Generalized Algebraic Difference Approach Derivation of Dynamic Site Equations with Polymorphism and...

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SH-1-10-08

Generalized Algebraic Difference Approach Derivation of Dynamic Site Equations with Polymorphism and Variable Asymptotes fromExponential and Logarithmic Functions

Cieszewski

3298798

Generalized Algebraic Difference Approach Derivation of Dynamic SiteEquations with Polymorphism and Variable Asymptotes fromExponential and Logarithmic Functions

Chris J. Cieszewski and Mike Strub

Abstract: The generalized algebraic difference approach (GADA) uses both: two-dimensional functions ofexplicit time and two-dimensional functions of explicit site to derive a single dynamic equation that is athree-dimensional function of explicit time and implicit site. In 2004 Cieszewski advanced dynamic siteequations with polymorphism, and variable asymptotes (Cieszewski, C.J. 2004. GADA derivation of dynamic siteequations with polymorphism and variable asymptotes from Richards, Weibull, and other exponential functions.PMRC Tech. Rep. 2004-5, p. 16.) are built with GADA using one general function of time T and two generalfunctions of site X. The function of time is the exponential function Y � MTb, and the functions of site are thequadratic function Z � b1 � b2X � b3X2 and the inverse half-saturation function Z � b1 � (b2/X � b3). Wediscuss the new dynamic equations based on five exponential substitutions (Richards, Gompertz, Korf, logistic,and log-logistic) and one new logarithmic model substitution in the exponential function of time and 10 specialcases of the two functions of site corresponding to their different parameter values. The new dynamic equationspresented offer breakthrough flexibility in modeling of the self-referencing dynamics with the exponential andlogarithmic functions considered in this article. FOR. SCI. ●●(●):000–000.

Keywords: base-age invariant, dynamic equations, site models, site index, polymorphism, variable asymptotes

Background and Objectives

Height Predictions

THE AVERAGE HEIGHT OF DOMINANT AND CODOMI-

NANT TREES or top height at a given age is a critical

component of growth and yield models for most

even-age plantations. Height growth is little affected by the

stand densities that are normally encountered in managed

stands. The site quality estimation procedures based on

stand height data are the most commonly used techniques

for evaluating site productivity (Clutter et al., 1983). Most

of these height-based techniques for evaluation of site qual-

ity rely on the development of site quality dependent height-

age models, hereafter called site models, which define an

infinite number of site index curves for different arbitrary

site qualities by modeling three-dimensional site-height-age

continuous surfaces. Each site index curve defines an ex-

pected height-age relationship referenced by the expected

height at a specified base age. Because the site models are

somewhat controversial, their background and development

is described below in more detail.

Single Curve Modeling

Height development on any individual site can be de-

scribed by a number of relatively simple functions of two

variables [i.e., Y � f(t)], which are hereafter called base

functions. The equations may be linear or nonlinear, but the

latter are far more relevant and desirable for height growth

modeling; linear functions are practically nonexistent in

height growth modeling because they are not parsimonious

and have poor extrapolation properties. The literature con-

tains a diverse suite of nonlinear base functions to choose

from. Most of the available functions are named after their

proponents. Frequently the sole difference between various

base functions is a single parameter. The examples include

the following:

➤ the Korf function (i.e., Y � Aeb/te

), which is the ex-

panded by one parameter Schumacher function (i.e.,

Y � Aeb/t) (see Korf and Prispevek 1939 versus Schu-

macher 1939);

➤ the Yang function [i.e., Y � A(1 � e�b � te

)], which is the

multiplication of the Weibull distribution [i.e., Y � (1 �

eb � te

)] by a constant (see Yang et al. 1978 versus

Weibull 1939);

➤ the Bailey function [i.e., Y � (1 � eb � te

)d], which is the

Yang function expanded by one parameter (see Bailey

1980 versus Yang et al. 1978),

➤ the Richards [i.e., Y � (1 � eb � t)d]) function, which is

the Mitscherlich monomolecular function expanded by

one parameter [i.e., Y � (1 � e)b � t] (see Richards 1959

versus Mitscherlich 1930); and

➤ the Hossfeld (1822) function [Y � A/(1 � B/tj)], which

is the expanded by one parameter half-saturation func-

tion [i.e., Y � A/(1 � B/t)] that can be found, for

example, in earlier works of Galileo two centuries ago

and that according to some mathematicians has an un-

traceable origin going way back to the roots of

mathematics.

Not only are there many nonlinear base functions avail-

able for modeling single curves but also it usually makes

little difference what model is used for single line trends

Chris J. Cieszewski, University of Georgia, Warnell School of Forestry and Natural Resources, Athens, GA—Phone: (706) 542-8169; Fax: (706) 542-8356;[email protected]. Mike Strub, Weyerhaeuser Company.

Manuscript received October 19, 2007, accepted November 9, 2007 Copyright © 2008 by the Society of American Foresters

rich5/for-fs/for-fs/for00308/for2291d08a mcquinne S�16 4/22/08 12:57 Art: FS-07-133 Input-md

Forest Science ●●(●) 2008 1

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because many of the nonlinear models are so flexible that

they greatly overlap with each other in their capabilities of

defining various curve shapes. For this reason modeling

single curves is a relatively easy task. Yet, although, the

base functions are fundamental constructs of mathematics

that have been in use since the beginning of quantitative

science and are extremely well-known and developed, there

still are occasional publications announcing new forms of

the base functions.

