Journal of Forest Research

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A simultaneous estimation procedure for theparameters of the maximum size-density line andself-thinning curve to predict stand developmentof fast-growing tropical species
Susumu Kurinobu & Tomiyasu Miyaura
To cite this article: Susumu Kurinobu & Tomiyasu Miyaura (2011) A simultaneous estimation
procedure for the parameters of the maximum size-density line and self-thinning curve to predict
stand development of fast-growing tropical species, Journal of Forest Research, 16:6, 522-525,
DOI: 10.1007/s10310-010-0232-7
To link to this article: https://doi.org/10.1007/s10310-010-0232-7

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Published online: 30 Oct 2010.

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J For Res (2011) 16:522–525
DOI 10.1007/s10310-010-0232-7

SHORT COMMUNICATION

A simultaneous estimation procedure for the parametersof the maximum size–density line and self-thinning curve
to predict stand development of fast-growing tropical species
Susumu Kurinobu • Tomiyasu Miyaura

Received: 2 July 2010 / Accepted: 29 September 2010 / Published online: 30 October 2010
Ó The Japanese Forest Society and Springer 2010

Abstract A simultaneous estimation procedure for theparameters of two functions, i.e., the maximum size–density line and the self-thinning curve, is presented to predict
stand development for fast-growing tropical species. This
procedure assumes that the rate of periodic reduction instand density with increasing quadratic mean diameter on alogarithmic scale (r) will increase inversely proportionallyto the distance from the maximum size–density line andconsequently equals the slope of the line at distance 0.
Under this assumption, the maximum size–density line canbe incorporated into the self-thinning curve to form an
integrated equation with three parameters: k and m, theslope and constant of the maximum size–density line, anda, the rate of reduction of r of the self-thinning curve.
These parameters are estimated simultaneously using
measurement data on stand density, quadratic mean
diameter, and the corresponding r. This procedure was
evaluated by application to two data sets: 186 measurements of Acacia mangium and 95 measurements ofParaserianthes falcataria, for which the parameters k andm have previously been calculated. The parameters estimated using this procedure were in good agreement withprevious ones based on the A. mangium data set, and theElectronic supplementary material The online version of thisarticle (doi: 10.1007/s10310-010-0232-7) contains supplementarymaterial, which is available to authorized users.
S. Kurinobu (&)
Forest Tree Breeding Center,
Forestry and Forest Product Research Institute,Ishi, Juo, Hitachi, Ibaraki 319-1301, Japane-mail: kurinobu@affrc.go.jp
T. Miyaura
Faculty of Science and Technology, Ryukoku University,
Seta, Ohstu, Shiga 520-2194, Japan123

differences found for the P. falcataria data set were also
small, within the error variances. Therefore, it is concluded
that the proposed procedure would give almost the same
estimates from a single calculation step as the previous
procedure that required two separate calculation steps.
Keywords Acacia mangium  Maximum size–density
line  Paraserianthes falcataria  Quadratic mean
diameter  Self-thinning curve

Introduction
An intentional shift toward solid timber production has
become evident in the planting of fast-growing tropical
species, not only Paraserianthes falcataria but also Acacia
mangium and Eucalyptus, which have been planted
to supply raw material for pulp and paper (Cossalter and
Pye-Smith 2003). This trend may be primarily due to thehigher profitability of solid timber production, but is also
due to increasing demand for substitutes for reduced timber
supply from natural forests. Therefore, the role of standdensity control in plantations has become more important
than ever before to ensure sustainable supply of target sizeof logs to meet diversified production purposes, such as
furniture, finger-joining wood, and veneer.
Stand density control, aiming to optimize tending
regime in the plantation, requires a growth model able topredict stand development based on the size–density relationship. This type of growth model, although it has been
widely used in Japan (Yoda et al. 1963; Tadaki 1964; Ando
1968), is still limited for fast-growing tropical species.
However, a few of them have been applied for A. mangium
and P. falcataria, using the relationship between standdensity and quadratic mean diameter (Kurinobu et al.

J For Res (2011) 16:522–525

523

2006, 2007); first the maximum size–density line wasdetermined, then the self-thinning curve was derived. The
size–density relationship has been recognized as variable
even within a species, because of the controversy regarding
the generality of the 3/2 power low (Weller 1987; Zeide
1987; Osawa and Sugita 1989; Sackville Hamilton et al.
1995). Therefore, in practice, a simplified procedure for
estimating the parameters of the maximum size–density
line and the self-thinning curve is desirable, so that themodel can be easily revised for target environments forplantation establishment.
In this paper, a simultaneous estimation procedure forthe parameters of the maximum size–density line and selfthinning curve was derived by assuming that the rate ofperiodic reduction in stand density with increasing quadratic mean diameter on a logarithmic scale will increase
inversely proportionally to the distance from the maximum
size–density line and consequently equals the slope of theline at distance 0. Then, the procedure is examined byapplication to A. mangium and P. falcataria data sets
whose parameters were previously calculated (Arisman
et al. 2004; Kurinobu et al. 2008).

