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Discoveries in Agriculture and Food Sciences - Vol. 12, No. 5

Publication Date: October 25, 2024

DOI:10.14738/dafs.125.15549.

Egonmwan, Y. I., & Orukpe, W. O. (2024). Comparison of Non-Linear Height-Diameter Models of Tectona grandis L.f. in a Rainforest

Site in Southern Nigeria. Discoveries in Agriculture and Food Sciences, 12(5). 20-28.

Services for Science and Education – United Kingdom

Comparison of Non-Linear Height-Diameter Models of Tectona

grandis L.f. in a Rainforest Site in Southern Nigeria

Y. I. Egonmwan

ORCID https://orcid.org/0000-0001-6516-8881

Department of Forest Resources and Wildlife

Management, University of Benin, Benin City, Nigeria

W. O. Orukpe

Department of Forest Resources and Wildlife

Management, University of Benin, Benin City, Nigeria

ABSTRACT

This study was carried out to select the best model for predicting height for effective

management of Tectona grandis stand in the study area. Five 3-parameters well- established non-linear models were used to compare their suitability in predicting

height, evaluate and select the best H-D model based on test statistics criteria such

as R2, RMSE AIC, and BIC. Models with the lowest RMSE, AIC and BIC and highest R2

values are judged the best. One hundred (100) trees were enumerated and were

subjected for analysis in the study. A plot size of 0.04ha (20m x 20m) with an area

of 0.64ha was chosen for this study and 10 sample plots were purposively selected

from the study area. The result indicated that schnute model had the highest R2

value of 99.9% but it also had the highest RMSE, AIC and BIC values, hence the worst

candidate in predicting height in the study area. Weibull model although did not

have the largest R2 of 97.2% when compared to schnute, had the smallest RMSE, AIC

and BIC (1.37596E-14, -832.6836 and -823.9054 respectively) and hence is the best

candidate model. The Chapman-Richards, Logistic and Korf models also performed

well based on the evaluation criterion. Weibull function with 3-parameters has

been confirmed to provide a secure estimate of total tree height for Tectona grandis

in the rainforest site of the study area of Southern Nigeria.

Keywords: Comparison, fit statistics, Weibull model, parameter-estimates.

INTRODUCTION

The relationship between tree height and diameter at breast height (DBH) is an important

element of forest structure that has long been used to characterize important aspects of tree

growth, stand development and that helps define relationships among individuals. Height

diameter models are used to calculate measurable quantities needed to characterize forest

growth and yield [1] especially volume, weight and biomass. For inventory purposes, it is

common practice to measure DBH on all trees in a plot while limiting the measurement of height

to a subsample of trees because measurements of height can be more difficult and time

consuming [2]. The relationship between tree DBH and total height is a structural characteristic

of a tree that describes key elements of stem form, and thus the volume of the harvestable stem.

[3]

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Egonmwan, Y. I., & Orukpe, W. O. (2024). Comparison of Non-Linear Height-Diameter Models of Tectona grandis L.f. in a Rainforest Site in Southern

Nigeria. Discoveries in Agriculture and Food Sciences, 12(5). 20-28.

URL: http://dx.doi.org/10.14738/dafs.125.15549

The height-diameter relationship describes the correlation between height and diameter of the

tree in a stand on a given site and this can be represented by a linear or non-linear mathematical

model. Developing forest inventory estimates often involves predicting tree volumes from only

diameter at breast height (dbh.) and/or merchantable height. [3]. Tree height and diameter at

breast height (DBH; that is, diameter outside of the bark of tree taken at 1.3m above ground

level) are fundamental tree characteristics used in forest measurements in order to come up

with estimates for timber volume, site index and other important variables related to forest

growth and yield, succession and carbon budget models [4]

DBH can be determined easily and accurately at little cost and time using such instruments as:

diameter tapes, callipers, Biltmore stick and bark gauge. “Information on tree heights is

essential in forest inventories for computing tree volumes. However, compared to measuring

tree diameter, field measurement of tree height is rather tedious. This is why many forest

inventories save time and effort by predicting tree heights using height–diameter (H–D) models

instead of direct measurements. To improve prediction of the local H–D curve, height

measurements from a subsample of trees on each sample plot or sample plot cluster may be

utilized. Predicting total tree-height based on observed diameter at breast height outside bark

is routinely in practical management and silvicultural research work” [5]

