全 文 ：第 49 卷 第 6 期
2 0 1 3 年 6 月
林 业 科 学
SCIENTIA SILVAE SINICAE
Vol. 49，No. 6
Jun．，2 0 1 3
doi: 10．11707 / j．1001-7488．20130611
Received date: 2012 － 08 － 06; Revised date: 2012 － 10 － 23．
Foundation project: Supported by the Scientific Research Funds for Forestry Public Welfare of China(201004026) ; Ministry of Education“Overseas
Experts and Scholars”project．
* Li Fengri is corresponding author．
树木胸径和树高二元分布的建模与预测*
金星姬1 李凤日1 贾炜玮1 张连军2
(1. 东北林业大学林学院 哈尔滨 150040; 2. 美国纽约州立大学环境科学和林业学院 锡拉丘兹 NY13210)
摘 要: 采用二元广义 β 分布(GBD-2)和 Johnsons SBB分布对美国东北部云冷杉林的水平和垂直分布进行拟合。
拟合优度检验结果表明: 无论是胸径、树高边缘分布和联合分布，还是树高和材积的预估，GBD-2 分布的拟合效果
均比 SBB分布要好。使用常用的林分变量如林分密度、公顷断面积、林分平均胸径、平均树高、平均冠幅和冠长建立
回归模型对 GBD-2 分布的参数进行估计，表明未来林分水平和垂直结构可以由这些变量进行预测。所建立的二元
分布模型可以为研究树木胸径、树高的实际关系和动态提供信息。
关键词: 二元广义 β 分布(GBD-2); Johnsons SBB分布; 拟合优度; 林分水平和垂直分布; 回归模型
中图分类号: S757 文献标识码: A 文章编号: 1001 － 7488(2013)06 － 0074 － 09
Modeling and Predicting Bivariate Distributions of Tree Diameter and Height
Jin Xingji1 Li Fengri1 Jia Weiwei1 Zhang Lianjun2
(1. School of Forestry，Northeast Forestry University Harbin 150040;
2. College of Environmental Science and Forestry，State University of New York ( SUNY-ESF) Syracuse，NY13210，USA)
Abstract: The horizontal and vertical structures of the spruce-fir stands in the northeast，USA were modeled by the
bivariate generalized beta distribution (GBD-2) and Johnsons SBB distribution． The goodness-of-fit tests indicated that
GBD-2 performed better than did Johnsons SBB in fitting both marginal and joint distributions of tree diameter and height，
and in predicting tree height and volume． Regression models were developed for predicting the parameters of the GBD-2
distributions using ordinary stand variables as predictors，such as stand density，basal area，mean tree diameter，mean
tree height，and mean crown length and width． Thus，the future stand horizontal and vertical structures can be predicted
when the future values of these stand variables were available． The bivariate distribution models developed in this study
will provide useful information on the realistic relationships and dynamics of tree diameter and height．
Key words: bivariate generalized beta distribution ( GBD-2 ); Johnsons SBB; goodness-of-fit; stand horizontal and
vertical structures; regression model
The horizontal structure of a forest stand can be
characterized by the distribution of tree diameter at
breast height ( at 1. 37 m，DBH ) and the relative
locations of trees，while the vertical structure of the
same stand can be represented by the distribution of
tree height ( HT) and diameter-height relationships．
Various probability density functions ( pdf) have been
used to describe the frequency distributions of tree
diameter， such as Weibull ( Bailey et al．，1973 )，
lognormal ( Bliss et al．，1964 )， gamma ( Nelson，
1964 )， beta ( Clutter et al．，1965 )， Johnsons SB
(Hafley et al．，1977)，and logit-logistic (Wang et al．，
2005) ． However，few efforts have been made to fit the
frequency distributions of tree height (Schreuder et al．，
1977) ． Thus，in modeling the vertical structure of a
stand，the most common practice is to fit a distribution
function to the diameter frequency data，and then use
an empirical diameter-height relationship to estimate
the mean height per diameter class． Although this
approach is satisfactory in many situations，it ignores
the natural relationships between tree diameter and
