The Base‑Age Invariance of Equations

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\
markright{
\
underline{PMRC Technical Report 1999
-
4; Cieszewski, Bailey; BAI Poly. Site Eqs.
\

w.
\

Var.
\

Asympt.
\

\
&
\
ldots }}


\
title{

\
underline{T
he Method for Deriving Theory
-
Based Base
-
Age Invariant}

\
underline{Polymorphic Site Equations with Variable Asymptotes}

\
underline{and other Inventory Projection Models}
\
\

\

\
\

\

\
\

\

\
\

\

\
\

\

\
\


Plantation Management Research Cooperative
\
\


Warnell
School of Forest Resources
\
\


University of Georgia
\
\

\

\
\

\

\
\

\

\
\

\

\
\

\

\
\


PMRC Technical Report 1999
--

4

\
\

\

\
\

\

}

\
author{Prepared by
\
\

\

\
\


C.J. CIESZEWSKI and R.L. BAILEY
\
\

\

\
\

\

\
\

\

\
\


}

\
maketitle


\
begin{abstract}


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baselineskip=0
.7truecm

Biologically realistic site models require the ability to concurrently express variable asymptotes
and polymorphism in curve shapes. Moreover, it is only logical and rational to require these
models be invariant to changes in the index or base ag
e. This manuscript explains the Generalized
Algebraic Difference Approach that can be used effectively to derive truly base
-
age invariant
difference equations capable of describing concurrent polymorphism and variable asymptotes.
This new generic methodolo
gy for derivation of even the most complex dynamic equations is
mathematically sound. The equations derived with it can be extremely flexible and may generate
intricate patterns of concurrent polymorphism and variable asymptotes. This methodology is
relev
ant to all situations in which the dependent variable is a function of an unobservable
variable, and the models can be implicitly defined by their initial conditions. It is equally useful
for derivation of new equations and for improvement of existing base
-
age specific equations.

\
vspace{.5cm}


\
noindent {
\
bf Keywords:} site index; base
-
age invariance; mixed effects; inventory updates;
growth and yield.


\
end{abstract}

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\
section*

{Dynamic Equation Modeling}

%
\
markright{
\
underline{PMRC Technical Report 1999
-
4; Cieszewski,

%Bailey; BAI Poly. Site Eqs.
\

w.
\

Var.
\

Asympt.
\

\
&
\
ldots }}


\
subsection
*

{Background}

The earliest efforts in growth and yield modeling concentrated on two
-
dimensional relationships
such as for example height over age. Both, hand drawn curves and the earliest equations that
were capable of consistently generating more intric
ate shapes approximated two
-
dimensional
relations (e.g., Hosfeld~1822). These site models, at times, were developed separately for
different sites or even individually for different stands. Until today many users of site models
prefer to consider them as t
wo
-
dimensional relationships and rarely discuss them in a context of
three
-
dimensional systems; sometimes sites are denoted as discrete quality or productivity
classes, e.g., {
\
bf A}, {
\
bf B},
\
ldots, or {
\
bf I}, {
\
bf II},
\
ldots.


Historically, site model
s were presented as graphs or tables representing a four
-
variable height
prediction system for a discrete collection of sites, or stands (e.g., USDA 1929). The variables
were reference
-
height (discrete); age of reference
-
height (discrete or continuous); pr
ediction age
(continuous); and prediction height (continuous). However, these systems describe
three
-
dimensional relationships between height, age and growth intensity, although, the growth
intensity is seemingly defined by two implicit variables.


For som
e applications, guide curves were proportionally or otherwise (e.g.,
Osborne and
Schumacher 1935
) adjusted for individual stands by a simple means of multiplication. For
example, in an anamorphic system the curve closest to an observation was multiplied by

a ratio of
the observation and the corresponding height on the curve at the same age. The new curve
generated in this way would pass through the known height
-
age pair. Algebraically adjusting a
single base model to specific situations or stands by scaling
, improves the parsimony, consistency
and utility of the system over the other approach of multiple
-
models developed individually for
each stand. This approach reduces the number of models involved in the prediction system and, in
the analysis phase, allow
s data from different stands to be combined in a complementary system.
It extends the discrete reference
-
variable to a continuous reference
-
variable through the simple
multiplication and is therefore more functional.


Newer approaches to site
-
dependent mo
deling almost exclusively involve three
-
dimensional
functions. Usually, the models describe relationships between a response variable, time or age,
and at least one more variable representing intensity of the modeled processes (e.g.,
Garc
\
'{
\
i}a~1983
). T
he response variable can be height, diameter, basal area, volume, number of
trees, weight or any other measurable quantity. The variable representing intensity of the
processes is usually for convenience expressed as an implicit measure in a form of a sam
ple
observation of the response variable. For example, a height model can use site index ($ S $) at a
given base
-
age ($ A_b $). An early algebraic inclusion of $ S $ into simple anamorphic equations
was followed by increasing equation complexities necessa
ry to describe numerous desirable
characteristics. Some of these characteristics include

\
begin{description}


\
item[1)] curves through the origin;


\
item[2)] polymorphism;% (Fig.~
\
ref{fig:poly});


\
item[3)] variable asymptotes;% (Fig.~
\
ref{fig:polyvar}
and
\
ref{fig:nondis});


\
item[4)] equality of $ S $ and predicted height at base
-
age; and


\
item[5)] theoretical justifications and interpretations.


\
end{description}



Bailey and Clutter~(1974) introduced the concept of {
\
bf base
-
age invariance} in whic
h a
dynamic equation can compute predictions directly from any age
-
height pair without
compromising consistency of the predictions. Dynamic equations can be viewed as four
-
variable
relationships that are continuous over all variables. Yet, dynamic equation
s simply describe
three
-
dimensional surfaces much the same as the other fixed
-
base
-
age site equations. In these
equations, one of the dimensions uses two implicit variables.


The predictions of dynamic equations are unaffected by arbitrary changes in base
-
age. Bailey and
Clutter~(1974) applied a technique that has become know as the algebraic difference approach
(ADA). Site equations derived with this approach are mathematically sound (i.e., it cannot lead to
$1=0$) and they always compute consistent numbe
rs.


We present here the Generalized Algebraic Difference Approach (GADA)
---

a new generic
methodology for the derivation of very flexible dynamic equations that are truly
\
underline{base
-
age invariant},
\
underline{polymorphic}, have
\
underline{variable a
symptotes}
and other desired properties, such as a
\
underline{theoretical basis} and $ S $ equal height at
base
-
age which could not all be derived with the traditional ADA. We will show how to
systematically derive these equations, as opposed to creating t
hem in {
\
it ad hoc} ways, while
neither compromising their mathematical soundness nor sacrificing any logical relationships
among the equation's variables. The derivation proposed here is suitable for all kinds of growth,
yield, and decline or oscillation
models that could be considered for any pooled cross
-
sectional
and longitudinal data with one unobservable variable and mixed effects modeling.







\
subsection*

{
The Base
-
Age Invariance of Equations}

The Concise Oxford Dictionary of Mathematics (Clapha
m 1996) defines {
\
bf invariant} as {
\
sl ``A
property or quantity that is not changed by one or more specified operations or transformations.''}
Other dictionaries contain consistently similar definitions. The notion of invariance in mathematics
has an une
quivocal meaning consistent with its dictionary and encyclopedic (e.g., Gellert {
\
it et
al.}~1977) definitions; it means that the elements of interest called ``invariants'' in a given system
will remain
\
underline{unaffected} while other elements in the sy
stem are varied. In base
-
age
invariant equations the {
\
bf ``invariants''} are the computed heights and, as a result, the shapes of the
height over age curves. The varied elements are the base
-
ages and the reference heights.


Site curves are base
-
age inva
riant if and only if they are unequivocally unaffected by all choices of
base
-
ages
---
rather than slightly affected, unaffected within a certain range, or similar to unaffected.
This means that, in a dynamic equation, any arbitrary age
-
height pair on a curv
e must define the very
same curve, not merely ``similar curves''. The base
-
age
-
invariant equations also have the path
invariance property (Clutter {
\
it et al.} 1983). This means that one
-
step predictions or yearly or
decadal iterations will all predict th
e same values at a given final age. When curves generated by
using different base
-
ages are not positively identical, the equation is simply not base
-
age invariant.


True base
-
age invariant equations constitute initial condition difference equations, or dyn
amic
equations, that are the most modern and the most advanced forms of integral basic site
equations
---
predicting height as a function of age and site
---
used in forest biometrics. They represent
a continuous four
-
variable prediction system directly interp
reting three
-
dimensional surfaces without
explicit knowledge of the third dimension, which depends on an unobservable variable. They are
applicable to
\
underline{integral modeling},
\
underline{dynamic equation modeling}, and to
\
underline{state space model
ing} (Garcia 1994) as well as to descriptions of complex
infinite
-
dimensional processes in
\
underline{periodic} or
\
underline{yearly}
\
underline{iterations}
(e.g., Cieszewski and Bella 1993). Unlike yearly difference or differential equations, they can be
used directly for forward and backward estimations.