Site-Height-Age Modeling

The site models have been in common use for several

decades. In general, they are not as well known than the

base functions. The main ambiguity about them is that they

cannot be operationally defined by usual algebraic con-

structs of explicit functions [e.g., Y � f(t, X)] because the

site quality (e.g., X) is unobservable in the sense of Judge et

al. (1985). Instead, they are defined by implicit functions

[i.e., Y � f(t, y)] of themselves (y, which is substituting for

X, is a subset of Y and as such it can be observed, but the

function has the same variable on both sides of the equa-

tion). Because of their self-referencing (Northway 1985)

nature, they are much more complicated than the base

functions discussed earlier; to be defined and usable they

have to contain a known value of the function at a known

reference point. In essence, they do not model any specific

values, but rather they model the relative dynamics of

changes for existing arbitrary observations. Similar to the

base functions, the site models are also usually named after

their proponents, with some examples being the following:

➤ the Ek-Payandeh model [i.e., Y � M1 � SM2(1 �e�a � t)b1 � Sb2]

(Ek 1971, Payandeh 1974);

➤ the Biging (1984) model [i.e., Y � M1 � SM2(1 �e�a � t)b],

➤ the Monserud (1984) log-logistic model [i.e., Y � M1 �

SM2(1 �ea1�a2 �t�a

3 � ln S)]; and

➤ the McDill and Amateis (1992) model [i.e., the alge-

braic difference approach polymorphic form of Y � a/(1

� S/tc)].

➤ The two most common approaches to development of

site models are based on fixed- versus variable base-age

reference points and static versus dynamic site equa-

tions. The static site equations are Y � f(t, S), where S is

a subset of Y at a fixed reference point, such as site

index base age (e.g., 50 years). The dynamic site equa-

tions are of the form: Y � f(t, t0, y0), where y0 is a subset

of Y at any variable t � t0. The dynamic site equations

evolved in the course of history much more slowly than

the static site equations (Figure 1) that are much easier

to manipulate. Development of static site equations is

frequently simplified using the assumption that a height

measurement at a base age is independent from heights

at other ages. Accordingly, the advocates of the simpler

static site equations attempted to model more compli-

cated patterns much earlier than the advocates of the

dynamic equations (Figure 1) who were facing much

more difficult algebraic challenges.

➤ A strong need for flexible site equations with direct

input of heights and ages triggered yet another approach

to derivation of base-age variant dynamic site equations

with use of the methods for formulating static site equa-

tions without rigorous mathematical derivations similar

to those discussed in this article. The base-age variant

approach is based on arbitrary assignments of various

implicit variables, such as y0 and t0, to different parts of

dynamic site equations. The result of this approach was

a number of internally inconsistent ill-conditioned [1]

approximations with the appearance of an equation

(placed “�” sign in between the dependent variable and

the right-hand side of the model). An early example of

such a model is described in Goelz and Burk (1992).

Assuming that the Goelz and Burk (1992) model is an

equation we can obtain from it the following example of

an algebraic contradiction. The model computes height

of 24.2 m at age 150 using for site definition, say, height

13.0 at base age 20, but it computes 7.2 m at age 20

using for site definition the very height 24.2 m at base

age 150, which implies that 13.0 � 7.2 or, after sub-

tracting 7.2 from both sites and then dividing both sites

by 5.8, it implies that 1 � 0. See Bailey and Cieszewski

(2000) for other examples of the models in this category

of the base-age variant models.

Historical Evolution of Site Models

The basic goal in research of new site models has always

been the same as the basic goal in searching for new base

functions. This goal is to achieve a parsimonious model

with the ultimate flexibility in describing trends defined by

various data. The flexibility of a base function is measurable

by its capabilities of defining different curves and its adapt-

ability to various algebraic operations. The flexibility of a

site model is measured by the same criteria plus the degree

of broadness in diversity of curve shapes—in the generated

families of curves—and by the model’s solvability for the

unobservable site variable, which needs to be substituted by

direct height-age measurements. Thus, the ultimate three-

fold goal in development of site models is to assure that the

generated site curves are flexible along the time axis (i.e.,

they are based on flexible base functions of time), the

generated families of site curves are flexible along the site

quality axis (i.e., they are capable of modeling strong

changes in growth patterns across different site qualities

generating curves with diversely changing polymorphic

shapes and magnitudes having multiple asymptotes), and

the derived site equations are able to maintain full algebraic

integrity while directly using height and age for the site

quality definition (base-age invariant dynamic equations).

Site curves are base-age invariant if they are unequivo-

cally unaffected by all choices of base ages. This means that

in a base-age invariant site equation any arbitrary age-height

pair on a curve must define a single curve. The base-age

invariant equations also have the path invariance property

(Clutter et al. 1983), which means that one-step predictions

or yearly or decadal iterations will all predict the same

values at a given final age. The evolution of static site

equations was relatively quick (Figure 1) at the price of

failing the last goal. Thanks to the simplicity of formulating


2 Forest Science ●●(●) 2008

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static site equations including the lack of algebraic chal-

lenges in formulating these equations they historically

evolved quickly (Figure 1) from simple anamorphic equa-

tions to advanced polymorphic equations with multiple as-

ymptotes (e.g., King 1966, Ek 1971, Payandeh 1974, Mon-

serud 1984, Newnham 1988) and were greatly exploited by

the exponential functions, which are less suitable for the

dynamic equation derivation. However, the attractive sim-

plicity of static site models also had its drawbacks, because

these models use a single base age for site variable defini-

tion, whereas the actual trees and stands for which the

models are applied are at various ages. Because site models

need to be used with site quality described by heights at

different ages, the static site models are not readily usable in

operations without some kind of site index solutions exter-

nal to the model, such as numerical searches or table look-

ups. Worse yet, the fixed-base-age site models may lead to

development of incompatible site index prediction models

(e.g., Curtis et al. 1974, Monserud 1984), which in turn

cause erroneous height predictions, such as shrinking

heights (Rose et al. 2003) or unreasonable growth

overestimates.

