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Forest Ecology and Management 243 (2007) 83–93

New dynamic site equation that fits best the Schwappach data for

Scots pine (Pinus sylvestris L.) in Central Europe

Chris J. Cieszewski a,*, Mike Strub b, Michał Zasada c

a Warnell School of Forestry and Natural Resources, University of Georgia, Athens, GA 30602, USAb Weyerhaeuser Company, Hot Springs, AR, USA

c Faculty of Forestry, Warsaw Agricultural University, Nowoursynowska 159, Building #34, 00-776 Warsaw, Poland

Received 31 March 2006; received in revised form 20 February 2007; accepted 21 February 2007

Abstract

Using historical growth series data of Scots pine (Pinus sylvestris L.) in Central Europe we examine all the dynamic site equations previously

used for modeling the height growth of this species as well as a new dynamic site equation that has not been used previously in the context of this

forestry data. The tested models included two groups of anamorphic and polymorphic dynamic site equations (three-dimensional site–height–age

models, such as Y = f(t,t0,y0)). One group of the models is based on the algebraic difference approach (ADA) implementation of different,

preexisting base equations (two-dimensional equations, such as Y = f(t)). The other group of models is based on newer generalized algebraic

difference approach (GADA) formulations of new site–height–age relationships that may use older models only as a part of their structure. The

models were selected because they were relevant to Scots pine height growth modeling in other studies. We compared all the models with each

other in terms of the sum of square deviations associated with fitting them simultaneously to all sites represented by the Scots pine data. All the fits

were based on base-age invariant stochastic regressions, in which the global model parameters that are common to all growth series are estimated

simultaneously with the site-specific effects that are different for each of the site productivity series. Cieszewski’s model [Cieszewski, C.J., 2005. A

new flexible GADA based dynamic site equation with polymorphism and variable asymptotes. PMRC Technical Report 2005-2] best described

the data.

# 2007 Elsevier B.V. All rights reserved.

Keywords: Yield tables; Site productivity; Site index model; Growth model; Dynamic equations; Initial condition models

1. Background

1.1. Species range and morphology

Scots pine (Pinus sylvestris L.), also called Scotch pine, is

the most widely distributed pine in the world. Its native range

includes mostly northern Europe and northern Asia (Fig. 1). It

spreads from Norway to southern Spain latitudinally, and from

northwest Spain to the Pacific coast longitudinally (Carlisle and

Brown, 1968). The species has been also introduced, and

sometimes naturalized, in many areas in the United States and

Canada (Burns and Honkala, 1990). Its wide distribution is

caused in large part by its ability to adapt to various climates,

* Corresponding author. Tel.: +1 706 542 8169; fax: +1 706 542 8356.

E-mail addresses: biomat@forestry.uga.edu (C.J. Cieszewski),

Michal.Zasada@wl.sggw.waw.pl (M. Zasada).

URL: http://www.growthandyield.com/chris/

0378-1127/$ – see front matter # 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.foreco.2007.02.025

soil and topographic conditions. It grows in areas with an

annual precipitation varying from 200 to over 1780 mm. It

survives in extremely high (as in the Mediterranean region), and

low temperatures (as in eastern Siberia at �65 8C). Its

altitudinal range is from 0 to about 2500 m above sea level.

However, the primary distribution of Scots pine indicates that it

is a tree of the continental climates, creating mostly pure, even-

aged stands in the boreal zone, sometimes mixed with Norway

spruce (Picea abies L.), common oak (Quercur robur L.), and

silver birch (Betula pendula Roth.). In Europe Scots pine plays

an important role in many countries where its areas exceed over

1 million ha. Scotch pine occupies about 43 million ha in

Russia, 12.9 million ha in Finland, 9.3 million ha in Sweden,

5.9 million ha in Poland, 4.9 million in Belarus, 3 million ha in

Germany, 2.2 million ha in Ukraine, 1.6 million in Norway,

1.1 million ha in France, about 1 million in Spain, and less than

1 million in each of the remaining European countries. In some

of these locations it makes up more than half of the total

forested area (CILP, 2000; Pisarenko et al., 2001; Podolyako

Fig. 1. Native range of Scots pine in Europe (based on Białobok, 1970; Carlisle

and Brown, 1968; Critchfield and Little, 1966; Staszkiewicz, 1970) and

approximate location of Schwappach sample plots.