Considering the utility and popularity of models for the dynamics of stands and forests, the
literature on deriving dynamic equations is rather limited. Such derivations have been applied in
different c
ontexts. In a broad classification, the approaches to these derivations can be listed as:



\
begin{itemize}


\
item initial condition solutions to differential equations, e.g., Lenhart (1968, 1972);


\
item the algebraic difference approach of Bailey and Clu
tter~(1974);


\
item equating sub
-
defined ratios of base equations, e.g., Amaro {
\
it et al.}~(1998); and


\
item initial
-
condition site index substitution in expanded dynamic equations, e.g., Cieszewski
and Bella (1989).


\
end{itemize}


We discuss further o
nly the algebraic difference approach.



\
subsection*

{The Algebraic Difference Approach}

The dynamic equations depend on their entangled self
-
definition, i.e., the inverse function of their
underlying base equation that is entangled into the base equation
. They are extremely sensitive to
any {
\
it ad hoc} algebraic operations that are otherwise harmless with all explicit equations. This
sensitivity goes far beyond what ``usual'' operations with explicit equation based common sense
may dictate. For example,
these dynamic equations cannot be added by sides or altered in other
ways by adding arbitrary exponentiation or multiplication within the existing structures. They
cannot be created by assuming or relaxing any implicit assumptions or in other {
\
it ad hoc}
ways.
All of the above would be perfectly admissible with explicit equations but not with these
implicitly defined dynamic equations.


The Algebraic Difference Approach (ADA) is similar to the standard mathematical procedure
applied in calculus for boundar
y solutions to differential equations. It consists of replacing an
arbitrary parameter in a function $ Y $ of $ t $ with its solution using specific values of $ Y_0 $ and
$ t_0 $ instead of $ Y $ and $ t $.


Writing the base function $ Y $ as a function o
f $ t $ and $ n $ parameters $
\
rho_1
\
ldots
\
rho_n $
as:


\
begin{equation}


Y(t) = f(t,
\
rho_1
\
ldots
\
rho_{n
-
1},
\
rho_n),

\
label{eq:adabase}


\
end{equation}


its solution for an arbitrary parameter $
\
rho_n $ is a function of two independent variables

$ Y $
and $ t $ and the $ n
-
1 $ remaining parameters. In this solution, $ Y $ and $ t $ are independent
variables and therefore can be assigned arbitrary values $ Y_0 $ and $ t_0 $:


\
begin{equation}


\
rho_n = u(t, Y,
\
rho_1
\
ldots
\
rho_{n
-
1}) = u(t_0, Y
_0,
\
rho_1
\
ldots
\
rho_{n
-
1})

\
label{eq:adasol}


\
end{equation}


where $ Y_0 $ denotes a given value of $ Y $ for an arbitrary $ t_0 $.


The above solution can be used in place of $
\
rho_n $ in the base function to define a new dynamic
function of time $

t $, an arbitrary time $ t_0 $, a given function value $ Y_0 $ at $ t_0 $ and the
remaining $ n
-
1 $ parameters:


\
begin{equation}

Y(t, t_0, Y_0) = w(t, t_0, Y_0,
\
rho_1
\
ldots
\
rho_{n
-
1})

\
label{eq:adabai}


\
end{equation}

Since the above function is un
defined without the arbitrary values of $ t_0 $ and $ Y_0 $, these
values are called the initial conditions of this function. With $ t_0 $ and $ Y_0 $ assuming any
values, the equation represents a dynamic equation. Using this approach, a site index equati
on may
be developed that produces a site index curve unchanging under all choices of base
-
age. Hence, the
term base
-
age invariant equation.


The ADA was introduced to forestry literature using a logarithmic transformation of the
Schumacher~(1939) equation,

i.e.,


\
begin{equation}


\
ln Y(t) =
\
alpha
-

\
beta / t


\
label{eq:schbase}


\
end{equation}


where $Y$ was height but it could be any variable of interest (e.g., volume, density, basal area, etc.),
and $ t $ was defined as a simple exponential function
of age ($ Age ^ c $ where $c$ was a
parameter). The solution for the proportionality constant of this function is


\
[


\
alpha =
\
ln Y_0 +
\
beta /t_0

.
\
]

Using ADA, the anamorphic equation with variable asymptotes is


\
begin{equation}


\
ln Y(t, t_0, Y_
0) =
\
ln Y_0 +
\
beta (1/t_0
-

1/t).


\
label{eq:anam74}


\
end{equation}

A polymorphic equation (with a single asymptote) may be obtained by applying ADA to the $
\
beta
$ parameter of the equation. The solution of eq.~(
\
ref{eq:schbase}) for the slope paramet
er is


\
[


\
beta = t_0 (
\
alpha
-

\
ln Y_0)


\
]

so the polymorphic dynamic equation is


\
begin{equation}


\
ln Y(t, t_0, Y_0) =
\
alpha + (
\
ln Y_0
-

\
alpha )t_0 / t


\
label{eq:poly74}


\
end{equation}


The above equations derived by ADA represent true equali
ties. They are consistent in predictions
and any point on a curve will always unequivocally define the very same curve. The equations also
have the path invariance property described earlier. In short, the curves are indifferent under choices
of base
-
ages;

they are base
-
age invariant.


The algebraic difference approach is, in principle, similar to the method used for
solutions of differential equations with initial conditions, but as applied to base equations
it moves outside of integration theory. For exam
ple, a general solution to the differential
equation (4) of McDill and Amateis~(1992):


\
[

{ d Y
\
over d t } =
\
alpha {Y
\
over t}
\
left( 1
-

{Y
\
over M}
\
right)


\
]








M
H
A
H
a
dA
dH
1

should be presented with an intercept defined by the integration consta
nt $ C $:


\
[

Y(t) = {M
\
over 1 +
\
beta / t^
\
alpha } + C


\
]
C
A
B
M
H
a



/
1

However, defining the initial conditions based on $
C $,
as integration theory would
suggest
,
is unreasonable in those cases where biological
interpretation

requires that $Y(0
)
= 0 $ (i.e., $
C = 0 $).
Thus, McDill and Amateis~(1992)
applied

the algebraic difference
approach to the parameter $
\
beta $ and using its initial condition solution:


\
[

\
beta = {t_0} ^
\
alpha
\
left( {M
\
over Y_0 }
-

1
\
right)


\
]










1
0
0
0
H
M
A
B
a

they have derived their polymorphic dynamic equation [5]
\
footnote{The reference [5] is
for the original publication not this text.}, i.e.,


\
[

Y(t) = {M
\
over 1 +
\
left( {M
\
over Y_0}
-

1
\
right)
\
left( { t_0
\
over t}
\
right) ^
\
alpha }


\
]

a
A
A
H
M
M
H

















0
0
1
1


The ADA has been applied successfully in various modeling contexts by B
\
'{e}gin and
Sch
\
"{u}tz~(1994); Borders {
\
em et al.}~(1984 and 1988);

Cao~
{
\
em et al.}~
(1993, 1997);
Clutter
{
\
em et al.}~(1983); Clutter {
\
em et al.}~(1984); DuPlat and Tran
-
H
a~(1986);
Lappi and
Bailey~(1988); McDill and Amateis~(1992);
Ramirez {
\
em et al.}~(1987); and others cited therein.
These workers have used the ADA and the dynamic equations for modeling growth and yield of
height, diameter and basal area, as well as tree

survival in forest populations.




%
\
clearpage

\
section*

{
The New Methodology}

%
\
markright{
\
underline{PMRC Technical Report 1999
-
4; Cieszewski,

%Bailey; BAI Poly. Site Eqs.
\

w.
\

Var.
\

Asympt.
\

\
&
\
ldots }}


\
subsection*

{Theoretical Foundation and Symb
olic Definition }

To facilitate the Generalized Algebraic Difference Approach (GADA) formulation we identify a
theoretical variable labeled the growth intensity factor $ {
\
cal X} $ and define it to be the
quantification of those particular growth dynamics
that are uniquely associated with a site and
individual characteristics of growth or survival capabilities.


$ {
\
cal X} $ is used consistently in all equation formulations to describe the rules of changes in curve
shapes across different sites. It can be
either a variable or a function of any number of variables. Such
variables can include climate, water availability, organic soil depth, leaf area or rates of
photosynthesis, measure of ozone and genetic components. $ {
\
cal X} $ is continuous, monotonic
and

relevant to the modeled dynamics; it can describe the relative rates of change in terms of direct
functional relationship. We can assume that, for example, small values for $ {
\
cal X} $ represent low
growth intensity and high values for $ {
\
cal X} $ repre
sent high growth intensity. Not being
practically obtainable, $ {
\
cal X} $ is eventually replaced with the initial conditions that are
measurable so that the equation can be operationally useful. However, this happens only after the
equation is explicitly
formulated in a satisfactory way when it already contains all the desired
properties of a site equation, such as, polymorphism and variable asymptotes.



