Page 1

International Journal of Ecology and Environmental Sciences 32(2): 221-225, 2006

© INTERNATIONAL SCIENTIFIC PUBLICATIONS, NEW DELHI

Nonlinear Statistical Model for Culm Growth of Muli Bamboo,

Melocanna baccifera

GITASREE DAS1*, ASHESH K. DAS2 AND SUBRATA NANDY2

1 Department of Statistics, North-Eastern Hill University, Shillong, Meghalaya, 793003, India.

2 Department of Ecology and Environmental Science, Assam University, Silchar, Assam, 788011, India.

* Corresponding author; Post Box No. 7, Laitumkhrah, Shillong, Meghalaya, 793003, India. E-mail:

gitasree@sancharnet.in

ABSTRACT

Melocanna baccifera or Muli bamboo is an important non-timber forest product (NTFP) of North-East India. The

study on growth pattern of bamboos, especially the commercially most important parameter viz. the culm height,

is important in developing scientific management systems for optimum yield. In this article nonlinear statistical

models were used to describe the culm height growth of Melocanna baccifera. The adequacy of the models were judged

by testing the validity of the assumptions of randomness and normality of residuals by using one-sample run test

and Shapiro-Wilk test respectively. Durbin-Watson test was used to examine the presence or absence of

autocorrelation. To assess the goodness of fit of the suggested models, statistics viz. R2, root mean square error

(RMSE), mean absolute error (MAE) etc. were computed for each model. The estimated parameters can be treated

as a summary of the growth pattern. These values may be used for comparison across species or varieties or same

species growing under different environmental conditions. The physical interpretation of the parameters is also being

discussed.

Key Words: Bamboo Culm, Growth Model, Nonlinear, Logistic, Gompertz, Diagnostics

INTRODUCTION

Nonlinear statistical models have been used to describe

growth behaviour, as it varies in time. The type of

model needed in a specific area and specific situation

depends on the type of growth that occurs. A nonlinear

model is one in which at least one of the parameters

appears nonlinearly. Generally growth models are

mechanistic, arising as a result of making assumptions

about the type of growth, writing down differential or

difference equations that represent these assumptions

and then solving these equations to obtain a growth

model. Most of the mechanistic modeling has been

done in a biological context. Biologists are interested in

the description of growth and are interested to

understand its underlying mechanism. For example it is

very important from management point of view to

know how large the plants will grow, how fast they grow

and how these factors respond to environmental

conditions or treatments. Unlike empirical models, here

the parameters have meaningful biological inter-

pretation. If the curve is changing in a nonlinear fashion

it may be possible to find a simple mathe-matically

defined curve which describes the change and which can

be estimated from the data. The estimated parameters

can then be treated as a summary of the growth pattern.

These values may be used for comparison across species

or varieties or same species growing under different

environmental conditions. While single growth curves

may be of interest in their own right, very often growth

curve data are collected to determine how growth

responds to various treatments or other covariate

information. Therefore it is essential to reduce each

individual growth curve to a small number of

parameters so that changing patterns of growth can be

understood in terms of changes in these parameters.

Page 2

Das et al.: Non-Linear Statistic Models

Int. J. Ecol. Environ. Sci.

222

Each growth curve may be summarised by its parameter

estimates as a single low-dimensional multivariate

observation. These observations may then be subjected

to an analysis of variance or to a regression analysis

(Seber and Wild, 1989).

Richards (1959) discussed application of growth

functions viz. monomolecular, logistic, gompertz, von

Bertalanffy’s extended form etc. for plant data.

Venugopal and Prajneshu (1997) studied generalized

allometric growth model for length-weight relationship

of Indian pearl oyster, observed for a period of 36

months. They also tested the validity of the various

assumptions and explored the possibility of

autocorrelation in error term. The use of sigmoidal

growth models viz. logistic, gompertz etc. has been

discussed by Gore and Paranjpe (2000) for describing

the growth of single species population.