height by treating them separately since the tree height
can vary considerably for a given diameter ( Scheuder
et al．，1977; Tewari et al．，1999) ．
第 6 期 金星姬等: 树木胸径和树高二元分布的建模与预测
An alternative is to model the distributions of tree
diameter and height simultaneously using a bivariate
distribution． This approach is desirable because it
allows us to predict the tree height by diameter from the
fitted bivariate distribution through the conditional
expectation of tree height given DBH． Over the last
decades，Johnsons SBB have been utilized for modeling
the bivariate tree diameter-height frequency data
(Schreuder et al．，1977; Hafley et al．，1985; Knoebel et
al．，1991; Tewari et al．，1999; Zucchini et al．，2001) ．
Li et al． (2002) applied a bivariate generalized beta
distribution (GBD-2) to model the joint distributions of
tree diameter and height of Douglas-fir stands in the
northwest，USA， and found that GBD-2 performed
better than Johnsons SBB in goodness-of-fit statistics
and stand volume predictions． Wang et al． ( 2007;
2008; 2010 ) used a general copula approach to
modeling multivariate distributions， with numerical
examples for bivariate tree diameter-height data or
trivariate tree diameter-height-volume data．
The purposes of this study were 1 ) to fit two
bivariate distribution functions ( i． e．， GBD-2 and
Johnsons SBB) to the joint distributions of tree diameter
and height of spruce-fir stands in the northeast，USA，
2) to compare the model fitting and performance
between the two bivariate distribution functions，and
3) more importantly，to develop regression models for
predicting the parameters of the GBD-2 distribution
using available stand variables．
1 Data and methods
1. 1 Data
Fifty ( 50 ) plots were collected from the even-
aged， unmanaged natural spruce-fir forests in
northwestern Maine， USA， located in the region
between 69°W and 71°W in longitude，between 45°N
and 46. 5° N in latitude and between 750 m and
1 200 m in elevation (Kleinschmidt et al．，1980) ． The
plot area ranged from 0. 002 5 to 0. 02 hm2 in size
(mean plot size was 0. 008 hm2 with standard deviation
0. 004 hm2 ) ． In these plots， balsam fir ( Abies
balsamea) and red spruce ( Picea rubens) accounted
for about 95% of total number of trees and 94% of
total volume． Other minor species included black
cherry ( Prunus serotina)，eastern white pine ( Pinus
strobus)，white spruce (Picea glauca)，black spruce
(Picea mariana)，and other hardwoods． In each plot，
tree DBH， H， crown length ( top to the base of
crown)，and average crown width were recorded for
each tree ( Solomon et al．，2002 ) ． The numbers of
trees in each plot ranged from 12 to 109． Mean tree
diameters were from 2. 2 to 19. 2 cm and mean total
heights were from 2. 7 to 17. 1 m (Tab． 1) ．
Tab. 1 Summary statistics of the stand
variables across the 50 plots
Stand variables Mean Std． Min． Max．
Number of trees per plot 34. 4 17. 7 12 109
Stand density /( tree·m － 2 ) 0. 61 0. 54 0. 18 2. 52
Stand basal area /(m2·hm － 2 ) 50. 2 14. 3 10. 0 79. 6
Stand mean DBH /cm 11. 5 3. 9 2. 2 19. 2
Stand mean height /m 11. 6 3. 7 2. 7 17. 1
Stand mean crown length /m 4. 4 1. 5 1. 6 8. 4
Stand mean crown width /m 1. 1 0. 3 0. 5 1. 9
1. 2 Methods
The plot of “Solution Available” region was
constructed for the 50 spruce-fir plots ( Li et al．，
2002 ) ． Most of the tree diameter and height
distributions of all 50 plots fell in the region of