The Tectona grandis L.f. is a fast-growing tropical hardwood tree species belonging to the family

of Verbenaceae. Teak is one of the most widely cultivated exotic species in Nigeria because of

its good anatomical and physical properties [6]. It is also multipurpose tree species and as such,

there is continuous demand for its products [6]. The teak plantation in the study area is being

managed for research purpose. There are dearth of information on non-linear H-D models for

teak plantation in Nigeria which is important for the management of the plantation. The main

aim of the study therefore was to compare five non-linear height-diameter models for height

prediction and prescribing silvicultural option for the effective management of the teak

plantation stand.

METHODOLOGY

Study Area

The study area is located in the University of Benin which was formerly called the institute of

technology. It was founded on Saturday 23rd of November 1970. It occupies an area of about

13000 hectares. The study area is located between latitudes 6o24’08 N and longitudes 5o37’38

E. It is under the Ovia North East Local Government Area, Edo State, Nigeria. The climate

condition around the area varies all year round. In the wet season, it is warm, oppressive and

overcast and the dry season is hot and muggy. Over the course of the year, the temperature

typically varies from 67oF to 88oF and is rarely below 60oF or above 91oF. It has a tropical

climate. During most months of the year, there is significant rainfall. The average annual

temperature is 25.7oC. About 2679mm/ 105.5 inch of precipitation falls annually.

Data Collection

A total of 30 sample plots of 0.04ha (20m x 20m) with an area of 0.64ha were chosen for this

study. Ten sample plots were purposively selected from the total sample plots. Within each plot,

the following measurements were taken; diameter at breast height (dbh), diameter at top (dt),

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Discoveries in Agriculture and Food Sciences (DAFS) Vol 12, Issue 5, October- 2024

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midpoint diameter (dm), total height (tht), merchantable height (mht) were taken with spiegel

relaskop. Observed volume by Newton and basal area were computed.

Data Cleaning

Data cleaning is the process of detecting and correcting inaccurate records from a data set. It

involves fixing or removing outliers, incorrectly formatted, duplicate or incomplete data within

a dataset. The duplicated data were cleaned and incorrect computations were adjusted. After

inputting the data into the system, each datum was crosschecked to ensure that there no wrong

data was inputted and also to ensure that there were no duplicate data.

The Applied Nonlinear Models

From past published studies on H–D modeling (Table 1), five (5) sets of 3- parameters nonlinear

models were chosen for this study. In other to avoid problems with back transformation bias,

only models that expressed tree height without transformations were used. The models were

parameterized so that parameter b0 defines the scale of the H–D models, which is the major

dimension of variability among sample plots. To generate parameter estimates, the datasets

were handled with R-script using package ‘nlme’ of R-environment [7].

Table 1: The five non-linear 3-parameters height-diameter models chosen are;

Model Form Reference Eq.

Chapman Richards HT = 1.3 + bo(1 − e

−b1.DBH)

b2 Chapman 1961, Richards 1959 [1]

Weibull HT = 1.3 + bo (1 − e

−b1.DBHb2

)

Yang et al. 1978 [2]

Schnute; HT = [1.3

b1 + (b2

b1 − 1.3

b1

)

1−e

−bo(DBH−DBH0)

1−e

−bo(DBH2−DBH0)

]

1

b1

Schnute 1981 [3]

Logistic HT = 1.3 +

bo

(1+b1

−1

.DBH−b2)

Hung et al., 1992 [4]

Korf/Lundqvist; HT = 1.3 + bo. e

(−b1.DBH−b2

) Stage 1963, Zeide 1989 [5]

bo, b1, b2 = model parameters to be estimated, e = base of the natural logarithm, HT = total tree height (m); DBH =

diameter at breast height (cm)

Evaluation Statistics

The models used in this were adjudged base on four test statistics; Root Mean Square Error

(RMSE), Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC). The best

model was adjudged on the criterion that the smaller the values of the evaluation statistics, the

better the fitting model. The model evaluation criterion is indicated below:

R

2 = 1 −

SSres

SStot

[6]

RMSE = √

∑ (Yi−Ŷ

i

n )2

i=1

n

[7]

AIC = n ln (

RSS

n

) + 2p [8]

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Egonmwan, Y. I., & Orukpe, W. O. (2024). Comparison of Non-Linear Height-Diameter Models of Tectona grandis L.f. in a Rainforest Site in Southern

Nigeria. Discoveries in Agriculture and Food Sciences, 12(5). 20-28.

