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Forest Ecology and Management 204 (2005) 145–158

A finite mixture distribution approach for characterizing

tree diameter distributions by natural social class

in pure even-aged Scots pine stands in Poland

Michał Zasadaa,b,*, Chris J. Cieszewskia

aWarnell School of Forest Resources, University of Georgia, Athens, GA, USAbDivision of Dendrometry and Forest Productivity, Faculty of Forestry, Warsaw Agricultural University,

Nowoursynowska 159, 02-776 Warsaw, Poland

Received 6 June 2003; received in revised form 7 July 2003; accepted 4 December 2003

Abstract

The overall diameter distribution of an even-aged stand can be treated as a compound of distributions of trees belonging to

various classes. The presented study shows the applicability of a finite mixture distribution approach for describing diameter

structure of such classes on an example of pure, even-aged Scots pine stands. We performed our study under an assumption that

all starting values for parameter fitting should be available as a result of a routine forest inventory. When the parameters of the

mixture are fitted to the unimodal distributions without any constraints, this approach often fails. This is because components of

such distributions strongly overlap, and the fact of overlapping falsifies information about individual components. For a mixture

forming a unimodal distribution, due to difficulties in fitting, it is necessary to use functions that exhibit certain skewness and are

relatively flexible, such as lognormal or gamma. There is also a need for providing better start information, keeping certain

parameters fixed, or using a multi-stage fitting. Despite possible problems with fitting parameters to the strongly overlapping

distributions, this approach shows much potential in practical applications in silviculture and growth and yield modeling.

# 2004 Elsevier B.V. All rights reserved.

Keywords: Biosocial classification; Crown class; Kraft classification; Stand structure; Diameter distribution; Mixture distribution; Compound

distribution; Univariate distribution; Growth model

1. Background

In many cases, especially during dbh measure-

ments, foresters classify trees for more detailed stand

* Corresponding author. Tel.: +48 22 593 8086.

E-mail address: mzasada@dendro.sggw.waw.pl (M. Zasada).

0378-1127/$ – see front matter # 2004 Elsevier B.V. All rights reserved

doi:10.1016/j.foreco.2003.12.023

descriptions. In 1884, Kraft created what would

become one of the most widely used tree classifica-

tions (Kraft, 1884). This classification, often referred

to as ‘‘classic’’, is based on tree position (or

dominance) in the stand social structure and its crown

development and extent (Lonnroth, 1925; Assmann,

1961). Kraft recognized the following classes of trees:

.

M. Zasada, C.J. Cieszewski / Forest Ecology and Management 204 (2005) 145–158146

1. p

Fig

redominant/superior—reaching above the main

stand layer, with the biggest crowns,

2. d

ominant—creating the main canopy of the stand,

having well developed crowns,

3. c

o-dominant—co-creating the main canopy of the

stand, but having more weakly developed crowns,

4. d

ominated/intermediate/suppressed—having

stunted crowns located (a) in the middle story or (b)

partially in the understory,

5. o

vertopped/completely suppressed (a) still alive or

(b) dying or already dead.

The classes are often combined into broader gr-

oups. For example classes 1–3 are often called the

upper story, or the dominant stand, and classes 4–5

the lower story, or the suppressed stand. Sometimes

classes 1–3 refer to the upper story, class ‘‘4’’ forms

a middle story, and class ‘‘5’’ is the lower story

(Assmann, 1961). Dominant and co-dominant trees

have been grouped as a single class by Champion and

Griffith (1948) or even divided into more detailed

classes (Gevorkiantz et al., 1943). Kraft categories

are also a starting point for some other tree classi-

fications, such as thinning, natural, Assmann,

Schadelin-Hausrath and many others (e.g., Hawley,

1921; Assmann, 1961).

. 1. Even-aged Scots pine stand (BD150) diameter distribution consis

The described social structure exists in natural,

mostly even-aged stands. It is affected by stocking,

and it also depends strongly on human activity,

especially thinnings. Belonging to a given social

class reflects a position of a tree in a stand, and

through this, its growth potential (Oliver and Larson,

1996). Therefore, this additional information can be

used for example when planning selection thinning

in the stand (for choosing and marking the most

promising/crop trees in the stand). Also when

modeling growth and yield of stands, especially

using an individual-tree approach, it might be

desirable to use the social position of the tree as

an additional variable in the height growth model

(Goff and West, 1975; Monserud and Ek, 1977,

1979; Ritchie and Hann, 1986), as part of the

between-tree competition and tree mortality assess-

ment (Keister and Tidwell, 1975; Monserud, 1976),

or as a part of the thinning algorithm in the model

(Bruchwald, 1986, 1988b).