The first step in the GADA is to select a base equation and to identify in it any desired number of
si
te
-
specific parameters. Then, define explicitly how the site
-
specific parameters change across
different sites by replacing them with explicit functions of $ {
\
cal X} $ and new parameters. In this
way the initially selected two
-
dimensional base equation ex
pands into an explicit three
-
dimensional
site equation describing both cross
-
sectional and longitudinal changes with two independent
variables $ t $ and $ {
\
cal X} $. In the final step a solution for $ {
\
cal X} $ replaces all $ {
\
cal
X}
\
mbox{s}$ with impli
cit definitions using the equation's initial conditions $ t_0 $ and $ Y_0 $.


Symbolically, the base equation is


\
begin{equation}


Y(t) = f(t,
\
rho_1
\
ldots
\
rho_{n
-
1},
\
rho_n)

\
label{eq:genbase}


\
end{equation}


\
noindent

where $
\
rho_1
\
ldots
\
rho_n
$ are the equation parameters.



If in the base equation~(
\
ref{eq:genbase}) a given site
-
specific parameter $
\
rho_i $ is defined as a
function $ g_i $ of $ {
\
cal X} $ and any number of $ j $ new parameters, viz., $
\
rho_i
\
equiv g_i
\
!
\
left( {
\
cal X},
\
rho_{i_1}
\
ldots
\
rho_{i_j}
\
right) $, the base equation~(
\
ref{eq:genbase}) with
multiple site
-
specific parameters is changed to the explicit three
-
dimensional site equation with two
independent variables $ t $ and $ {
\
cal X} $:


\
begin{equation}

Y(t, {
\
c
al X} ) = f
\
!
\
Bigl( t,
\
rho_1
\
ldots
\
rho_{m
-
1}, g_m
\
! ({
\
cal X},
\
rho_{m_1}
\
ldots
\
rho_{m_k})
\
ldots g_n
\
! ({
\
cal X},
\
rho_{n_1}
\
ldots
\
rho_{n_l})
\
Bigr)

\
label{eq:genmix}


\
end{equation}

\
noindent where $ Y(t, {
\
cal X} ) $ is a function of $ t $,

$ {
\
cal X} $, and $ m + k + l
-
1 $
parameters.



If eq.~(
\
ref{eq:genmix}) can be solved for $ {
\
cal X} $, the RHS of this solution, with initial
condition values for $ t $ and $ Y $, i.e.,


\
begin{equation}


{
\
cal X} = u(t, Y,
\
rho_1
\
ldots
\
rho_{n_l}) =

u(t_0, Y_0,
\
rho_1
\
ldots
\
rho_{n_l}),
\
label{eq:gensol}


\
end{equation}

can be substituted in eq.~(
\
ref{eq:genmix}) in place of $ {
\
cal X} $ so the dynamic equation


\
[ Y(t, t_0, Y_0) = f
\
!
\
Bigl( t,
\
rho_1
\
ldots
\
rho_m, u
\
!
\
left( t_0, Y_0,
\
rho_1
\
ldots
\
rho_{n_l}
\
right)
\
Bigr),


\
]

after reformulation and elimination of redundant parameters, becomes the dynamic equation with an
implicitly defined initial condition:


\
begin{equation}

Y(t, t_0, Y_0) = f(t, t_0, Y_0,
\
rho_1
\
ldots
\
rho_w)

\
label{
eq:genbai}


\
end{equation}


where


\
begin{equation}

n
-
1
\
leq w
\
leq m + k +
\
ldots + l
-
1
\
qquad
\
mbox{and}
\
quad k
\
lesseqqgtr
\
ldots
\
lesseqqgtr l.


\
label{eq:lesscoef}


\
end{equation}



The result in eq.~(
\
ref{eq:lesscoef}) means that equation~(
\
ref{eq
:genbai}) has a
\
underline{smaller
or equal} number of parameters than the equation~(
\
ref{eq:genmix}).



Practical applications of the GADA involve different levels of complexity and difficulty in equation
derivations. We classify the equations as simple o
r complex depending on whether the solutions
involved are based on just a reformulation of an equation (simple) or on finding its roots (complex).





\
subsection*

{Specific Cases}


\
subsubsection*

{Simple Equations}

In the simplest applications the adva
ntage of introducing $ {
\
cal X} $ is not immediately obvious.
For example, to replicate the Bailey and Clutter~(1974) derivation based on two equations with the
GADA, we write eq.~(
\
ref{eq:schbase}) in two ways:


\
begin{eqnarray}


\
ln Y(t, {
\
cal X}) & = &

{
\
cal X}
-

\
beta_a / t
\
label{eq:schbasanam}
\
\

&
\
mbox{and} &
\
nonumber
\
\


\
ln Y(t, {
\
cal X}) & = &
\
alpha _p
-

{
\
cal X} / t
\
label{eq:schbaspoly}


\
end{eqnarray}

where $
\
beta_a $ is the slope parameter of the anamorphic equation and $
\
alpha _p $ is

the
asymptote parameter of the polymorphic equation. Applying the ADA with respect to $ {
\
cal X} $ in
either of these two equations completes the process and concludes this application of the GADA. The
two dynamic equations in Bailey and Clutter~(1974) ma
y be derived in this way.



The greatest advantage of introducing $ {
\
cal X} $ manifests itself when more than one simultaneous
site
-
specific parameter is necessary to adequately describe changes in curve shapes across different
sites. For example, in a s
imple assumption of concurrent polymorphism with varying asymptotes
both $
\
alpha $ and $
\
beta $ in


\
[


\
ln Y(t) =
\
alpha
-

\
beta / t


\
]

could be dependent on $ {
\
cal X} $ while $ {
\
cal X} $ could define the limiting size, i.e.,


\
begin{equation}


\
ln

Y(t, {
\
cal X}) = {
\
cal X} +
\
beta {
\
cal X} / t


\
label{eq:simp1base}


\
end{equation}


The solution for $ {
\
cal X} $ would then be


\
[

{
\
cal X} = {
\
ln Y
\
over 1
-

\
beta / t }= {
\
ln Y_0
\
over 1
-

\
beta / t_0 }


\
]


and applying the GADA to eq.~(
\
ref{eq:
simp1base}) with respect to $ {
\
cal X} $ would result in a
dynamic equation based on Schumacher's equation that provides polymorphic base
-
age invariant
curves with variable asymptotes:


\
begin{equation}


\
ln Y(t, t_0, Y_0) =
\
ln Y_0 {t_0 (t
-

\
beta)
\
ove
r t (t_0
-

\
beta ) }
\
label{eq:simpbai1}


\
end{equation}



The assignment of $ {
\
cal X} $ to $
\
alpha $ means that given an objective measure of growth
intensity the upper production limit would be increasing with increasing innate growth potential. This
w
ould result in variable asymptotes. The assignment of $ {
\
cal X} $ to $
\
beta $ means, in simple
terms, that the shapes of curves change with changing growth intensity which defines a polymorphic
equation. Clearly, both variable asymptotes and polymorphism

occur if $ {
\
cal X} $ affects both $
\
alpha $ and $
\
beta $.



Alternatively, the objective could be a single equation that concurrently expresses 1)~a similar
polymorphism to that of the Bailey and Clutter~(1974) polymorphic equation; and 2)~similar
asym
ptotic properties to those of the Bailey and Clutter~(1974) anamorphic equation. The advantage
of introducing $ {
\
cal X} $ becomes most evident here as this objective is accomplished simply by
adding eqs.~(
\
ref{eq:schbasanam}) and~(
\
ref{eq:schbaspoly}) by
sides, i.e.,


\
begin{equation}

2
\
ln Y(t, {
\
cal X}) =
\
left({
\
cal X}
-

\
beta_a / t
\
right) +
\
left(
\
alpha _p
-

{
\
cal X} / t
\
right)
\
label{eq:simp2bas}


\
end{equation}


\
noindent


Thus, the solution


\
[

{
\
cal X} = {t (
\
ln Y
-

\
alpha _p' ) +
\
beta_a'
\
ove
r t
-
1 } = { t_0 (
\
ln Y_0
-

\
alpha _p' ) +
\
beta_a'
\
over
t_0
-
1}


\
]

substituted for $ {
\
cal X} $ in eq.~(
\
ref{eq:simp2bas}) produces a single dynamic equation
exhibiting concurrently both of the desired properties:


\
begin{equation}


\
ln Y(t, t_0, Y_0) =

\
alpha _p'
-
{
\
beta_a'
\
over t} + {(t
-
1 ) t_0
\
over (t_0
-
1 )t }
\
left(
\
ln Y_0
-

\
alpha
_p' + {
\
beta_a'
\
over t_0 }
\
right)
\
label{eq:simpbai2}


\
end{equation}


The ability to combine the properties of two different dynamic equations into one dynamic equat
ion
by adding their
\
underline{explicit} forms by sides is a unique advantage of the new methodology.
However, this must be done in the explicit stage with the base equations. If the sides of two dynamic
equations are added directly the result is a degener
ated
\
footnote{

``improper'', ``inadmissible'', ``degenerate'', and so on, are operations and/or formulations that can
lead to $ 1=0 $, after Simmons~(1972) (p.~160, l.~17 and bottom of the page) who writes ``{
\
ldots
This terminology follows a time
-
honored
tradition in mathematics, according to which situations that
elude simple analysis are dismissed by such pejorative terms as ``improper,'' ``inadmissible,''
``degenerate'', ``irregular,'' and so on.
\
ldots }''

}

relationship that has neither the property o
f base
-
age invariance nor that of equality. Adding
equations by sides is improper when applied directly to dynamic equations. It is admissible only
when applied to base equations before entangling the equations with initial conditions. Although not
discern
ible with the basic algebraic difference approach the above dynamic equation becomes clearly
apparent with the Generalized Algebraic Difference Approach.





