Content uploaded by Maurizio Marchi

Author content

All content in this area was uploaded by Maurizio Marchi on Sep 05, 2019

Content may be subject to copyright.

ORIGINAL PAPER

Nonlinear versus linearised model on stand density model ﬁtting

and stand density index calculation: analysis of coefﬁcients

estimation via simulation

Maurizio Marchi

1,2

Received: 20 August 2018 / Accepted: 25 January 2019 / Published online: 4 May 2019

The Author(s) 2019

Abstract The stand density index, one of the most

important metrics for managing site occupancy, is gener-

ally calculated from empirical data by means of a coefﬁ-

cient derived from the ‘‘self-thinning rule’’ or stand density

model. I undertook an exploratory analysis of model ﬁtting

based on simulated data. I discuss the use of the logarith-

mic transformation (i.e., linearisation) of the relationship

between the total number of trees per hectare (N) and the

quadratic mean diameter of the stand (QMD). I compare

the classic method used by Reineke (J Agric Res

46:627–638, 1933), i.e., linear OLS model ﬁtting after

logarithmic transformation of data, with the ‘‘pure’’ power-

law model, which represents the native mathematical

structure of this relationship. I evaluated the results

according to the correlation between N and QMD. Linear

OLS and nonlinear ﬁtting agreed in the estimation of

coefﬁcients only for highly correlated (between -1 and

-0.85) or poorly correlated ([-0.39) datasets. At

average correlation values (i.e., between -0.75 and

-0.4), it is probable that for practical applications, the

differences were relevant, especially concerning the key

coefﬁcient for Reineke’s stand density index calculation.

This introduced a non-negligible bias in SDI calculation.

The linearised log–log model always estimated a lower

slope term than did the exponent of the nonlinear function

except at the extremes of the correlation range. While the

logarithmic transformation is mathematically correct and

always equivalent to a nonlinear model in case of predic-

tion of the dependent variable, the difference detected in

my studies between the two methods (i.e., coefﬁcient

estimation) was directly related to the correlation between

N and QMD in each simulated/disturbed dataset. In gen-

eral, given the power law as the ‘‘natural’’ structure of the

N versus QMD relationship, the nonlinear model is pre-

ferred, with a logarithmic transformation used only in the

case of violation of parametric assumptions (e.g. data dis-

tributed non-normally).

Keywords Ordinary least squares Power law Reineke

function Silviculture Ecological mathematics Forest

mathematics

Introduction

The maximum degree of competition that forest popula-

tions of a species can sustain is described by the principle

of self-thinning (Yoda et al. 1963; Westoby 1984). This

rule, a power function linearised by a logarithmic trans-

formation, states that environmental resources can satisfy

only a limited number of trees within a stand, and this

number grows progressively smaller as tree age and size

Project funding: The project was fully funded by the Italian National

Rural Network project, in the framework of the European Network for

Rural Development (ENRD) and also by EU, in the framework of the

Horizon 2020 B4EST project ‘‘Adaptive BREEDING for productive,

sustainable and resilient FORESTs under climate change’’, UE Grant

Agreement 773383 (http://b4est.eu/).

The online version is available at http://www.springerlink.com

Corresponding editor: Zhu Hong.

&Maurizio Marchi

maurizio.marchi85@gmail.com

1

CREA - Research Centre for Forestry and Wood, Arezzo,

Italy

2

geoLAB - Laboratory of Forest Geomatics, Department of

Agriculture, Food, Environment and Forestry, Universita

`

degli Studi di Firenze, Florence, Italy

123

J. For. Res. (2019) 30(5):1595–1602

https://doi.org/10.1007/s11676-019-00967-0

increase. The proximity of a population to this asymptotic

boundary indicates the intensity of intraspeciﬁc competi-

tion in a monospeciﬁc population. After the boundary is

exceeded, competition between individuals begins and, in

the absence of disturbance, growth trends of dominated

individuals progressively slow in favour of more compet-

itive species. This process leads the population to a max-

imum number of plants of a given size that can coexist

within a given unit area of land (Liira et al. 2011; Vos-

pernik and Sterba 2015).

