Goodness-of-fit analysis for lumber data

Summary Four different probability distributions were studied to evaluate their relative goodness-of-fit in describing the modulus of rupture (MOR) and modulus of elasticity (MOE) of populations of dimension lumber. The distributions under consideration were the normal, lognormal, Weibull and Johnson's S_B. The populations of lumber consisted of 96 data sets of various species groups, mechanical properties, sizes, structural grades and growth regions. The goodness-of-fit criteria selected in this study were the log likelihood, Kimball and Kolmogorov-Smirnov (K-S) tests. The K-S statistic was also calculated at the value of the random variable associated with the lower five percent exclusion limit of the empirical cumulative distribution. This value indicated the degree of goodness-of-fit at the lower tail of the distribution. The results indicated that the S_B distribution generally provided the best fit to the data. The maximum likelihood test overwhelmingly recommended the S_B distribution. The Kimball and Kolmogorov-Smirnov tests gave milder endorsements of the S_B distribution. No distribution proved to be superior to the others in modeling the lower five percent exclusion limit of the populations.The author would like to thank the Engineering Data Management (EDM) Inc. of Fort Collins, Colorado, for the use of their parameter evaluation software, STAtistical Data MANager     

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.     

Modeling diameter distributions of Gmelina arborea plantation in Omo Forest Reserve,Nigeria with Johnson’s SB

Diameter distribution modeling is an essential tool for obtaining reliable information on the structure, growth, and yield of forests. However, this tool has not been duly utilized in many forest plantations, especially in Nigeria. Thus, this article presents the effectiveness of using the Johnson S_B function for modeling diameter distributions of a Gmelina arborea plantation in Omo Forest Reserve, Nigeria. Fifty plots of 0.04 ha were randomly selected across three age series. All trees with DBH ≥ 5.0 cm in the selected plots were measured. Three fitting methods for the S_B distribution were compared: conditional maximum likelihood (CML), moments (MOM), and Knoebel and Burkhart (KB) methods. The parameter recovery model (PRM) was used to recover the parameters of the distribution. The assessment was based on Kolmogorov-Smirnov (K-S), mean square error (MSE), mean absolute error (MAE), and bias. The results showed that the underlying diameter distribution of the Gmelina arborea followed the Johnson’s S_B distribution fitted with MOM. The number of rejections by the K-S test was zero for MOM and CML. The mean K-S value for MOM was statistically different from CML and KB methods. The results obtained after recovering the parameters of the S_B distribution were comparable with the real distributions of the Gmelina arborea stand.     

Species abundance distribution models of Toona ciliata communities in Hubei Province,China

The study of plant species abundance distribution(SAD) in natural communities is of considerable importance to understand the processes and ecological rules of community assembly.With the distribution of tree,shrub and herb layers of eight natural communities of Toona ciliata as research targets,three different ecological niche models were used:broken stick model,overlapping niche model and niche preemption model,as well as three statistical models:log-series distribution model,log-normal distribution model and Weibull distribution model,to fit SAD of the different vegetation layers based on data collected.Goodness-of-fit was compared with Chi square test,Kolmogorov-Smirnov(K-S) test and Akaike Information Criterion(AIC).The results show:(1) based on the criteria of the lowest AIC value,Chi square value and K-S value with no significant difference(p 0.05) between theoretic and observed SADs.The suitability and goodness-of-fit of the broken stick model was the best of three ecological niche models.The log-series distribution model did not accept the fitted results of most vegetation layers and had the lowest goodness-of-fit.The Weibull distribution model had the best goodness-of-fit for SADs.Overall,the statistical SADs performed better than the ecological ones.(2) T.ciliata was the dominant species in all the communities;species richness and diversity of herbs were the highest of the vegetation layers,while the diversities of the tree layers were slightly higher than the shrub layers;there were fewer common species and more rare species in the eight communities.The herb layers had the highest community evenness,followed by the shrub and the tree layers.Due to the complexity and habitat diversity of the different T.ciliata communities,comprehensive analyses of a variety of SADs and tests for optimal models together with management,are practical steps to enhance understanding of ecological processes and mechanisms of T.ciliata communities,to detect disturbances,and to facilitate biodiversity and species conservation.     