The properties of dynamic site equations are not plagued

by the above concerns. However, the development of these

equations, with polymorphism and variable asymptotes, is

much more difficult and was considered impossible for

many years (e.g., Bailey and Clutter 1974, Clutter et al.

1983, Borders et al. 1984). The difficulty stems from the

fact that reformulating a base function to a simple polymor-

phic dynamic site equation with a single asymptote in-

creases the algebraic complexity drastically (compare Equa-

tion 1 versus Equations 7 and 8 or Equation 2 versus

Equations 9 and 10). Consequently, until very recently all of

the base-age invariant dynamic equations published in the

literature were either anamorphic or simple polymorphic

with a single asymptote (Figure 1).

Despite many earlier attempts to develop flexible site

models (Schumacher 1939, Bailey and Clutter 1974, Clutter

et al. 1983), including those that were splining segments of

multiple equations (Borders et al. 1984), the first well-be-

haved advanced polymorphic base-age invariant dynamic

site equation was published in 1989 (Cieszewski and Bella

1989), 50 years after the first development of dynamic site

Figure 1. Examples of historical evolution of the static site models (dashed frames and underlined citations), thebase-age invariant dynamic site equations (clean citations), and the base-age variant site models (solid frame citations).The site model flexibility is assigned an average index values of 1 for anamorphic models, 2 for simple polymorphicmodels, 3 for advanced polymorphic models, and 4 for exponential function-based advanced polymorphic models.



equations (i.e., Schumacher 1939), 22 years after well-es-

tablished advanced polymorphic static site equations (e.g.,

King 1966), and 15 years after a published claim that such

a derivation was impossible with the technology available at

the time (Bailey and Clutter 1974).

Current Development of Site Equations andObjectives of This Study

Since the first advanced polymorphic dynamic site equa-

tion (Cieszewski and Bailey 1989) was introduced, it has

been found useful in describing many data sets around the

world (e.g., Cieszewski and Bella 1991b, Elfving and

Kiviste 1997, Eriksson et al. 1997, Kiviste 1997, Trincado

et al. 2003, Johansson 1999, Palahi et al. 2004, Rivas et al.

2004, Krumland and Eng 2005). After the introduction of

GADA (Cieszewski and Bailey 2000) several more ad-

vanced polymorphic dynamic site equations emerged (Du-

plat and Tran-Ha 1997; Cieszewski 2000, 2001, 2002, 2003,

2004, 2005, Krumland and Eng 2005, Cieszewski et al.

2006, 2007) starting a new era of advanced dynamic site

equations. Although most of the newly developed advanced

dynamic site equations are based on fractional base func-

tions, the exponential base functions such as the Richards

function are considered by many biometricians to be the

paradigm of growth and yield modeling (Lundgren and

Dolid 1970, Ek 1971, Pienaar and Turnbull 1973, Payandeh

1974a, 1974b, Hegyi 1981, Garcia 1983, Biging 1985,

Newnham 1988). Because such models lead to particularly

difficult derivation of dynamic site models, the published

literature contains only simple dynamic equations based on

the single parameter response of these models to site vari-

ation (e.g., Bailey and Clutter 1974, Clutter et al. 1983,

Cieszewski and Bella 1991a, Amaro et al. 1998).

In this article we present derivations of new advanced

dynamic site equations with polymorphism and multiple

asymptotes, with multiple parameter responses to site vari-

ation, from the exponential class of models including the

Chapman-Richards, Weibull, Yang, Bailey, and other func-

tions using various assumptions about site quality impact on

curve shapes. The new derivations are a demonstration of

the GADA application to derivation of advanced dynamic

site equations from base functions that traditionally were

used as bases for advanced polymorphic static site equations

(e.g., Ek 1971, Payandeh 1974a, 1974b, Newnham 1988)

but only as simple polymorphic dynamic site equations

(e.g., Bailey 1980, Clutter et al. 1983, Amaro et al. 1998).

The new models are more advanced and more flexible than

any previously published dynamic site equations based on

the exponential base functions. They are advanced polymor-

phic dynamic site equations with multiple asymptotes, and

they accomplish what in the past was impossible using only

the earlier algebraic difference approach (ADA) (Bailey and

Clutter 1974).

Assumptions and Methods

Base Functions

Let us assume that the modeled phenomenon Y is a

function of T(Y � f(T)), such that the basic model can be

defined as

Y � M Tb, (1)

where M and b are the base-model estimable parameters or

their arbitrary transformations. Table 1 contains various

examples of different functions defined by model 1, depend-

ing on the assumed definitions of T. Following are examples

of derivations of various dynamic site equations, based on

the above base model 1 and the definitions of the generic

variable T in Table 1 defined by Equations 2–6, as well as

any other possible transformations of the variable T, which

can be easily added by the reader.