MATERIALS AND METHODS

In the present article an attempt has been made to

search for an appropriate statistical model to describe

the culm height growth of Melocanna baccifera as it varies

with time. Growth of culms was studied in a stand of

Muli bamboo (Melocanna baccifera) during August 1999

to May 2000 (Nandy et al, 2004) in Hailakandi district

of Barak Valley in North-East India. Bamboo being a

quick growing plant and its height getting stabilized

over a comparatively short span of time, longitudinal

study was preferred rather than considering different

random samples at different time (Hills 1974). Forty

newly sprouted culms were selected randomly and

identified with numbered aluminium foil. The culms

emerge during the month of August-September and it

approximately takes 37 weeks for height to get

stabilized. Since all the forty culms began their life

together, their average height was considered to study

the behaviour of growth. The height growth curve of

culms appears to have a smooth S shape or sigmoidal

pattern (Figure 1).The shape of the growth curve is

described by the rate of change of growth at different

times. Even though the height growth stabilises within

a short period of eight months, it does not grow at the

same rate throughout this period. During the first three

months the growth rate is increasing with maximum

growth of 17.7 cm per day in November. Nearly sixty

per cent of the total height growth is attained during

this period. From December onwards growth rate starts

decreasing. During February, March and April it

decreases very slowly, finally showing zero growth by

the end of April, i.e. the culms seize to grow in height

any further at this point of time (Figure 1).

Figure 1. Culm height growth showing fitted Gompertz model

(----) and sample values (P) (upper) and culm height

growth rate (lower) of Melocanna baccifera.

Sigmoidal Growth Model

The shape of the growth curve is described by the rate

of change of growth at different time t. For certain types

of growth data, the growth rate does not steadily

decline, but rather increases to a maximum before

declining to zero. This criterion, as discussed in detail

Page 3

32: 221-225 Das et al.: Non-Linear Statistic Models

223

[]

df

dt

g f

( ) ( )

h kh f

( )

∞−

df

dt

r

kf k

f

=−

()

f t

( )

k

−

e

r t q

()

=

+

−

1

df

dt

rfkfrk

=−>>

(log log ), 00

()

()

f t

( )

k r t

(

q

expexp)

=∗−−−

f t

( )

k

−

e

t

r t q

(

( )

)

=

+

+

−

1

ε

()

[]

f t

( )

k r t

(

qt

exp exp) ( )

=∗−−−+ ε

by Nandy et al. (2004), is observed in the growth-rate

curve for Melocanna baccifera (Figure 1). Sigmoidal

growth models can best represent the growth curves in

such cases. For such models the position of the point of

inflection being the time when the growth rate is

maximum. Sigmoidal behaviour can be achieved by

modeling the current growth rate as proportional to the

product of functions of the current size and remaining

growth, namely

(1)

where g and h are increasing functions with g(0) = h(0)

=0, f is the size at time t and k is the maximum possible

growth size.

Logistic model

The simplest form of (1) is with g(f)= h(f) = f, i.e. the

growth rate is proportional to the product of the present

size and the future amount of growth. so that

(2)

where r>0 and 0<f <k and constant of proportionality

being used is r / k, One may observe from (2) that the

relative growth rate f -1df / dt decreases linearly in f as f

approaches k. The equation (2) has a general solution as

follows:

(3)

known as logistic or autocatalytic model. From (2) one

can obtain the second derivative. It is easily seen that

growth rate is maximum when f = k / 2 and from (3)

f = k / 2 occurs when t = q. The maximum growth rate

is rk / 4 and the growth rate is symmetrical about t = q,

giving a symmetrical sigmoidal growth curve. The curve

increases to an upper limit k when t is large. The

constant k is known as the ‘carrying capacity’ of the

environment.

Gompertz Model

The gompertz model is often used where the growth is

not symmetrical about the point of inflection. The

growth rate is

(4)

and the relative growth rate now declines non-linearly

in f, or more precisely linearly with log f. From (4) we

obtain

(5)

The point of inflection is at time t = q with f = k / e, at

which point the maximum growth rate rk / e occurs.

Additive Error Structure

The above models have been presented determin-

istically, which is obviously unrealistic. We replace

these nonlinear deterministic models with nonlinear

statistical models by including additive error structure

(t) with appropriate assumptions, on the right hand

side of the equations (3) and (5). Thus the logistic and

gompertz models respectively appear as follows:

(6)

(7)

Estimation of Parameters

In order to estimate the parameters k, q, and r by the

method of least squares, (t) is assumed to be indepen-

dently and identically distributed (i.i.d) following

normal distribution. However, normal equations

obtained by minimization of residual sum of squares are

nonlinear in parameters. Since it is not possible to

obtain explicit solutions for nonlinear equations, the

next alternative is to obtain approximate solutions by

employing iterative numerical procedures. In this work,

the most commonly used method known as Levenberg-

Marquardt non-linear iterative procedure (Seber and

Wild 1989) was used for fitting the models to the data.