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.     

Mechanical stress grading of tropical timbers without regard to species

Some reports have shown that for single species the correlation between modulus of elasticity (MOE) and modulus of rupture (MOR) in bending is quite high. Tropical timbers consist of hundreds of species that are difficult to identify. This report deals with the mechanical stress grading of tropical timber regardless of species. Nine timber species or groups of species with a total number of 1094 pieces measuring 60 × 120 × 3000 mm, were tested in static bending. The MOE was measured flat wise, while MOR was tested edge wise. Statistical analysis of linear regression with a dummy model and analysis of covariance were used to analyze the role of MOE and the effect of species on prediction of MOR. The analysis showed that using MOE as a single predictor caused under/overestimation for one or more species and/or groups of species. The accuracy of prediction would be increased with species identification. An allowable stress and reference resistance for species and/or groups of species were provided to compare with the prediction of strength through timber grading. The timber strength class for species and/or groups of species was also established to support the application of mechanical timber grading.     

Effects of moisture content and specific gravity on static bending properties and hardness of six wood species

This study was designed to investigate the effects of moisture content (MC) and specific gravity (SG) on the bending strength and hardness of six wood species including Japanese cedar (Cryptomeria japonica D. Don), China fir (Cunninghamia lanceolata), western hemlock (Tsuga heterophylla), red meranti (Shorea spp.), Selangan batu (Shorea spp.), and red oak (Quercus spp.). The experimental results are summarized as follows: Effects of MC and SG on the strength (MOR), stiffness (MOE), and hardness (H _B) could be represented by a multiregression formulas. A negative correlation existed between these properties and MC, whereas a postive correlation showed between them and the SG. The changing rate of these properties induced by 1% MC changes varied with the wood species: 2.6% change in MOR was observed in Japanese cedar, China fir, western hemlock, red meranti, and Selangan batu; and 3.9% was found in red oak. For MOE, a 0.58% change was observed in Japanese cedar, China fir, and red meranti; western hemlock and Selangan batu exhibited 1.2% and red oak 2.5%. For hardness, a 1.1% change was observed in Japanese cedar, western hemlock, and red oak; red meranti and China fir exhibited 3.3%; and Selangan batu 1.8%.A part of this report was presented at the 48th annual meeting of the Japan Wood Research Society in Shizuoka, Japan, April 3-5, 1998     

Static bending strength performances of cross-laminated woods made with five species

Thirty types of three-ply parallel- and cross-laminated woods were prepared from five species, and their static bending strength performance were investigated. The modulus of elasticity (MOE), proportional limit stress, and modulus of rupture (MOR) perpendicular to the grain were increased by cross-laminating, and the extent of the increase increased with decreasing density of the species. The measured values of MOE parallel and perpendi-cular to the grain of parallel-laminated woods and perpendicular to the grain of face laminae of cross-laminated woods were approximately equal to those calculated from true MOEs of individual laminae. However, the MOE parallel to the grain of face laminae of cross-laminated woods was much lower than the calculated MOE owing to the effect of the deflection caused by shear force on the MOE. The percentage of deflection caused by shear force versus total deflection (Y _s) showed high values, from 16.1% (buna) to 40.5% (sugi), and it decreased linearly with increasing shear modulus in the cross section of the core. In addition, there was an extremely high positive correlation between the MOR and the measured MOE parallel to the grain of face laminae of cross-laminated woods. The MOR was also highly dependent on the shear modulus in cross section of the core.Part of this paper was presented at the 50th Annual Meeting of the Japan Wood Research Society, Kyoto, April 2000     

Static bending properties of Finnish birch wood

The purpose of this study was to determine the modulus of elasticity (MOE) and the modulus of rupture (MOR) in the radial bending test for small, clear specimens of Finnish birch (Betula pendula Roth and B. pubescens Ehrh) wood originating from mature trees. The dependency of MOE and MOR on the specific gravity of birch wood was studied, and the relationship between MOE and MOR was modelled at the different heights and at the different distances from the pith of the tree. For B. pendula, the mean values for MOE and MOR were 14.5 GPa and 114 MPa, whereas B. pubescens had means of 13.2 GPa and 104 MPa, respectively. At the corresponding specific gravity, the bending stiffness and strength values did not differ between the two species. The results indicated a linear relationship between the MOE and MOR, irrespective of the birch species or the within-stem location. Both MOE and MOR increased clearly from the pith towards the surface of the tree and decreased slightly from the base to the top of the tree. It seems that if products with as high stiffness and bending strength as possible are wanted, sorting of raw materials into different grades according to their within-tree origin can be of value.     