C.J. Cieszewski et al. / Forest Ecology and Management 243 (2007) 83–9384

et al., 2001; Statistics Norway, 2003; Buksha and Pasternak,

2004; FCGB, 2004).

1.2. Site models

Historically, growth and yield of forest stands were

determined using tables containing averaged inventoried values

of height, diameter, number of trees, volume, etc., for different

site and age classes. Initially such relationships had been

developed as sets of curves drawn manually for given sites,

usually referred to as site classes, or site index classes, and

marked with symbols such as I, II, III or A, B, C. Examples of

such earlier models are yield tables developed at the beginning

of the 20th century by Schwappach (1908) and Graves (1910).

These tables generally consisted of averaged measurement data

divided into various classes. Subsequently, growth curves were

described using simple mathematical equations (e.g., two-

dimensional equations, such as Y = f(t)) created with basic

statistical analyses performed for the entire set of measure-

ments or individual site classes (Husch et al., 1972, p. 356;

Bruchwald, 1977; Jarosz and Kłapec, 2002). Currently, the

majority of the newly developed site models are based on more

complicated self-referencing (Northway, 1985) functions (i.e.,

Y = f(t,y), where y is a subset of Y) with parameters estimated by

the simultaneous fit to all growth curves, without isolating any

individual sites classes. These kinds of site models can be based

on static equations (three-dimensional site–height–age models,

such as Y = f(t,S)) with a fixed base age (e.g., Bruchwald, 1988;

Curtis et al., 1974; Monserud, 1984; Stage, 1963), or dynamic

equations (three-dimensional site–height–age models, such as

Y = f(t,t0,y0)) with a variable base age (e.g., Schumacher, 1939;

Bailey and Clutter, 1974; Borders et al., 1984; Cieszewski and

Bella, 1989; Cieszewski and Bailey, 2000). Given equal

flexibility and the ability to describe data, the dynamic site

equation forms are more desirable than the static forms, and are

the only type of site equations we consider in our study.

In the beginning the self-referencing functions were based

on anamorphic (proportional) models, which have a single

shape for all productivity sites, but are scaled up and down

having different asymptotes for different productivity sites

(e.g., Schumacher, 1939; Bailey and Clutter, 1974; Bruchwald,

1988). The anamorphic site models were superseded by

polymorphic site models, which generate different shapes of

growth patterns for different sites. Initially, all the dynamic-

equation-based polymorphic site models were derived by the

procedure named the algebraic difference approach (ADA) by

Bailey and Clutter (1974). The essence of the ADA is to

designate one parameter in a base equation to vary between

different site qualities. Accordingly, all ADA models are

essentially equivalent to their base equations and do not offer

any new original site response definitions, which could involve

any increased levels of complexity. For this reason we denote

the ADA-based polymorphic site models as ‘‘simple poly-

morphic’’.

Over time the need for advanced polymorphic site models,

exceeding the ADA capabilities and having an arbitrary level of

polymorphism and also variable asymptotes, became increas-

ingly more evident. At first this need was manifested only by

the emergence of various static site equations with site quality

responses far more flexible than the ADA-based dynamic site

equations. For decades, leading static site equations (e.g., Ek,

1971; Payandeh, 1974; Monserud, 1984) were preferred due to

having site responses governing both the asymptotes and the

relative growth rates, which is more flexible than what can be

achieved by a single parameter response in any of the

traditional base functions.