\
subsubsection*

{Complex Equations}

We label dynamic equations as simple when they can be derived

through direct reformulation as
shown above. All dynamic equations that require in their derivations the roots of an equation in order
to determine $ {
\
cal X} $ we label complex because the type of solutions required may constitute a
considerable barrier
in practical applications.




Even with few parameters, an equation can be complex with solutions that involve roots. Examples
occur in formulations involving quadratic relationships or combinations of direct and inverse
proportionality. Such a relationsh
ip could be between an equation characteristic, e.g., a variation in
asymptotes or polymorphism, and a growth intensity measure $ {
\
cal X} $. For example, the
derivation of a complex dynamic equation could follow from a theory that asymptotes are
exponenti
ally proportional to growth intensity, and that polymorphism is inversely proportional to
growth intensity, i.e.,


\
begin{equation}


\
ln Y(t, {
\
cal X}) =
\
alpha {
\
cal X}
-
{
\
frac {
\
beta /{
\
cal X}}{t}}
\
label{eq:comp1base}


\
end{equation}


\
noindent


For th
is base equation, the solution for $ {
\
cal X} $ involves finding roots of a quadratic equation
and a selection of the most appropriate root to entangle into the dynamic equation. The selection of
the most appropriate expression for $ {
\
cal X} $ may depend
on the equation parameters that in turn
depend on the data and the domain of the applicable ages. The solution for $ {
\
cal X} $ in
eq.~(
\
ref{eq:comp1base}) is


\
[


{
\
cal X} =
\
left
\
{
\
begin{array}{l}
\
left
\
{
\
begin{array}{l}

0.5 (
\
ln y + {
\
cal R} ) /
\
alp
ha = 0.5 (
\
ln Y_0 + {
\
cal R}_0 ) /
\
alpha
\
cr
\
mbox{or: }
\
cr

0.5 (
\
ln y
-
{
\
cal R} ) /
\
alpha = 0.5 (
\
ln Y_0
-
{
\
cal R}_0) /
\
alpha
\
end{array}
\
right.
\
cr


\
mbox{where: }
\
cr


\
left
\
{
\
begin{array}{l}

{
\
cal R} =
\
sqrt {(
\
ln Y) ^ 2 + 4
\
,
\
alpha
\
,
\
beta
/ t}
\
cr


\
mbox{and: }
\
cr

{
\
cal R}_0 =
\
sqrt {(
\
ln Y_0) ^ 2 + 4
\
,
\
alpha
\
,
\
beta / t_0}
\
cr


\
end{array}
\
right.
\
end{array}
\
right.
\
label{eq:comp1sol}


\
]

Selecting the root more likely to be real (as opposed to complex) and positive, i.e., the on
e involving
addition rather than subtraction of the square
-
root, with the usual initial conditions and substituting it
into eq.~(
\
ref{eq:comp1base}) results in the following dynamic equation:


\
begin{equation}


\
ln Y(t, t_0, Y_0) = {
\
ln Y_0 + {
\
cal R}_0
\
over 2}
-
{2
\
gamma / t
\
over
\
ln Y_0 + {
\
cal R}_0 }
\
label{eq:comp1bai}


\
end{equation}

where $
\
gamma =
\
alpha
\
beta $.


Another situation requiring a root
-
finding solution arises when the cross
-
sectional changes are
described by polynomial functions of $
{
\
cal X} $.


The pursuit of a best equation form may become a tedious procedure depending on many factors
including the data analysis. For each explicit or base equation, several possible approaches may be
used to derive the implicit dynamic equation. How
ever, at any time a new implicitly defined equation
is considered, the formulation of proper relationships in the explicit equation should be completed
prior to the entangling of implicit solutions. For this work, a good understanding of the explicit
equat
ion's mathematical structure and the biological expectations of growth differences over different
sites give the modeler a distinct advantage. However, absent an understanding or knowledge of the
expected growth relationships, curve shapes desired, or the
functional changes wanted, one may
explore a formulation of generic relationships such as we discuss in subsequent sections. After
parameter estimation with such a model, one may then exercise hypothesis testing based on estimates
of model parameters and t
heir error structures to resolve questions about biological behavior of the
system.




\
subsubsection*

{Multiple and Stepwise Regression Equations}

The methodology we advocate defines a rigorous mathematical procedure facilitating the
derivation of equati
ons with implicitly defined initial conditions from explicit theoretical bases
relating to biological, geometric or algebraic theories. This methodology emphasizes the role of
the modeler in formulating the hypothesis upon which the equations are built pri
or to their final
restructuring into dynamic equations. The equations are formed by the modeler rather than by
default via statistical analysis. However, the ``generic equations'' discussed later are intended to
provide an excessive amount of flexibility i
n anticipation that statistics of fit will determine the
final forms of the dynamic equations.


It may be necessary in practice for statistics rather than the modeler to determine forms of final
equations. This may apply not only to stepwise and permutati
onal regressions but also to any
other type of linear or nonlinear regression analysis or model fitting in which the criteria for
model selection depend on residual analysis or statistical results. Given such a situation it may
seem that the GADA is antith
etical to regression theory. We believe otherwise.



The methodology presented here can be used to improve existing regression equations even if
they are produced by step
-
wise regressions. With stepwise regression in particular, the GADA
may have consider
able value in equation improvement efforts. Consider, for example, the
following four
-
parameter equation based on stepwise
-
regression:


\
begin{equation}

Y(t, S) =
\
alpha
\
sqrt {t} +
\
beta t ^ 2
\
ln ^ {32} t
-
\
gamma {
\
frac {t ^ {5

/2} }{
\
ln t}} + S
\
delta
\
sqrt {t}


\
label{eq:stepwise1H}


\
end{equation}


\
noindent


with solution


\
begin{equation}

S(t, Y) =
-
{
\
frac {
\
alpha
\
sqrt {t_0}
\
,
\
ln t_0 + t_0
\
beta t_0 ^ 2
\
ln ^ {33}
-
t_0 ^ {5/2}
-
\
gamma
Y_0
\
,
\
ln

t_0}{
\
delta
\
,
\
ln t_0
\
sqrt {t_0} }}


\
label{eq:st
epwise1S}


\
end{equation}

giving rise to the two
-
parameter dynamic equation


\
begin{equation}

Y(t, t_0, Y_0) =
\
beta
\
,
\
left (t ^ 2
\
ln ^ {32} t
-

t_0 ^ {

3/2} t_0
\
ln ^ {32}
\
sqrt {t}
\
right ) +
\
gamma
\
,
\
left ({
\
frac {


t_0 ^ 2
\
sqrt {t}}{
\
ln t_0}}
-
{
\
fra
c {t ^ {5/2}}{
\
ln t}}


\
right ) + Y_0
\
,
\
sqrt {{
\
frac {t}{t_0}}}


\
label{eq:stepwise1bai}


\
end{equation}

The GADA methodology has been used to convert the model into a dynamic relationship that:



\
begin{description}



\
item[1.] Generates identical curv
es as those produced by eq.~(
\
ref{eq:stepwise1H})



\
item[2.] Predicts heights at base
-
age equal to site indexes



\
item[3.] Can compute site index and height from the same equation



\
item[4.] Can use heights and ages directly instead of fixed
-
base
-
age
site indexes and



\
item[5.] Can be easily fitted and applied with use of any base
-
age.



\
end{description}


All these improved properties accrue with a reduction by half in the number of parameters and no
contradiction to regression theory or practice.






\
subsubsection*

{Generic Equations}

In this section, we present the most advanced category of dynamic base equations. They may be
considered the epitome of equation
-
based modeling with dynamic equations as discussed herein.
Generic equations are form
ulated in the absence of explicit expectations about the final model form.
A modeler may want to cover a wide range of possible equations during a single analysis to save
time and make equation selection more efficient. Schnute~(1981) discusses an excellen
t example of
such a practice. Generic equations have been considered here as a separate category because of their
potentially large number of parameters and complicated appearance resulting from either simple or
complex derivations. Development of generic
equations should be considered with caution because it
can easily lead to over
-
parameterization and model instability as well as difficulties with parameter
estimation.