Crown classification or tree social position can be

also translated into tree diameters. The total diameter

distribution of trees can be broken down into sub-

distributions for each social class (e.g., Lonnroth,

1925; Assmann, 1970; Wroblewski, 1984; see Fig. 1).

However, Assmann (1961), describing a typical

structure of stands according to tree classes and

ting of (a) Kraft classes (left) and (b) natural social classes (right).

M. Zasada, C.J. Cieszewski / Forest Ecology and Management 204 (2005) 145–158 147

stories noticed that complete correspondence between

the classes in the diameter distribution and the social

or biological tree classes is unattainable because the

social tree classes usually overlap within a few

diameter ranges. This overlapping exists partially as a

result of errors in classification, and partially because

of a stochastic relationship between tree heights and

diameters and spatial variability of tree dimensions in

the stand. Assmann claims also that the diameter

distribution curves of height layers in the stand as well

as social classes can be closely approximated by the

normal distribution. Wroblewski (1984, 1993) pro-

vides a detailed description of parameters of tree

social classes as well as equations for determining

attributes of diameter distributions in Kraft biosocial

classes in Scots pine stands in Poland.

For at least a hundred years foresters have written

about how to describe stand diameter distributions

using theoretical models. Various such models have

been used, among others, for silvicultural purposes

(e.g., Meyer, 1952; Leak, 1964), to determine a stand’s

developmental stages (e.g., Poznanski, 1997; Goelz

and Leduc, 2002), and to build stand yield tables and

growth models (e.g., Lenhard and Clutter, 1971;

Bailey and Dell, 1973; Clutter et al., 1984; Borders et

al., 1987; Shiver, 1988; Borders and Patterson, 1990).

Modeling of the diameter distribution of pure even-

aged stands is a relatively simple issue and it is

possible using almost any theoretical distribution

(such as normal, beta, gamma, Weibull, etc.).

However, the history of mathematical modeling of

diameter distributions started not from the simplest

cases. In 1898, de Liocourt published a paper showing

the diameter distribution of uneven-aged forests as a

geometrical series. Meyer and Stevenson (1943) used

this relationship in an exponential form for modeling

of mixed stand structure in Pennsylvania. Meyer

(1952) and Leak (1964, 1965) used the Liocourt curve

also in their later studies.

Meyer (1930) developed probably the first math-

ematical model of the diameter distribution for even-

aged stands based on shortleaf pine stand data. We can

find much research done later on the utilization of

various theoretical functions for diameter distribution

modeling, such as Pearson distributions (Schnur,

1934), Pearl-Reed growth curves (Osborne and

Schumacher, 1935; Nelson, 1964), gamma distribu-

tion (Nelson, 1964), lognormal distribution (Bliss and

Reinker, 1964), beta distribution (Clutter and Bennett,

1965; McGee and Della-Bianca, 1967; Lenhard and

Clutter, 1971), Weibull distribution (Bailey and Dell,

1973), SB Johnson distribution (Hafley and Shreuder,

1977; Siekierski, 1992), double-normal distribution

(Bruchwald, 1988a; Siekierski, 1991), and much

more.

The idea of describing the tree diameter structure

by a set of overlapping component distribution is not

new. Lonnroth (1925) described examples of tree

diameter distributions consisting of compound layer

distributions. He used a few normal distributions to

describe separate social classes for Scots pine and

obtained the distribution of the whole stand as a

combination of these components. Assmann (1961)

also claimed that the diameter distribution curves of

height layers in the stand as well as social classes

could be closely approximated by the normal

distribution. Also, more than 30 years ago Bailey

and Dell (1973) noted that in many cases the diameter

structure of the stand could be too complicated to be

modeled by a single function. They proposed using a

mixture of distributions in such situations and even

provided a couple of references on the subject (Falls,

1967; Mason, 1967). Similar propositions can be

found also in publications by other authors (e.g., Goff

and West, 1975). Maltamo (1997) modeled distribu-

tions of Scots pine and Norway spruce components in

a mixed stand. He obtained more accurate results for

individual species distribution parameters when the

distributions were formed using separate models for

each species rather than when using models for the

entire stand.