While the ADA was too limited to derive flexible dynamic

site equations due to its focus on the base functions, a new

approach of changing the focus towards introduction of explicit

site quality response functions opened the possibilities for

derivation of more flexible dynamic site equations and

introduced the first advanced dynamic site equation with

polymorphism and variable asymptotes (Cieszewski and Bella,

1989). The methodology for derivation of such models was

subsequently named (Cieszewski and Bailey, 2000) the

generalized algebraic difference approach (GADA). Unlike

in the case of the ADA-based models the GADA-based models

are not equivalent to their base equations. In addition to the

temporal model, defined by the function of time, and implied by

the base model, the GADA-based models contain also an added

explicit cross-sectional model, defined by a function of site

quality that is different from any direct proportionality to the

site quality of the individual base model parameters. The

polymorphic GADA-based models are typically more

advanced in modeling both polymorphism and variable

asymptotes, and we refer to them as the ‘‘advanced

polymorphic’’ dynamic site models. The development of the

new methodology was a major breakthrough in dynamic

equation modeling, and allowed the practitioners to use the

preferred dynamic equations for modeling even the most

C.J. Cieszewski et al. / Forest Ecology and Management 243 (2007) 83–93 85

difficult data trends. Subsequent to the development of GADA,

many studies (e.g., Cieszewski et al., 1999, 2006; Cieszewski,

2003; Cieszewski and Nigh, 2002; Elfving and Kiviste, 1997;

Eriksson et al., 1997; Johansson, 1999; Kiviste, 1997, 1998;

Krumland and Eng, 2005; Trincado et al., 2003; Rivas et al.,

2004; Dieguez-Aranda et al., 2005, 2006a, 2006b, 2006c)

reported that the advanced polymorphic models described the

growth patterns in their data better than either anamorphic or

simple polymorphic models.

1.3. Objectives of the study

The objective of this study was to find the best fitting

dynamic site equation for the historical Schwappach (1908)

data for Scots pine height growth in Central Europe.

2. Data

In this study we used the data from permanent sample

plots described by Schwappach (1908) and summarized in the

form of yield tables. The measurement results from these

plots describe the structure of Scots pine stands growing on

various sites divided into five site productivity classes. These

data are discussed in different sources (Schwappach, 1908;

Erteld, 1958; Pirogowicz, 1978; Cieszewski et al., 2006) and

we reiterate here only some relevant essentials. Locations

of the plots are shown in dark gray in Fig. 1 on the

background (in light gray) of the natural Scots pine range in

Europe.

The data were published in its original form in Schwappach

(1896). Several years later, after some additional measurements

and processing, the data were expanded and published again

(Schwappach, 1908), representing 144 plots that were

measured one to five times each with a total number of 588

measurements for each of the monitored variables. The

variables measured on each plot included mean height based

on the Lorey formula (basal area weighted mean height). The

calculated Lorey heights from each plot formed fragments of

height growth curves representing moving weighted averages

Table 1

Considered base equations, their derivatives, and sample ageless differential equations

to these equations

of changing populations. These fragments of curves were used

to draw guide curves for each of five separate site productivity

classes denoted as I, II, III, IV and V. The average height at the

age of 100 years for each of the classes equaled 28.0, 24.1, 20.3,

16.3, and 12.5 m, respectively. The measurement procedures

and compilation of the tabular data are described in detail in

Schwappach (1908). According to the data summary, four of

the classes (I–IV) were relatively well represented in the data,

while one (the poorest site class V) was based on scant data.

The lowest productivity class was based on measurements of

only 8 plots with a total of 34 measurements (about 5% of the

entire dataset). Due to the unreliable nature of these

measurements, low importance and slight representation of

the lowest sites in operational forestry (Bruchwald and

Kliczkowska, 1997, 2000), we excluded the data for site class

V from our analysis.

3. Methods

3.1. Selection of the dynamic site equations

We considered various base equations (Table 1) and, derived

from them, anamorphic and simple polymorphic (Table 2), and

advanced polymorphic (Tables 3A–3D), dynamic site equations

used by different authors modeling Scots pine height growth in

the past (e.g., Jarosz and Kłapec, 2002; Bruchwald, 1985;

Bruchwald et al., 2000; Cieszewski, 2001; Elfving and Kiviste,

1997; Cieszewski and Zasada, 2002; Palahı et al., 2004; Rivas

et al., 2004 describes Cieszewski et al., 2006; Dieguez-Aranda

et al., 2006b). Cieszewski et al. (2006) describes the

comparison of nine of the above models tested on the basis

of properties of base equations and implications of the methods

used to derive the dynamic equations from these base

equations, including all the equations proposed by Jarosz

and Kłapec (2002), Bruchwald et al. (2000), Elfving and

Kiviste (1997), and implicitly also by Palahı et al. (2004).