An example founded on the Schumacher base equation might represent a lack of strong co
mmitment
to either the asymptote or the shape parameter being the only or most predominant expression of
growth intensity. In other words, it may be appropriate to derive an equation that might, but does not
have to, have asymptotes affected by site factor
s and might, but does not have to, have curve shapes
varying across different sites. In addition, these effects could occur in flexible proportions. A simple
base equation that satisfies such requirements is a generalization of eq.~(
\
ref{eq:simp2bas}):



\
begin{equation}


\
ln Y(t, {
\
cal X}) = (
\
alpha +
\
alpha ' {
\
cal X} )
-
(
\
beta +
\
beta' {
\
cal X} ) / t


\
label{eq:gen1base}


\
end{equation}


\
noindent


where $
\
alpha ' $ and $
\
beta' $ are the weighting parameters. This generic form of the explicit

three
-
dimensional equation can easily examine what proportions of eqs.~(
\
ref{eq:schbasanam}) and
(
\
ref{eq:schbaspoly}), or eqs.~(
\
ref{eq:schbase}) and~(
\
ref{eq:simp1base}), are best blends for any
given data. This equation also allows one to examine if eq.~(
\
ref
{eq:simp1base}) should indeed be
directly proportional to $ {
\
cal X} $ or if it should be only linearly (or partially) proportional to $
{
\
cal X} $. To illustrate these three alternative hypothesis, eq.~(
\
ref{eq:simp2bas}) can be written as
a weighted sum
of eq.~(
\
ref{eq:schbasanam}) and eq.~(
\
ref{eq:schbaspoly}). This produces a
reparameterized version of eq.~(
\
ref{eq:gen1base}):


\
[


\
ln Y(t, {
\
cal X}) =
\
alpha '
\
left({
\
cal X}
-

\
beta_a / t
\
right) +
\
beta'
\
left(
\
alpha _p
-

{
\
cal X} / t
\
right)


\
]

(wh
ere: $
\
alpha _p =
\
alpha /
\
beta' $ and $
\
beta_a =
\
beta /
\
alpha ' $); it can be written as a
weighted sum of eq.~(
\
ref{eq:schbase}) and eq.~(
\
ref{eq:simp1base}), which is also equivalent to
eq.~(
\
ref{eq:gen1base}):


\
[


\
ln Y(t, {
\
cal X}) =
\
alpha ' {
\
cal X}
\
left(
\
alpha
-

\
beta / t
\
right) +
\
beta'
\
left(
\
alpha
-

\
beta / t
\
right)


\
]

and it can be written as a linear generalization of eq.~(
\
ref{eq:simp1base}):


\
[


\
ln Y(t, {
\
cal X}) =
\
left(
\
alpha ' {
\
cal X} +
\
beta'
\
right)
\
left(
\
alpha
-

\
beta /

t
\
right)


\
]


where: $
\
alpha ' $ and $
\
beta' $ are the weights of the anamorphic and polymorphic forms and $
\
alpha _p =
\
alpha /
\
beta' $ and $
\
beta_a =
\
beta /
\
alpha ' $ and both $
\
alpha '
\
neq 0 $ and $
\
beta'
\
neq 0 $. The solution for $ {
\
cal
X} $ in eq.~(
\
ref{eq:gen1base}) is


\
[


{
\
cal X} = {
\
ln Y
-

\
alpha
-

\
beta /t
\
over
\
alpha '
-

\
beta' / t} = {
\
ln Y_0
-

\
alpha
-

\
beta /t_0
\
over
\
alpha '
-

\
beta' / t_0}


\
]


and after applying the GADA to eq.~(
\
ref{eq:gen1base}) and using this solution
, the resulting
simple generalized dynamic equation based on eq.~(
\
ref{eq:schbase}) has the following form:


\
[


\
ln Y(t, t_0, Y_0) =
\
alpha
-

{
\
beta
\
over t} + {
\
alpha '
-

\
beta' / t
\
over
\
alpha '
-

\
beta' / t_0}
\
left(
\
ln
Y_0
-

\
alpha + {
\
beta
\
over t_
0}
\
right)
\
label{eq:gen1bai1}


\
]


However, this equation is clearly over
-
parameterized to the extent of being undefinable. This can be
rectified by combining the parameters $
\
alpha ' $ and $
\
beta' $ into one parameter. Depending on
which of the two pa
rameters in eq.~(
\
ref{eq:gen1base}) is more likely to be equal to zero the
corresponding dynamic equation could have one of the two forms:


\
begin{equation}


\
ln Y(t, t_0, Y_0) =
\
left
\
{
\
begin{array}{l}


\
alpha
-

{
\
beta / t} +
\
left(
\
ln Y_0
-

\
alpha + {
\
beta / t_0}
\
right){
\
left(1
-

\
gamma / t
\
right) /
\
left( 1
-

\
gamma / t_0
\
right)}
\
cr


\
mbox{for expected: }
\
alpha '
\
neq 0
\
cr


\
mbox{or: }
\
cr


\
alpha
-

{
\
beta / t} +
\
left(
\
ln Y_0
-

\
alpha + {
\
beta / t_0}
\
right) {
\
left(
\
delta
-

1 / t
\
right)
\
le
ft(
\
delta
-

1 / t_0
\
right) }
\
cr


\
mbox{for expected: }
\
beta'
\
neq 0
\
cr
\
end{array}
\
right.
\
label{eq:gen1bai2}


\
end{equation}


where: $
\
gamma =
\
beta' /
\
alpha ' $ and $
\
delta =
\
alpha ' /
\
beta' $ and at least one of the two
parameters must be dif
ferent from zero. If both $
\
alpha ' = 0 $ and $
\
beta' = 0 $ there is no site
equation defined by eq.~(
\
ref{eq:gen1base}) but rather a simple two
-
dimensional single
-
line
equation that does not involve the concept of site index or base
-
age invariance. That

is to say, the
data either represent a single site or a series of different sites containing excessive amounts of
crossing or noise rendering unique identification of separate sites impossible.


Hypothesis testing on equation~(
\
ref{eq:gen1bai2}) may be ca
rried on by means of simple tests of
significance for different model parameters. Some potential outcomes from such tests could be:



\
begin{itemize}



\
item $
\
beta' = 0 $: The equation is anamorphic with variable asymptotes.



\
item $
\
alpha ' = 0 $: Th
e equation is polymorphic with a single asymptote.



\
item $
\
beta'
\
neq 0 $ and $
\
alpha '
\
neq 0 $: The equation is polymorphic and has variable
asymptotes.



\
item $
\
left |
\
alpha'
\
right |
\
ll
\
left |
\
beta'
\
right | $: The equation exhibits relativel
y strong
polymorphism.



\
item $
\
left |
\
alpha '
\
right |
\
gg
\
left |
\
beta'
\
right | $: The equation exhibits relatively strong
identification of variable asymptotes.


\
end{itemize}


An example of a complex generic equation can be developed from a gener
alization of
eq.~(
\
ref{eq:comp1base}):


\
begin{equation}


\
ln Y(t, {
\
cal X}) =
\
alpha +
\
alpha ' {
\
cal X}
-
{
\
beta + {
\
beta' /{
\
cal X}}
\
over t}
\
label{eq:gen2base}


\
end{equation}


\
noindent


The solution involves solving a quadratic equation in $ {
\
cal

X} $. Since there are two roots, careful
consideration must be given to which is most appropriate in the final equation. The selection may
depend on the model parameters, which in turn depend on the data and the domain of the applicable
ages. In the above

example the root most likely to be real and positive, and therefore more likely to
be useful is


\
[

{
\
cal X} =
\
left
\
{
\
begin{array}{l}

0.5
\
left({
\
cal R}_0
-

\
alpha
\
right)/{
\
alpha ' }


\
cr
\
mbox{where: }
\
cr

{
\
cal R}_0 = { {
\
beta } / {t_0 }} +
\
ln Y_0

+
\
sqrt {
\
left (
\
ln Y_0
-

\
alpha + {
\
beta / t_0 }
\
right ) ^ 2 +
4
\
, {
\
gamma / t_0}}


\
end{array}
\
right.
\
label{eq:gen2sol}


\
]

Substituting this root for $ {
\
cal X} $ (eq.~(
\
ref{eq:gen2base})) results in a generalization of
eq.~(
\
ref{eq:comp1bai}), i.
e., the following complex generic dynamic equation:


\
begin{equation}


\
ln Y(t, t_0, Y_0) = {{
\
cal R}_0 +
\
alpha
\
over 2} + { 2
\
,
\
gamma / t
\
over {
\
cal R}_0
-

\
alpha
}
-
{
\
beta
\
over t }
\
label{eq:gen2bai}


\
end{equation}

where: $
\
gamma =
\
alpha '
\
bet
a' $.






\
subsection*

{Properties of the Approach}

\
subsubsection*

{Parsimony}

The Generalized Algebraic Difference Approach is more parsimonious than most traditional
approaches to site equation derivations or formulations and can derive more complex eq
uations
than the traditional Algebraic Difference Approach. In terms of the potential for final equation
flexibility, it exceeds the capabilities of fixed
-
base
-
age modeling approaches. This new approach
can, in various cases, produce equations that are mor
e flexible and have fewer parameters than
the corresponding to them fixed
-
base
-
age equations. An example is in
eq.~(
\
ref{eq:stepwise1bai}).