generalized beta distribution ( GBD ) or the overlap
region of GBD and generalized lambda distribution
( GLD ) ． Therefore，GBD is appropriate to fit the
marginal distributions of tree diameter and height for
the 50 plots and GBD-2 is applied to model the joint
distribution of tree diameter and height．
The functions and methods in Li et al． (2002 )
were used to 1) fit the marginal GBD distributions to
tree diameter and height，respectively; 2) fit GBD-2 to
the joint distribution of tree diameter and height． The
probability density functions of GBD and GBD-2 are
provided in Appendix 1; 3 ) fit the marginal SB
distributions to tree diameter and height，respectively;
4) fit Johnsons SBB to the joint distribution of tree
diameter and height． The probability density functions
of SB and SBB are provided in Appendix 2; 5) test the
goodness-of-fit for the marginal GBD or Johnsons SB
distributions of tree diameter and height，as well as the
two bivariate distributions; and 6) compare tree height
and volume predictions between GBD-2 and Johnsons
SBB(Appendix 4) ． Prediction bias was defined as the
difference between observed and predicted tree height
57
林 业 科 学 49 卷
or volumes，and relative bias was calculated as the
percentage of the prediction bias over the predicted tree
height or volume， respectively． An average relative
bias was then obtained for each of the 50 plots． To
examine the prediction bias across tree sizes，all trees
in the 50 plots were grouped into 2-cm diameter
classes． An average relative bias was calculated for
each diameter class．
In forestry practice，it is desirable to be able to
predict the parameters of the GBD-2 distributions based
on conventional forest inventory variables． One
intuitive approach is to build empirical regression
models between these parameters and stand variables．
However，the correlations between the parameters of a
frequency distribution and stand variables are generally
low due to the fact that parameter estimates are highly
data-specific， stands with similar conditions can
produce highly variable parameter estimates．
Therefore， it is difficult to directly predict the
parameters based on stand characteristics (Schreuder et
al．，1977; Knoebel et al．，1991 ) ． Alternatively，
parameter recovery approach can be employed，i． e．，
instead of predicting the GBD-2 parameters directly
from stand variables，some moments of tree diameter
and height can be predicted from the stand variables
because these moments are highly correlated with other
stand variables． In this study， the seven linear
regression models were developed to predict the
response variable Y which included ψ ( an association
measure between two marginal distributions )，
minimum of DBH， maximum of DBH， standard
deviation of DBH，minimum of H，maximum of H，and
standard deviation of H:
Y = α0 + α1(Density) + α2(Basal area) +
α3(Mean DBH) + α4(Mean H) +
α5(Mean crown length) +
α6(Mean crown width) + ε ．
where α0 － α6 are regression coefficients to be
estimated， and ε is the model error term．
Consequently，the parameters of GBD-2 can then be
solved by the relationships among the parameters using
equations ( 8 ) － ( 11 ) in Appendix 3 ( Li et al．，
2002) ．
2 Results and discussion
2. 1 Goodness-of-fit test for fitting marginal and
joint distributions
Of the 50 plots none ( 0% ) of the diameter
distributions and one ( 2. 0% ) of the height