The first practical applications of the mixture

distribution approach for tree diameter distributions

were published by Liu et al. (2002) and Zhang et al.

(2001). Liu et al. (2002) provide an example of a finite

mixture application for bimodal diameter distributions

of mixed stands in the northeastern United States.

They used the Weibull function as a component of the

mixture. Analyzed mixtures were flexible enough to fit

irregular, multimodal, and extremely asymmetric

diameter distributions. Zhang et al. (2001) used a

mixture of two Weibull distributions to model rotated-

sigmoid tree diameter distributions of four mixed

stands. In comparison to the single Weibull function,

only the mixture was able to fit the overall distribution,

and gave a standard error at least four times smaller

M. Zasada, C.J. Cieszewski / Forest Ecology and Management 204 (2005) 145–158148

than the other analyzed distributions. The authors

proved that the mixture distribution is a good

alternative for modeling diameter distributions of

mixed and uneven-aged stands. Hessenmoller and von

Gadow (2001) used a different approach, showing

applicability of two joined distributions to describe

bimodal diameter distributions of beech stands in

Germany. The vertical structure of those stands

strongly depends on applied thinnings. During

thinning from above, there are only thicker trees

taken from the stand, while thin trees can survive for a

long time, forming a typical structure consisting of

two constituent populations. The authors fitted two

Weibull distributions using minimum and maximum

stand dbh, stand height and basal area as explanatory

variables. The joining point of the distributions was

estimated using the proportion of trees in the lower

layer of the stand. Another approach for using mixture

distributions was described by Zucchini et al. (2001).

The authors used a multivariate mixture model to

describe diameters and heights of trees in uneven-aged

beech stands. They used a combination of two two-

dimensional normal distributions. One of the advan-

tages of the developed model is that the model

parameters have a known practical interpretation.

Different authors tried also to apply other methods for

the tree diameter distribution modeling, such as seg-

mented distributions (Cao and Burkhart, 1984; Goelz

and Leduc, 2002), percentile approach (Borders et al.,

1987) or neural networks (Leduc et al., 1999, 2001).

Even though the social structure of even-aged

stands of Scots pine is fairly well known (e.g.,

Lonnroth, 1925; Assmann, 1961; Wroblewski, 1984,

1993), as are the effects of tree social position on its

growth (e.g., Assmann, 1961; Krumland and Wensel,

1988; Tarasiuk and Zwieniecki, 1990; Hanus et al.,

1999), this additional information was not yet

extensively used in growth and yield modeling. This

is partially because there are practically no methods

for post-measurement classification of trees into

appropriate groups.

The objective of the presented study was to analyze

the social structure of the even-aged stands using finite

mixture distributions and broad tree classes of trees

based on the original Kraft classification. This

development included analyses of various theoretical

distributions, different estimation methods and start-

ing assumptions. Analyzed classes were limited to two

groups of trees only. The first class, comprised of trees

belonging to classes 1–3 according to Kraft, was

referred to as a dominant stand. The remaining trees

(classes 4 and 5) were referred to as a suppressed or

dominated stand (Assmann, 1961). Such distinguished

groups can be also approximately defined using the

crown classification used in North America (Hawley,

1921, p. 155): dominant and co-dominant trees form

the dominant stand, and intermediate and overtopped/

suppressed trees compose the suppressed/dominated

stand. Our goal was to propose the best way of

handling the overall diameter distribution as a sum of

natural social classes by suggesting applicable

theoretical distributions and analysis methods. We

assumed that all starting values for operational

parameter fitting should be available as a result of a

routine forest inventory, which means dbh measure-

ments only, without any additional classification of

trees or their crowns.

2. Data

In our study, we used the diameter distribution data

of 43 well stocked, pure, even-aged Scots pine (Pinus

sylvestris L.) stands provided by the Department of

Forest Productivity at the Faculty of Forestry in

Warsaw, Poland. The sample plots were located on

typical pine sites (coniferous forest habitats) in two

large forest complexes: Bory Dolnoslaskie (Brody,

Lubsko and Gubin Forest Districts) in western Poland,

and Pisz Primeval Forest (Spychowo and Krutyn

Forest Districts) in the northeastern part of the country.