Accordingly, the site models discussed here were either derived

from base equations that best fitted our data or from base

equations that other researchers recommended for other Scots

corresponding

Table 2

ADA dynamic site equations based on one varying site parameter in the modified Gompertz (1982) base equation [1] (Jarosz and Kłapec, 2002) and two

Hossfeld (1822) base equations [2] and [3], and difference equations corresponding to these dynamic site equations

Source—Eqs. [7] and [8]: Cieszewski et al. (2006); Eq. [9]: Bruchwald et al. (2000); Eq. [10]: Cieszewski and Zasada (2002); Eq. [11]: Cieszewski (2002);

Eq. [12] Mcdill and Amaties (1992).

C.J. Cieszewski et al. / Forest Ecology and Management 243 (2007) 83–9386

pine data. Thus, for example, even though there are many

advanced dynamic site equations (e.g., Cieszewski, 2004)

derived from the Richards (1959) base equation (i.e.,

Y(t) = a(1 � e�bt)c) none of them are considered here because

this equation fitted our data poorly (SSR = 0.7246) even when

fitted separately to individual series.

The anamorphic and simple polymorphic site models

(Table 2) compared here are essentially variations of three

height–age base functions (Cieszewski et al., 2006). Table 1

contains the three base functions, their derivatives, and their

related ageless differential equations. Subsequently, Tables 2,

Table 3A

GADA advanced polymorphic dynamic site equation [13] (SSR = 1.2) based on C

(2006), and related difference equations corresponding to it

‘‘(*)’’ indicates citation of this article. Source—Eq. [13]: Cieszewski et al. (2006)

3A–3D contain all the considered dynamic site equations, their

derivatives, and ageless differential equations related to them,

which might be useful in various modeling situations, such as

simulating height–density interactions using age-dependent

difference equations as in Cieszewski and Bella (1993) or

simulating age-independent volume–density interactions

using a system of differential equations, such as in Tait

et al. (1988).

The base function [1] (Jarosz and Kłapec, 2002) is a

modification of the Gompertz function (Gompertz, 1825)

expanded by the additional intercept parameter, where Y(t) is

ieszewski et al.

.

Table 3B

GADA advanced polymorphic dynamic site equation [14] (SSR = 1.8) based on Cieszewski et al. (2006), and the related

difference equations corresponding to it

‘‘(*)’’ indicates citation of this article. Source—Eq. [14]: Cieszewski et al. (2006).

C.J. Cieszewski et al. / Forest Ecology and Management 243 (2007) 83–93 87

the dependent variable (such as height) as a function of t, t the

prediction age, a, b, c, d, are the estimation parameters.

The base function [2], which was used by Bruchwald et al.

(2000) for modeling height growth of Scots pine in Poland,

originates from Hossfeld (1822), where a and b are the

estimable parameters and all other symbols are as previously

defined. Finally, the base function [3], which was used by

Elfving and Kiviste (1997) for modeling height growth of Scots

pine in Sweden, also originates from Hossfeld (1822), where a,

b, and c, are the model parameters and all other symbols are as

previously defined.

According to Cieszewski et al. (2006) the best model for

describing the data was a GADA dynamic site equation (i.e.,

Cieszewski, 2001) based on combining the height–age base

function [3] and the following height–site base function:

YðXÞ ¼ a1X þ a2X2

1þ X[4]

Table 3C

GADA advanced polymorphic dynamic site equation [15] (SSR = 0.53) based on

difference equations corresponding to it

‘‘(*)’’ indicates citation of this article. Source—Eq. [15]: Cieszewski (2001).

where Y(X) is the dependent variable value (such as height) as a

function of X (i.e., site quality), X the any independent variable

describing site quality, and a are the model parameters that may

vary for different ages.

Models [3] and [4] are both base models based on fractional

functions and describing simple two-dimensional relationships.