ADA Derivations

Because model 1 is defined in a flexible manner, we can

derive from it various dynamic equations with different

levels of complexity. Let us assume that Y0 defines the

reference point at time T0. The ADA-based dynamic equa-

tion based on solutions for M is then an anamorphic model

of the following form:

YM � Y0�T/T0�b. (7)

The ADA-based polymorphic dynamic equation (with a

single asymptote) based on the solutions for the parameter b

is

Yb � M T�ln�Y0/M�/lnT0�. (8)

An example of another polymorphic model with a single

asymptote, based on the parameter a1 varying with site

productivity is the ADA reparameterization of model 1,

which, for example, for the Bailey (1980) function and

Cieszewski and Bella (1991a) function (Equation 2) is

Ya � M�1 � ��Y0

M��1/b�

� 1��t0��a2�

ta2�b

, (9)

where, according to Equation 2, T � 1 � e��a1 ta2�.

An equivalent polymorphic dynamic equation (Equation 2)

with a single asymptote for the same base function based on

using the solution for parameter a2 has the form,

Ya � M�1 � exp�� a1

� �ln�-�Y0/M�1/b � 1�

ln�e�a1�ln� x�/ln�t0��b

. (10)

In the above examples, the ADA solutions are based on

assumptions of single parameter responses to the site pro-

ductivity variation. This approach results in either anamor-

phic models with variable asymptotes or polymorphic mod-

els with a single asymptote. To derive models with both

polymorphism and variable asymptotes, more than one pa-

rameter has to be a function of site productivity. The site

productivity responses in M and in b can be mixed, and in

the next section several derivations of models with both of

these parameters responding to site quality are presented.

Responses in parameter a of model 1 cannot be com-

bined with any other parameters. When the exponentiated

parameter a varies with site productivity, none of the other

two parameters M or b can be used (together with a) to



AQ: C

T1

AQ: D

model the site productivity responses because when M or b

is a function of X and the exponentiated a is also a function

of X, the model cannot be solved for X in a closed-form

solution. For this reason the only useful model with param-

eter a varying with site productivity is the ADA reparam-

eterization of model 1 similar to Equations 9 and 10 above

for the examples of a Chapman-Richards special case re-

sulting from combining Equation 1 and any of the special

cases of Equation 2.

GADA Derivations

To derive polymorphic models with variable asymptotes

it is necessary to apply the GADA presented by Cieszewski

(1994) and Cieszewski and Bailey (2000), which allows for

definition of site responses in multiple parameters. With use

of this approach the site responses in the Richards function

can be modeled using both parameters m and b. To facilitate

such derivation model 1 is reparameterized into a form more

suitable for manipulation of these two parameters (using em

instead of M and taking the log of the function) as follows:

Y � emTb, (11)

LY � m � b LT,

where LY is the natural logarithm of Y and LT is the natural

logarithm of T defined by any of the Equations 2–6.

If we assume that the site productivity can be defined by

an unobservable variable X, a few examples of assumptions

about the relationships between the parameters m and b and

the unobservable variable X are the parameters m and b are

linear functions of X, which means m can be proportional to

b or linearly inversely proportional to b; the parameter m is

a linear function of X, whereas the parameter b is an inverse

fractional function of X, which means that the parameters

are nonlinearly proportional to each other’s inverse, or the

parameters m and b are quadratic functions of X, which

means they can be in various flexible relationships to each

other.

Linear Assumptions

The simplest assumption is that both parameters are

linear functions of the unobservable site variable X:

m � m1 � m2 X, (13)

b � b1 � b2 X, (14)

which requires the solution for the unobservable variable X

in the equation,

LY � ml � m2 X � (b1 � b2 X)LT. (15)

The general solution for the above relationship is

X � ��LY � m1 � b1 LT

m2 � b2 LT. (16)

The self-referencing dynamic equation can be defined

using one known reference point, but to make the equation

parsimonious one of the parameters multiplying X should be

fixed. Because X is arbitrary, we can redefine it to make theTab

le1.

Dif

fere

nt

def

init

ion

sof

Tre

sult

ing

invar

ious

exponen

tial

funct

ions

def

ined

by

Equat

ion

1

Equat

ion

Nam

eof

model

/cit

atio

nD

efin

itio

nof

TR

esult

ing

subst

ituti

on

tom

odel

1

2(a

llfo

rb

�0)

Cie

szew

ski

and

Bel

la(1

991a)

;S

chnute

(1981):

a2

�1;

for

only

a1

�0,

also

:B

aile

y(1

980):

a2

Bail

ey

�1/a

2;

Ric

har

ds

(1959):

a2

�1;

Wei

bull

(1939):

b�

1,

M�

1;

Yan

get

al.

(1978):

b�

1

T�

�1�

e�a

1•t1

/a2

for

a1�

0

t1/a

2 for

a1

�0

Y�

�M�1

�e

�a1•t

1/a

2

�b

for

a1

�0

M1tb

/a2

ifa1

�0

3L

ogis

tic

(Rober

tson

1923)

T�

1

1�

e�

at

Y�

M�

1

1�

e��

at��b

4K

orf

and

Pri

spev

ak(1

939);

Sch

um

acher

(1939):

b�

1T

�e

�1

/tY

�M

(e1

/t)b

5G

om

per

tz(M

edaw

ar1940)

T�

e�

ea•t

Y�

M(e

�e

�a

t )b

6L

og-l

ogis

tic

(Monse

rud

1984)

T�

1

1�

e��

aln

�t��

Y�

M�

1

1�

e��

a1ln

�t��

b



model more parsimonious in one of the following ways,

according to which the Equations 1, 13, 14, and 16 offer the

following specific cases of dynamic equations.