Materials and method
The maximum size–density line for quadratic mean
diameter (dq) is expressed with the following linear equation on a log–log scale of stand density [N = log(n), n:number of stems per ha] and diameter [D = log(dq)](Reineke 1933):
N ¼ m þ kD;

ð1Þ

where k and m are the slope of the line and a constant,respectively. The distance between the line and themeasurement (L) is given by the following formula(Arisman et al. 2004):
L ¼ xy=ðx2 þ y2 Þ1=2 ;ð2Þwhere x is the distance between the measurement(D, N) and the size–density line in the horizontal
direction: x = (N - m)/k - D, and y is the distance in thevertical direction: y = m ? kD - N. Thus the relationshipbetween x and y is expressed as y = -kx, and replacing y in
Eq. 2 with this relationship gives the following equation:
L ¼ kx=ð1 þ k2 Þ1=2 :ð3ÞMortality caused by self-thinning is expressed as a rateof periodic reduction in stand density (DN) with increasing
quadratic mean diameter (DD) on a logarithmic scale(Kurinobu et al. 2006). The trend for the rate (r = DN/
DD) as a function of distance (L) was fitted with an
exponential function (Eq. 4), instead of the Mitscherlich

function used in the previous study. This is because thesimple exponential function can satisfy the assumption thatthe rate will equal the slope of the line at distance 0,
although both functions can express the general trend thatthe rate will increase inversely proportionally to the
distance from the maximum size–density line.
r ¼ k expðaLÞ;ð4Þ

where a is the rate of reduction of r in the self-thinning
curve. Equation 4 can be rearranged by replacing L with
Eq. 3 and x with (N - m)/k - D. In this way, the maximum
size–density line is incorporated into the self-thinning
curve, leaving parameters k and m, as shown in Eq. 5.
h1=2 ir ¼ k exp aðm þ kD  N Þ= 1 þ k2:
ð5Þ
The three parameters in Eq. 5 (a, k, and m) can be
estimated simultaneously using measurement data on standdensity (N), quadratic mean diameter (D), and thecorresponding r (=DN/DD) by using the Gauss–Newton
iteration method with a set of partial derivatives of theequation with respect to the parameters (Snedecor andCochran 1989; see Appendix online).
This procedure was evaluated using two data sets: 283
diameter measurements from 51 permanent plots of
A. mangium, and 128 measurements from 32 plots of
P. falcataria (Table 1). The permanent plots were 0.1 ha insize for both species, and diameter at breast height (DBH)
was measured for all surviving trees. Periodic measurement, twice per year, was conducted for 3 years forP. falcataria, while the measurement for A. mangium wasrepeated almost annually until 8–9 years old. The parameters of size–density lines of both data sets, k and m, werepreviously calculated by minimizing the average distance
between measurements and the maximum size–density
line, i.e., the minimum distance boundary method(Arisman et al. 2004; Kurinobu et al. 2008).
The periodic reduction in stand density (DN) on a logarithmic scale and the increment of quadratic mean diameter (DD) were calculated using 183 pairs of consecutive

Table 1 A. mangium and P. falcataria data sets for application of
the simultaneous estimation procedureSpecies/trait

A. mangium
Mean

P. falcataria
Mean

Number of plots

51 (283)

32 (128)

Age (years)

5.9 (3–10)

5.4 (3–8)

DBH (cm)
Stem density (ha-1)

14.5 (6.4–23.1)
1,168 (540–2,210)

18.0 (10.2–31.4)
878 (300–1,570)

The number in parenthesis associated with the number of plots is thetotal number of measurements, and those for age, DBH, and stem
density are measurement ranges

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J For Res (2011) 16:522–525

distance

2500
2000
1500

-0.1
0

0.0

0.1

0.2

0.3

0.4

1000

rate

-1
500

-2
-3

10

15

20

25 30

-4

Quadratic mean diameter (cm)(a) Maximum size-density line of A. mangium(b) Self-thinning curve of A. mangium
distance

2500
2000
1500

-0.1
0

0.0

0.1

0.2

0.3

0.4

1000
-1

rate

The three parameters estimated by the simultaneous estimation procedure for the two data sets are presented
together with the previous estimates of k and m in Table 2.
For A. mangium, the parameters estimated using this procedure agreed well with the previous ones. The differences
found for P. falcataria were also small, within the error
variances. These results suggest that the proposed procedure would give almost the same estimates as the previous
method. However, the coefficients of determination weregenerally lower, being 0.472 for A. mangium and 0.335 forP. falcataria, due to instability of the rate, which varies
from plot to plot depending on conditions.
These results were further examined by plotting therespective maximum size–density line and self-thinning
curve for the two species (Fig. 1). There were a few
measurements slightly above the maximum size–density
line estimated by the proposed procedure for both species(Fig. 1a, c). These results may be regarded as a possible
consequence of the parameter estimation, because thisprocedure simply determines the three parameters tomaximize the correlation between the rate and the estimates according to Eq. 5. The self-thinning curve showed
fairly good agreement with the average rates for the eightgroups of measurements that were sorted by their distances(Fig. 1b, d). Thus, the proposed procedure will capture a