Maximum biomass stocking is a fundamental indicator

to balance forest management. Growth trends, timber

quality and provision of ecosystem services from forests

are highly inﬂuenced by tree density and competition for

resources (Solomon and Zhang 2002; Corona 2015; Mason

and Connolly 2016; Marchi et al. 2018). Competition

indices ofplant interactions (Pommerening and Sa

¨rkka

¨

2013; Cabon et al. 2018) or allometric relationships (An-

fodillo et al. 2013; Marchi et al. 2017a,b) are valuable

tools that support decision-makers. Forest management

options that promote sustainable use (Fabbio et al. 2018)

can reduce competition between trees and aid selection of

preferred trajectories for forest stands in view of predicted

climate change effects (Wang et al. 1998; Ray et al. 2017).

The stand density index (SDI), widely accepted as a

powerful tool for evaluating growth sustainability of even-

aged forest systems, was proposed by Reineke (1933) for

tree species in the United States and later was adopted in

many other countries (Pretzsch and Biber 2005;Shaw

2006; Castagneri et al. 2008; Poschenrieder et al. 2018).

SDI is calculated as:

SDI ¼NQMD

a

k

;ð1Þ

where N is tree density, QMD is the quadratic mean

diameter, ais a constant whose value can be 10 or 25

according to the use of imperial or metric measuring sys-

tem, respectively, and kis a coefﬁcient. The biological

meaning of this index is the maximum number of trees per

unit area a forest population (stand) can sustain for a pre-

determined QMD. For instance, an SDI of 400 means that

400 trees per hectare are expected when the QMD of the

stand is 25 cm. Although originally implemented for

monospeciﬁc, even-aged stands, generalized use in uneven-

aged stands and in multi-species stands has often been

proposed (Rivoire and Le Moguedec 2012). The core of the

function (1) is represented by the coefﬁcient k. The value

of this parameter is the relationship between the total

number of trees per unit area (N) and the mean tree volume

or mass of the stand. This rule is parameterized for forests

as the stand density model where N is related to the QMD

of the stand. Even when approximated with a negative

exponential function (Fonseca and Duarte 2017), this

relationship follows a power law (PWL), which represents

the ‘‘pure’’ and mathematically-corrected shape of an N

versus QMD relationship that is expressed as:

N¼bQMDk

;ð2Þ

when QMD is either large or small, being asymptotic to

zero and ??, respectively. The use of PWL is generally

dropped in favour of logarithmic transformation of the data

ﬁtted using an ordinary least squares (OLS) function

(Reineke 1933; Pretzsch and Biber 2005;Shaw2006;Ge

et al. 2017). In this case, Eq. (2) becomes:

log N ¼logbþklog QMD:ð3Þ

This basic equation, ﬁrst used by Reineke (1933),

enabled the analysis of the size–density relationship for

many forest species in both pure and mixed stands. Using

Eq. 3, Reineke (1933) derived a slope (kcoefﬁcient) equal

to -1.605 that was consistent across 15 datasets repre-

senting 14 species (13 coniferous) and was identical for 12

species. Although the slope was taken as constant, the

intercept term bwas considered to be species-speciﬁc. This

rule was supported by Yoda et al. (1963) but researchers in

Europe have often questioned it (Pretzsch and Biber 2005;

Vacchiano 2005; Castagneri et al. 2008). Recent studies

demonstrated the need to evaluate the slope of plotted

curves according to site quality (Ge et al. 2017), and biases

could result from the log-transformation of the data (Smith

1993), and questions have been raised on its biological

validity (Packard 2014). Despite this, the use of log-

transformed data versus nonlinear regression for analysing

biological power laws remains a common approach. Other

methodologies have been tested for estimating coefﬁcients,

including stochastic frontier functions (Zhang et al. 2005;

Weiskittel et al. 2009), reduced major axis regression

(Solomon and Zhang 2002; Zhang et al. 2005), quantile

regression (Vospernik and Sterba 2015; Ducey et al. 2017),

bisector regression (Newton 2006), hierarchical Bayesian

models (Zhang et al. 2015) and mixed modelling (Zhang

et al. 2005).