CASE A (b1 � 0 and b2 � 1: the model must be

polymorphic):

Y � e�m1�m2 X0)TX0, (17)

where

X0 � �� LY0 � m1

m2 � LT0

. (18)

CASE B (m1 � 0 and m2 � 1: the asymptote must vary with

site productivity):

Y � eX0T�b1�b2 X0), (19)

where

X0 � �� LY0 � b1 LT0

1 � b2 LT0

. (20)

Inverse-Saturation Assumptions

The site parameters can be conditioned to be consistently

proportional to each other’s inverse over the site productiv-

ity dimension by the following definitions:

m � m1 � m2 Xi (21)

b � b1 �b2

X � b3, (22)

which define

LY � m1 � m2 X � �b1 �b2

X � b3�LT. (23)

There are two general solutions to Equation 23. The solution

more likely to be positive is

X � 0.5� � � 2 m2 b3 � ��2 �4 m2 LT b2

m2,

(24)

where

� � m1 � LY � m2 b3 � LT b1. (25)

Equation 23 defines a polymorphic model with variable

asymptotes. Similar to the previous example, because X is

arbitrary we can redefine this model to make it more par-

simonious, getting rid of redundant parameters. Two spe-

cific examples of parsimonious dynamic equations with

polymorphism and variable asymptotes implied by Equa-

tions 1, 21, 22, and 24 are presented below.

CASE A (b2 � 1 and b3 � 0: the model must be

polymorphic but may have a single asymptote if m2 � 0):

Y � e�m1�m2 X0)T�b1�1/X0), (26)

where the specific point solution using a known reference

point is

X0 � 0.5�� 2 � 4 LT0 m2

m2, (27)

and

� � LT0 b1 � LY0 � m1. (28)

Case B (m1 � 0 and m2 � 1: the model may be anamorphic

if b2 � 0):

Y � eX0T�b1�b2/X0�b3) (29)


point is

X0 � �b3 � 0.5� � 0.5��2 � 4 LT0b2, (30)

and where

� � LT0b1 � LY0 � b3. (31)

Quadratic Assumptions

Finally, we assume a generic case of site productivity

influence on the model parameters, in which the site param-

eters can be inconsistently proportional over time, or in-

versely proportional, to each other, and their relationship

may be changing over time more than once. Such a com-

plicated relationship can be modeled, for example, by de-

fining m and b as proportional to different quadratic func-

tions of the unobservable site variable X:

m � m1 � m2 X � m3 X2, (32)

b � b1 � b2 X � b3 X2, (33)

which used in Equation 12 implies the general relationship,

LY � m1 � m2 X � m3 X2

� (b1 � b2 X � b3 X2)LT . (34)

The general solution for X in Equation 34 has two cases of

which that more likely to be greater than zero is

X �� 2 � 2�b1 LT � m1 � LY�w

w, (35)

where

� � b2 LT � m2, (36)

and

w � 2 b3 LT � 2 m3. (37)

As in the earlier examples the self-referencing dynamic

equations can be defined using one known reference point,

but to assure that the final dynamic equation is parsimoni-

ous at least one of the parameters multiplying X should be

fixed. Six examples of specific variants of parsimonious

dynamic equations implied by the above Equations 1, 32,

33, and 35 are presented below.

CASE A (b � X: the model must be polymorphic without

the quadratic term or the intercept in b):

Y � e�m1�m2 X0�m3 X0

2

TX0, (38)



X0 �1

2

�� 2 � 4�m1 � LY0�m3

m3, (39)


point is

and

� � m2 � LT0. (40)

CASE B (b2 � 1: the model must be polymorphic but the

quadratic term in b may be eliminated):

Y � e�m1 � m2 X0 � m3 X0

2)T(b1�X0�b3 X

0

2), (41)


point is

X0 �� 2 � 2�b1 LT0 � m1 � LY0�w

w,

(42)

and where

� � m2 � LT0, (43)

and

w � 2 b3 LT0 � 2 m3. (44)

CASE C (b3 � 1: the model must be polymorphic with fixed

quadratic term in b):

Y � e�m1�m2 X0�m3 X0

2)T(b1�b2 X0�X

0

2), (45)


point is

X0 �� 2 � 2�b1 LT0 � m1 � LY0�w

w,

(46)

and where

� � LT0b2 � m2, (47)

and

w � 2 LT0 � 2 m3. (48)

CASE D (m � X: the model must have variable asymptotes

with quadratic site response function in b):

Y � eX0T�b1�b2 X0�b3 X02) (49)


point is

X0 �� 2 � 2�b1 LT0 � LY0�w

w, (50)

and where

� � LT0b2 � 1, (51)

and

w � 2 LT0b3. (52)

CASE E (m2 � 1: the model must have variable asymptotes

with potentially either linear or quadratic site response

function in m):

Y � e�m1 � X0 � m3 X0

2

T(b1�b2 X0�b3 X0

2), (53)


point is

X0 �� 2 � 2�b1 LT0 � m1 � LY0�w

w, (54)

and where

� � LT0b2 � 1, (55)

and

w � 2 LT0b3 � 2 m3. (56)

CASE F (m3 � 1: the model must have variable asymptotes

with quadratic site response functions in m and potentially

in b but not necessarily):

