Open Journal of Forestry
2013. Vol.3, No.4, 115-121
Published Online October 2013 in SciRes (http://www.scirp.org/journal/ojf) http://dx.doi.org/10.4236/ojf.2013.34019
Copyright © 2013 SciRes. 115
Modeling the Distribution of Marketable Timber Products of
Private Teak (Tectona grandis L.f.) Plantations
Noël H. Fonton1*, Gilbert Atindogbé1, Arcadius Y. Akossou2, Brice T. Missanon1,
Belarmain Fadohan1, Philippe Lejeune3
1Laboratory of Study and Research in Applied Statistics and Biometry,
University of Abomey-Calavi, Abomey-Calavi, Benin
2Faculty of Agronomy, University of Parakou, Parakou, Benin
3Unit of Forest and Nature Management, Gembloux Agro-Bio Tech, University of Liege, Gembloux, Belgium
Received May 16th, 2013; revised July 2nd, 2013; accepted July 22nd, 2013
Copyright © 2013 Noël H. Fonton et al. This is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the
original work is properly cited.
Management of marketable products of private plantations will not be sustainable without class girth be-
ing identifiable readily. Modeling marketable products is a key to obtain good fitness between observed
and theoretical girth distribution. We determine the best parameter recovery method with the Weibull
function for two sylvicultural regimes (coppice and high forest). Data on stand variables were collected
from 1101 sample plots. The three Weibull function parameters were estimated with three parameters re-
covery methods: the maximum likelihood method, the method of moments and the method of percentiles.
Stepwise regression and the simultaneously re-estimated parameter using the Seemingly Unrelated Re-
gression Estimation were applied to model each parameter. The results indicated that the three methods
successfully predicted girth size distributions within the sample stands. The method of moments was the
best one with lowest values of Reynolds error index and Kolmogorov-Smirnov statistic however the syl-
vicultural regimes. The Weibull parameter distribution model developed for each of the two sylvicultural
regimes was quite reliable.
Keywords: Weibull; Parameter Recovery Method; Reynolds Index; Sylvicultural Regime; Poles; Logs
The multipurpose management of small woodlots by small-
holder forestry has been gaining more importance (Harrison et
al., 2002). The growing of the demand for forest products
(Scheer, 2004) explained the importance of their management
mainly for the smallholder farmer to generate substantial in-
come (Aoudji et al., 2012). Teak (Tectona grandis L.f.) is the
most important reforestation and commercial plant species in
coastal West Africa due to its fast growing potential (Niskane,
1998), good-quality timber (Louppe, 2008). Reforestation with
this specie has increased the above ground biomass and carbon
stock at 10-year-old about 45% higher than a nearby degraded
secondary forest (Odiwe et al., 2012). In Benin, the success of
state-owned plantations has encouraged farmers to invest in
teak sylviculture, establishing plantations on small plots rang-
ing from 0.05 ha to 28.10 ha (Atindogbé, 2012). Various prob-
lems constrain both traders and smallholder farmers (Aoudji et
al., 2012). These include the lack of market information, high
transaction costs, difficulties for traders to get timber supplies
(Anyonge and Roshetko, 2003; Nawir et al., 2007), and the low
return to smallholder farmers (Maldonado & Louppe, 1999;
Nawir et al., 2007). According to the above problems, efforts
were needed to have information on the different classes of the
merchants products on stand before harvesting. Forest owners
and managers have no reliable tools to provide them with a
comprehensive scheme of resources available and monitoring,
harvesting and sales operations. Therefore, one challenge is to
determine the minimum level of information required to char-
acterize harvests (Lafond et al., 2012).
Stand tables for total or marketable volume are based on the
distribution of tree diameters using traditionally probability
density functions (PDFs) (Parresol et al., 2010). Many func-
tions have been suggested for establishing tree diameters size
class distribution (e.g., normal, exponential, beta, Johnson’s
S, Gamma, Weibull, logit-logistic). However, the Weibull
function appears the most often used (Little, 1983; Rennolls et
al., 1985; Rondeux et al., 1992; Lindsay et al., 1996; Liu et al.,
2004; Newton & Amponsah, 2005; Lei, 2008) owing to its
flexibility (Hafley & Schreuder, 1977; Kilkki et al., 1989) and
the best description of diameter structure. This function can
also model many types of failure rate behaviors when appropri-
ate parameters are included.
