Nonlinear statistical model for culm growth of muli bamboo, Melocanna baccifera
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Article: Phenology and culm growth of Melocanna baccifera (Roxb.) Kurtz in Barak Valley, NorthEast India
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ABSTRACT: The phenology and growth of culms of Muli bamboo (Melocanna baccifera) were studied in the Hailakandi district of Barak Valley in NorthEast India. The culms emerge during the months of August and September, and the growth curve is Sshaped. The growth continues for a period of 245 days with rapid growth attained after 45 days. The leafing pattern is characterized by periodic growth leafexchange type. The adaptive strategy of this growth pattern is discussed in the context of restoration of degraded lands.Journal of Bamboo and Rattan 12/2003; 3(1):2734.  [Show abstract] [Hide abstract]
ABSTRACT: Wellknown allometric model is critically reviewed for determining the lengthweight relationship. Procedure to be followed for applying this model under the assumption of multiplicative as well as additive error is discussed. The possibility of autocorrelations in error term is explored. It has been demonstrated that the usual allometric relationship does not always work satisfactorily. Accordingly, a generalization of this model is studied in detail. As an illustration, this generalized version of allometric model is used to determine the lengthweight relationship of some data on the Indian pearl oyster.Biometrical Journal 01/1997; · 1.15 Impact Factor  Journal of Experimental Botany  J EXP BOT. 01/1959; 10(2):290301.
Page 1
International Journal of Ecology and Environmental Sciences 32(2): 221225, 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, NorthEastern 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. Email:
gitasree@sancharnet.in
ABSTRACT
Melocanna baccifera or Muli bamboo is an important nontimber forest product (NTFP) of NorthEast 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 onesample run test
and ShapiroWilk test respectively. DurbinWatson 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 mathematically
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.
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Das et al.: NonLinear Statistic Models
Int. J. Ecol. Environ. Sci.
222
Each growth curve may be summarised by its parameter
estimates as a single lowdimensional 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 lengthweight 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 NorthEast 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 AugustSeptember 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
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32: 221225Das et al.: NonLinear 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
( )
kr t
(
q
expexp)
=∗−−−
f t
( )
k
−
e
t
r t q
(
( )
)
=
+
+
−
1
ε
()
[]
f t
( )
kr t
(
qt
exp exp) ( )
=∗−−−+ ε
by Nandy et al. (2004), is observed in the growthrate
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 nonlinearly
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 nonlinear iterative procedure (Seber and
Wild 1989) was used for fitting the models to the data.
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Das et al.: NonLinear 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 onesample run test for inferring
about randomness (Draper and Smith, 1981) and
ShapiroWilk 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 DurbinWatson
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 twostage 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. ShapiroWilk 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 DurbinWatson 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. ShapiroWilk statistic W (0.968)
lies in the acceptance region at 5% level. However the
value of runtest 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
Statistic Logistic
iid
Gompertz
iid AR(1)AR(1)
Parameter estimates
k
r
q
AR
Residual analysis
Run test: z
ShapiroWilk test: W
DurbinWatson 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
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32: 221225Das et al.: NonLinear 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 DurbinWatson
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 runtest statistic
(1.808) is less than the tabulated value 1.96 at 5%
level. ShapiroWilk 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 DurbinWatson 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, NorthEast 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: 290300.
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 lengthweight
relationship. Biometrical Journal 39: 733739.