Modelling irregular and multimodal tree diameter distributions by finite mixture models: an approach to stand structure characterisation

Tree structural diversity is assessed by modelling stand diameter at breast height (DBH) distribution. The aim of this study was to verify the suitability of a mixture of two- and three-component Weibull and gamma models for describing irregular and multimodal DBH distributions. Investigations were carried out in natural Abies alba Mill. and Fagus sylvatica L. stands, representing the growing-up stage, in the Świętokrzyski National Park (Central Poland) and in the Pieniny National Park (Southern Poland). Sample plots (0.25–0.5 ha in area) were selected from a database used for forest dynamics analysis. The parameters of the mixture model were estimated using maximum likelihood estimation (MLE) and a combination of an expectation–maximisation (EM) algorithm with a Newton-type (NT) method. The multistart method (MM) of choosing initial values was used. The DBH distributions in mixed Abies–Fagus forests of complex structure were accurately approximated by two- and three-component mixture models. The root mean square error (RMSE) values varied from 3.4515 to 7.6682 for the Weibull mixture and from 3.6457 to 7.1419 for the gamma mixture. The results indicated that the mixture Weibull and gamma models were, in general, equally suitable for modelling irregular and multimodal DBH distributions. Finite mixture models may be used to characterise tree structural diversity, especially in uneven-aged stands of complex structure.     

Juvenile diameter distributions of loblolly pine characterized by the two-parameter Weibull function

Diameter distributions of juvenile loblolly pine (Pinus taeda L.) were characterized utilizing a two-parameter Weibull distribution to aid in forecasting and simulation of young stands. Juvenile diameter distributions were studied to gain insight into the effects of various stand-level factors. Results show that diameter distributions in juvenile loblolly pine stands can be successfully characterized with the two-parameter Weibull function. Repeated measures analysis detected significant planting density, age, and age by planting density interaction effects for the scale and shape parameter estimates from the two-parameter Weibull distribution. Using parameter recovery techniques, estimated diameter distributions were derived from easily attainable stand-level characteristics (i.e. basal area per hectare, planting density, age, and quadratic mean diameter). A thorough understanding of juvenile diameter distributions should prove especially useful for operational planning of stands on short rotations that require estimates of productivity at early ages.     

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.     

Modeling diameter distributions of mixed-species forest stands

This study compares three diameter distribution models to fit mixed-species forest stands using four example plots with two or three species components in Daxing'an Mountain, PR China. The methods include (1) a finite mixture model (FMM) to fit two or three species components simultaneously, (2) a single Weibull function to fit the whole plot only, and (3) a single Weibull function to fit each species component separately and the summation of the individual species produced the whole plot. Our results indicated that Method 2 is only suitable to regular and unimodal diameter distributions with a balanced reversed J-shape. Method 3 may be able to fit each species component well if its frequency distribution is known and available in the data. However, Method 3 ignores the interspecies relationships within a given plot. Thus, the summation of the species components may not produce a good fit for the whole plot. In contrast, Method 1 (FMM) fits the species component distributions simultaneously with the constraint that the individual components add up to the whole plot, without requiring the observed frequencies for each species across the diameter classes. The FMM models are more flexible to describe highly skewed and irregular diameter distributions for the whole plot, as well as provide the acceptable estimation for each species component and the mixing proportions. Thus, the FMM models can be a useful tool for effectively managing mixed-species forest stands.     

A Structural Alternative to Chronosequence Analysis for Uneven-Aged Northern Hardwood Forests

Abstract

Structural changes over time in forests of shade-tolerant species are difficult to study because these stands are often broadly uneven-aged and not suitable for traditional chronosequence analysis. Yet because of past disturbances of variable intensity and frequency, there is often substantial structural variation among stands of a given site quality class. In this study, diameter distributions from 70 primary northern hardwood stands were analyzed to determine if certain structural indices might allow placement of stands in a developmental sequence.