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.     

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.     

Radial and between-family variations of the microfibril angle and the relationships with bending properties in Picea jezoensis families

With emphasis on tree breeding for wood quality in Picea jezoensis, we aimed to evaluate radial and between-family variations in the microfibril angle (MFA) of the S₂ layer in the latewood tracheids in 10 open-pollinated families of 43-year-old P. jezoensis trees. In addition, the relationships between MFA/wood density with the modulus of elasticity (MOE) or modulus of rupture (MOR) were investigated. Significant differences in MFA between families were found from the pith toward the bark. MFA showed higher values around the pith area, although some families showed relatively lower values than others around this area. In addition, due to a larger coefficient of variations of MFA near the pith, the potential for juvenile wood MFA improvement may be greater compared with mature wood. MOE was correlated with MFA in juvenile wood and with wood density in mature wood, whereas MOR was mainly correlated with wood density at radial positions in both woods. Therefore, to improve the MOE and MOR of P. jezoensis wood, both MFA and wood density would be factors to consider in both juvenile and mature woods. On the other hand, there are indications that, only wood density would be an important criterion for improving mature wood properties.     

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 flexible modeling of irregular diameter structure for the volume estimation of forest stands

Simple distribution models (SDMs) have deficiencies in portraying irregular forest diameter structure. This paper introduces mixture distribution models (MDMs) to improve the estimation accuracy of stand volume of Siberian larch (Larix sibirica Ledeb.) forests. Stand volume was estimated by combining the suitable diameter mixture model and diameter–height model. Appropriate mixture models are derived by integrating multiple SDMs of Burr III and XII, Johnson S_B, Weibull or lognormal probability density functions (pdf) that satisfied the criteria of goodness of fit tests. Results showed that the average bias of volume estimation for all of the study plots using SDM and MDM approaches are underestimated by 6.93 and 2.42 m³, respectively. Each of the estimates is equivalent to an estimation error of 25.59 ± 18.18 and 7.08 ± 2.97 %. This suggests that the MDM approach is a more flexible and suitable modeling technique for forest volume estimation, in particular for forests that have been frequently disturbed by natural events.     

Genetic variation in wood stiffness and strength properties of hybrid larch (Larix gmelinii var. japonica × L. kaempferi)

Genetic parameters for wood stiffness and strength properties were estimated in a 29-year-old hybrid larch stand (Larix gmelinii var. japonica × Larix kaempferi). The study included 19 full-sib larch families from Hokkaido, northern Japan. Implications of these genetic parameters in wood quality improvement are subsequently discussed. Traits included in the analyses were the dynamic modulus of elasticity of green logs (E _log), the modulus of elasticity (MOE), the modulus of rupture (MOR), compression strength parallel to the grain (CS) in small clear specimens, wood density (DEN), and diameter at breast height (DBH). DEN had the lowest coefficients of variation and MOE the highest. The narrow-sense heritability estimates of E _log, MOE, MOR, and CS were 0.61, 0.44, 0.60, and 0.43, respectively, and those of DEN and all mechanical properties increased from an inner to outer position within the stem. E _log and DEN had high positive phenotypic (0.52–0.83) and genetic (0.70–0.92) correlations with MOE, MOR, and CS. The mechanical properties of the inner position of the stem had rather high phenotypic and genetic correlations with those of the outer position and overall mean. The predicted gains in wood stiffness (E _log and MOE) were higher than those of the strength properties (MOR and CS). The predicted correlated responses in MOE, MOR, and CS when selecting for E _log and DEN were 72.6%–97.8% of a gain achievable from direct selection of these traits. DBH showed an insignificant correlation with all mechanical properties, although selection of this trait had a slightly negative effect on the mechanical properties.     

