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Cross-correlations among
Single Tree Growth Models
Hubert Hasenauerl, Robert A. Monserud2,
Timothy G. Gregoire3
Abstract. - Single tree growth and yield models basically consist of a
number of equations to update tree parameters over time. Although it
seems reasonable to assert that these equations are interrelated from a
biological standpoint, it is customary to consider them independently and
apply linear or nonlinear regression techniques separately rather than
jointly. Using more than 7,500 Norway spruce (Picea abies) trees, we
compare an individual tree basal area increment model, a height
increment model, and a crown model using least square methods
separately and jointly by applying three stage least square (3SLS)
regression techniques. Results indicate a strong cross-equation correlation
between the basal area and the height increment model and the height
increment and the crown model. This suggests that the use of joint
regression techniques would be superior.
INTRODUCTION
Individual tree forest growth and yield models usually employ a set of
equations to describe stand development over time. A typical single-tree stand
simulator (Monserud 1975, Wykoff et al. 1982, Burkhart et al. 1987, Hasenauer
1994) consists of different equations for predicting periodic diameter or basal area
increment, height increment, and the probability of mortality for each sample
tree. These equations are usually developed separately.
From a biological standpoint, it seems reasonable to assert that the change
in a tree's basal area, height, and risk of mortality are not uncorrelated
phenomena (Dixon et a1. 1990). Depending on their interrelationships, joint
estimation of the models' parameters may be necessary in order to provide
estimates that are consistent, or it may be desirable in order to provide more
precise estimates than can be obtained otherwise.
Seminal work by Aitken (1934-35), Haavelmo (1943), Theil (1953),
' Biornetrician, Institutfur Waldwachsturnsforschung,Universitatfur Bodenkultur, Wien, Austria.
~iornetrician,Intermountain Research Station, USDA Forest Service, Moscow, ID 83843, USA.
Biometrician, Dept. Forestry, Virginia Polytechnic Institute & St. Univ., Blacksburg, VA, USA.
Zellner (1962), and Zellner and Theil (1962) resulted in almost all methods
currently available for estimating the parameters in intercorrelated (simultaneous)
systems of equations.
The objective of this paper is to evaluate and compare joint versus
separate regression techniques for single tree growth and yield modeling. We
estimate the parameters of individual tree models for basal area increment, height
increment, and crown ratio using least squares methods separately and jointly by
applying two-stage (Theil 1953) and three-stage least squares (Zellner and Theil
1962) techniques. We specifically investigate (1) the differences in the estimated
coefficients and (2) the correlation between the predictions.
METHODS
Independent Regressions
We begin with a system of three individual tree growth equations for stand
conditions in Austria: basal area increment, height increment, and crown ratio.
These equations were developed independently using the same dataset.
Basal area increment: After eliminating the qualitative site descriptors
chosen by Monserud and Sterba (1996), which reduced the variance explained
only by 2.6 % , we are left with the following model for Y, :
with MA the 5-year basal area increment (outside bark), D the diameter at breast
height (1.3 m) in cm, C= (1ICR)-1 where CR is the crown ratio, BAL the basal
area (m2/ha) of trees larger in diameter than the subject tree, CCF the crown
competition factor of Krajicek et al. (196 I), ELEV the elevation in hectometers,
and SL the tangent of the slope angel (%/loo).
Height increment: Hasenauer and Monserud (1996a) used a similar
formulation to predict 5-year height increment AH, where H is the tree height,
and all other parameters as previously defined. The second equation (Y,) in the
system is:
I.(aH) = a + bl-h(D) + b 2 - ~ +2 b3*h(C)+ c l * M L + c2-CCF + 4.+ 4 * S L + e,
(2)
Crown ratio: To ensure that the predictions of crown ratio (defined as the
crown length divided by tree height) are bounded between 0 and 1, Hasenauer
and Monserud (1996b) chose a logistic function. After linearizing the logistic and
rearranging, we are left with the following logarithmic transformation of crown
ratio (Y,=ln(C)):
where C= (I/CR)-1, H/D is the heightldiameter ratio (mlcm), AZ is the azimuth
in radians, and all other parameters are as previously defined. Change in crown
ratio is not available, because height to crown base was not remeasured after the
initial inventory.
Simultaneous Equation Systems
In the system above, it is likely that the errors c in equations (1) - (3) are
intercorrelated because they are associated with various attributes of the same
tree. If this is the only common influence among the three equations, then
Zellner's (1962) seemingly unrelated regression (SUR) procedure would be
appropriate because the equations are related through contemporaneous
correlations in the variance-covariance matrix. We write the multivariate
regression model as
(4)
Y=xb+e
where Y is a 3n x 1 vector of dependent (endogenous) variables, X is the 3n x
+pZ+ p 3 design matrix, /3 is the @, +p, +p,) x 1 vector of coefficients to be
estimated, and t is the 3n x 1 error vector. The errors E have fixed mean E [ E ]=O
and variance
/ W l
lqe]
=
~ [ e t ? ]=
w12
w;, w, w,
&I
I
Wl,
w32
w3
=
D
(5)
where W,=o:~ are the main diagonal viriances and VVj=oJ are the covariances,
with I the n-dimensional identity matrix.