The aims of this study were to evaluate the performance

of OLS and PWL methods for estimating kin SDI calcu-

lations and to analyse the discrepancies between them.

Comparison between these two methods for stand density

model ﬁtting is based on simulated data with a known

degree of correlation between N and QMD. The subsequent

stand density management trajectories that were generated

were evaluated according to the results of these

comparisons.

1596 M. Marchi

123

Materials and methods

To analyse OLS and PWL robustness in coefﬁcient esti-

mation, I generated an artiﬁcial dataset composed of

100,000 replicates of 200 records each. The aim was to

simulate empirical records derived from a hypothetical

distribution of monitored stands with a known degree of

correlation between N and QMD. A ﬁve-step procedure

was followed for this simulation (Fig. 1).

First, a ‘‘perfect’’ dataset with nonparametric correlation

between N and QMD of -1.0 was built with the normally

distributed dependent variable N by extracting the 200

records representing the total number of trees per hectare

(N) from a normal distribution with mean = 2700 tree-

sha

-1

and a standard deviation = 685 treesha

-1

. Those

starting values were judged as adequate to obtain a broad

range of simulated stands of density from 300 (mature

stand) to 5000 (young high forest stand before self-thinning

starts) and a theoretical SDI around 680. Then, the

corresponding quadratic mean diameter (QMD) for each of

the 200 normally distributed N values was calculated with

the following equation, derived from Eq. (2) and assuming

k=-1.605 and b= 1.210

5

:

QMD ¼ﬃﬃﬃﬃ

N

b

k

s:ð4Þ

This ﬁrst simulated dataset represented the hypothetical

and perfect situation of a forest stand in self-thinning

mode. Afterward, this original vector of QMD values (200

records) was iteratively modiﬁed to generate 99,999

alternative artiﬁcial stands. A series made by 200 random

multipliers was then generated 99,999 times, to increase or

reduce QMD with an artiﬁcial degree of noise. Such mul-

tipliers were derived from a second normal distribution

with mean = 1 and with a variable and randomly generated

standard deviation between 0.01 and 0.3. As a result, the

Spearman correlation coefﬁcient (Spearman 1987) between

the original vector N and the 100,000 artiﬁcial vectors of

5

Fig. 1 Flowchart of the simulation process

Nonlinear versus linearised model on stand density model ﬁtting and stand density index…1597

123

QMD values ranged from -1.0 to -0.2 (Fig. 2). The

main idea behind this operation was to control the degree

of correlation between N and QMD to determine whether

this information might be used as ancillary data to evaluate

the coefﬁcients estimated by the two methods. Summary

statistics of 100,000 generated plots are reported in

Table 1.

With the generated and log-transformed data, the stand

density model of the simulated 100,000 records was ﬁtted

using the classical linear ordinary least squares (OLS) ﬁt and

a power law was ﬁtted using the Gauss–Newton algorithm

(PWL). The degree of equivalence between the estimated b

and kcoefﬁcients was evaluated using a simple linear model

analysis (expected results were slope =1 and inter-

cept =0). The difference between coefﬁcients, i.e. kor b

estimated by PWL minus kor bestimated by OLS, was also

evaluated in light of the correlation between N and QMD in

each of the simulated plots. A parametric ANOVA based on

correlation coefﬁcient classes was ﬁnally performed to

assess differences within the estimated kvalues using the

Duncan test post hoc to identify possible groups. The degree

of correlation between N and QMD was the factor targeted

for evaluation, and eight classes were drawn dividing the

dataset into eight groups. Every class was deﬁned including

all the plots with a qvalue between xand x?0.1 (e.g., class

7 included all the records with -0.4 Bq\-0.3, while

class 6 was -0.5 Bq\-0.4).

Results

Given the nature of generated datasets, comparison

between ﬁtted models was possible in the absence of ref-

erence values. Indeed, the original kand bvalues were

assumed to be starting points that were only calculable with

the initial perfect dataset that contained no random noise.