Page 4

Das et al.: Non-Linear Statistic Models

Int. J. Ecol. Environ. Sci.

224

ε ρε

( )

t

() ( )

u tt

=−+

1

To start the iterative procedure, initial estimates of the

parameters of the models were required. Many sets of

initial values were tried to ensure global convergence.

The iterative procedure was stopped when the reduction

between the successive residual sums of squares was

found to be negligible.

Model Adequacy

It is important to remember that our confidence in

statistical inference procedures is related to the validity

of the assumptions about them. A mechanically made

inference may be misleading if some model assumption

is grossly violated. An examination of the residuals is an

important part of regression analysis, because it helps to

detect any inconsistency between the data and the

postulated model. If no abnormalities are exposed in

this process, then we can consider the model adequate

and proceed with the relevant inferences. Otherwise we

must search for a more appropriate model.

The adequacy of the models can be judged by

testing the validity of the underlying assumptions of

randomness and normality of residuals. Residuals were

examined by using one-sample run test for inferring

about randomness (Draper and Smith, 1981) and

Shapiro-Wilk test for testing normality.

Growth data will often contain correlated errors if

the data are collected as repeated measurements over

time on the same experimental units. Growth models

with uncorrelated errors become untenable because of

long runs of residuals with the same sign (Seber and

Wild, 1989). When the models suggested are found to

be inappropriate for a data set, using Durbin-Watson

test one can examine the presence or absence of

autocorrelation (Chatterjee and Price, 1977). This test

examines the presence or absence of first order

autocorrelations, i.e. AR (1), among the residuals. If this

test reveals presence of AR(1) error structure, the term

(t) in equation (6) and (7) shall be replaced by

(8)

where is the autoregressive model parameter and u(t)

is assumed to be i.i.d. normally distributed with mean

zero and variance

together with the AR parameter can be estimated by

using two-stage estimation procedure. Finally, to assess

the goodness of fit of the suggested models, statistics

viz. R2, root mean square error (RMSE), mean absolute

error (MAE), were computed for each model.

2. The unknown model parameters

RESULTS AND DISCUSSION

In the first instance, the logistic model with additive

error term given by eqn. (6) is fitted to the data and the

results are presented in Table 1. A perusal of the first

column of this table shows that the assumption of

randomness of residuals is not satisfied, since the

calculated value of the run test statistic (4.995) lies in

the critical region at 5% level. Shapiro-Wilk statistic W

(0.933) for testing the assumption of normality being

smaller than the tabulated value (0.936) at 5% level

also lies in the rejection region. Thus this model is

inappropriate for the present data set. However, the

calculated value of Durbin-Watson statistic (0.148),

being less than the tabulated value 1.41 at 5% level,

indicates possibility of AR(1) error structure. Therefore

eqn. (6) with errors at two consecutive time epochs

having AR(1) structure given by eqn. (8) was fitted to

the data and the results are shown in the second

column of Table 1. Shapiro-Wilk statistic W (0.968)

lies in the acceptance region at 5% level. However the

value of run-test statistic (3.178) though less than

before, but still lies in the critical region. This shows

that the modification is not adequate. Thus, the logistic

model with additive error and AR(1) structure is not

found satisfactory for the data. And as such functional

form of the model needs to be altered.

Table 1. Summary statistics of fitting logistic and

gompertz model to culm height growth data of

Melocanna baccifera.

Model

StatisticLogistic

iid

Gompertz

iid AR(1)AR(1)

Parameter estimates

k

r

q

AR

Residual analysis

Run test: z

Shapiro-Wilk test: W

Durbin-Watson test: d

Goodness of fit statistic

R2

Mean absolute error

Root mean square error 55.231

1651.227 1660.977 1697.267 1698.855

0.2800.268

13.03213.085

0.945

0.184

10.853

0.182

10.839

0.833

4.995

0.933

0.148

3.178

0.968

4.303

0.940

0.354

1.808

0.950

1.799

0.9926

44.304

0.9991

13.689

18.791

0.9988

17.551

22.731

0.9996

9.576

13.164

Page 5

32: 221-225Das et al.: Non-Linear Statistic Models

225

Next, another form of sigmoidal growth model,

where the growth is not symmetrical about the point of

inflection viz. gompertz model with additive error

structure (eqn. (7)) was considered and the results are

given in the third column of Table 1. Now Shapiro-

Wilk statistic W (0.940) lies in the acceptance region at

5% level. But a high value of run test statistic (4.303)

violates the assumption of randomness of residuals.