Many techniques for estimating Weibull function parameters
have been developed: the graphical methods and the analytical
methods. The accuracy of the estimate depends on the size of
the sample and the method used. Graphical methods tend to
provide crude estimates, while analytical methods provide bet-
ter estimates that include confidence limits (Murthy et al., 2004)
and are reported to be more accurate (Razali et al., 2009). The
common analytical methods are the method of moments (MOM),
N. H. FONTON ET AL.
the maximum likelihood method (MLM), the method of percen-
tiles (MOP) and the method of least squares (MLS). However,
the most suitable method depends on the stands characteristics
(Liu et al., 2004; Lei, 2008).
The aim of this study was to determine the best estimator
method for Weibull function parameters for two different syl-
vicultural regimes: the coppice and the high forest. The Mod-
eled parameters were then used to predict the distribution of
marketable products of the private teak plantations as a useful
Study Site and Data
This study was carried out in the Guinea-Congo zone of Be-
nin (West Africa) located between 6˚17' and 6˚58'N, and 1˚56'
and 2˚31'E. The region has a bimodal rainfall regime, with a
mean annual precipitation of 1100 mm and a daily mean tem-
perature of 29.9˚C over the period 1971-2009 (www.World-
clim.org, 2005). Clayey-sand and vertisol are the dominant soil
types. The original native vegetation, a semi-evergreen dry
forest, was strongly influenced by human activities and is now
reduced to a few relict forests and forest reserves.
Data were collected using a snowball sampling method
which yielded 1101 private teak plantations: 844 coppices and
257 high forests. The size of each plantation (area) was meas-
ured. Then five (for plantations <0.5 ha) or ten (for plantations
≥0.5 ha) replicates strips of five trees were randomly sampled.
On each strip, the planting space between trees (e) and between
lines (l), the survival rate (t), and the girth at breast height (cbh)
for all trees over 10 cm (lowest girth size of the marketable
products) were measured. Timber merchants use height classes
of marketable products based on girth classes: small poles (10 -
19 cm), medium poles (20 - 39 cm), large poles (40 - 49 cm),
small posts (50 - 64 cm), large posts (65 - 79 cm), small logs
(80 - 109 cm) and large logs (≥110 cm).
Statistical Parameter Modeling
The complete three-parameter Weibull probability density
function of trees girth x is given by (Bailey & Dell, 1973)
for xα, α0, β0, γ0
where , α is a location parameter, β is a scale pa-
rameter, and γ is a shape parameter. The recovery methods
based on maximum likelihood, the maximum likelihood
method (MLM), on moments, the method of moment (MOM)
and on percentiles, the method of percentiles (MOP) were
In relation to the unknown parameters α, β, γ and n the num-
ber of trees, the logarithm of the likelihood function,
of Equation (1) is given by:
log θlog exp
For estimating these parameters with the MLM, the Equation
(2) was maximized with a three-equation system as follows:
where n is the number of trees in the plantation and the
girth of tree i. The SAS software (SAS 9.2) was used to solve
iteratively the equation system (3).
The moment order k (μ) of the Weibull function is given
with the gamma function written for a real value s as:
The parameters α, β and were
estimated by MOM with two processes. The moments of order
1 (μ), order 2 (σ2) and order 3 (μ3) (Razali et al., 2009) of the
equation 4 were computed as follows:
μβ 131 121
The system (5) was solved using the R package rootSolve
The parameter recovery method based on percentiles (MOP)
requires computation of the 0th (minimum girth), 25th, 50th, and
95th percentiles of the distribution of the girth as x0, x25, x50, and
x95, respectively. The three parameters were estimated by solv-
ing the following three equations simultaneously (6) (Borders
et al., 1987):
ln ln 10.25
ln αln α
where n is the number of trees in the plantation, is the gamma
function, xq is the quadratic mean girth of the plantation, and
ln is the natural logarithm.
Copyright © 2013 SciRes.
N. H. FONTON ET AL.
Copyright © 2013 SciRes. 117
d to assess the goodness of fit of the
Comparison Criteria eraged 21.3 cm, 5170 stems/ha and 18.4 m2·ha−1, respectively
(Table 1). While the mean girth was larger in high forests than
in coppices, the reverse trend was obtained for the density and
basal area of trees.
Two statistics were use
ree methods: the Kolmogorov–Smirnov statistic (KS) and the
prediction index error of Reynolds (e) (Reynolds et al., 1988).