Y � em1�m2 X0�X0

2)T(b1�b2 X0�b3 X

0

2) (57)


point is

X0 �� 2 � 2�b1 LT0 � m1 � LY0�w

w,

(58)

and where

� � LT0b2 � m2, (59)

and

w � 2 LT0b3 � 2. (60)

Discussion

Site-dependent modeling of self-referencing relation-

ships is more than a century old. Traditionally, forest

growth and yield modelers have used fixed base-age site

equations requiring iterative numerical searches for opera-

tional use (e.g., Ek 1971, Payandeh 1974b, Monserud

1984). Others tried to overcome the limitations of the earlier

simple dynamic equations by joining (splining) predictions

from multiple models into a common prediction system

(e.g., Bruce et al. 1984). Others (see examples in Bailey and

Cieszewski 2000 and in Haight 2001) have attempted to

derive base-age invariant dynamic equations producing in-

stead base-age variant models, such as Goelz and Burk

(1992), that generate different curves depending on selec-

tion of different base ages, as we discussed earlier in the

Background and Objectives section.

Derivation of advanced dynamic equations with poly-

morphism and variable asymptotes from the Richards-like

exponential functions is nontrivial. In recent years the dy-

namic equations have been becoming more frequently used,

although only a few modelers have so far used advanced

dynamic site equations with both polymorphism and vari-

able asymptotes based on the exponential functions (i.e.,



AQ: E

Duplat and Tran-Ha 1997, Cieszewski 2004, Krumland and

Eng 2005). The main challenge in developing dynamic

equations from the exponential functions lies in finding

suitable combinations of exponential and logarithmic func-

tions that can be mixed with other multiparameter functions

while providing appropriate close form solutions for the site

variable and allowing for derivation of proper base-age

invariant, closed form dynamic equations. The number of

possible base-age invariant dynamic equations that can be

derived with GADA is tightly controlled by the availability

of close form analytical solutions for the cross-sectional

component of the three-dimensional site equations. No

proper dynamic equations can be derived without such

solutions, and, therefore, the number of possible derivations

in this method is rather limited. Most of the practically

viable GADA-based solutions that are applicable to the

Richards-like family of exponential functions are presented

here. Although the solutions offered here were not possible

before the introduction of GADA, future improved methods

of dynamic equation derivation may offer other ways of

deriving base-age invariant dynamic equations with many

more solutions.

The advanced dynamic equations presented here have

broad potential for applications with various types of

growth data. Because these equations have been published

previously only in an internal report (i.e., Cieszewski 2004),

they have been practically unknown and have found rela-

tively little use so far. Nevertheless, several scientists have

explored the use of these new models for different species in

the United States and Europe. For example, Barrio Anta et

al. (2006) have found the Cieszewski (2004) model based

on the hybrid of a linear inverse function mixed with the

Korf function, which is defined by Equations 4 and 29 with

b3 � 0 the best for development of a basal area growth

system for maritime pine in northwestern Spain, and Adame

et al. (2006) found the Cieszewski (2004) model based on a

hybrid of the linear function mixed with the Bailey (1980)

model, which is here defined by Equations 2 and 19, the

best for modeling dominant height growth and site index

curves for rebollo oak (Quercus pyrenaica Willd.) in north-

west Spain. Barrio Anta et al. (2006) have considered some

of the equations presented here for modeling loblolly pine

height growth but in the end concluded that the Cieszewski

(2003) model fitted their data best.

Model selection among the multitude of possible equa-

tions depends on the data analyzed and the modeler’s ex-

perience and preferences. Given a high level of nonlinearity

in the modeled trends, there is no simple procedure leading

to finding and identifying the most suitable model. Gener-

ally, there are two main traditional approaches to model

selection, which can be described as top down and bottom

up. The top-down approach is based on fitting the most

flexible model and narrowing down its parameter space to

identify the most parsimonious solution. The most suitable

approaches for this are Equations 41, 45, 53, and 57. This

approach is typically very difficult and unstable with many

problems associated with lack of convergences, local con-

vergences, instability of parameters and excessive comput-

ing time required for finding the best parameter estimates.

However, its advantage is that in the hands of a skilled

practitioner it allows exploration of the greatest range of

possibilities in a relatively short time and identification of a

simpler model without missing its true trends, which can be

easily hidden by data artifacts associated with drawbacks of

various simple models. Schnute (1981) gives a good over-

view example of this type of approach and the consider-

ations associated with it. Because of the large potential for

local convergences, this approach should be repeated mul-

tiple times by using various initial parameter values.

The bottom-up approach is generally simpler. It starts out

dealing with multiple models that are simpler and easier to

fit than those used in the top-down approach. However, the

chance of missing the best possible fit in analysis is higher

because it never fully explores the whole multidimensional

space of the general model functionality. This approach can

also be very time-consuming because it requires fitting a

multitude of models, which will tempt the practitioner to

follow different paths of exploration and pose various chal-

lenges in choosing the best model of those with equal fit but

distinctly different behavior. In the end there is no golden

rule making developing self-referencing models easy. The

bottom line is the experience of the researcher in under-

standing numerical limitations of computer searches and in

familiarity with the modeled phenomena and what are their

most distinguishing and essential characteristics.

The functions defined in Table 2 contain two main

classes of GADA models, of which one is based on the

half-saturation function and one on the quadratic function.

The 10 different formulas in this table correspond to differ-

ent variants of the same models, which will behave similarly

within well-defined ranges of their parameters, but which

will have different convergence behavior at the limits of

definability of their parameter. Selection of the most suit-

able model form depends on data and model fitting behav-

ior. For example, if the initial analysis were done with

model 41 and the parameters m3, b1, and b3 turned out to be

redundant, the final selection would be model 17, which

would be more parsimonious than the original model.