Nevertheless, the low value (0.354) of Durbin-Watson

statistic indicates possibility of AR(1) error structure.

Therefore gompertz model with additive error (Eqn.

(7)) and AR(1) structure (eqn.(8)) was tried and the

results are shown in the fourth column of Table 1. The

residual analysis now clearly shows that with this

modification the assumption of randomness is not

rejected, since the calculated value of run-test statistic

(1.808) is less than the tabulated value 1.96 at 5%

level. Shapiro-Wilk statistic W (0.950) also lies in the

acceptance region at 5% level. Thus assumption of

normality of residuals is not violated. Also the

calculated value of Durbin-Watson statistic (1.799)

being greater than the tabulated value 1.52 at 5% level

indicates absence of autocorrelation. Hence, gompertz

model with additive and AR(1) error structure appears

to be the most suitable model for the present data.

Finally, to assess the goodness of fit, statistics viz.

R2, root mean square error (RMSE) and mean absolute

error (MAE) were computed for each model. Gompertz

model with additive and AR(1) error structure gives the

highest value of R2 (0.9996) and the lowest value of

MAE (9.576) and RMSE (13.164). This finally

confirms that the gompertz model with additive and

AR(1) error structure is the most suitable model for

describing the culm growth of Melocanna baccifera. The

AR(1) error structure implies that in addition to

random fluctuations, the error at a particular time

epoch is directly proportional to the error at the

immediate past time epoch. The fitted values of culm

height by gompertz model with additive and AR (1)

error structure along with the observed values are shown

in Figure 1.

Regarding physical interpretation of the para-

meters of the gompertz model, that was found to be the

most suitable for the present data, one may refer to eqn.

(5) and the interpretation of its parameters. The

parameter q gives the position of the point of inflection

i.e. the time when the growth rate is maximum. From

Figure 1 one may notice that the peak growth rate is

observed for nearly seven weeks, and an average of

these time periods is 10.5 weeks, which is very close to

the estimated value of q = 10.839 weeks. Also average

growth rate during this peak period is 116.42 cm per

week, again very close to the estimated maximum

growth rate rk / e = 113.75 cm per week. Finally the

growth attained at this point is estimated as f = k / e =

624.97 cm is also not very far from the observed growth

attained during this period as 602.38 cm. Hence the

observed culm height growth curve of Melocanna

baccifera may be summarised by its parameter estimates

as k = 1698.855, q = 10.839 and r = 0.182. The role of

k, q and r is very important in developing scientific

management systems for optimum yield of bamboo.

ACKNOWLEDGEMENTS

One of us (G.D.) would like to acknowledge Prof.

Debasis Sengupta, Indian Statistical Institute, Kolkata

for helping her to understand the basic nuances of

diagnostics in regression analysis. She acknowledges

also Prof. Prajneshu, Principal Scientist, Indian

Agricultural Statistics Research Institute, New Delhi,

whose numerous articles on nonlinear growth models

and stimulating discussion with him on several

occasions has immensely helped shape this work. The

study was partially supported by an ongoing project

“Application of Statistical Modelling on Management

of Bamboos” sponsored by the Department of Science

and Technology, India.

REFERENCES

Chatterjee, S. and Price, B. 1977. Regression Analysis by

Example. John Wiley, New York. 227 pages.

Draper, N.R. and Smith, H. 1981. Applied Regression

Analysis. John Wiley, New York. 709 pages.

Gore, A.P. and Paranjpe, S.A. 2000. A Course on Mathe-

matical and Statistical Ecology. Kluwer Academic

Publishers, Dordrecht. 304 pages.

Hills, M. 1974. Statistics for Comparative Studies. Chapman

and Hall, London. 194 pages.

Nandy, S., Das, A.K. and Das, G. 2004. Phenology and culm

growth of Melocanna baccifera (Roxb.) Kurz in Barak

Valley, North-East India. Journal of Bamboo and Rattan

3: 27 - 34.

Richards, F. J. 1959. A flexible growth function for empirical

use. Journal of Experimental Botany 10: 290-300.

Seber, G.A.F. and Wild, C. J. 1989. Nonlinear Regression.

John Wiley, New York. 768 pages.

Venugopalan, R. and Prajneshu. 1997. A generalized

allometric model for determining length-weight

relationship. Biometrical Journal 39: 733-739.