Modal stand diameter and the fraction of total overstory crown area in large trees (> 46 cm dbh) were among the most useful measures for distinguishing stages of development. Modal diameter ranged from 10 cm to 62 cm for stands on good sites. Correlations between diameter and age were highly significant (P < 0.0001) for the three principal species on all three site quality classes, suggesting that stands with a greater proportion of large trees are generally in a later stage of development. Stands with < 45% of the crown area in large trees have predominantly unimodal size distributions. As modal stand diameter increases, the shape of the size distribution changes from positively skewed to nearly symmetric. As the proportion of crown area in large trees exceeds 45% and the ratio of crown area in large to mature (26-45 cm dbh) trees exceeds 1.5, the form of the size distribution changes to multimodal, irregular, or descending monotonic.     

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.     

Modeling diameter distribution of Austrian black pine (<Emphasis Type="Italic">Pinus nigra</Emphasis> Arn.) plantations: a comparison of the Weibull frequency distribution function and percentile-based projection methods

The main purpose of the present investigation is to examine and compare three methods for diameter distribution modeling in terms of their fitness to predict from stand level variables the diameter distributions of even-aged Austrian black pine (Pinus nigra Arn.) plantations in Bulgaria. The percentile-based projection method involving empirical probability density function based on 12 percentiles was the first method tested. A new modified approach based on the first method was proposed as the second alternative. The third method was the 2-parameter Weibull functional model in which parameters were recovered from the first and the second raw moments and the second central moment of the empirical distributions. The Kolmogorov–Smirnov test was applied to compare the experimental distributions with the predicted ones, and estimation of the error indices was employed to evaluate the total absolute deviation of the predicted numbers from the actual ones by diameter class. The two-parameter Weibull function proved superior to the examined alternative percentile-based projection methods and the newly proposed percentile method, without a driver percentile showed improved precision compared to the classical percentile method (with a driver percentile). The parameters of the Weibull frequency distribution function can be easily recovered from the stand quadratic mean diameter. Consequently, this diameter distribution model could be incorporated as a sub-model for stand horizontal structure characterization within the Stand Density Management Diagram modeling framework.     

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.     

检查法林分生长预测及择伐模拟研究

检查法是一种适合于异龄林经营的集约作业方法，由于目前试验时间还较短（第１择伐周期１９８７－１９９２年），需要对其长期经营效果作出预测。文中提出了一种修正参数预测模型（ＭＰＰＭ）和线性规划方法相结合对于林分进行生长预测和择伐模拟研究的方法。拟合林分直径分布采用了形状灵活的Ｗｅｉｂｕｌｌ分布形式，使得这种方法很适合于异龄林分（反Ｊ形直径分布）的生长预测（当然也适于同龄林），特别是当径阶株数出现突然变化     

四川江安县人工经营毛竹林分结构特征

通过对江安县毛竹人工林的典型样地调查,利用Weibull分布函数对其年龄、胸径和枝下高的密度分布进行了模拟。结果表明:(1)毛竹人工林株数分布较均匀,且结构较合理,1度竹达598株·hm-2,占林地总株数的16.5%;(2)毛竹人工林分的年龄分布符合三参数Weibull分布函数,其所有样地拟合的可决系数R2达0.99812,各年龄组误差均在3%以内;(3)毛竹人工林分胸径和枝下高分布符合二参数Weibull分布函数,其全部样地拟合的可决系数R2分别为0.99951和0.99819;由于人工干扰的原因,其胸径分布的前4个径阶组和枝下高分布的前3个高阶组的拟合误差较大,最大值分别为31.69%和-30.02%,而其余各组的拟合误差则多在3%以下;(4)毛竹人工林的胸径-年龄和枝下高-年龄具有相似的结构特征,即,成熟龄组的胸径分布呈正态分布,对应的竹高分布则呈现左截尾的非完整的正态分布,而其余龄组则无明显分布特征,这与人为经营活动及其水平是相关的。     

Using optimization for fitting individual-tree growth models for uneven-aged stands

