated using the method of maximum likelihood. Models for these parameters were derived using regression analysis. For the beta distribution, regression models were formed for the minimum, maximum and standard deviation of diameters within individual stands. These models were used when the exponents of the beta distribution were calculated analytically. Also, some parameter models for beta and Weibull distributions from previous studies were compared with the measured diameter distributions. The distributions obtained were compared using diameter sums and an estimate of the proportion of sawtimber. The results did not reveal any major differences between the suitability of the beta and two‐parameter Weibull distributions. There are appropriate models available for both of the distributions and the more similar the original data is to the data of an application, the better are the results. The two‐parameter approach of the Weibull distribution gave better results than the three‐parameter approach. The poorest results for all the predicted distributions were obtained at the extremes of the distributions.     

Comparison of selected statistical distributions for modelling the diameter distributions in near-natural <Emphasis Type="Italic">Abies</Emphasis>–<Emphasis Type="Italic">Fagus</Emphasis> forests in the Świętokrzyski National Park (Poland)

The aim of the study is to compare selected theoretical distributions (normal, lognormal, Weibull, gamma, logistic, and exponential) in describing the tree diameter (DBH) distributions of mixed near-natural forests consisting of fir Abies alba Mill. and beech Fagus sylvatica L. growing in various vertical structures. Tree DBH data were collected between 1997 and 2008 from 51 sample plots established in the Świętokrzyski National Park in Poland. The empirical data represent differentiated DBH distributions, ranging from almost symmetric to extremely asymmetric ones. The chi-square test and the modified Kolmogorov–Smirnov test were chosen for the goodness-of-fit testing. In addition to the test statistics, the bias (B), the root mean square error (RMSE) and the graphical method (quantile–quantile plots) were used. In one-storied stands, the most suitable distributions were the normal and logistic distributions; in two-storied and multilayered stands, the Weibull and gamma distribution were the most suitable; and in selection stands, the exponential distribution was the most appropriate to describe the DBH distribution. The order of precision of the tested distributions (from the highest to the lowest) was Weibull, gamma, logistic, normal, exponential, and lognormal. The normal and exponential distribution should be applied only to one-storied and selection forests, respectively. The least suitable distribution for DBH distribution modelling was the lognormal one.