Model [3] is one of the oldest base models used in forestry

(Hossfeld, 1822) and is frequently referred to as Hossfeld IV

most likely after Kiviste (1988). When the two-dimensional

(Y = f(t)) Hossfeld base model [3] is used directly as a three-

dimensional (Fig. 2) site equation (Y = f(t,t0,y0)), which can be

done only in the context of the ADA, the resulting dynamic

equations can be only anamorphic (i.e., Eq. [9], Table 2) or

simple polymorphic (i.e., Eq. [10], Table 2). On the contrary,

incorporating the definition of temporal changes defined by Eq.

[3] into the definition of cross-sectional changes defined by Eq.

[4] as merely a part of a site model definition results in a much

more powerful algebraic formulation of the three-dimensional

Cieszewski (2001), and the related

Table 3D

GADA advanced polymorphic dynamic site equation [16] (SSR = 0.43) based on Cieszewski (2005), and the related

difference equations corresponding to it

‘‘(*)’’ indicates citation of this article. Source—Eq. [16]: Cieszewski (2005).

C.J. Cieszewski et al. / Forest Ecology and Management 243 (2007) 83–9388

site–height–age relationship (Fig. 2). Accordingly, the basic

GADA formulation, which can be used for deriving a dynamic

equation based on combining the two response models [3] and

[4] into a common site–height–age equation is

Yðt;XÞ ¼ Xð f þ XÞtc

bþ Xtc[5]

or

Yðt;XÞ ¼ f þ X

1þ ðb=XtcÞ [6]

where Y(t,X) is the dependent variable value (such as height) as

a function of both: X (i.e., site quality), and t (e.g., age); c, f , and

b are the new model parameters (different than those in Eq. [3]),

and other symbols are as previously defined.

Solving Eqs. [5] or [6] for X and substituting the initial

condition solution in place of X in Eqs. [5] or [6] results in a

GADA-based dynamic site equation. When the solution for X is

substituted in Eq. [5], the resulting model is Cieszewski (2001)

(Eq. [13], Table 3A). When the solution for X is substituted in

Eq. [6], the resulting model is Cieszewski and Bella (1989).

Either of these models can be reformulated to the form of the

other. The models described in Cieszewski and Bella (1989) are

identical while Cieszewski (2001) describes a generalization of

those models in an improved algebraic form with multiple

solutions and dynamic site model forms. The variant of this

model that was used here is presented as Eq. [15] (Table 3C),

and it is equivalent to the model used by Elfving and Kiviste

(1997) with Scots pine data (i.e., Eq. [13], Table 3B), except it is

defined for t = 0.

Finally, Cieszewski (2005) has recently proposed another

new flexible dynamic site equation (i.e., Eq. [16], Table 3D),

which in a study of non-forestry data had better performance

than many other models. We have added this new model to our

analysis to compare its performance against the performance of

all the models described above.

3.2. Parameter estimation

We estimated the parameter values for all the models

using the Scots pine data (Schwappach, 1908) described

above. Most of the fittings were done in Excel because of

the simplicity of parameter estimation with small data

sets, requiring estimation of only several nonlinear para-

meters and well-defined growth trends in averaged series

data. All site equations were fitted to all the data pooled

together.

Parameters of self-referencing models are usually estimated

using one of two general approaches known as ‘‘base-age-

specific’’ or ‘‘base-age-invariant’’. The most common is the

base-age-specific approach, which assumes that the model

reference values (y0), such as site indices, are equal to

arbitrarily selected data points (e.g., Curtis et al., 1974;

Monserud, 1984). In such an approach only the global model

parameters are estimated. The base-age-specific approach is

relatively simple in implementation, but it can result in

erroneous identification of growth curves (Goelz and Burk,

1996; Cieszewski, 2003). Parameters of such models can be

referred to as ‘‘base-age-specific’’, which means that they are

influenced by the arbitrary selection of the base age, and are

Fig. 2. Site model as a three-dimensional space defined by continuous axes of

age, height, and site index class; the model can define an individual growth

curve as an intersection of the response surface with the plane defined by a fixed

site quality value.