A convenient model type for simulating the dynamics of uneven-aged and uneven-sized stands of Finland is individual-tree model. This is because the stand structures are complex due to the presence of several tree species and irregular size distributions of trees. The required minimum set of models in this approach consists of species-specific individual-tree diameter increment models, individual-tree survival models, and ingrowth models. The development of these models needs data in which the diameter and survival of each tree of the sample plots is known for at least two time points. For this, the trees need to be numbered, which is tedious in uneven-aged forests due to the great number of small trees and the continuous ingrowth process. This study proposes a modelling approach that fits the above models but requires only the diameter distributions of the plots in the beginning and at the end of the measurement interval. The method uses non-linear optimization to derive such values for model parameters that, when the models are applied to the initial diameter distribution, the simulated stand development results in a diameter distribution which agrees with the measured ending distribution. The study showed that the method produces similar models and model parameters as regression analysis. Since the method is less demanding in terms of modelling data, it brings new data sets available for modelling the dynamics of uneven-aged stands and reduces the cost of collecting new data. The models fitted by the proposed optimization method were rather similar to the models developed earlier for Finnish uneven-aged forests.     

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.     

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.     

樟树苗期直径结构Weibull分布的研究

应用Weibull分布函数对樟树苗期直径分布进行了拟合，结果表明Weibull分布的拟合效果很好。探讨直径分布规律对于樟树的科学管理具有重要的意义。     

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。     

Use of Gumbel and Weibull functions to model extreme values of diameter distributions in forest stands

Context

Families of the Gumbel (type I), Fréchet (type II) and Weibull (type III) distributions can be combined in the generalized extreme value (GEV) family of distributions. Maximum and minimum values of diameters in forest stands can be used in forest modelling, mainly to define parameters of the functions used in diameter class models as well as in some practical cases, such as modelling maximum diameters for sawing and processing purposes.

Aims

The purpose of this study was to examine and compare two extreme value distribution functions (the Gumbel and the Weibull functions) in modelling the distribution of the minimum and the maximum values of representative sets of tree diameter samples. Both of these functions were applied to the lower and upper values of the diameter distributions of the main forest species in northwest Spain: Quercus robur L., Betula pubescens Ehrh., Pinus radiata D. Don, Pinus pinaster Ait. and Pinus sylvestris L.

Methods

Parameters of the Gumbel function were estimated using the mode and the moments of the distributions, and parameters of the Weibull function were estimated using the moments method.

Results

In general, the Weibull distribution was the most suitable model for describing the maximum diameters. The mode method of the Gumbel yielded the best results for minimum diameters of birch and Monterrey pine. The Gumbel distribution, fitted by either the mode- or moments-based methods, proved more suitable than the Weibull distribution for describing the minimum diameters in maritime pine and Scots pine stands.

Conclusion

In some cases, better results were obtained with the Gumbel than the Weibull distribution for describing the distribution of extreme diameter values in forest stands in northwest Spain. This is the first example of the application of the Gumbel distribution in forest modelling.     

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。     

Modelling stand structure in young Scots pine dominated stands

The purpose of this study was to construct models for predicting the structure of young Scots pine (Pinus sylvestris L.) stands. The two-parameter Weibull function characterized the height distribution of the stands. In young stands height was preferred to dbh as a random variable because of its continuous feature. Tree diameters were predicted using a multiplicative model, fitted as a linearized mixed-effect model. The modelling data consisted of repeatedly measured Scots pine dominated juvenile stands, carried out on a sub-sample of the 7th National Forest Inventory. The data covered a dominant height range from 0.2 up to 17 m. Two independent data sets were used to validate the models. The Weibull function was fitted using the maximum likelihood method. Four methods for predicting the distributions were compared: (1) parameter prediction models (PPM) consisting of seemingly unrelated regression equations, (2) a generalized linear model (GLM) which was a one-stage distribution and model fitting procedure, (3) a hybrid method including PPM for the shape parameter together with moment-based parameter recovery for the scale parameter, and (4) inclusion of moment-based parameter recovery for the scale parameter in the estimated GLM. Goodness-of-fit were tested in terms of Kolmogorov–Smirnov and error index statistics. Parameter recovery showed no improvement when used with PPM, but it improved GLM and gave the overall best performance for this new method. The constructed diameter–height model showed quite flexible and unbiased behaviour. Models are recommended as practical tools for Finnish forest management planning purposes.