Number of stems [ha -1]

Results and discussion

trend for the rate reasonably well on an average basis evenwith large plot-to-plot variation.
The assumption for the trend of the rate (DN/DD), whichbecomes steeper as it gets closer to the maximum size–
density line and consequently equals the slope of the line at
distance 0, is known as density-dependent mortality(Tadaki 1963; Osawa and Sugita 1989). Unlike in otherstudies, this trend was accounted for by the distance
between measurements and the maximum size–density
line, and then the distance itself was absorbed, yielding asingle equation (Eq. 5) with three unknown parameters: a,k, and m, for simultaneous solution. Therefore, the procedure presented herein is regarded as one of the mathematically integrated solutions to account for density-related

Number of stems [ha -1]

measurements of A. mangium and 95 pairs for P. falcataria, respectively. Then, the simultaneous estimation
procedure was applied to these pairs of data, i.e., the rate
[r = DN/DD = (Nt?1 - Nt)/(Dt?1 - Dt)] and the corresponding stand density (Nt) and quadratic mean diameter(Dt), to estimate the parameters a, k, and m, as shown in
Eq. 5. In the P. falcataria data set, there was a measurement with abrupt reduction in stand density due to illegal
logging by encroachment, which was thus omitted from theanalysis.

500

-2
-3

10

15

20

25 30

Quadratic mean diameter (cm)(c) Maximum size-density line of P. falcataria-4(d) Self-thinning curve of P. falcataria

Fig. 1 Maximum size–density line and self-thinning curve plottedusing the parameters resulting from the simultaneous estimation
procedure (Table 2). a, c Agreement of maximum size–density line
for A. mangium and P. falcataria; solid line simultaneous estimation
procedure, dashed line minimum distance boundary method. b,
d Trend between the rate (r = DN/DD) and the distance between themeasurement and the maximum size–density line. Closed circles are
the average rates for the eight groups of measurements that were
sorted by their distances, and the estimated trend is shown by the solidline

Table 2 Parameters estimated by the simultaneous estimation procedure and by the minimum distance boundary method for the A. mangium
and P. falcataria data sets
Parameters

A. mangium
Simultaneous estimation
procedure

P. falcataria
Minimum distanceboundary method

Simultaneous estimation
procedure

Minimum distanceboundary methoda-15.397 (1.698)

k-1.638 (0.109)-1.67-2.060 (0.138)-1.912m

5.129 (0.138)

5.204

5.625 (0.193)

5.445

2

R

0.472

-15.230 (2.920)

0.335

Parameters a, k, and m are shown in Eq. 5. k and m by minimum distance boundary method were from Arisman et al. (2004) for A. mangium andKurinobu et al. (2008) for P. falcataria. Standard errors for parameter a, k, and m are given in parenthesis. R2 is the coefficient of determination

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J For Res (2011) 16:522–525

mortality, focusing on the relationship between quadratic
mean diameter and stand density.
Density-related mortality was well described with thisprocedure on an average basis, as shown in Fig. 1b, d.
However, the coefficients of determination were generally
low, due to the nature of the rate, which is prone to beaffected by plot-to-plot variation in periodic growth andmortality (Table 2). Thus, this procedure requires either adata set with sufficient number of measurements to offset
plot-to-plot variation of the rate or proper stratification ofplots by site index classes, etc. to reduce the variation inorder to estimate parameters suitable for growth modelingwork.
A practical advantage of this procedure is its simplicity,
estimating the three parameters a, k, and m with a single
calculation step using the rate as a response variable and
D and N as independent variables in Eq. 5. Even with thissimplified procedure, the estimated parameters showed
fairly good agreement with previous results (Table 2).
Therefore, this procedure would be a convenient tool tomeet the immediate need for developing growth models aswell as for their repeated update based on rapidly accumulating data, as is usually the case with fast-growing
tropical species.
Another desirable property of this procedure is that it isfree from data editing when determining the maximum
size–density line, because it utilizes all data for the rate andcorresponding D and N values to estimate the threeparameters in Eq. 5. On the contrary, the previous procedures, such as principal component analysis or reduced
major axis (RMA) regression (Sackville Hamilton et al.
1995), both require data sets from overcrowded stands
selected from hundreds of measurements (Osawa andSugita 1989). The minimum distance boundary method
may alleviate this problem, but it may be biased by datawith measurement error due to its assumption that allmeasurements are below the line. Hence, this procedure
has good potential to provide more consistent estimates ofthe maximum size–density line as long as the data set
contains the rates of overcrowded stands.
In conclusion, a simultaneous estimation procedure forthe parameters of two functions, i.e., the maximum size–
density line and the self-thinning curve, was derived toaccount for density-related mortality. The parameters
estimated using this procedure agreed well with thosefor the maximum size–density line derived by previous

525

methods, and also described the average trend of selfthinning reasonably well. Therefore, due to its simplicity,
this procedure would be suitable for use in growth modeling work for fast-growing tropical species where frequent
update is anticipated.

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