Our contention that the Generalized Algebraic Difference Approach is more parsimonious than
the fixed
-
base
-
age
approach is justified by three points.



\
begin{description}



\
item[1. The GADA does not require any new parameters] in addition to the ones existing in the
explicit
-

or
fixed
-
base
-
age

site equation to which it is applied, which is evident from the
metho
dology definition symbolized by equations~(
\
ref{eq:genmix}) to~(
\
ref{eq:genbai}).



\
item[2.
The conclusion in eq.~(
\
ref{eq:lesscoef})] applied to derivations of dynamic equations
from
fixed
-
base
-
age

site index equations demonstrates unequivocally that th
e final dynamic
equation~(
\
ref{eq:genbai}) has a
\
underline{smaller} or
\
underline{equal} number of parameters
than the initial
fixed
-
base
-
age

equations used in eq.~(
\
ref{eq:genmix}).



\
item[3. The lack of dimensionality or range definition on $ {
\
cal X}
$, ] assures that any
multi
-
parameter expression involving the unobservable, multidimensional variable $ {
\
cal X} $ will
always be reparameterized into the most parsimonious form. For example, the three parameter
relationships $
\
alpha {
\
cal X} ^
\
gamma $
and $
\
beta {
\
cal X} ^
\
gamma $ are automatically
equivalent to the one parameter relationships $
\
alpha ' {
\
cal X}' $ and $ {
\
cal X}' $ or $ {
\
cal X}' $
and $
\
beta' {
\
cal X}' $.



\
end{description}


Clearly, if the approach
\
underline{never} uses
\
under
line{more} parameters {
\
bf (1.)} but
\
underline{sometimes} uses
\
underline{fewer} parameters {
\
bf (2.}~and {
\
bf 3.)} than another
approach then it is, in general, a more parsimonious approach.


Point (1.) is based on the fact that the only step in the GADA

that adds parameters is the
formulation of the explicit base site equation. In a special case of the GADA, the base site
equation can be formulated as a fixed
-
base
-
age site index equation and still be applicable for the
dynamic equation derivation. Thus,
there is no disadvantage involved in this step of the GADA.


Point (2.) accounts for such situations as the four parameter fixed
-
base
-
age
eq.~(
\
ref{eq:stepwise1H}) been re
-
derived with the GADA as the two parameter dynamic
eq.~(
\
ref{eq:stepwise1bai}) with

increased flexibility.


Finally, point (3.) accounts for situations in which the explicit base site equation is
unintentionally over
-
specified, a fact that cannot be easily identified with the more traditional
approaches. An example can be the
Schumacher
~(1939) equation
with $
\
alpha
\
propto
\
alpha '
{
\
cal X} ^
\
gamma $ and $
\
beta
\
propto
\
beta' {
\
cal X} ^
\
gamma $. Since {
\
cal X} is an
\
underline{unobservable} variable and, unlike site index, has only a theoretical meaning not
intended for explicit prac
tical use, it can be freely redefined as either $ {
\
cal X}' =
\
alpha ' {
\
cal
X} ^
\
gamma $ or $ {
\
cal X}' =
\
beta' {
\
cal X} ^
\
gamma $, vis.,


\
begin{equation}


\
ln Y(t, {
\
cal X}) =

\
alpha {
\
cal X} ^
\
gamma + (
\
beta{
\
cal X} ^
\
gamma ) / t =

{
\
cal X}' +
\
beta' {
\
cal X}' / t
\
equiv

{
\
cal X} +
\
beta {
\
cal X} / t
\
equiv


\
alpha {
\
cal X} + {
\
cal X} / t


\
label{eq:parsimony}


\
end{equation}

Even if the modeler does not notice this opportunity for parameter reduction in
eq.~(
\
ref{eq:parsimony}), the derivati
on defined by the GADA automatically reduces the number
of parameters by cancellation of terms during routine algebraic operations. Such is not as likely to
happen when dealing with fixed
-
base
-
age site index equations.




\
subsubsection*

{Robustness}

Two
aspects of the GADA approach to deriving models based on dynamic equations assure a high
degree of robustness in applications. First, the theoretical variable ${
\
cal X}$ has no restrictions in
interpretation. Second, the unobservable variable ${
\
cal X}$ is

eliminated during the derivations.
Not only are the dynamic equations derived with the GADA generalizations of many functional
forms of the unobservable variable as shown above, but they are also generalizations of many, at
times contradictory, theories b
ehind the model. For example, if the applied theory were based on a
proportional relationship the final dynamic equation would include this proportional relationship as
a special case but would not be limited to it. The same dynamic equation would consolid
ate many
various theories as numerous special cases that include a competing theory based on a
corresponding inverse
-
proportional relationships.




\
section*

{
Example of Application of the GADA to Comparing
Base
-
Age Specific Fitting

Methodologies
\
footnot
e{This section describes a part of study conducted in 1990 by the first
author in collaboration with Dr.
\

R.O. Curtis, USDA Forest Service, who inspired the
investigation described here through his questioning of different fitting methodologies for site
in
dex models and who also provided the data for such analysis.

}
}

A number of authors have addressed questions related to the various options for fitting site models.
Yet, various issues remain unresolved that may be examined with the aid of the GADA.



Cur
tis~(1990
\
footnote{

Curtis, RO, 1990. Site Index Curves From Stem Analyses
---
Methodology Effects and a New
Technique. Talk presented on Western Mensurationist Meeting, June 20
-
22, 1990, Bend, Oregon,
USA. Results also contained in an unpublished manuscript

(rev.~5/07/1990) by R.O. Curtis.

})

compared curves generated by two fixed
-
base
-
age site index equations fitted with base
-
age
specific methodologies using base
-
ages 50 and 100 years. Comparison of fitting methodologies
should be conducted with a common ma
thematical expression so the effects of the methodologies
are not confounded with the effects resulting from using different mathematical formulations for
each method. For this reason, Curtis derived equations that are very similar from a common base
equat
ion.


Curtis'~(1990) equation for base
-
age 100 has a form:

\
begin{equation}

Y(t, S_{100}) =
\
exp
\
left( {
\
ln S_{100} +
\
alpha (
\
ln t
-

\
ln 100) +
\
beta (
\
ln t
-

\
ln 100) ^ 2
\
over

1.0 +
\
gamma (
\
ln t
-

\
ln 100 ) +
\
delta (
\
ln t
-

\
ln 100) ^ 2}
\
right)


\
label{eq:curtisH100}


\
end{equation}

with the inverse site index prediction equation:


\
begin{equation}

S_{100}(t, Y) =
\
exp
\
left( {
-

\
alpha (
\
ln t
-

\
ln 100)
-

\
beta (
\
ln t
-

\
ln 100) ^ 2


+
\
ln Y ( 1.0 +
\
gamma (
\
ln t
-

\
ln 100) +
\
delta (
\
ln t
-

\
ln
100) ^ 2 )}
\
right)


\
label{eq:curtisS100}


\
end{equation}

and the equation for base
-
age 50 has the form:


\
begin{equation}

Y(t, S_{50}) =
\
exp
\
left( {
\
ln S_{50} +
\
alpha (
\
ln t
-

\
ln 50) +
\
beta (
\
ln t
-

\
ln 50) ^ 2
\
over

1.0 +
\
gamma (
\
ln t
-

\
ln 50 )
+
\
delta (
\
ln t
-

\
ln 50) ^ 2}
\
right)


\
label{eq:curtisH50}


\
end{equation}

with the inverse site index prediction equation:


\
begin{equation}

S_{50}(t, Y) =
\
exp
\
left( {
-

\
alpha (
\
ln t
-

\
ln 50)
-

\
beta (
\
ln t
-

\
ln 50) ^ 2


+
\
ln Y ( 1.0 +
\
gamma (
\
ln

t
-

\
ln 50) +
\
delta (
\
ln t
-

\
ln 50) ^ 2 )}
\
right)


\
label{eq:curtisS50}


\
end{equation}


\
noindent



Based on his analysis with the above equations Curtis concluded, among other things, that:
\
newline



\
begin{minipage}{6truein}


%
\
baselineskip=0.4true
cm

``
\
ldots Reversibility, relative insensitivity to choice of reference age, and the uncertainties
associated with inconsistency in errors in the predictor variables used in derivation versus those
used in practical applications of conventional regressio
ns all suggest that the structural
relationship is a reasonable compromise and a plausible alternative to the more commonly used
regression procedures. The structural equation has great practical advantage of providing a single

equation for both site inde
x and height growth estimates.''


\
newline


\
end{minipage}


Two notable themes of Curtis' work, which can be markedly enhanced by an application of the
Generalized Algebraic Difference Approach, are:



\
begin{description}



\
item[i)] comparison of base
-
age

dependent fitting methodologies using different base
-
ages; and



\
item[ii)] development of models using a ``single'' equation for height and site index predictions.