URL: http://dx.doi.org/10.14738/dafs.125.15549

BIC = n ln(

RSS

n

) + p ln n [9]

Map of the Study Area

Plate 1: Satellite image of the study area.

RESULTS

The estimated parameters of the five fitted H-D models and their corresponding evaluation test

statistics criterion are summarized in table 2 below. Models with the lowest RMSE, AIC and BIC

and highest R2 values are judged the best. Schnute model had the highest R2 value of 99.9% but

it also had the highest RMSE, AIC and BIC values (see table 2), hence performed poorly in

predicting height in the study area. Weibull model although did not have the largest R2 of 97.2%

when compared to schnute, it had the smallest RMSE, AIC and BIC (1. 37596E-14, -832.6836

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Discoveries in Agriculture and Food Sciences (DAFS) Vol 12, Issue 5, October- 2024

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and -823.9054 respectively) and hence is the best fit model. The Chapman Richards, Logistic

and Korf models also performed well based on the evaluation criterion.

Table 2: Parameters and fit indices of the height prediction models for the teak data

Parameters Fit statistics

H-D Models b0 b1 b2 R

2 RMSE AIC BIC

Chapman- Richards

27.846 0.636416 18311.12 0.911 3.149 35.894 29.894

Korf 30.4928 0.063406 9.066469 0.9219 5.3368 49.6369 45.4151

Logistic 0.0000004 0.075 0.635506 0.9953 16.2201 78.6031 74.3813

Weibull 27.846 0.0000003 0.612322 0.9719 1. 37596E-14 -832.6836 -823.9054

Schnute 27.846 0 1.0635 0.999 2.23593E+15 15.3494 922.7465

bo, b1, b2 – parameter estimates, R2 – coefficient of determination, RMSE – root mean square error, AIC – Akaike

information criteria, BIC- Bayesian information criterion.

Fig.1: Scatter plot of total height (HT) against diameter at breast height (DBH) for Tectona

grandis

Figure 2: Scatter plot and fitted height trajectories of the five H-D models.

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Egonmwan, Y. I., & Orukpe, W. O. (2024). Comparison of Non-Linear Height-Diameter Models of Tectona grandis L.f. in a Rainforest Site in Southern

Nigeria. Discoveries in Agriculture and Food Sciences, 12(5). 20-28.

URL: http://dx.doi.org/10.14738/dafs.125.15549

Figure 3: Residual plot of the five H-D models.

Table 3: Summary statistics of the data

STATISTICS DBH (cm) THT(m) BA(m2) VOL.(m3)

Mean 44.17 33.10 0.17 4.34

Standard Error 1.38 0.52 0.01 0.26

Standard Deviation 13.78 5.20 0.10 2.60

Kurtosis -0.57 4.32 0.02 0.61

Skewness 0.38 -1.36 0.91 0.83

Minimum 15 8 0.0177 0.136

Maximum 75.5 40.5 0.4478 13.609

DISCUSSION

Summary of the growth variables are listed in table 3. The minimum and maximum diameter at

breast height was 15 cm and 75 cm respectively. The minimum height was 8m while the

maximum height was 40.5m. The standard error for the various parameters ranged from 0.01

to 1.69. The result revealed a strong relationship between the growth parameters for the

pooled data.

The results of goodness for predicted height are shown in table 2 along with the performance

criteria (R2, RMSE, AIC and BIC) for the five models. Models with the lowest RMSE, BIC and AIC

values closest to unity are known to perform best (Ahmadi et al. 2013). The R2 values as

indicated in table 2 showed that Logistic, Weibull and Schnute models had the highest R2 values

of 99.5%, 97.2% and 99.9% respectively, hence they were selected as the best models for the

H-D prediction which also agrees with the findings of [8]. This implies that 99.5%, 97.2% and

99.9% variation in height were explained by the diameter. Also, the curve was best captured by

the Chapman-Richards, Logistics and Schnute fit (figure 3). The predicted value follows the

same nonlinear pattern and fall close to the line of best fit without much outliers.