distributions fitted by the GBD distribution were
significantly different from the observed distribution at
α = 0. 05 level． Correspondingly，three (6. 0% ) of the
diameter distributions and seven ( 14. 0% ) of the
height distributions fitted by the Johnsons SB
distribution were significantly different from the
observed distribution． Further， none ( 0% ) of the
predicted bivariate GBD-2 distributions was
significantly different from the observed distributions at
α = 0. 05 level，while nine (18. 0% ) of the predicted
Johnsons SBB distributions were significantly different
from the observed ones． Therefore， the GBD-2
distribution was better than the Johnsons SBB
distribution in fitting the joint distribution of tree
diameter and height for the spruce-fir stands in the
Northeast，USA．
2. 2 Model comparison by tree height and volume
predictions
For predicting tree height from the two joint
distributions，the predicted tree height using GBD-2
was significantly different ( α = 0. 05 ) from the
observed tree height in only 1 ( 2. 0% ) of the 50
plots，but this occurred in 9 (18. 0% ) of the 50 plots
for the Johnsons SBB ． Again， the GBD-2 performed
better in predicting tree height， although numerical
methods must be employed for its use．
Fig. 1 Average relative bias (% ) of volume prediction from the
GBD-2 and Johnsons SBB distributions for
diameter classes over all of the 50 plots
67
第 6 期 金星姬等: 树木胸径和树高二元分布的建模与预测
Fig． 1 shows the average relative biases (% ) of
volume predictions across 2 cm diameter classes． It
appears that the Johnsons SBB distribution produces
relatively larger positive biases ( underestimation) for
small-sized trees ( e． g．， ＜ 10 cm in diameter) and
for large-sized trees ( e． g．， ＞ 22 cm in diameter)，
but relatively larger negative biases ( overestimation)
for middle-sized trees than did the GBD-2 distribution．
2. 3 Development of regression models for
predicting the GBD-2 parameters
The correlations between the GBD-2 parameters
and available stand variables in the data were low in
general，and about 52% (28 /54) of these correlation
coefficients were not significantly different from zero at
α = 0. 05 level ( Tab． 2 ) ． On the other hand， the
estimated association parameter ψ^ of GBD-2， the
minimum values，maximum values and the standard
deviation of tree diameter and height were significantly
correlated with majority ( 88% or 37 /42 ) of the
available stand variables ( Tab． 3) ． Therefore，seven
linear regression models for the ψ^， the minimum
values，maximum values and the standard deviation of
tree diameter and height were constructed using these
stand variables including stand density，basal area，
mean tree diameter， mean tree height， and mean
crown length and width． Since the stand variables
themselves were highly correlated， the principal
component regression approach was employed (Neter et
al．，1990 ) ． Tab． 4 provides the final regression
coefficients and model fitting statistics for the seven
regression models． The model R2 were ranged from 0. 5
to 0. 95 for the models developed for predicting the
minimum and maximum values and the standard
deviation of tree diameter and height，while the model
R2 for the model of the parameter ψ^ of the GBD-2 was
relatively low (0. 28) ．
Tab． 2 Correlations of the parameters of GBD-2 with the available stand variables
GBD-2 parameters
Density /
( tree·m － 2 )
Basal area /