Sample Applications

The two most typical applications of the material pre-

sented here are the fitting of the models to other data or to

other models (data) and the use of the methodology pre-

sented for derivation of other models not presented here. We

give some examples of such applications below.

Example of Top-Down Approach to ModelFitting Using Monserud (1984) Data

The Richards (1959) function is considered the paradigm

of growth and yield modeling, and it has been used for many

different species and variables around the world. Yet, Mon-

serud (1984) found that this model was not able to describe

inland Douglas-fir height growth satisfactorily. Site models

can fail to describe site-dependent height growth well be-

cause of lack of description of the temporal changes in the

height growth as a function of time, lack of description of

the cross-sectional changes in height as a function of site, or

both of these.



AQ: F

T2

Because it is not clear which of the above was the reason

that Monserud (1984) found the Richards base function

unsuitable for his site model development, we choose the

temporal model definition equation (Equation 2) in Table 1

as one of the more general variants based on the Cieszewski

and Bella (1991a) function, which includes the Richards

(1959) function as a special case submodel. For the cross-

sectional definition of the model we choose Equation 49 in

Table 2 as one of the more general variants of the Cieszew-

ski (2004) models.

Next, we use the Monserud (1984) model to generate the

pseudo-data in Table 3 with the following equation and

parameters:

H ��S�

1 � exp�� ln�Age) � ln�S��, (61)

where � � 42.397, � � 0.3,488, � � 9.7,278, � � 1.2934,

and � 0.9779, H is height at age, and S is the site index

at base age 50 years.

In the course of model fitting it became evident that the

parameter b3 in Equation 49 was not contributing to the fit,

and, therefore, it was set to b3 � 0. As a result, the model

was collapsed to the form of Equation 19 (Table 2). We

made attempts to reduce the model further by trying to force

one of the parameters a1 or a2 to equal 1, but these attempts

appeared to have a relatively high cost in the sum or squared

residuals, which was the fitting criterion. On the other hand,

to improve the estimation properties for the model we had

reparameterized Equation 2 to have parameters a1 and a2 as

receptacles (i.e., T � 1 � e�ta2/a1

) of their values in Table 2.

The final model fit the Monserud (1984) model data well

Table 2. General GADA formulations of polymorphic dynamic equations with variable asymptotes based on the exponential class of base modelsincluding Richards, Weibull, Yang, and Bailey functions as special cases

Equation The main formulation The corresponding subequations

16 Y � e�m1 � m2 X0) TX0

X0 � �� LY0 � m1

m2 � LT0

18 Y � eX0T�b1 � b2 X0�

X0 � �� LY0 � b1 LT0

1 � b2 LT0

25 y � e�m1 � m2 X0�T�b1 � 1/X0�

X0 � 0.5� v � �v2 � 4 LT0m2

m2v � LT0b1 � LY0 � m1

28 Y � eX0T�b1 � b2/X0 � b3)X0 � � b3 � 0.5� � 0.5��2 � 4 LT0b2

v � LT0b1 � LY0 � b3

37 Y � e�m1 � m2 X0 � m3 X02�TX0

X0 �1

2

� v � �v2 � 4�m1 � LY0�m3

m3v � m2 � LT0

40 Y � e�m1 � m2 X0 � m3 X02)T(b1 � X0 � b3 X0

2)

X0 �� v � �v2 � 2�b1 LT0 � m1 � LY0�w

wy � m2 � LT0

w � 2b3 LT0 � 2 m3

44 Y � e�m1 � m2 X0 � m3 X02

)T(b1 � b2 X0 � X02)

X0 �� v2 � 2�b1 LT0 � � LY0�w

wv � LT0b2 � m2

w � 2 LT0 � 2 m3

48 Y � eX0T�b1 � b2 X0 � b3 X02

X0 �� v � �v2 � 2�b1 LT0 � LY0�w

wv � LT0b2 � 1

w � 2 LT0b3

52 Y � e�m1 � X0 � m3 X02)T(b1 � b2 X0 � b3 X0

2

X0 �� 2 � 2�b1 LT0 � m1 � LY0�w

wv � LT0b2 � 1

w � 2LT0b3 � 2 m3

56 Y � e�m1 � m2 X0 � X02�T�b1 � b2 X0 � b3 X0

2�

X0 �� v � �v2 � 2�b1 LT0 � m1 � LY0�w

w

v � LT0b2�m2

w � 2 LT0b3 � 2



AQ: G

T3

(Figure 2) having a root mean square error equal to 0.205

foot (0.138 m), which is much below the usual measurement

error and the regression prediction error. The final model

had the form,

H � e��V�W b2)/(1�W b1)](1 � e(a2/a1)(b2-b1V)/1�W b1)

, (62)

where

W � ln�1 � e�t0a2

/a1�� and V � ln� 1

H0�;

H is the prediction of height at age t; H0 is a known

reference height at age t0; a1, a2, b1, and b2 are the model

estimable parameters with the estimated values of 9.998,

0.6399, �1.519, and 10.216; and the code for the model is

H � exp{-(ln(1/H0)�ln(1-exp[-1/a1�t0�a2])�b2)/�1

� ln(1-exp(-1/a1�t0�a2))�b1)} � (1-exp(-1/a1�

t�a2))�[-(b1�ln(1/H0)-b2)/(1�ln(1-exp(-1/a1�t0�a2))�b1)].