\
end{description}




Some observations regarding the themes in Curtis' work and approa
ches to their investigation are in
order. First, equations~(
\
ref{eq:curtisH100}) and~(
\
ref{eq:curtisH50}) are similar but not the same.
They will not generate identical curves even if fitted to data without error. Since they have four
nonlinear parameters,

they are flexible enough to appear on a graph as visually similar even though
algebraically and numerically each produces a different set of co
-
ordinates. Ironically, only
dynamic equations can provide principally pure grounds with a single common equatio
n for directly
testing base
-
age specific fitting techniques using various base
-
ages.


Second, only dynamic equations can actually provide one single equation for estimating both height
and site index if such a principle is accepted within an adopted stati
stical framework. Although
equations~(
\
ref{eq:curtisH100}) and~(
\
ref{eq:curtisS100}) are derived from each other and are
merely inverse functions of each other, they are, in fact, two separate equations. The same applies to
the equations~(
\
ref{eq:curtisH50
}) and~(
\
ref{eq:curtisS50}). The derivation of dynamic equations
allows one to specifically address these principle points and thus, enhance any study similar to
Curtis'.


To analyze any or all of the four equations~(
\
ref{eq:curtisH100}) to~(
\
ref{eq:curti
sS50}) with
different base
-
ages one can apply the Generalized Algebraic Difference Approach to derive a
dynamic generalization of these equations. First, we define two variables, the site variable $ {
\
cal
X} $ to take the place of $
\
ln S $ and the measure
ment base
-
age $ {
\
cal Z} $ to take the place of
the constants 100 and 50. Inclusion of these and simplification produces the following
generalized explicit equation that can be base
-
age specific:


\
begin{equation}


\
ln Y(t, {
\
cal X}, {
\
cal Z}) = {{
\
cal X}
+
\
left (
\
alpha +
\
beta
\
,
\
ln t
\
right )
\
ln ({{t}/{
\
cal Z}})


\
over 1 +
\
left (
\
gamma +
\
delta
\
,
\
ln t
\
right )
\
ln ({{t}/{
\
cal Z}})}


\
label{eq:curtisgenH}


\
end{equation}

where:

$ Y $ is any applicable variable of interest such as height or volume;

$ {
\
cal X} $ is the GADA ``universal'' unobservable site variable;

$ {
\
cal Z} $ is a constant or a parameter equal to

100 for eq.~(
\
ref{eq:curtisH100}) and 50 for eq.~(
\
ref{eq:curtisH50}). The parameters are unique
to this equation and the equation is pr
esented as the logarithmic transformation for the sake of
simplicity of presentation.


Just as its special cases (i.e., eqs.~(
\
ref{eq:curtisH100}) and~(
\
ref{eq:curtisH50})),
eq.~(
\
ref{eq:curtisgenH}) is base
-
age specific and cannot be directly analyzed wi
th different
base
-
ages. Following the GADA, the initial condition solution for $ {
\
cal X} $ in
eq.~(
\
ref{eq:curtisgenH}) is:


\
begin{equation}

{
\
cal X} =
\
ln ({{t_0}/{
\
cal Z}})
\
left (
\
ln ({Y_0} ^ {


\
gamma}t_0 ^ {
\
delta
\
ln Y_0
-

\
beta})
-

\
alpha
\
right ) +

\
ln Y_0


\
label{eq:curtisgenS}


\
end{equation}

and substitution into eq.~(
\
ref{eq:curtisgenH}) with some simplifications leads to the following
dynamic generalization of the four equations~(
\
ref{eq:curtisH100}) to~(
\
ref{eq:curtisS50}):


\
[


\
ln Y(t, t_0,
Y_0, {
\
cal Z}) = {
\
ln ( Y_0
\
,
\
left ( t_0 / {
\
cal Z}
\
right ) ^ {
\
ln Y_0
\
left

(
\
gamma +
\
delta
\
,
\
ln t_0
\
right )
-

\
beta
\
,
\
ln t_0
-

\
alpha}
\
left ( t/{
\
cal Z}
\
right ) ^ {
\
alpha +
\
beta
\
,
\
ln t

})


\
over

1 + (
\
gamma +
\
delta
\
,
\
ln t )
\
ln ( t/{
\
ca
l Z} )

}


\
]

or


\
begin{equation}

Y(t, t_0, Y_0, {
\
cal Z}) =
\
exp
\
left({
\
ln ( Y_0
\
,
\
left ( t_0 / {
\
cal Z}
\
right ) ^ {
\
ln Y_0
\
left

(
\
gamma +
\
delta
\
,
\
ln t_0
\
right )
-

\
beta
\
,
\
ln t_0
-

\
alpha}
\
left ( t/{
\
cal Z}
\
right ) ^ {
\
alpha +
\
beta
\
,
\
ln t

})


\
over

1 + (
\
gamma +
\
delta
\
,
\
ln t )
\
ln ( t/{
\
cal Z} )

}
\
right)


\
label{eq:curtisbai}


\
end{equation}


\
noindent



The dynamic equation~(
\
ref{eq:curtisbai}) includes an infinite number of different equations with
the constant (or parameter) $ {
\
cal

Z} $ equal to any arbitrary or estimated real number. Four
special cases of this equation are the four equations~(
\
ref{eq:curtisH100}), (
\
ref{eq:curtisS100}),
(
\
ref{eq:curtisH50}) and~(
\
ref{eq:curtisS50}).



Depending upon imposed constraints as shown be
low, eq.~(
\
ref{eq:curtisbai}) simplifies to one
of the following four special cases:



\
begin{description}



\
item[1)]

When $ t_0={
\
cal Z}=100 $ then $ Y_0 $ is equivalent to $ S_{100} $, $ Y $ is equivalent to
predicted height, eq.~(
\
ref{eq:curtisbai}) i
s equivalent to eq.~(
\
ref{eq:curtisH100}), and it
becomes:


\
[

Y(t, S_{100}) =
\
exp
\
left({
\
ln (S_{100}
\
left (.01 t
\
right ) ^ {
\
alpha +
\
beta
\
,
\
ln t })


\
over

1 + (
\
gamma +
\
delta
\
,
\
ln t )
\
ln (.01 t )

}
\
right)


\
]



\
item[2)] When $ t={
\
cal Z}=100

$ then $ Y $ is equivalent to $ S_{100} $, $ Y_0 $ is
equivalent to direct height measurements, eq.~(
\
ref{eq:curtisbai}) is equivalent to
eq.~(
\
ref{eq:curtisS100}), and it simplifies to:


\
[

S_{100}(t_0, Y_0) =
\
exp
\
left({
\
ln ( Y_0
\
,
\
left (.01 t_0
\
ri
ght ) ^ {
\
ln Y_0
\
left

(
\
gamma +
\
delta
\
,
\
ln t_0
\
right )
-

\
beta
\
,
\
ln t_0
-

\
alpha})

}
\
right)


\
]



\
item[3)]

When $ t_0={
\
cal Z}=50 $ then $ Y_0 $ is equivalent to $ S_{50} $, $ Y $ is equivalent to
predicted height, eq.~(
\
ref{eq:curtisbai}) is equ
ivalent to eq.~(
\
ref{eq:curtisH50}), and it
becomes:


\
[

Y(t, S_{50}) =
\
exp
\
left({
\
ln (S_{50}
\
left (.02 t
\
right ) ^ {
\
alpha +
\
beta
\
,
\
ln t })


\
over

1 + (
\
gamma +
\
delta
\
,
\
ln t )
\
ln (.02 t )

}
\
right)


\
]



\
item[4)] When $ t={
\
cal Z}=50 $ then $

Y $ is equivalent to $ S_{50} $, $ Y_0 $ is equivalent
to direct height measurements, eq.~(
\
ref{eq:curtisbai}) is equivalent to eq.~(
\
ref{eq:curtisS50}),
and it simplifies to:


\
[

S_{50}(t_0, Y_0) =
\
exp
\
left({
\
ln ( Y_0
\
,
\
left (.02 t_0
\
right ) ^ {
\
ln

Y_0
\
left

(
\
gamma +
\
delta
\
,
\
ln t_0
\
right )
-

\
beta
\
,
\
ln t_0
-

\
alpha})

}
\
right)


\
]



\
end{description}


Equation~(
\
ref{eq:curtisbai}) is base
-
age invariant with respect to its initial conditions, i.e., the
internal base
-
age of the equation, $ t_0
$ and $ Y_0 $. However, as with any other equation, the
values calculated with the given formula are subject to the values of the parameters and constants
of the formula. The value of $ {
\
cal Z} $ addresses differences between
eq.~(
\
ref{eq:curtisH100}) and

eq.~(
\
ref{eq:curtisH50}) as well as many other similar equations.
However, the fact that eq.~(
\
ref{eq:curtisbai}) is base
-
age invariant allows each of these
equations to be fitted with a base
-
age specific regression methodology using any arbitrary
base
-
ag
e selection.