The optimal recovery parameter method is the one with low
value for the two criteria. The prediction index error of Rey-
nolds (Equations (7) and (8)) was computed (Pauwels, 2003) as:
Descriptive statistics of the estimated parameters for cop-
pices are presented (Table 2). For MLM, the parameters ,
averaged 8.14, 12.78, and 2.34, respectively. For MOM,
mean values were 8.97, 11.79, and 2.20, respectively. For MOP,
the three parameters averaged 9.20, 12.17, and 2.41, respec-
tively. Statistics on the estimated parameters for high forests
were presented (Table 2). For MLM, mean values for the pa-
rameters , and
varied 9.89, 15.97, and 3.07, respec-
tively. For MOM, these values were 11.28, 13.94, 2.52 respec-
tively. For MOP, the three parameters means were 10.14, 18.50,
where is the observed and ˆ
is the estimated fre-
quencytrees in girth size class j, of
is the total number of
trees, and m is the number of classesˆ
is estimated as fol-
where uj and lj are the upper and lower limits of class j.
Modeling Weibull Parameters
sed to establish the rela-
, density and basal area of trees av-
For coppices, the percentages of plantations that fitted the
Weibull distribution were 0.95% for MLM, 0.97% for MOM,
and 0.91% for MOP. For high forests, these percentages were
0.94% for MLM, 0.99% for MOM, and 0.96% for MOP. The
MOM method showed the lowest values for the error index of
Reynolds (e%) and the Kolmogorov-Smirnov statistic (KS) for
both coppices and high forests (Table 3).
Multiple regression method was u
nship between the estimated parameters α, β, and (depend-
ent variables) and dendrometric characteristics of stands. The
predictor variables were density (N·ha−1), surface, and basal
area of stand (G·m2·ha−1); the girth of the tree of mean basal
area (xg cm); the mean, maximum, and minimum of girth; and
the 25th (P25), 50th (P50), and 75th (P75) percentiles of girth dis-
tribution. Models were established and tested for each set of
dendrometric characteristics. A stepwise regression was then
used to select the best subset of two variables. The estimated
parameters were simultaneously re-estimated using the seem-
ingly unrelated regression estimation (SURE). This can account
for correlation errors between equations and is asymptotically
efficient in the absence of specific errors (Liu et al., 2004).
Density, N (/ha), quadratic mean of girth, xg (cm) and basal area (G
m2·ha−1) of the study plantations.
Sylvicultural regimesVariablesMean Min. Max.SE
(cm)20.2 10.7 51.60.2
N·ha−1 5952 289 22,458104
(n = 844)
G·m2/ha−120.1 1.0 133.60.50
xg (cm)24.4 11.3 58.40.43
N·ha−1 2798.9 632.5 8590.169.1
(n = 257)
G·m2·ha−1 13.2 1.8 58.10.45
Min. and Max. are minimal and maximal values of the dendrometric parameters;
SE is the standard error of the mean.
Overall, values for girth
parameters of Weibull for the two sylvicultural regimes; α, β, and γ are the Weibull position, scale, and shape parameters, respectively.
MLM is the maximum likelihood method; MOM the method of moments and MOP the method of percentiles.
α β ˆ
MeMan Min. Mx. SE Mean MiMx. SE Men Min. Mx. SE thod ean. aaa
a) Coppicgime es re
MLM 8.14 0.00 17.05 0.08 12.78 0.01 55.58 0.23 2.34 0.06 8.43 0.03
MLM 9.89 0.00 19.88 0.22 15.4765.11 0.53 3.07 0.92 6.30 0.06
MOM 8.97 0.00 19.00 0.10 11.79 1.18 46.70 0.22 2.20 0.56 3.50 0.02
MOP 9.20 0.00 20.00 0.12 12.17 0.66 129.38 0.35 2.41 1.00 32.00 0.08
MOM 819.96 0.25 13.94 2.29 67.08 0.49 2.52 0.68 3.52 0.03
MOP 10.14 0.00 19.62 0.32 18.50 1.32 180.15 1.31 3.72 1.00 39.98 0.27
MLM aximihoood, Mmeth omMOP thod of s; Miax arinimuhe m resy;
SE is the standard error.
is the mum likeld methOM the od of ments and the mepercentilen and me the mm and taximumpectivel
N. H. FONTON ET AL.
Comparison of the efficiency of the three parameters estimation methods: values of the error index of Reynolds (e%) and the Kolmogorov-Smirnov
es High forests
statistic (KS) for the two sylvicultural regimes.
e% KS e% KS
Metds Mean SE Mean SE Mean SE Moy SE ho
MLM 13.03 0.40 0.29 0.01 12.60 0.77 0.27 0.02
MOM 11.35 0.31 0.26 0.01 10.12 0.56 0.21 0.01
MOP 19.26 1.15 0.43 0.03 13.12 1.28 0.28 0.03
Consequently, MOM was chosen as the most appropriate
arameters’ Model Development
oppices are presented
method for modeling the distribution parameters of the Weibull
function for private teak plantations. The models revealed sig-
nificant differences between coppices and high forests for all
three parameters estimated using MOM (Table 4).