Based on this assumption, the introduced changes to the

QMD vector brought the two models to calculate a wide

range of kand bvalues. The kcoefﬁcient for OLS ranged

from -1.61 to -0.11 while branged from 810

3

to

1.310

5

. Similarly, with PWL, kranged from -1.62 to

-0.08, while branged from 8.510

3

to 1.410

5

. With the

linear model analysis, a common trend resulted between

the estimated kand bvalues. Models appeared to be similar

but with very high (-1.0 Bq\-0.8) and very low

correlations (-0.3 \q) as shown in Fig. 3. The OLS

model generally estimated higher k-values (Fig. 3, left

side) and lower bvalues (Fig. 3, right side) than did PWL,

especially when correlation coefﬁcients ranged from

-0.75 to -0.4.

The relationship between the difference inkestimates

from OLS versus PWL and the correlation between N and

QMS in each artiﬁcial plot is shown in Fig. 4. Here, a

simple local polynomial regression ﬁtting was also added

as a trend line and to characterize the behaviour of the

phenomenon. Concerning k(on the left), the core of the

stand density index calculation, the difference increased

quickly, when the correlation coefﬁcient was between

-1.0 and -0.75, becoming almost ﬂat at -0.6. Then the

difference decreased, becoming smaller until correlation

values exceeded -0.3. The difference in calculated max-

imum SDI values was also evaluated and is reported in the

right panel of Fig. 4.

Seven statistically different correlation groups were

detected, including border crossing classes (e.g., ‘‘ab’’,

‘‘bc’’ and ‘‘abc’’) (Table 2). The smallest differences sup-

ported by statistical evidence in kestimation were detected

for the class 8 (-0.3 Bq), class 7 (-0.4 Bq\-0.3)

and class 6 (-0.5 Bq\-0.4), all of which differed

from each other and from the other groups. Then class 1

(-1.0 Bq\-0.9) was intermediate between classes 6

Fig. 2 Histogram of Spearman correlation coefﬁcients between the

generated number of trees per hectare (N), and the 100,000 quadratic

mean diameter (QMD) simulations

Table 1 Summary statistics of the 100,000 randomly generated

forest monitoring plots

Dataset N QMD

Original

Minimum 522 10.3

Mean 2512 25.2

Maximum 5097 35.7

SD used to

modify QMD

Correlation

coef. (q)

Adjusted

Minimum 522 0.01 -0.99

Mean 2512 0.15 -0.71

Maximum 5097 0.35 -0.22

N is number of trees per hectare; QMD is the quadratic mean diameter

of the stand; SD is standard deviation of the normal distribution used

to generate multipliers

1598 M. Marchi

123

and 2 (-0.9 Bq\–0.8), 3 (-0.8 Bq\-0.7) and 4

(-0.7 Bq\-0.6), and was ranked with an average

difference in kbetween 0.2769 and 0.2968. An ‘‘admixture

zone’’ was comprised of class 2 (‘‘abc’’ group) and classes

3 and 4 (‘‘ab’’ group) where differences were less pro-

nounced. The highest values were recorded for class 5

(-0.6 Bq\-0.5) with an average difference around

0.3. Class 5, which might also represents the most common

case in empirical data, was used to generate Fig. 5, where a

simulation of an elementary stocking chart was drawn with

two SDI values (1000 and 500) and two possible kvalues

differing from ±0.3. As a result, a discrepancy between

the two lines was evident (Fig. 5).

Fig. 3 Relationship between k(left) and b(right) coefﬁcients

calculated by OLS and PWL models in each of the 100,000 simulated

plots. Each dot represents a single record and is coloured according to

the correlation class between N and QMD. N = number of trees/ha;

OLS = ordinary least squares; PWL = power law; QMD = quadratic

mean diameter of the stand b

Fig. 4 Relationship between the difference in k(left) and maximum

SDI (right) estimated with ordinary least squares and power law

models in each of the 100,000 simulated plots (y-axis) and the

correlation between N and quadratic mean diameter of the stand

(QMD) in each (x-axes). Each dot represents a single record and is

coloured according to the correlation class between N and QMD. The

black line is a local polynomial regression model and describes the

average trend

Nonlinear versus linearised model on stand density model ﬁtting and stand density index…1599

123

Discussion

The degree of equivalence between OLS and PWL coef-

ﬁcients were highly dependent on the ‘‘quality’’ of the data,

i.e. on the degree of random noise during QMD and N

calculation for each plot. Even if logarithmic transforma-

tion is a primary tool for OLS linear regression of power

law relationships (Anfodillo et al. 2013), the results showed

that this mathematical adjustment should be carefully

evaluated when the primary goal is to study the model’s

coefﬁcients rather than its predicted values. Transformation

might introduce biases into the ﬁtting procedure, resulting

in quite different values for kand b. This bias could then

directly be transmitted to the stand density index calcula-

tion where kis the only coefﬁcient.