Estimation of the internal shear strength distribution of the element for laminated veneer lumber by nonlinear least-squares method

Until now we developed an estimation method for strength distributions of laminated veneer lumber (LVL) element by nonlinear least-squares method (NLM). Estimated strengths by this method were modulus of elasticity (MOE) and modulus of rupture (MOR) in the horizontal use direction and the vertical use direction, tensile strength and compression strength. But to use LVL for structural members, shear strength was also needed. Therefore, we tried to estimate the shear strength distribution of LVL element by NLM same as MOE and MOR in the horizontal use direction and the vertical use direction, the tensile strength of LVL and the compression strength of LVL in the previous reports. We conducted shear strength test for LVL and estimated element shear strength distribution by LVL strength data in the horizontal and vertical use direction. Next, we simulated LVL shear strength distribution using element shear strength distribution and compared with experimental ones in each use direction. They were overlapped in both use direction. Therefore, we could validate NLM for estimating element shear strength distribution.     

A comparison of diameter distribution models for Khaya ivorensis A.Chev. plantations in Brazil

The purpose of this study was to compare Beta, Gamma, Johnson's S_B and Weibull functions fitted by different methods for describing the horizontal structure of Khaya ivorensis (African mahogany) plantations in Brazil. The database comprised 128 plots from six plantations at varying ages. The function fits were compared using the Kolmogoroff–Smirnoff test, mean bias and mean absolute error for the number of trees and basal area per hectare per diameter class. Johnson's S_B outperformed the other functions, although all functions provided an adequate fit. The best methods were method of moments and maximum likelihood fitted using 25% of the minimum observed diameter as the location parameter for the Johnson's S_B function. The errors were greater in diameter classes with higher frequencies. Location and scale parameters were highly correlated with mean diameter and age for the Weibull and Johnson's S_B functions, respectively, which is convenient for diameter prediction. Gamma's scale parameter had medium correlation with age. Beta's parameters had low correlation with stand attributes assessed.     

Flexural and visual characteristics of fibre-managed plantation Eucalyptus globulus timber

ABSTRACT

The main goal of this study was to investigate the visual characteristics, recovery rate, and flexural properties of sawn boards from a fibre-managed plantation Eucalyptus globulus resource as a potential raw material for structural building applications. The impacts of the visual characteristics, strength-reducing features, and variation in basic density and moisture content on the bending modulus of elasticity (MOE) and modulus of rupture (MOR) of the boards were investigated. The reliabilities of different non-destructive methods in predicting MOE and MOR of the boards were evaluated, including log acoustic wave velocity measurement and numerical modellings. The MOE and MOR of the boards were significantly affected by the slope of grain, percentage of clear wood, and total number of knots in the loading zone of the boards. The normal variation in basic density significantly influenced the MOE of the boards while its effect on the MOR was insignificant. The numerical models developed using the artificial neural network (ANN) showed better accuracies in predicting the MOE and MOR of the boards than traditional multi-regression modelling and log acoustic wave velocity measurement. The ANN models developed in this study showed more than 78.5% and 79.9% success in predicting the adjusted MOE and MOR of the boards, respectively.     

Variation in tensile properties and relationship between tensile properties and air-dried density for moso bamboo

This research investigated the variation in tensile properties and the relationship between the tensile properties and the air-dried density for the moso bamboo (Phyllostachys pubescens) by sampling at different heights and radial positions. Results showed that the variation of the longitudinal tensile properties in the radial direction was greater than that in the longitudinal direction. The longitudinal tensile modules of elasticity (MOE) ranged from 8.49 to 32.49 GPa. MOE for the outermost layer was 3–4 times as high as that for the innermost layer. The longitudinal tensile strength (MOR) ranged from 115.94 to 328.15 MPa. MOR for the outermost layer is 2–3 times as high as that for the innermost layer. Linear and curvilinear regressions were done from tested data of MOE, MOR and air-dried density in this paper. The linear equation worked a little better than the curvilinear one to predict the longitudinal MOR and MOE from air-dried density. __________ Translated from Scientia Silvae Sinicae, 2006, 42(3): 72–76 [译自: 林业科学]     

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.