The ordinary least squares (OLS) estimator of P = (pi ,f$,pi) is
(6)
b = (x'x)" xiy
with variance
(7)
In situations where 0,. + 0 and 0,.+ d for some constant d,then b is
inefficient and the usual estimator of (7), namely
cSv(b) = (x'x)-' 8'
(8)
is biased. Zellners' (1962) seemingly unrelated regression procedure is a form of
feasible generalized least squares (Judge et al. 1980) in which Q is estimated by
h , where has the same form as (5) but with moment estimators, say 8; and
cov(b) = (x!x)-'
(X'QX)
(x'x)-l
blj., in place of the unknown parameters of and oy . With
efficient estimator of
0 is
8
The variance of
B
6 , an asymptotically
= (x/Q-lx)-l
is estimated by
X'Q-ly
(9)
++
For our set of single-tree equations there are p, +p, +p, =8 8 10=26 estimated
parameters, plus the main diagonal partition of the variance-covariance matrix for
each model (d1ye2yd3).The remaining off diagonal p a r t i t i ~ n s ( d , , , ~ ~ , , ~ ~ ~ )
contain the cross-equation covariances between each estimated parameter.
If X, =X2=X3then SUR is identical to OLS, even when the cross-equation
covariances are nonzero. Otherwise, SUR provides an asymptotically more
B
precise estimator of p than OLS, and the comparative efficiency of increases
with increasing cross equation correlation and with increasing dissimilarity among
the regression matrices X,, X, and X,.
The use of in (C) = Y, in the models for in (BM) and In (AH), eqns. (1) and
(2) respectively, precludes the straightforward use of SUR as outlined. Whenever
a response or endogenous variable from one model appears as a regressor variable
in another model, both the OLS and SUR estimator of P will be biased and
inconsistent (Judge et al. 1980). In this case two stage least squares or 2SLS (see
Judge et al. 1980) can be used to provide consistent and asymptotically unbiased
estimates of p .
An even more efficient estimator was invented by Zellner and Theil
(1962) : three-stage least square (3SLS). By combining Thiel 's (1953) 2SLS
procedure with Zellner's (1962) SUR procedure, the resulting 3SLS estimators
of p are consistent and asymptotically more efficient than 2SLS. We denote the
3SLS estimates of p as
and its estimated variance as
P"
e
It is convenient to partition
where
= C&(B)
as
e.. is the pj x p, estimated covariance matrix of (li and p,.
rJ
For details
concerning the 3SLS estimator in a forestry context see Murphy and Beltz (1981),
Borders and Bailey (1986), and Gregoire (1987).
In an individual tree model, estimates of E[YJ , E[Y2] , and E[YJ are
needed for each tree. Let x[ denote the p, x 1 row vector of covariate values for
a particular tree for which the estimated coefficients will be applied, such that
$1 =
serve as an estimate of E [ ~ , , X Let
~.
4 ijl
y,
and
(13)
y3 be similarly defined.
The distributional properties of the random errors e in conjunction with
the estimator of pi determine the statistical properties of b. and p1 . The scalar
covariance between random variables
x p, covariance matrix of
p
1
and
yl and F,
B,.
is x1212
x:, where
Given an estimate of C12, say,
y2 can be estimated by
Therefore, the correlation between yl and 5, can be expressed as
(12), then the covariance between
"
Y1
12
and
is the p,
e12as in
2;
1 12 2 '
between El and j3, and the correlation i2,3 between 9, andf3
The correlation i13
can be estimated similarly.
DATA
Data were obtained from the Austrian National Forest Inventory
(Forstliche Bundesversuchsanstalt 198I), a systematic hidden permanent sample
plot design over the whole of Austria, with a 5-yr remeasurement interval. In a
given year a fifth of the plots are remeasured, ensuring a representative sample
of all Austrian forests each year. The total inventory comprises 22,000 permanent
plots. We restricted ourselves to the 4,135 forested plots containing remeasured
Norway spruce (Picea abies), not crossed by roads, and in a single ownership.
Permanent sample plots were established from 1981 to 1985. Trees with
a diameter at breast height (DBH, 1.3 m) larger than 10.4 cm were selected by
angle count sampling using a BAF of 4 m2/ha. Trees with a DBH between 5 and
10.4 cm were measured within a circle of 2.6 m radius located at plot center;
smaller trees were not recorded. At plot establishment, the following data were
recorded for every sample tree: species, DBH to the nearest mm, and distance
and azimuth from plot center. Total height and height to the crown base were
measured to the nearest decimeter on every fifth tree. Plot descriptors were
evaluated within a circle of 300 m2. Elevation is measured to the nearest 100 m,
slope is measured to the nearest l o % , and aspect to the nearest 45".Additional
site descriptors were measured but not used in this comparison study.