(m2·hm － 2 )
Mean DBH /
cm
Mean height /
m
Mean crown length /
m
Mean crown width /
m
ψ^
(P-value)
0. 344 9
(0. 012 3)
－ 0. 105 8
(0. 455 3)
－ 0. 413 0
(0. 002 3)
－ 0. 398 6
(0. 003 4)
－ 0. 392 8
(0. 004 0)
－ 0. 167 3
(0. 255 7)
DBH
β^1
(P-value)
－ 0. 339 9
(0. 013 7)
0. 163 2
(0. 247 7)
0. 354 4
(0. 009 9)
0. 421 6
(0. 001 9)
0. 299 8
(0. 030 8)
0. 131 0
(0. 374 8)
β^2
(P-value)
－ 0. 414 4
(0. 002 3)
0. 544 4
(0. 000 0)
0. 462 7
(0. 000 6)
0. 391 1
(0. 004 1)
0. 398 4
(0. 003 4)
0. 061 0
(0. 680 4)
β^3
(P-value)
－ 0. 204 9
(0. 145 1)
0. 154 8
(0. 273 2)
0. 248 3
(0. 075 9)
0. 224 8
(0. 109 1)
0. 067 3
(0. 635 4)
0. 047 4
(0. 749 1)
β^4
(P-value)
0. 031 5
(0. 824 6)
0. 048 1
(0. 734 9)
－ 0. 049 4
(0. 728 1)
－ 0. 050 6
(0. 721 6)
－ 0. 088 3
(0. 533 5)
－ 0. 258 8
(0. 075 7)
Height
β^1
(P-value)
－ 0. 156 5
(0. 267 8)
0. 093 5
(0. 509 6)
0. 095 4
(0. 501 2)
0. 229 6
(0. 101 6)
0. 127 7
(0. 366 9)
－ 0. 048 6
(0. 742 7)
β^2
(P-value)
－ 0. 442 5
(0. 001 0)
0. 364 4
(0. 007 9)
0. 526 4
(0. 000 1)
0. 416 3
(0. 002 1)
0. 332 4
(0. 016 1)
0. 335 0
(0. 020 0)
β^3
(P-value)
－ 0. 261 0
(0. 061 6)
0. 099 3
(0. 483 7)
0. 344 0
(0. 012 5)
0. 311 0
(0. 024 8)
0. 077 3
(0. 586 1)
0. 183 7
(0. 211 2)
β^4
(P-value)
0. 379 6
(0. 005 5)
－ 0. 378 8
(0. 005 6)
－ 0. 420 5
(0. 001 9)
－ 0. 350 7
(0. 010 8)
－ 0. 464 0
(0. 000 5)
－ 0. 092 6
(0. 531 4)
To evaluate the performance of the prediction
models，the univariate and bivariate χ2 tests (Li et al．，
2002) were used for testing the goodness-of-fit between
the observed and the predicted marginal and joint
distributions of tree diameter and height of each plot．
The results indicated that 69% of the plots for the
diameter distribution，63% of the plots for the height
distribution，and 67% of the plots for the bivariate
distribution were predicted satisfactorily by the
prediction models．
2. 4 Three example plots
Three plots were arbitrarily selected as the
examples to show the application of the prediction
models． The estimated and predicted model parameters
as well as the stand variables of these three plots were
listed in Tab． 5． The estimated model parameters were
obtained from fitting the GBD-2 distribution to each
plot， while the predicted model parameters were
77
林 业 科 学 49 卷
obtained from the seven prediction models using the
available values of the six predictor stand variables of
each plot． The scatter plots of the observed and the
simulated tree height versus diameter，as well as the
histograms of the observed and the simulated tree
diameter and height are shown in Fig． 2，3 and 4． It
appears that for all three example plots the predicted
height-diameter relationships cover the similar ranges of
the observed ones， and the predicted diameter and
height frequency histograms have similar patterns and
trends to the observed ones．
Tab． 3 Correlations of the parameter ψ^ of GBD-2，the minimum values，maximum values
and the standard deviations of tree diameter and heights with the available stand variables
GBD-2 parameters
Density /
( tree·m － 2 )
Basal area /
(m2·hm － 2 )
Mean DBH /
cm
Mean height /
m
Mean crown
length /m
Mean crown
width /m
ψ^
(P-value)
0. 344 9
(0. 012 3)
－ 0. 105 8
(0. 455 3)
－ 0. 413 0
(0. 002 3)
－ 0. 398 6
(0. 003 4)
－ 0. 392 8
(0. 004 0)
－ 0. 167 3
(0. 255 7)
DBH
Minimum
(P-value)
－ 0. 682 7
(0. 000 0)
0. 497 2
(0. 000 2)
0. 797 2
(0. 000 0)
0. 838 5
(0. 000 0)
0. 549 1
(0. 000 0)