The equivalent of eq.~(
\
ref{eq:curtisH100}) can be fitted using eq.~(
\
ref{eq:curtisbai}) as a
base
-
age specific site index equation. For a base
-
age of 100 years, $ t_0={
\
cal Z}=100 $ and $
Y_0 $ is assigned the values of $ S_{100} $ during t
he fitting process. At the same time, the
equivalent of eq.~(
\
ref{eq:curtisH100}) can be fitted as a base
-
age specific site index equation
using base
-
age 50 years if $ {
\
cal Z}=100 $, $ t_0=50 $ and $ Y_0 $ is assigned the values of $
S_{50} $ during the f
itting process, i.e.,


\
[

Y(t, S_{50}) =
\
exp
\
left({
\
ln ( Y_0
\
, 2 ^ {
\
alpha +
\
beta
\
,
\
ln t_0
-

\
ln Y_0
\
left (
\
gamma +
\
delta
\
,
\
ln

t_0
\
right ) }
\
left (.01 t
\
right ) ^ {
\
alpha +
\
beta
\
,
\
ln t })


\
over

1 + (
\
gamma +
\
delta
\
,
\
ln t )
\
ln (.01 t
)

}
\
right)


\
]

Furthermore, both of these equations, either with base
-
age 100 or with base
-
age 50 years, could in
the application phase be used directly
---
without reformulation
---
to calculate site indexes at either
of the two base
-
ages or any other base
-
a
ge. For example, the model could be base
-
age 50 specific
($ {
\
cal Z} = 50 $) but be used with base
-
age 100 site indexes ($ t_0 = 100 $) for height
predictions:


\
[

Y(t) =
\
exp
\
left({
\
ln ( Y_0
\
, 2 ^ {
\
ln Y_0
\
left (
\
gamma +
\
delta
\
,
\
ln t_0
\
right )
-

\
be
ta
\
,
\
ln t_0
-



\
alpha}
\
left (.02 t
\
right ) ^ {
\
alpha +
\
beta
\
,
\
ln t })


\
over

1 + (
\
gamma +
\
delta
\
,
\
ln t )
\
ln (.02 t )

}
\
right)


\
]

Similarly, the equivalent of eq.~(
\
ref{eq:curtisH50}) can be fitted using eq.~(
\
ref{eq:curtisbai})
as a base
-
age

specific site index model using base
-
age 100 years if $ {
\
cal Z}=50 $, $ t_0=100 $
and $ Y_0 $ is assigned the values of $ S_{100} $ during a fitting process. The equivalent of
eq.~(
\
ref{eq:curtisH50}) can be fitted as a base
-
age specific site index model

using base
-
age 50
years if $ {
\
cal Z}=50 $, $ t_0=50 $ and $ Y_0 $ is assigned the values of $ S_{50} $ during the
fitting process. Furthermore, both of these models, based on either base
-
age 100 or base
-
age 50
years, could be used directly without reform
ulation to calculate site indexes at either of the two
base
-
ages or any other base
-
age from any height ($ Y_0 $) and age ($ t_0 $) measurements. For
example, the following equation is equivalent to eq.~(
\
ref{eq:curtisS100}) ($ t = 100 $) but is
base
-
age 50

specific ($ {
\
cal Z} = 50 $):


\
[

S_{100}(t_0, Y_0) =
\
exp
\
left({
\
ln ( Y_0
\
,
\
left (.02 t_0
\
right ) ^ {
\
ln Y_0
\
left

(
\
gamma +
\
delta
\
,
\
ln t_0
\
right )
-

\
beta
\
,
\
ln t_0
-

\
alpha} 2 ^ {
\
alpha +
\
beta
\
,
\
ln t })


\
over

1 + (
\
gamma +
\
delta
\
,
\
ln t
)
\
ln 2

}
\
right)


\
]



Equation~(
\
ref{eq:curtisbai}) can be used for analysis of many other fitting techniques, such as
those described in Borders {
\
it et al.}~(1988) and Furnival {
\
it et al.}~(1990), i.e., all possible
combinations of data measurements,
non
-
overlapping growth intervals, etc. The differences in
curves from different methodologies as discovered by others support Curtis' conclusion that all
base
-
age specific methodologies as tested on his data produce different curves. This holds also for
th
e methods described by Borders {
\
it et al.}~(1988) and Furnival {
\
it et al.}~(1990) and some
other methodologies tested on the same data
\
footnote{Personal communication: First author's
correspondence with Dr.
\

R.O. Curtis and Dr.~B.E. Borders, July 13, 199
0.

}.


Finally, since equations~(
\
ref{eq:curtisH100}) and~(
\
ref{eq:curtisH50}) are, in fact, two
different equations having different properties and varying by the arbitrary constants, 100 vs.
\

50,
one may well ask the question:


\
begin{quote}


What value

of the constant, 100, 50 or some other, results in the best curves given an arbitrary
base
-
age specific fitting using, say, the base
-
age 75 years or some other base
-
age?


\
end{quote}


Such a question cannot be answered with either eq.~(
\
ref{eq:curtisH100
}) or
eq.~(
\
ref{eq:curtisH50}). There is no a direct way to fit eq.~(
\
ref{eq:curtisH100}) or
eq.~(
\
ref{eq:curtisH50}) with base
-
age 75 years, or other base
-
ages. Nor are these equations
conditioned to predict heights equal to site indexes at base
-
age 75 ye
ars, or other base
-
ages.



The derivation of eq.~(
\
ref{eq:curtisbai}) allows one to answer these and other similar questions.
Equations~(
\
ref{eq:curtisH100}) and (
\
ref{eq:curtisH50}) can be fitted and compared directly
with each other using base
-
age 75 ye
ars ($ t_0=75 $; $ Y_0
\
equiv S_{75} $; and $ {
\
cal Z}=100
$ vs.~$ {
\
cal Z}=50 $) or any other base
-
age. Moreover, an infinite number of equations similar
to eq.~(
\
ref{eq:curtisH100}) and eq.~(
\
ref{eq:curtisH50}) with various (different than 100 or 50)
con
stants can be analyzed simultaneously in one regression run using eq.~(
\
ref{eq:curtisbai}) by
simply defining $ {
\
cal Z} $ as an estimable regression parameter. Such a parameter ($ {
\
cal Z}
$) can be estimated by any base
-
age specific regression regardless

of the value of the regression
base
-
age ($ t_0 $). Furthermore, the predicted values from any model based on this generalized
equation~(
\
ref{eq:curtisbai}) will always give height equal to site index at any base
-
age $ t_0 $
and for any value of $ {
\
cal Z}

$.




\
section*

{Discussion
}

The focus of this manuscript is on a methodology for algebraic derivation of dynamic equations
that are suitable for modeling pooled cross
-
sectional and longitudinal data and that more flexible
than other methods given in the

forestry literature on dynamic equations. The equations derived
can be fitted to data with any technique suitable for dynamic or fixed
-
base
-
age equations.
Furthermore, these equations can be used, if desired, in ways consistent with the other more
tradit
ional fixed
-
base
-
age equations. We recommend the methodology as a tool and not as an
ideology. It does not pre
-
empt any statistical assumptions on error structures or criteria of fitting.
We do not claim that all site models must be based on dynamic equati
ons. Yet, we have provided
evidence (e.g., compare eq.~(
\
ref{eq:stepwise1bai}) vs.
\

eq.~(
\
ref{eq:stepwise1H}) and
(
\
ref{eq:stepwise1S}) ) that given certain curve shapes, the dynamic equations are superior to the
fixed
-
base
-
age equations and other explicit

equations. They are generally more parsimonious and
flexible. They will predict appropriate heights when age equals base
-
age and will be easier to fit
with scant data or data from young trees.


We include examples of dynamic equations used for fitting bas
e
-
age
-
independent and
base
-
age
-
specific parameters. The purpose of these examples is to demonstrate the advantages of
the Generalized Algebraic Difference Approach to equation derivation over more traditional
approaches. These advantages arise from greate
r flexibility in the model analysis and its
applications.


{Acknowledgements

}

We dedicate this work to Dr.
\

J.L. Clutter who died much too early. Nonetheless, he inspired and
advocated the use of implicitly defined dynamic equations in modeling of all sit
e dependent forest
characteristics.


We are grateful to Drs.
\

I.E. Bella, J. Beck, H.E. Burkhart, R.O. Curtis, V. Lieffers, S. Northway,
R.A. Monserud, S. Titus, C. Tomas, and
L.V. Pienaar
, for their reviews of other unpublished
manuscripts describing the
generalized algebraic difference methodology and to Dr.
\

D. Tait for
his feedback relating to the concept of unobservable variables


We extend special thanks to Dr.
\

R.O. Curtis. Bob reviewed one early manuscript several times,
entertained extensive discu
ssions on related issues, and shared the noble fir data and unpublished
works. With his permission, some outcomes from this collaboration are included here. It was
Bob's idea, and a very helpful one, to use just one specific base model (Schumacher 1939) to

illustrate the various derivations.


All help has been greatly appreciated but any shortcomings are our own doing.



%
\
clearpage

\
section*

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%
\
baselineskip=0.8truecm



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end{description}


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end{document}