Plots were remeasured from 1986 to 1990, 5 years after establishment. In
the remeasurement, height and diameter were measured on the same trees, but not
height to crown base. In summary, observations of 7,797 Norway spruce trees
from 4,135 different permanent sample plots are available throughout Austria.
RESULTS
Coefficient estimates by OLS vs. 3SLS
Using the SYSLIN procedure in the Econometrics-Time Series module of
SAS (SAS Institute 1988), the parameters in equations (1) to (3) were first
estimated independently by applying ordinary least squares (OLS) and then jointly
by using 2SLS and 3SLS.
Attention was immediately focused on the in(C) and CCF terms in the
height increment model (2). These two terms were both significant (a=0.05) with
OLS. With 2SLS, the in (C) term remained significant but the CCF term became
strongly non-significant. With 3SLS, both the in(C) and CCF terms were nonsignificant. Thus the height increment 'model (2) was reduced from 8 parameters
to 6. The size of the matrix partitions in the variance-covariance matrix in (12)
is a 6x6 variance-covariance matrix for f2,and
is reduced accordingly. Now
&,
the off-diagonal partitions E;':,and E;':, are 8x6 and 6x10 matrices containing the
cross-equation covariances between the estimated parameters, respectively.
Because there are now p, +p, +p, =8 6 10=24 estimated parameters in the
three growth equations, there are 576 variance-covariance elements in (12), a
reduction of 100 elements.
++
Correlation between predictions
To investigate the strength of the interrelationships among our system of
equations and evaluate their importance, the correlations between the predictions
of in(ABA), in(AH) and in(C) (eqns. (I), (2), and (3)) are calculated for each
observation according to equation (14).
Fig. 1 displays the cross-equation correlations " between each pair of
rij
predictions vs. the respective predicted variables
In Fig. la and lb, " is
5 2
massed around 0.23, with a maximum correlation of 0.33 between the predictions
for in(BAI) ( " ) and i n ( M ) ( " ). This indicates that the first two equations in the
&.
Y1
3'2
system are fairly interdependent. The correlations
the first ( " ) and third equations
Y1
"
5 3
between predictions from
(y3) are weakly correlated, with a mean of
-0.07, and extrema at 0.05 and -0.1 1. The correlations
-
"
r23
between predictions
from the second ( ) and third equations ( " ) have a mean of zero (0.01), but
y2
Y3
are more dispersed than those for " ranging from 0.26 to -0.10 with a standard
r12 '
deviation of 0.08. These correlations between the basal area and height
increment models and between the height increment and crown models confirm
our assumption of cross-equation correlations.
;between each pair of -
predictions; YI
the predicted in(AH), and jj3 the predicted
tig. 1 : The cross equation correlation
r/
indicates the predicted h ( A B A ) ,
111( ( I /CR)-I ) .
-
Y2
Y1
DISCUSSION
Because individual tree forest growth models are based on multivariate
attributes observed on the same individuals (e-g. basal area increment, height
increment, crown ratio), the resulting set of growth equations can be considered
a silnultaneous system. Therefore, joint regression techniques should be
considered for silnultaneously determining the parameter estimates. If endogenoub
variables do not appear on the right hand side (RHS), the seemingly unrelated
regression (Zellner 1962) will still improve the efficiency of the parameter
estimates. If endogenous variables are used as predictor variables then multi-stage
estimation techniques (2SLS or 3SLS) are necessary to obtain parameter estimates
that are consistent; such estimates will also be asymptotically efficient. Ignoring
the simultaneous nature of the system by separately applying OLS to each model
can result in estimates that are biased and inconsistent.
Using 3SLS with our system of equations allowed the detection and
tleletion of two non-significant terms (CCF and ln(C) in eq. (2)) that OLS had
determined as significant. This led to a simplification of the si~nultaneous
structure of our system because in(C) was originally considered as an endogenous
variable on the RHS.
One of the advantages of joint regression techniques is that both the
correlation among the predictor variables within a certain equation and across all
equations are available. Fig. 1 indicates that cross-equation may change thei 1degree of dependency in the system and that these correlations can be used i l l a
simulator to better predict stand and tree dynamics. Predictions between the first
two growth equations (basal area and height increment) are fairly well correlated.
while correlations between predictions from the first and third (height increment
and crown ratio) are rather weak.
ACKNOWLEDGEMENTS
T h ~ research
s
was conducted when Hubert Hasenauer was a Visiting Scientist at Virginia
Tech Depart nien t of Forestry In Blacksburg, and at the Intermountain Research Station's Forestry
S c ~ e n ~ Laboratory
es
in Moscow, Idaho. Hasenauer was working on a Schriidinger research ?rant
trom tht: Austrlan Science Foundat~on.We are g r a t e f ~ ~tol Karl Schieler and Klemens Schadauer
ot the Fecieral Forest Research Center In Vienna for making the Forest Inventory data available.
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