Table 4

The sum of square errors in fitting models [7]–[15] to the Schwappach data by

site model types and by base function type

Model type Base functions Range Min. BRge/

MinEq. [1] Eq. [2] Eq. [3]

Anamorphic 66.1 9.9 8.3 57.80 8.30 6.96

Simple polymorphic 33.7 2.0 29.0 31.70 2.00 15.85

Advanced polymorphic 1.2 1.8 0.52 1.28 0.52 2.46

Range 64.90 8.10 28.48

Minimum 1.20 1.80 0.52

TRge/Min. 54.08 4.50 54.77

C.J. Cieszewski et al. / Forest Ecology and Management 243 (2007) 83–93 89

defining potentially biased estimates through the propagation of

error in the data that was used to assign their values.

The less common, base-age-invariant approach treats the

reference values (y0) as varying parameters of the model (e.g.,

Bailey and Clutter, 1974; Garcia, 1983), or site-specific

parameters. During the model fitting, X may also be estimable

directly as the model varying parameter, but this approach will

not always work since the domain of X values and the way they

enter the estimation process is unknown. In the base-age-

invariant parameter estimation the local site-specific para-

meters, unique for each site growth series, and the global

parameters (e.g., a, b,. . .), common to all sites/series, are

estimated simultaneously. The parameters estimated by the

base-age-invariant approach are called base-age-invariant

(Bailey and Clutter, 1974) because their values are not

affected by any arbitrary selections of base ages—they are

always the same. The base-age invariant approach is relatively

more complicated and difficult to implement than the base-age

specific approach. Yet, this approach is theoretically and

practically more desirable because it does not violate

regression assumptions and is more likely to reveal the true

growth trends in the data (Cieszewski et al., 2000). For this

reason we use only this approach in our study. The fit criterion

was the minimum sum of square residuals, and given the

simplicity of the considered data it was the only criteria of fit

used for comparison of the different equation forms and

their capability to generate the curves best fitting the Scots

pine data.

4. Results

The results fell into two categories. Only two of the models

fitted the data well. The rest of the models fitted the data poorly.

The worst value ranges of SSEs were for the anamorphic and

simple polymorphic models (Table 4). The anamorphic models

were generally the worst fitting with SSE ranging from 8.3 to

66.1. The simple polymorphic models also fitted poorly with

the SSE range of 2.0–33.7.

The results showed huge differences between different

model types (i.e., anamorphic and simple polymorphic versus

advanced polymorphic). The differences were the most

noticeable with the two most flexible base equations.

Accordingly, models [7], [8], and [13] had SSEs ranging from

1.2 to 66.1, while models [11], [12], and [15] had SSEs ranging

from 0.52 to 29.0. The different site model types [9], [10], and

[14] that are based on the least flexible base function generated

much more moderate responses to the changes in the site

response definitions, ranging from 1.8 to 9.9.

The advanced polymorphic models had SSEs ranging from

0.42 to 1.8 (Table 4). Models [15] and [16] produced less than

half the SSEs of the next best fitting advanced polymorphic

model [13] and a quarter of the SSE of the best simple

polymorphic model [10]. Models [15] and [16] produced very

similar residuals (Fig. 3) with almost perfect fits to the data

(Fig. 4a) and virtually identical curves within the range of the

fitting data. The residuals of both models are quite similar with

the exception that model [16] has better behaving residuals in

that they have smaller extremes in young ages, a more uniform

range of values across different age classes, and about six times

smaller bias. The deviations of model [15] ranged from �0.31

to 0.22 with a mean of 0.006 while the deviations of model [16]

ranged from �0.25 to 0.19 with a mean of 0.001.

The parameters of the best model [16] were a = 0.568791952,

b = �47.20668703, c = 1282.848702, and j = 1.445843461. The

parameters of the second best model [15] were b = 10.19710114,

f = �5.154278306, and j = 1.446039881. The models appear to

have different extrapolation properties (Fig. 4b). In the

extrapolation areas, outside of the range of the data that were

used to fit the models, model [15] has faster initial growth rates

and lower later growth rates in older ages. On the very low site the

difference between the two models is very small and almost

unnoticeable. On very high sites the difference is greater and

quite noticeablewith model [16] showing longer sustained height

growth in the older ages.