Field data collection often is the most expensive com-

ponent of research projects but is necessary to obtain high

quality data. Similarly, the contributions of robust models,

analytical procedures, and statistics must not be underes-

timated (Fassnacht et al. 2014; Ferrara et al. 2017; Marchi

et al. 2017a,b). Important adjustments are sometimes used

to handle biases introduced to datasets, especially when

using inverse functions (Smith 1993). For SDI calculation,

the use of a log-transformed version of the canonical SDI

formula, Eq. (1), cannot be a possible solution. This would

be true even if Eq. (1) could be written as:

log SDI ¼log N þklog QMD

a

:ð5Þ

Using Eq. (5) with a value for kestimated from Eq. (3)

is worthy of note. An additional test highlighted that the

difference in SDI curves shown in Fig. 5remains. It is well

known that the logarithmic transformation of an arithmetic

dataset often results in slightly biased estimates when

values are predicted and transformed back to arithmetic

units from OLS (Baskerville 1972; Newman 1993; Packard

2013). This issue is mainly due to an estimation based on a

geometric mean of the dependent variable rather than on

the arithmetic mean at that value of the independent vari-

able (Smith 1993).

Accurate determination of the self-thinning trajectory

for any population is a difﬁcult task, whatever the data

source and for many biological issues. Despite taking the

appropriate analytical steps, some problems are inherent in

the data where the actual area of a sampled stand that is in

self-thinning mode is often poorly deﬁned. Indeed, despite

the intensity of sampling efforts, only some stands are

actually in self-thinning mode (Weiskittel et al. 2009;

Vospernik and Sterba 2015). Pests, insects, disease, and

wind storms can reduce tree densities more quickly than

natural mortality (Ray et al. 2017). Forests are already

adapting to a changing climate with a plastic reaction

across different ecological regions (O’Neill et al. 2008).

Many forest tree species are still growing under different

ecological regimes than those observed in the past (Pecchi

et al. 2019), shaping their spatial distribution and inﬂu-

encing their ecological dynamics. This could generate

unexpected stresses which might alter the canonical self-

thinning rule. In all the cases, it is necessary to monitor

forest resources carefully as well as to adjust statistical

analyses and mathematical models that are insensitive to

observations in under-stocked conditions. Although some

efforts have been made to address these problems (Bi et al.

2000), limitations still persist. Stochastic natural impacts to

stands might be represented by introduction of random

noise to the artiﬁcial dataset. Most recommended analytical

Fig. 5 SDI curves calculated with different kvalues for the same N

and QMD dataset; kvalues here were adopted according to detected

difference for class 5 (±0.03). N = number of trees/ha; QMD =

quadratic mean diameter of the stand; SDI = stand density index

Table 2 Results of Duncan test where the difference between Power

Law and Ordinary Least Squares in k estimation was statistically

evaluated

Correlation class Mean Group

5(-0.6 Bq\-0.5) 0.3013 a

4(-0.7 Bq\-0.6) 0.2968 ab

3(-0.8 Bq\-0.7) 0.2876 ab

2(-0.9 Bq\-0.8) 0.2810 abc

1(-1.0 Bq\-0.9) 0.2769 bc

6(-0.5 Bq\-0.4) 0.2627 c

7(-0.4 Bq\-0.3) 0.1496 d

8(-0.3 Bq) 0.0339 e

1600 M. Marchi

123

approaches would suggest cleaning the dataset prior to

modelling by removing the non-asymptotic size-density

values from the data sets. However, this issue is irrelevant

to this study where the equivalence between methods was

tested rather than the ability of a method to yield a real

value.

