Modeling the Distribution of Marketable Timber Products of Private Teak (<i>Tectona grandis</i> L.f.) Plantations

doi:10.4236/ojf.2013.34019

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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

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

Email: *[email protected]

Received May 16th, 2013; revised July 2nd, 2013; accepted July 22nd, 2013

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

Introduction

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),

*Corresponding author.

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

management tool.

Methods

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)



γ1γ

γxαxα

;θexp

ββ β

for xα, α0, β0, γ0



 





 



 











(1)

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

compared.



θα,β,γ



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 θL



γ1γ

γxx

log θlog exp

ββ β

nii







 













 



 





 (2)

For estimating these parameters with the MLM, the Equation

(2) was maximized with a three-equation system as follows:

 





 

γ1

γγ

ˆˆ

β1x α

ˆˆˆ

γxαlog αxα

log xα

ˆˆ

γ1xαγβ xα0

ii i















 



















 











(3)

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

by:



μβ 11

γγ





























(4)

with  the gamma function written for a real value s as:

 

Γd, 0.

sxexs





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:

μαβ 1ˆ

σβ11

ˆˆ

γγ

312 1

μβ 131 121

ˆˆˆ ˆ

γγγγ















 

















 











 

 









(5)

The system (5) was solved using the R package rootSolve

(R2.14.1).

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):





050

95 25

α1

ln 10.95

ln ln 10.25

γˆˆ

ln αln α

α1ˆ

β2

1ˆ

α12

ˆˆ

γγ

211

ˆˆ























 

































 





 

 

 



(6)

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.

116

N. H. FONTON ET AL.

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:

Optimal Method

Descriptive statistics of the estimated parameters for cop-

pices are presented (Table 2). For MLM, the parameters ,

and

αˆ

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

αˆ

βˆ

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,

3.72, respectively.



mNN



%100





(7)

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-

lows:



ˆd

NNfxx

(8)

where uj and lj are the upper and lower limits of class j.

Modeling Weibull Parameters

sed to establish the rela-

tio

Results

Data Summary

, 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).

Table 1.

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

Coppices

(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

High forests

(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

able 2.

parameters of Weibull for the two sylvicultural regimes; α, β, and γ are the Weibull position, scale, and shape parameters, respectively.

Estimated

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

High regim

MLM 9.89 0.00 19.88 0.22 15.4765.11 0.53 3.07 0.92 6.30 0.06

11.2 0.00

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

b) forestse

1.91

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.

Table 3.

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.

Coppic

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

γ with



50 50

ln xg

LP P. The

f nal regression equations are: i



25 75

75 50

α6.736 0.4050.167

β5.58950.2430.001

γ1.989 0.0248.116lnx

P P



 

The associated are less than 0.001. The model to

value

roduedict marketable pcts from coppices is:



25 75

x 6.736 0.4050.167P







 

exp5.8950.2430.001

FPG







 











where 75 50

1.989 0.0248.116PLP



 ,

n, x is the girth of the tree, G is



e basa

is the Weibull

th thth

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:

25 75

9.795.69x and 26.611.1x

PP  

with The results of the SURE analysis f high

value 0.000P.

are presented

forest in Table 6 and the best models equations

with value

P less than 0.023 are:



min

α5.249 0.476x

0.5

0.111

β10.567 0.00423.519lnx

γ1.8678.283lnx0.014



 

 

The equation to predict the marketable products for high for-



ts is then:



min

x 5.249 0.476x0.111

x1exp

10.567 0.00423.519

FGLP





 



 























where 50

1.867 8.2830.014wLPN



xF i  , s the Weibull

Table 4.

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

α, ˆ

β, and ˆ

are the Weibuition,shape parters, respec-

tively. M is thed ofents; statis ae

associatedrobabilit.

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

irn

over, for both sylvicultuppices and high forests,

the average value of the γ of the Weibull dis-

ll pos scale, and ame

metho

y value

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

Discussion

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

shape parameter

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-

118

N. H. FONTON ET AL.

Table 5.

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

α, ˆ

β, and ˆ

γ respectively the Weibull position, scale and shape parameters.

α ˆ

β ˆ

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

Table 6.

SURE (Seemi

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

α, ˆ

β, and ˆ

respectively the Weibull position, scale and shape parameters.

α ˆ

β ˆ

Constant xmin G Consnt G2 LP50 Constant LP50 1/2

ta N

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 

50 xg

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).

Conclusion

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

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

MOM.

Acknowledgements

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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