The Weibull model however, did not have the largest R2 but had the smallest RMSE, AIC and

BIC (1.37596E-14, -832.6836 and -823.9054 respectively) and hence was chosen as the best fit

model. Although Weibull model did not produce the highest R2 value, it was still chosen as the

best model because according to [9], R2 is a poor tool for model selection as it almost always

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Discoveries in Agriculture and Food Sciences (DAFS) Vol 12, Issue 5, October- 2024

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favors the most complex models. Also [10] stated that R2 cannot be used to select the best model

because it increases as predictors are added i.e. overfitting is guaranteed. Hence, R2 cannot be

used as a sole indicator for determining the best model. The Chapman Richards and the Korf

models were also well fitted to the tree H-D data.

The Logistics model yielded the maximum RMSE which implies that the estimator is not as

efficient compared to other growth curves used for the study, this is in sharp contrast in the

work by [8], who compared ten nonlinear H-D models and Logistic model out-performed all the

other models used in the study with the lowest RMSE. According to [11] determination of fixed

and random effects parameters in a model is a flexible decision subject to debate. While [12]

were of the opinion that all parameters in a model should first be considered mixed if

convergence is possible. However, [13] suggest those parameters with high variability and less

overlap in confidence intervals obtained by fitting at each plot should be considered mixed if

the convergence is not achieved when considering all the parameters. Hence the results of this

study suggest that the ability of DBH in determining height is strong enough as observed among

Chapman-Richards, Logistics and Schnute model’s ability for predictive purposes.

Weibull was the most suitable model for the teak plantation in this study, agreeing with the

work of [14]. The fitness of Weibull model in predicting tree height diameter distribution is well

documented. For example, [4] found the Weibull model to be adequate in predicting tree height

diameter data of nine major tree species in Ontario’s boreal forest. However, some researchers

such as [15] and [16] in their study had Weibull model as the least fit due to its inconsistency

in describing the tree height diameter relationship of trees. The analysis of the plot of residual

and predicted values demonstrated that there was little or no systematic bias towards over-or

underestimation of the tree total height (Fig. 3). Plotting the residuals also showed that the

models were randomly distributed and had heterogeneous residuals. The five H-D models

produced tree height trajectories that are consistent with biological realism: monotonic

increment, inflection point and asymptote (Fig. 2). A number of researchers found that adding

stand variables to the height-diameter equations and using the generalized height-diameter

models increase the precision [17]; [18]. Although Weibull model out-performed the others H- D models in this study, Chapman-Richards, Schnute and Logistics models explained over 90%

of the total variance in their corresponding response variables for the data sets. Hence, five

models used in this study have shown to be very flexible and have been used extensively in

growth and yield studies for describing height-age, diameter-age, and volume-age

relationships” [19]

CONCLUSIONS

From this study, we found out that the Weibull model was best suitable for the teak plantation

in the study site. When compared to previous studies, this H-D model was found to be more

accurate and applicable for predicting height-diameter relationship. However, height-diameter

relationship varies within a region, depending on local environmental conditions [20], [21]. The

models used in this study do not take into account the effects of climatic and ecological factors

on height-diameter relationships within different ecological sites. Hence further development

of ecoregion-based individual tree height-diameter models is critical for accurate models on

which to base forest management decisions [20], [21].

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Egonmwan, Y. I., & Orukpe, W. O. (2024). Comparison of Non-Linear Height-Diameter Models of Tectona grandis L.f. in a Rainforest Site in Southern

Nigeria. Discoveries in Agriculture and Food Sciences, 12(5). 20-28.

URL: http://dx.doi.org/10.14738/dafs.125.15549

Competing Interest

The authors have declared that no competing interests exist.

Financial Disclosure

The authors declared that this study has received no financial support.

References

[1]. Ahmadi K., Alavi S.J., Kouchaksaraei M.T., Aertsen W. (2013): Non-linear height-diameter models for oriental

beech (Fagus orientalis Lipsky) in the Hyrcanian forests, Iran. Biotechnology, Agronomy, Society and

Environment, 17: 431–440.