Concerning the use of classic modelling techniques (i.e.,

OLS or PWL) on longitudinal studies such as SDI calcu-

lation from long-term monitoring plots data, a possible

solution might call for mixed models, linear and nonlinear.

Such techniques, which uses the restricted maximum

likelihood algorithm, are known to be much more reliable

than OLS and the Gauss–Newton ﬁtting procedure when

autocorrelated data and large datasets are analysed, such as

those derived from long-term forest time series. Some

authors have tested this possibility (Solomon and Zhang

2002; Pourmajidian et al. 2010). In the current case, an

interesting solution could include the correlation class as a

random term with the QMD as a ﬁxed effect. In this case,

the mixed models ‘‘gain’’ could refer to the reduction of

mean squared error from estimating coefﬁcients rather than

from the use of separate models. However, possible biases

in kestimation should also be evaluated as stressed by our

results.

As regards possible solutions for the detected problem,

even if a simple mathematical correction could be thought

to be applied, no general rules could be provided. The bias

correction for log transformation is generally simple when

data are available. Anyway, this correction is aimed to

adjust predicted values and not for coefﬁcients. In addition,

there is a chance, probably quite high, that the ﬁndings will

not hold, meaning they are as they are just because of the

bias. In this view, the use of PWL is preferred, with a

logarithmic transformation just in case of violation of

parametric assumptions (non-normality of data) to deal

with this.

Conclusions

Understanding long-term dynamics is fundamental for

sustainable forest management. The stand density model is

a well-known way to evaluate stand loading and to derive

the maximum stand density index in even-aged forests.

Even if the 100,000 simulated plots could include a wide

spectrum of study cases, empirical results from long-term

monitoring networks are necessary to support the evidences

obtained via simulation. In conclusion, given the power

law as the ‘‘natural’’ structure of the N versus QMD rela-

tionship, the nonlinear method is preferred, thus avoiding

the logarithmic transformation.

Open Access This article is distributed under the terms of the

Creative Commons Attribution 4.0 International License (http://crea

tivecommons.org/licenses/by/4.0/), which permits unrestricted use,

distribution, and reproduction in any medium, provided you give

appropriate credit to the original author(s) and the source, provide a

link to the Creative Commons license, and indicate if changes were

made.

References

Anfodillo T, Carrer M, Simini F, Popa I, BanavarJR MaritanA (2013)

An allometry-based approach for understanding forest structure,

predicting tree-size distribution and assessing the degree of

disturbance. Proc Biol Sci 280:20122375. https://doi.org/10.

1098/rspb.2012.2375

Baskerville GL (1972) Use of logarithmic regression in the estimation

of plant biomass. Can J For 2:49–53. https://doi.org/10.1139/

cjfr-2018-0119

Bi H, Wan G, Turvey ND (2000) Estimating the self-thinning

boundary line as a density-dependent stochastic biomass frontier.

Ecology 81:1477–1483. https://doi.org/10.1890/0012-

9658(2000)081%5b1477:ETSTBL%5d2.0.CO;2

Cabon A, Mouillot F, Lempereur M, Ourcival J, Simioni G, Limousin

J (2018) Thinning increases tree growth by delaying drought-

induced growth cessation in a Mediterranean evergreen oak

coppice. For Ecol Manag 409:333–342. https://doi.org/10.1016/j.

foreco.2017.11.030

Castagneri D, Vacchiano G, Lingua E, Motta R (2008) Analysis of

intraspeciﬁc competition in two subalpine Norway spruce (Picea

abies (L.) Karst.) stands in Paneveggio (Trento, Italy). For Ecol

Manag 255:651–659. https://doi.org/10.1016/j.foreco.2007.09.

041

Corona P (2015) Forestry research to support the transition towards a

bio-based economy. Ann Silvic Res 38:37–38. https://doi.org/10.