0. 125 1
(0. 396 9)
Maximum
(P-value)
－ 0. 782 4
(0. 000 0)
0. 770 4
(0. 000 0)
0. 887 7
(0. 000 0)
0. 832 8
(0. 000 0)
0. 695 6
(0. 000 0)
0. 392 0
(0. 005 9)
StDev
(P-value)
－ 0. 639 0
(0. 000 0)
0. 727 9
(0. 000 0)
0. 709 8
(0. 000 0)
0. 622 7
(0. 000 0)
0. 651 5
(0. 000 0)
0. 349 5
(0. 014 9)
Height
Minimum
(P-value)
－ 0. 537 6
(0. 000 0)
0. 338 6
(0. 014 1)
0. 632 5
(0. 000 0)
0. 719 6
(0. 000 0)
0. 358 5
(0. 009 1)
0. 037 1
(0. 802 3)
Maximum
(P-value)
－ 0. 917 7
(0. 000 0)
0. 678 8
(0. 000 0)
0. 933 2
(0. 000 0)
0. 955 6
(0. 000 0)
0. 689 0
(0. 000 0)
0. 427 8
(0. 002 4)
StDev
(P-value)
－ 0. 445 7
(0. 000 9)
0. 512 0
(0. 000 1)
0. 395 9
(0. 003 7)
0. 318 5
(0. 021 4)
0. 448 9
(0. 000 8)
0. 278 7
(0. 055 1)
Tab． 4 Regression coefficients for the parameter ψ^ of GBD-2，the minimum (Min． ) values，maximum (Max． )
values and the standard deviation (Std． ) of tree diameter (DBH) and height (H) using the available stand variables
Dependent
variable
Intercept
Density /
( tree·m － 2 )
Basal area /
(m2·hm － 2 )
Mean DBH /
cm
Mean H /
m
Mean crown
length /m
Mean crown
width /m
Fit statistics
R2 P-value
ψ^ 128. 963 5 － 18. 546 8 1. 363 4 － 3. 836 9 － 4. 067 8 － 9. 662 4 － 2. 543 2 0. 277 0 0. 030 6
Min． H － 0. 621 5 1. 369 7 － 0. 059 5 0. 524 1 0. 637 9 － 0. 488 1 － 2. 970 9 0. 695 9 ＜ 0. 000 1
Max． H 8. 676 6 － 2. 268 5 0. 023 0 0. 279 9 0. 344 4 － 0. 033 9 0. 213 2 0. 951 1 ＜ 0. 000 1
Std． H 2. 659 2 － 1. 085 5 0. 034 5 － 0. 085 4 － 0. 102 1 0. 144 8 0. 329 2 0. 504 2 ＜ 0. 000 1
Min． DBH － 1. 062 0 0. 535 0 － 0. 017 3 0. 329 7 0. 388 0 － 0. 069 1 － 1. 825 4 0. 769 3 ＜ 0. 000 1
Max． DBH 2. 638 0 － 1. 565 1 0. 159 7 0. 345 8 0. 345 1 0. 324 6 1. 549 1 0. 868 4 ＜ 0. 000 1
Std． DBH 1. 117 7 － 0. 581 2 0. 046 0 － 0. 006 0 － 0. 024 0 0. 183 9 0. 463 0 0. 750 3 ＜ 0. 000 1
Tab． 5 Estimated (Est． ) and predicted (Pred． ) parameters of the regression model
and the stand variables for the three example plots
Plot
05 26 57
Stand
variables
Density 0. 720 0 0. 402 4 0. 755 1
Basal area 46. 895 2 62. 072 8 57. 645 7
Mean DBH 8. 553 5 13. 035 0 9. 140 5
Mean H 9. 607 4 12. 911 0 9. 526 8
Mean crown length 3. 128 8 5. 268 5 2. 930 8
Mean crown width 1. 114 4 1. 389 0 1. 194 4
GBD-2
Parameter Est． Pred． Est． Pred． Est． Pred．
ψ^ 64. 000 0 74. 581 8 81. 000 0 49. 158 1 68. 000 0 88. 372 3
β^2 -DBH 3. 250 0 2. 809 4 5. 364 3 4. 487 4 2. 900 0 2. 672 3
β^2 -DBH 13. 700 0 15. 204 6 25. 027 6 20. 256 8 15. 000 0 17. 238 0
β^3 -DBH 0. 377 8 0. 149 2 0. 234 1 0. 291 8 0. 249 9 0. 149 8
β^4 -DBH 1. 181 4 0. 892 7 1. 792 5 0. 769 6 0. 754 3 0. 914 4
β^1 -H 3. 350 0 3. 348 3 6. 370 0 4. 606 5 3. 700 0 2. 872 2
β^2 -H 10. 140 0 10. 609 8 12. 620 0 12. 799 9 10. 320 0 11. 414 2
β^3 -H 1. 368 3 0. 830 7 1. 929 2 0. 772 7 0. 462 5 0. 643 3
β^4 -H 0. 469 5 0. 272 6 1. 722 3 － 0. 040 4 0. 127 8 0. 175 4
87
第 6 期 金星姬等: 树木胸径和树高二元分布的建模与预测
Fig． 2 Observed versus predicted tree diameter and height of plot 05
Fig． 3 Observed versus predicted tree diameter and height of plot 26
Fig． 4 Observed versus predicted tree diameter and height of plot 57
3 Conclusion
The bivariate distributions of generalized beta
distribution (GBD-2) and Johnsons SBB were fit to a
wide range of tree diameter and height distributions of
the even-aged， unmanaged natural forest stands of
spruce-fir in the Northeast，USA． The goodness-of-fit
tests indicated that GBD-2 performed better than did