The results of the SURE analysis for c
Table 5. Stepwise regression revealed that the best subset of
two stand characteristics is P25 and P75 for ˆ
γ, P75 and G2 for
β, and LP50 and P75 for ˆ
LP P. The
f nal regression equations are: i
The associated are less than 0.001. The model to
roduedict marketable pcts from coppices is:
x 6.736 0.4050.167P
where 75 50
n, x is the girth of the tree, G is
is the Weibull
functio thl area, P25, P50,
and P75 are the 25, 50, and 75 are the th percentiles, and
LP50 the weighted percentile. Some dependent variables such as
P25 and P75 were computed with the best adjustment as follows:
9.795.69x and 26.611.1x
with The results of the SURE analysis f high
forest in Table 6 and the best models equations
P less than 0.023 are:
The equation to predict the marketable products for high for-
ts is then:
x 5.249 0.476x0.111
xF i , s the Weibull
Comparison of the sylvicultural regimes according to estimated pa-
rameters of Weibull function by the best estimator method MOM.
Coppices High forests
Param t P eters Mean SE Mean SE
α 8.97 0.10 11.28 0.25 8.580.000
β 11.79 0.2213.94 0.49 4.000.000
2.20 0.020. 02.52 0.03 8.8800
β, and ˆ
are the Weibuition,shape parters, respec-
tively. M is thed ofents; statis ae
distribution func is min e m
mum gth, G isas, Ne de th
percentile or median, and LP50 the weighted 50 percentile.
fficiency of the Parameter Recovery Methods
over, for both sylvicultuppices and high forests,
the average value of the γ of the Weibull dis-
ll pos scale, and ame
momt is the stic of tudent,nd P th
tion, xthe girth of the tree, x, is thini-
ir the bal area is thnsity,
P50 is the 50
Regardless of the parameter recovery method used, the ob-
served and theoretical distributions of the plantations wer
much closed according to the Kolmogorov-Smov test. More-
ral practices, co
tribution was lower than 3.6, suggesting that the distribution
ees is left-skewed. Using the optimal method (MOM), the
parameter α, whose value is associated with the minimum girth,
was 8.97 for coppices and 11.28 for high forest, both were close
to the minimum girth of this study (10 cm).
The parameter β, which gives an idea of the central value of
samples, had a maximum value of 46.70 for coppices and 67.08
for high forests, while the values measured, with the completed
inventory of 18 plantations, were 51.6 cm for coppices and 58.4
cm for high forests. These results are similar to previous find-
ings, which illustrated that α is a good predictor of the mini-
mum diameter (Frazier, 1981; Knoebel et al., 1986; Leduc et al.,
2001). They also support those of Lei (2008), who demon-
strated that the two-parameter Weibull distribution and MOM
provided the best estimation of the diameter distribution of
Chinese pine (Pinus tabulaeformis). These results are also con-
sistent with those found by Liu et al. (2004) in their study of the
diameter distribution of unthinned plantations of black spruce
(Picea mariana) in central Canada, although in their study MOP
was the preferred method. Zhang et al. (2003) previously dem-
Copyright © 2013 SciRes.
N. H. FONTON ET AL.
Regression coefficients of the predictors and statistics resulting from the RE (Seemingly Unrelated Regression Estimation) analysis for coppices
for the response variables i.e. ˆ
β, and ˆ
γ respectively the Weibull position, scale and shape parameters.
Constant 25 P75 Constant P75 G2 CPonstant P75 LP50
Coef. 6.74 0.405 −0.167 5.895 0.243 0.001 1.989 0.02 8.116
P <0.00 <0.001 <0.001 <0.001 <0.001 <0.001 <0.001 <0.001 <0.001
t 16.46 8.69 −6.100 7.720 7.130 4.650 25.410 6.810 10.060
SE 0.41 0.05 0.027 0.763 0.034 0.000 0.078 0.004 0.807
Coef. is the regressioients, G ard P75 are 75th ps, and n coefficis the basal ea, P25 an the 25th andercentile 50 lnLP 50 g
P xis ean
basal arnd P50 is thercentile, tstatistic of Snt and P the associated proba
where xg the girth of the tree of m
ea ae 50th p is the tudebility value.