12899/asr-1015

Ducey MJ, Woodall CW, Bravo-Oviedo A (2017) Climate and

species functional traits inﬂuence maximum live tree stocking in

the Lake States, USA. For Ecol Manag 386:51–61. https://doi.

org/10.1016/j.foreco.2016.12.007

Fabbio G, Cantiani P, Ferretti F, Di Salvatore U, Bertini G, Becagli C,

Chiavetta U, Marchi M, Salvati L (2018) Sustainable land

management, adaptive silviculture, and new forest challenges:

evidence from a latitudinal gradient in Italy. Sustainability

10:2520. https://doi.org/10.3390/su10072520

Fassnacht FE, Hartig F, Latiﬁ H, Bergerd C, Herna

´ndez J, Corvala

´nP,

Koch B (2014) Importance of sample size, data type and

prediction method for remote sensing-based estimations of

aboveground forest biomass. Remote Sens Environ

154:102–114. https://doi.org/10.1016/j.rse.2014.07.028

Ferrara C, Marchi M, Fares S, Salvati L (2017) Sampling strategies

for high quality time-series of climatic variables in forest

resource assessment. iForest - Biogeosci For 10:739–745. https://

doi.org/10.3832/ifor2427-010

Fonseca T, Duarte J (2017) A silvicultural stand density model to

control understory in maritime pine stands. iForest - Biogeosci

For 10:829–836. https://doi.org/10.3832/ifor2173-010

Ge F, Zeng W, Ma W, Meng J (2017) Does the slope of the self-

thinning line remain a constant value across different site

qualities? An implication for plantation density management.

Forests 8:355. https://doi.org/10.3390/f8100355

Liira J, Sepp T, Kohv K (2011) The ecology of tree regeneration in

mature and old forests: combined knowledge for sustainable

forest management. J For Res 16:184–193. https://doi.org/10.

1007/s10310-011-0257-6

Nonlinear versus linearised model on stand density model ﬁtting and stand density index…1601

123

Marchi M, Chiavetta U, Cantiani P (2017a) Assessing the mechanical

stability of trees in artiﬁcial plantations of Pinus nigra J.

F. Arnold using the LWN tool under different site indexes. Ann

Silvic Res 41:48–53. https://doi.org/10.12899/asr-1312

Marchi M, Ferrara C, Bertini G, Fares S, Salvati L (2017b) A

sampling design strategy to reduce survey costs in forest

monitoring. Ecol Indic 81:182–191. https://doi.org/10.1016/j.

ecolind.2017.05.011

Marchi M, Paletto A, Cantiani P, Bianchetto E, De Meo I (2018)

Comparing thinning system effects on ecosystem services

provision in artiﬁcial black pine (Pinus nigra J. F. Arnold)

forests. Forests 9:188. https://doi.org/10.3390/f9040188

Mason B, Connolly T (2016) Long-term development of experimental

mixtures of Scots pine (Pinus sylvestris L.) and silver birch

(Betula pendula Roth.) in northern Britain. Ann Silvic Res

40:11–18. https://doi.org/10.12899/ASR-1119

Newman MC (1993) Regression analysis of log-transformed data:

statistical bias and its correction. Environ Toxicol Chem

12:1129–1133. https://doi.org/10.1002/etc.5620120618

Newton PF (2006) Asymptotic size–density relationships within self-

thinning black spruce and jack pine stand-types: parameter

estimation and model reformulations. For Ecol Manag

226:49–59. https://doi.org/10.1016/J.FORECO.2006.01.023

O’Neill GA, Hamann A, Wang TL (2008) Accounting for population

variation improves estimates of the impact of climate change on

species’ growth and distribution. J Appl Ecol 45:1040–1049.

https://doi.org/10.1111/j.1365-2664.2008.01472.x

Packard GC (2013) Is logarithmic transformation necessary in

allometry? Biol J Linn Soc 109:476–486. https://doi.org/10.