Johnsons SBB in fitting both marginal and joint
distributions of tree diameter and height，as well as in
predicting tree height and volume． However，the better
predictions of tree height and volume by GBD-2 over
SBB may be partially due to the nature of the regression
relationships，i． e．， the expectation-type for GBD-2
and median-type for SBB ． Nevertheless，the results in
this study were very similar to the previous study for
the Douglas-fir stands in the northwest， USA ( Li
et al．，2002) ． The weakness of the using the GBD-2
distribution is that the relationship between tree
diameter and height is less obvious than the median
97
林 业 科 学 49 卷
regression relationship constructed by the Johnsons SBB
distribution． In addition， numerical methods are
required for fitting the frequency distributions and
predicting tree height using the GBD-2 distribution．
Other shortcomings of this study were 1) the size
of the available spruce-fir plots was relatively small．
Thus，the parameters of the bivariate distributions may
be estimated poorly for the plots with small number of
trees，and 2) there were no independent data or plots
available for validating the empirical regression models
for predicting the model parameters． Therefore，more
and large forest plots may be needed to further confirm
and validate the results in this study．
In this study empirical regression models were
developed for predicting the parameters of the GBD-2
distributions using ordinary stand variables as
predictors，such as stand density，basal area，mean
tree diameter and total height，and mean crown length
and width． Thus， the future stand vertical structure
can be predicted when the future values of these stand
variables are available．
The major advantage of the bivariate distributions
is that it allows for a consistent generation of a bivariate
diameter-height system，which can be used to improve
the prediction of tree heights and consequently tree
volumes． In recent years spatial models have been
developed to describe the spatial distribution patterns of
trees and stand canopy structure． When modeling
three-dimensional profile and changes over time of a
forest stand，a bivariate distribution of tree diameter
and height is desirable because it can provide a
realistic vertical profile and diameter-height dynamics
over time．
参 考 文 献
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Wang M，Rennolls K． 2007． Bivariate distribution modeling with tree
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tree diameters and heights: dependency modeling using copulas．
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of tree diameter， height， and volume． Forest Science， 56
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(责任编辑 石红青)
08
第 6 期 金星姬等: 树木胸径和树高二元分布的建模与预测
Appendix 1． Density functions (pdf) of GBD and GBD-2
The probability density function (pdf) of the GBD random variable x ( e． g．，tree diameter or height in this study)，if for the
parameters β1，β2，β3，β4 ＞ － 1，is
f( x) = Cβ － ( β3 + β4 +1)2 ( x － β1 )
β3 (β1 + β2 － x)
β4， (1)
on the interval (β1，β1 + β2 )，and 0 otherwise，where C =
Γ(β3 + β4 + 2)
Γ(β3 + 1)Γ(β4 + 1)
．
The distribution function (H) of the bivariate generalized beta distribution (GBD-2) does not have a closed form，and can be
constructed by Placketts copula (Plackett，1965) as follows:
H = S － S
2 － 4ψ(ψ － 1)槡 FG
2(ψ － 1)
． (2)