Overall, the results of this analysis demonstrate clearly that

the advanced polymorphic GADA formulations of dynamic site

equations are much superior to the simple ADA models. The

differences between the base model functions (e.g., Hossfeld IV

Fig. 3. Residuals in fitting the dynamic site Eqs. [15] and [16] with the base-age

invariant stochastic regression.

Fig. 4. Schwappach data for Scots pine (symbols in figure (a)) and site–height–

age curves obtained using the fitted dynamic site Eqs. [15] and [16] compared

against data (a) and between each other (b).

C.J. Cieszewski et al. / Forest Ecology and Management 243 (2007) 83–9390

versus modified Gompertz) had much smaller impact on

performance of the different site models than the differences

between the cross-sectional model definitions in those models

(i.e., anamorphic versus simple polymorphic, versus advanced

polymorphic). Accordingly, the worst of the advanced

polymorphic (GADA) site models was better than both the

best anamorphic and the best simple polymorphic models

regardless of their base function (e.g., Hossfeld IV versus

Gompertz) origin. These results clearly indicate that the base

functions alone are not sufficient for modeling self-referencing

phenomena, such as site–height–age relationships exhibiting

similar trends to those expressed by the data considered here.

5. Discussion

Dynamic site equations are self-referencing models based

on three-dimensional functions of three separate variables:

sites, ages, and heights (i.e., Y = f(t,X)), which define

continuous relationships (Fig. 2) between the height, the age,

and the site quality. Because the site quality is an unobservable

variable, it is computed from an inverse function of the

underlying site–height–age model (i.e., X = f�1(t,Y)) using a

hypothetical or known snapshot observation, such as an

inventory measurement (i.e., X = f�1(t0,y0)), which allows

using this solution in the main model (i.e., Y = f(t,f�1(t0,y0))).

This model structure provides an automatic non-linear

interpolation between any site and age classes. The simplest

of these models may be deemed as families of two-dimensional

functions, such as Y = F(t), in which one of the parameters

varies with sites. Among these the anamorphic models are the

simplest of the self-referencing functions because they define

the heights at different ages and sites as scaled values of an

average growth trend expectation. The algebraic height–site

definition in anamorphic models is always the simple

C.J. Cieszewski et al. / Forest Ecology and Management 243 (2007) 83–93 91

proportionality (i.e., Y = aX) between height and site quality

regardless of the complexity of the height–age definition of the

model.

The simple polymorphic models are somewhat similar to the

anamorphic models in the sense that they are also simple

scaling mechanisms of a guide curve along a single parameter

axis other than the dependent variable axis. When these models

are scaled along the age axis, their algebraic definitions of site

responses are analogous to their algebraic definitions of the age

responses, such as in model [8], which defines the site responses

as Y ¼ a e�be�cX þ d. When the simple polymorphic model is

based on some other form of the axis transformation, the site

response definitions depend on particular algebraic formula-

tions of the age base functions. For example, model [10] defines

the site responses as Y ¼ b=ð1þ aXÞ2, while model [12]

defines the site responses as: Y ¼ b=ð1þ aXÞ.The advanced polymorphic site equations are different from

the anamorphic and the simple polymorphic ADA-based site

models in the sense that they are hybrids of two arbitrarily

different functions of which one defines a height–age relation-

ship, that is Y = u(t), and one defines a height–site relationship,

that is Y ¼ wðXÞ. Typically the height–site function Y ¼ wðXÞhas more impact on the final pooled site equation performance

than the height–age function Y = u(t). Most practitioners can

easily relate to this fact since commonly the residuals in fitting

site–height models are one to two orders of magnitude greater

than the residuals in fitting individual height–age models.

Nonetheless, there is a strong synergy between the two site

model components, that is Y ¼ wðXÞ and Y = u(t). A good site

model can be realized only if both of the model components

(i.e., Y ¼ wðXÞ and Y = u(t)) are adequate and compatible to the

extent to which they can be incorporated into a common

algebraic formulation. If any of the two sub-models is lacking,

the combined site–height–age model will also be lacking. In

general, in a site–height–age model (Y = f(t,X)) with adequate

algebraic definitions of age (Y = u(t)) and site (Y ¼ wðXÞ)responses the definition of site responses (Y ¼ wðXÞ) will have

the greatest impact on the site–height–age model performance.