Regression coefficients values and their significant appreciation statistics resulting from thengly Unrelated Regression Estimation)
analysis for high forests with the response variables ˆ
β, and ˆ
respectively the Weibull position, scale and shape parameters.
Constant xmin G Consnt G2 LP50 Constant LP50 1/2
Coef 5.249 0.476 −0.111 10.567 0.004 −23.519 1.867 8.280.014
P <0.001 <0.001 < 0.001 <0.001 0.003 0.023 <0.001 0.003 <0.001
t 6.040 7.440 −4.060 15.000 3.640 −2.290 9.800 3.700 4.320
0.870 0.064 0.027 6.705 0.001 23.519 0.191 2.241 0.003
Coef.the regressficients, xe minih, G is trea, N is sity and is ion coefmin is thmum girthe basal a the den 50 lnLP
P xg is the tree ofsal
area aP50 is the 5 t is thistic of S P the ased probabilie, SE is the stadard error.
onstrated the effectiveness of the Weibull distribution for de-
f North America. Meanwhile, Bailey & Dell (1973) have
ribution were functions of
cs. In all cases the regres-
nt with P value less
forests. These find-
forests, a global model combining data from these two sylvi-
es would notb
e estimated parameters. The
parameter of the distribution shape, γ, is more influenced by
ed through theoretical dis-
tributions. Results indicated that the three methods compared
were generally suitablee distribution of mar-
ketable products. Howeverformance of each
where girth of mean ba
nd 0th percentile,e stattudent andsociatty valu
scribing the diameter distribution of natural stands of red spruce
(Picea rubens) and balsam fir (Abies balsamea) in the north-east
cultural regim be suitale. This is supported by the
observed differences between th
shown that MOM is more efficient than MOP for estimating
parameters of the Weibull distribution, but it requires very
complex calculations. Nanang (1998) in a study on the diameter
distribution of Azadirachta indica plantations in Ghana asserted
the same thing. It was also argued that MOM assures compati-
bility between the characteristics of the observed population
used in parameter recovery and those obtained through simula-
tion (Mateus & Tomé, 2011). The differences observed be-
tween coppices and high forests for all the three parameters
estimated by MOM confirm the need to build separate models
for different sylvicultural regimes.
Predicting the Weibull Parameters
Parameters of the Weibull dist
most of the dendrometric characteristi
sion coefficients were statistical significa
than 0.001 for coppices and 0.023 for high
gs are in agreement with Liu et al. (2004), who modeled the
three parameters of a Weilbull distribution using four charac-
teristics of black spruce stands (age, basal area, average height,
and site index). In most cases, the probability values associated
to the regressions were less than 0.0001. Since the distribution
model of marketable products from coppices differed from high
LP50 for both sylvicultural practices. The location parameter α
depends on the positional parameters P25 and P75 in coppices,
and on the minimum girth and basal area in high forests. β is
more influenced by the square of the basal area for both sylvi-
cultural regimes. These results differ from those of Torres-Rojo
et al. (2000), who found that the shape of the distribution is
strongly influenced by the diameter, mean basal area, density,
and dominant height of the trees; and that β is influenced by
diameter and mean basal area of trees. Moreover, several stud-
ies have previously found that the minimum girth most often
influences the value of α (Frazier, 1981; Knoebel et al., 1986;
Lejeune, 1994; Leduc et al., 2001).
In this study, models have been developed to assess market-
able teak resource produced by private teak plantations. The
main advantage of modeling parameter with stand characteris-
tics is the compatibility between the characteristics of the ob-
served populations and those obtain
for modeling th
er, the relative p
ethod depends on its ability to predict the observed girth size
Copyright © 2013 SciRes. 119
N. H. FONTON ET AL.
class frequencies. The method based on moment (MOM) appears
to be the most appropriate.
Distribution models for marketable products were developed
for coppices and for high forests using stand variables and
This study was sponsored by the Belgiun “Commission Uni-
versitaire pour le Développement (CUD)”—through le Projet
Interuniversitaire Ciblé: Contribution au développement d’une
filière du teck au départ des forêts privées du Sud-Bénin (Dé-
partement de l’Atlantique)”.
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