The top-down approach to model fitting described above

allowed us to find a parsimonious and reasonably well-fit-

ting model for the data considered and to confirm that even

in the presence of an overly flexible cross-sectional defini-

tion of the site model, the Richards (1959) base function is

not flexible enough to model the age-dependent height

growth changes in inland Douglas-fir stands identically to

the Monserud (1984) model.

Example of Extending the Approach to DeriveOther Models Based on Logarithmic Functions

The methodology presented here can be applied to any

base equations that can have a similar structure identified as

defined by Equation 1. Consider, for example, the following

logarithmic equation, which is particularly suitable for mod-

eling early rapid height growth of intensively managed

loblolly pine plantations:

H � a ln�Ctb � d�, (63)

that can be rewritten as

H � a ln�ectb � d�. (64)

By setting Y � e(H/a) � d, we define an equation similar to

Equation 1, that is,

Y � ectb. (65)

After we assume that t � T the above model can be

applied directly with the solutions presented in Table 2 to

formulate appropriate dynamic equation of the form Y �

ef1(X)tf2(X), which subsequently can be used with the solu-

tions for X as a dynamic equation Y � fL(t, t0, y0). After that

point the function fL needs to be reformulated to its original

untransformed form:

H � a ln�ef1�X�tf2�X� � d�, (66)

with an appropriate solution (Table 2) for X:

X0 � f3�t0, Y0�, (67)

where Y0 � e(H0/a) � d which completes the derivation of

the dynamic equation from the logarithmic function.

Summary

This report provides a description of a relatively simple

exercise in algebra that solves problems that lingered, un-

solved despite various attempts for most of the 20th century

(more than 70 years). We present some traditional ADA-

based dynamic equations that are either anamorphic or

Table 3. Pseudo-data of height growth above breast height generated using the Monserud (1984) height model

Age

Site index base age 50

40 50 60 70 80 90

10 17.75 15.32 12.96 10.67 8.46 6.3515 28.28 24.56 20.90 17.30 13.81 10.4420 38.62 33.72 28.86 24.04 19.30 14.6925 48.46 42.55 36.62 30.69 24.80 19.0030 57.70 50.92 44.06 37.14 30.19 23.2935 66.29 58.78 51.12 43.32 35.43 27.5040 74.23 66.11 57.77 49.22 40.48 31.6245 81.55 72.93 64.02 54.81 45.33 35.6250 88.30 79.26 69.86 60.09 49.95 39.4855 94.51 85.13 75.33 65.07 54.36 43.2060 100.23 90.58 80.43 69.76 58.55 46.7865 105.50 95.62 85.20 74.18 62.53 50.2170 110.36 100.31 89.65 78.34 66.30 53.5075 114.85 104.66 93.81 82.25 69.89 56.6680 119.01 108.70 97.70 85.93 73.28 59.6785 122.86 112.47 101.35 89.40 76.50 62.5590 126.44 115.98 104.76 92.66 79.56 65.3195 129.76 119.25 107.95 95.74 82.46 67.95100 132.86 122.32 110.96 98.64 85.21 70.47105 135.74 125.18 113.78 101.38 87.82 72.88110 138.44 127.87 116.43 103.97 90.31 75.19115 140.96 130.39 118.93 106.42 92.67 77.39



F2

polymorphic with single asymptotes and many new GADA-

based dynamic equations with both polymorphism and vari-

able asymptotes. Equations 7 and 8 represent ADA general

formulations, each of which defines five different models

depending on which of the relationships in Table 1 is used

to define the variable T in the model considered. Table 2

contains two main classes of GADA-based general formu-

lations that are based on the half-saturation and quadratic

functions. Each class has several variants of the models

relating to the limiting parameter values of the models and

a model convergence toward other models. Each of the 10

model variants in Table 2 can be applied with any of the five

different functions T in Table 1 to define a dynamic equa-

tion with polymorphism and variable asymptotes. The mod-

els presented are suitable for site-dependent height growth

modeling as well as for any other growth and yield model-

ing involving the use of unobservable variables substituted

by the self-referencing type of model definition. The models

presented can be used as implicitly defined integral equa-

tions, such as site index models, or as difference equations

(Tait et al. 1988, Cieszewski and Bella 1993) and applied in

a state-space approach to growth modeling (Garcia 1994),

which iterate on an annual basis interacting with dynami-

cally changing environmental factors.

Unlike the mathematically malformed and ill-condi-

tioned, internally inconsistent base-age variant models in

the works refuted in Bailey and Cieszewski (2000), the

equations provided here were derived with a process of rigid

algebraic operations, are mathematically sound, and can be

used in any mathematically sound implementation (e.g.,

iterated on an annual or periodic basis, in single- or multi-

ple-step predictions, as well as in forward or backward

computations). If used as site models, the equations pro-

vided can be treated as both height and site index equations

that are compatible with each other. All models presented

Figure 2. Fit of Cieszewski (2004) cross-sectional model 18 with time series model 2a (Fit Ch-R) to the Monserud (1984)height model data (Mon_H.



here are base-age invariant, path-invariant, parsimonious,

internally consistent, and mathematically sound.

Endnote

[1] This terminology follows a time-honored tradition in mathematics,according to which situations that elude simple analysis are dismissedby pejorative such terms as “improper,” “inadmissible,” “degener-ate,” “irregular,” and so on (Simmons 1972).

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