However, the site response definition (Y ¼ wðXÞ) will not make

up for an inadequate age response definition (Y = u(t)). This can

be easily seen in the results of this study (Table 4). The site

models derived with the height–age base function [2], which is

the worst of the three base functions (Cieszewski et al., 2006),

never improve much regardless of the complexity of the applied

height–site definitions. On the contrary, the other two height–

age base functions [1] and [3] do respond very strongly to

improvements in the incorporated height–site definitions.

In this study we used the permanent sample plot data

summarized by Schwappach (1908) into tabular representations

of discrete numbers of site and age classes to fit continuous

site–height–age relationships defined by different dynamic site

equations with various complexities. Such models should

provide better estimates than the direct use of the tabular data

because such a fitted model represents averaged growth trends

for each age class, in a given site class, corrected by the

additional information available in the other age classes; if all

data are considered together all the measurement and

environmental errors are more likely to zero out as larger

sample averages. Thus, for example, the tabularized summaries

documented in Schwappach (1908) still contain a significant

random component, which was partially reduced by averaging

data values in individual age and site index classes, but not

between them. Each value from the tabular compilation

represents information based only on one age and site class,

while each point of a curve generated by model [16] represents

combined information from all age and site classes. Similarly,

the interpolation of the tabular data between, say, ages 30 and

40, is based only on observations from these very age classes,

while such an interpolation by model [16] is based on data from

all age classes represented by the given site class. Because the

goal of modeling is generalization of the information

represented by the measurement data, the mathematical model

fitted to all observations from a given series is statistically more

efficient than any tabular data containing average values of the

individual age and site classes. A model fitted to all ages and

sites is more efficient than a series of sub-models fitted to

individual site classes (e.g., Jarosz and Kłapec, 2002). After

fitting separate models to individual age classes the pooling of

all the site and age classes together is the next natural step in

generalization of the measurement data. Modeling site classes

as a collection of disconnected individual models is analogous

to averaging height values in individual age classes such as it is

done in the Schwappach data. A model fitted simultaneously to

the whole dataset includes, in each of its points, pooled

information from analysis of all the data. Therefore, if the

mathematical expression is flexible enough to describe

modeled phenomena well, the model that is fitted to all pooled

data represents the modeled phenomenon better than any

number of individual sub-models fitted to subsets of the data

and better than the individual data themselves.

This is not true if the mathematical expression is not flexible

enough to describe the modeled phenomena well. In this study

an example of a model that is extremely flexible for modeling

age but inflexible for modeling site quality is the anamorphic

model [7]. This model can fit any site productivity class

individually very well but when fitted simultaneously to all site

and age classes it fails to describe all the data well (Table 4)

because it is lacking the flexibility in changing the curve shapes

between different sites. The SSE generated by model [7] is

more than 157 times larger than the SSE generated by the fit of

model [16].

6. Summary and conclusions

We have presented a study on the comparison of

performance between various dynamic site equations pre-

viously studied in the context of Scots pine data. Among 10

different dynamic site equations considered in this study, two

proved to be the best for the height growth data of Schwappach

(1908). Cieszewski (2005) model fit the data the best in terms of

SSE, which was 25% smaller than the next best model of

Cieszewski (2001). Yet, the later model may have better cross-

sectional extrapolation properties, as seems to be suggested by

its closer matching of the high productivity (i.e., IA class of the

C.J. Cieszewski et al. / Forest Ecology and Management 243 (2007) 83–9392

Szymkiewicz extrapolation data) series. However, this con-

jecture is rather inconclusive in light of the fact that the

Szymkiewicz IA class data was obtained at least partially

through manual extrapolations of the Schwappach (1908) data.

It seems merely peculiar that Eq. [6] would match the same

trend so exactly while Eq. [5] would not. Notwithstanding any

minor discrepancies in the fit of the two best dynamic site

equations and their extrapolation properties, both of them are

highly recommended for modeling other self-referencing

relationships.

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