where F = F( x) and G = G( y) are the GBD marginal cumulative distribution functions ( cdf) for the two random variables，x ( e． g．，
tree diameter) and y ( e． g．，tree height)，respectively，S = 1 + ( F + G) ( ψ － 1)，and ψ∈［0，∞ ) is a measure of association
between the two marginal distributions． Thus，the pdf of GBD-2 is
h( x，y) = ψfg［1 + (ψ － 1)(F + G － 2FG)］
(S2 － 4ψ(ψ － 1)FG) 3 /2
． (3)
where f = f( x) and g = g( y) are the GBD marginal pdfs of the two random variables，respectively (Li et al．，2002) ．
Appendix 2． Probability density functions (pdf) of SB and SBB
The pdf of a univariate Johnsons SB distribution ( Johnson，1949a) for a random variable x is given by
f( x) = δ
2槡 π
λ
( x － ε)(ε + λ － x)
exp － 1
2 γ
+ δln x － ε
ε + λ( )( )－ x[ ]
2
． (4)
where ε ＜ x ＜ ε + λ，δ ＞ 0，－ ∞ ＜ γ，ε ＜ ∞，λ ＞ 0，and Zx = γ + δln
x － ε
ε + λ( )－ x － N(0，1)，with ε the minimum value of x，
λ the range of x，γ and δ the shape parameters． From the definition，we know that Johnsons SB distribution is obtained from the
transformation on a standard normal variate．
The Johnsons SBB is the bivariate distribution of random variables x and y ( Johnson，1949b) when the standard normal variates，
Z1 and Z2，are defined as
Z1 = γ1 + δ1 ln
x － ε1
ε1 + λ1
( )－ x ， (5)
and
Z2 = γ2 + δ2 ln
y － ε2
ε2 + λ2
( )－ y ． (6)
where Z1 and Z2 have the joint bivariate normal distribution with correlation ρ，and the joint pdf is
f(Z1，Z2，ρ) = (2π 1 － ρ槡
2 ) －1 exp － 1
2
(1 － ρ2 ) －1 (Z21 － 2ρZ1Z2 + Z
2
2[ ]) ． (7)
Appendix 3． Estimation of the GBD parameters
One way for estimating the four parameters (β1，β2，β3，β4 ) of the marginal GBD distribution is as follows: set
β^1 = xmin， (8)
and
β^2 = xmax － β^1 ． (9)
where xmin and xmax are the minimum values and maximum values of the random variable x ( e． g．，tree diameter or height in this
study)，respectively． Then，β3 and β4 can be estimated as follows:
β^3 =
(珋x － β^1 )
2 ( β^1 + β^2 － 珋x) － s
2 ( β^2 － β^1 + 珋x)
s2 β^2
， (10)
β^4 =
β^2 ( β^3 + 1)
珋x － β^1
－ β^3 － 2 ． (11)
where 珋x and s2 are the sample mean and variance of the GBD random variable x，respectively (Li et al．，2002) ．
18
林 业 科 学 49 卷
Appendix 4． Prediction by bivariate distributions
For the Johnsons SBB distribution，the two marginal variables，diameter ( x) and height ( y)，have a simple median regression，
namely，
y =
λ^ y θ
ε^x + λ^ x － x
x － ε^( )x
φ
+[ ]θ
+ ε^y ． (12)
where θ = exp
ρ^γ^ x － γ^ y
δ^
[ ]
y
and  =
ρ^δ^x
δ^y
，ε^x，λ^ x，δ^x，^γ x and ε^y，λ^ y，δ^y，^γ y are the estimated parameters of the tree diameter and height
distributions，respectively． Hence，this relationship can be naturally used to predict the tree height for a given diameter．
For the GBD-2 distribution，there is no simple relationship between the two marginal variables available． After the parameters are
estimated，the conditional expectation of a tree height can be calculated for a given diameter using numerical methods． The expectation
of tree height ( y) given diameter ( x) is
E( y | x) = ∫
β1y + β2y
β1y
yhY| X( y | x)dy = ∫
β1y + β2y
β1y
y
hX，Y( x，y)
fX( x)
dy =
ψ∫
β1y + β2y
β1y
1 + (ψ － 1)(F + G － 2FG)
(S2 － 4ψ(ψ － 1)FG) 3 /2
ygY( y)dy ． (13)
where gY( y) = Cβ
－ ( β3y + β4y +1)
2y ( y － β1y)
β3y (β1y + β2y － y)
β4y is the marginal pdf of the tree height ( y)，with C =
Γ(β1y + β2y + 2)
Γ(β1y)Γ(β2y)
，
S = 1 + (F + G)(ψ + 1)，and F and G are the cumulative density functions of the two marginal variables x and y (Li et al．，2002) ．
28