1111/bij.12038

Packard GC (2014) On the use of log-transformation versus nonlinear

regression for analyzing biological power laws. Biol J Linn Soc

113:1167–1178

Pecchi M, Marchi M, Giannetti F, Bernetti I, Bindi M, Moriondo M,

Maselli F, Fibbi L, Corona P, Travaglini D, Chirici G (2019)

Reviewing climatic traits for the main forest tree species in Italy.

iForest - Biogeosci For 12:173–180. https://doi.org/10.3832/

ifor2835-012

Pommerening A, Sa

¨rkka

¨A (2013) What mark variograms tell about

spatial plant interactions. Ecol Modell 251:64–72. https://doi.

org/10.1016/j.ecolmodel.2012.12.009

Poschenrieder W, Biber P, Pretzsch H (2018) An inventory-based

regeneration biomass model to initialize landscape scale simu-

lation scenarios. Forests 9:212. https://doi.org/10.3390/f9040212

Pourmajidian MR, Jalilvand H, Fallah A, Hosseini SA, Parsakhoo A,

Vosoughian A, Rahmani A (2010) Effect of shelterwood cutting

method on forest regeneration and stand structure in a Hyrcanian

forest ecosystem. J For Res 21:265–272. https://doi.org/10.1007/

s11676-010-0070-7

Pretzsch H, Biber P (2005) A re-evaluation of Reineke’ s rule and

stand density index. For Sci 51:304–320

Ray D, Petr M, Mullett M, Bathgate A, Marchi M, Beauchamp K

(2017) A simulation-based approach to assess forest policy

options under biotic and abiotic climate change impacts: A case

study on Scotland’s National Forest Estate. For Policy Econ.

https://doi.org/10.1016/j.forpol.2017.10.010

Reineke LH (1933) Perfecting a stand-density index for even-aged

forests. J Agric Res 46:627–638

Rivoire M, Le Moguedec G (2012) A generalized self-thinning

relationship for multi-species and mixed-size forests. Ann For

Sci 69:207–219. https://doi.org/10.1007/s13595-011-0158-z

Shaw JD (2006) Reineke’s stand density index: where are we and

where do we go from here? Proceedings: Society of American

Foresters 2005 National Convention. October 19–23, 2005, Ft.

Worth, TX [published on CD-ROM]: Society of American

Foresters, Bethesda, p 13

Smith RJ (1993) Logarithmic transformation bias in allometry. Am J

Phys Anthropol 90:215–228. https://doi.org/10.1002/ajpa.

1330900208

Solomon DS, Zhang L (2002) Maximum size-density relationships for

mixed softwoods in the northeastern USA. For Ecol Manag

155:163–170. https://doi.org/10.1016/S0378-1127(01)00556-4

Spearman C (1987) The proof and measurement of association

between two things. By C. Spearman, 1904. Am J Psychol

100:441–471. https://doi.org/10.1037/h0065390

Vacchiano G (2005) Valutazione dello Stand density index in

popolamenti di abete bianco (Abies alba Mill.). Ital For e Mont

3:269–286

Vospernik S, Sterba H (2015) Do competition-density rule and self-

thinning rule agree? Ann For Sci 72:379–390. https://doi.org/10.

1007/s13595-014-0433-x

Wang Y, Titus SJ, LeMay VM (1998) Relationships between tree

slenderness coefﬁcients and tree or stand characteristics for

major species in boreal mixedwood forests. Can J For Res

28:1171–1183. https://doi.org/10.1139/x98-092

Weiskittel A, Gould P, Temesgen H (2009) Sources of variation in the

self-thinning boundary line for three species with varying levels

of shade tolerance. For Sci 55:84–93

Westoby M (1984) The self-thinning rule. Adv Ecol Res 14:167–225.

https://doi.org/10.1016/S0065-2504(08)60171-3

Yoda K, Kira T, Ogawa H, Hozumi K (1963) Self-thinning in

overcrowded pure stands under cultivated and natural conditions

(intraspeciﬁc competition among higher plants XI). J Biol

14:107–129

Zhang L, Bi H, Gove JH, Heath LS (2005) A comparison of

alternative methods for estimating the self-thinning boundary

line. Can J For Res 35:1507–1514. https://doi.org/10.1139/x05-

070

Zhang X, Zhang J, Duan A, Deng Y (2015) A hierarchical Bayesian

model to predict self-thinning line for Chinese ﬁr in Southern

China. PLoS ONE 10:e0139788. https://doi.org/10.1371/journal.

pone.0139788

Publisher’s Note Springer Nature remains neutral with regard to

jurisdictional claims in published maps and institutional afﬁliations.

1602 M. Marchi

123