[2]. Egonmwan I.Y., Ogana F.N. Application of diameter distribution model for volume estimation in Tectona grandis

Lf stands in the Oluwa Forest Reserve, Nigeria. Tropical Plant Research. 2020;7(3):573-80.

[3]. Fang, Z, Bailey R, L. (2001). Nonlinear mixed effects modelling for slash pine dominant height growth following

intensive silvicultural treatments. For. Sci. 47:287–300.

[4]. Gregoire T. G., Schabenberger O., Barrett. J. P. (1995). Linear modelling of irregularly spaced, unbalanced,

longitudinal data from permanent plot measurements. Can. J. For. Res. 25:137.

[5]. Huang, S., Titus, S., J and Wiens, D. P. (1992). Comparison of nonlinear height-diameter functions for major

Alberta tree species. Canadian Journal of Forest Research 22: 1297 - 1304.

[6]. Huang, S., Price, D., and Titus, S.J. 2000. Development of ecoregion-based height diameter models for white

spruce in boreal forests. For. Ecol. Manage. 129: 125–141. doi:10.1016/S0378-1127(99)00151-6.

[7]. Krisnawati, H., Wang Y., and Lui L. (2010). Generalized height-diameter models for Acacia mangium Wild.

Plantations in Sumatra. Journal of Forestry Research 7 (1):1 - 19.

[8]. Meyer HA. A mathematical expression for height curves. Journal of Forestry. 1940; 38:415–420.

[9]. Miranda I, Sousa V and Pereira H (2011) Wood properties of teak (Tectona grandisL.f.) from a mature

unmanaged stand in East Timor. Journal of Wood Science 57: 171–178.

[10]. Mkt updated July 1, 2019: The best model selected with AIC have lower R2 than the full/ global model.

[11]. Nugroho, B. (2014). Menuju KPH mandiri, apa yang harus dilakukan? In Sugiharto, Strategi pengembangan KPH

dan perubahan struktur kehutanan di Indonesia. Jakarta: Direktorat Jenderal Planologi Kehutanan Kementerian

Lingkungan Hidup dan Kehutanan

[12]. Osman E. l., Mamoun H., Idris El Zein A., Ibrahim E. l and Mugira, M. (2013). Height- Diameter Prediction

Models for Some Utilitarian Natural Tree species. Journal of Forest Products and Industries, 2013, 2(2), 31-39

ISSN 2325-453X.

[13]. Peng, C., Zhang, L., and Liu, J. (2001). Developing and validating nonlinear height-diameter models for major

tree species of Ontario’s boreal forests. Northern Journal of Applied Forestry 18:87 - 94.

[14]. Pinheiro J. C, Bates, D. M. (1998). Model building for nonlinear mixed effects model. Madison (WI): University of

Wisconsin.

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Discoveries in Agriculture and Food Sciences (DAFS) Vol 12, Issue 5, October- 2024

Services for Science and Education – United Kingdom

[15]. R Core Team (2017) R: A language and environment for statistical computing. R Foundation for Statistical

Computing, Vienna, Austria. Available from: http://www.R-project.org/. (accessed: 30 Jun. 2017).

[16]. Russel Pierce (2014): The best model selected with AICc have lower R2 than the full/ global model.

[17]. Scaranello, M. A da Silva, Luciana F.A., Simone A.V., Plinio B de Camargo, C.A J. and Luiz, A.M. (2012). Height- diameter relationship of Tropical Atlantic Moist Forest trees in south eastern Brasil Sci. Agric. 69 (1):26-37.

[18]. Somers GL, Farrar RM. Bio-mathematical growth equations for natural longleaf pine stands. Forest Science.

1991; 37:227–244.

[19]. Temesgen G, Amare B and Abraham Mahari (2014). Population dynamics and land use/land cover changes in

Dera District, Ethiopia. Global Journal of Biology, Agriculture and Health sciences. 3(1):137-140.

[20]. Tewari V.P., Álvarez-González J.G., García O. (2014): Developing a dynamic growth model for teak plantations in

India. Forest Ecosystems, 1: 1–9.

[21]. Y. I. Egonmwan (2022): Height-Diameter Models for Prediction of Teak Stand in Western Nigeria Asian Journal

of Research in Agriculture and Forestry AJRAF, 8(4): 293-300, 2022; Article no. AJRAF.93599.