Areas of plots were set such that they comprised a few

hundred trees. The breast height diameters (dbh) of all

trees were measured on the sample plots with an

accuracy of 1 mm. Each measured tree was classified

as belonging to the appropriate Kraft class. Each plot

was also characterized by its average total age

determined by counting rings on stumps of felled

trees and sample cores, average height defined using

the Lorey formula, and site index defined as the

average height of the 100 thickest trees per hectare at

base age 100 years and calculated using the site index

model for Scots pine in Poland (Bruchwald et al.,

2000; Cieszewski and Zasada, 2002). Table 1

summarizes the univariate statistics for the data used

in the analysis.

M. Zasada, C.J. Cieszewski / Forest Ecology and Management 204 (2005) 145–158 149

Table 1

Summary statistics of the 43 sample plots used in the study

Area Age SI N QMD H Ndom/N mudom/mu musup/mu Sdom/S Ssup/S

Minimum 0.063 21 12.7 299 52 6.0 0.50 1.01 0.69 0.71 0.41

Maximum 1.680 119 32.5 568 331 24.6 0.91 1.29 0.85 0.96 1.13

Average 0.553 63 23.0 436 181 16.5 0.70 1.10 0.75 0.85 0.60

Notation: Area, size of the sample plot; Age, average stand age; SI, site index; N, number of trees per plot; QMD, quadratic mean diameter of the

stand; mu, mean dbh of trees in the sample plot; S, standard deviation of trees’ dbh; dom, sup, subscripts indicating trees belonging to dominant

and suppressed stand, respectively.

3. Methods

Detailed description of the finite mixture approach

is available in the literature (e.g., Everitt and Hand,

1981; Titterington et al., 1985; McLachlan and Peel,

2000), and was provided by Liu et al. (2002), but since

this method is not yet commonly known among

foresters, we decided to provide a brief theoretical

introduction.

A mixture distribution can be obtained when

analyzing a heterogeneous population. As far as forest

stands are internally differentiated, we can talk about

their structure in terms of diameter, height, age,

species, layers, or social classes. The whole tree

population could be treated as a mixture of more or

less distinguishable groups.

Mathematically, the mixed probability density

function p is a weighted sum of k component densities

(Titterington et al., 1985):

pðxjcÞ ¼ p1f1ðxju1Þ þ � � � þ pkfkðxjukÞ or

pðxjcÞ ¼Xk

j¼1

pjfjðxjujÞ

where j = 1. . .k; 1 � pj � 0; p1 + � � � + pk = 1;

fj(x) � 0;P

fj(x) dx = 1; uj denotes parameters of

the fj(x) distribution, and C is a complete parameter

set for the overall distribution.

Is such a case, we say that the variable X has a finite

mixture distribution, and p(xjc) is a finite mixture

probability density function with a parameter vector

c. Parameters p1. . .pk are weights assigned to

distribution components, and f1(xju1). . .fk(xjuk) are

component densities of the mixture having their

parameters uj. For example, a function consisting of

two normal distributions (Fig. 2) can be written as:

pðxjcÞ ¼ pfðxjm1; s1Þ þ ð1 � pÞfðxjm2; s2Þ

where f(xjmj, sj) denotes the normal univariate dis-

tribution with mean mj and standard deviation sj,

p1 = p, p2 = (1 � p), u1 = (m1, s1), u2 = (m2, s2),

and finally c = (p, m1, m2, s1, s2).

Obviously, the components of the mixture model

can be described by any discrete or continuous

distribution. The parameters of the mixture can be

estimated using the method of moments (Pearson,

1894), the graphical approach (Kao, 1959), or the

maximum likelihood method (Rao, 1948; Rider, 1961;

Falls, 1967; Mason, 1967). Recent studies that use

finite mixture models rely on the maximum likelihood

method and its EM algorithm (Dempster et al., 1977).

Analyses of distributions expected to be mixtures

rely on finding a set of overlapping components that

provide the best fit to the summary distribution

(Figs. 2a and 3). A complete set of parameters of a

mixture consists of parameters of the individual

distribution components as well as the proportion of

the components. Analyses of the mixture distributions

and estimation of their parameters are always

performed under an assumption of a certain number

of constituent distributions. The problem arises when

comparable results can be obtained using different

numbers of components, as shown, e.g., by Basford et

al. (1997). Some researchers proposed using statistical

tests for choosing a number of components (e.g.,

Wolfe, 1970; Seidel et al., 2000). We did not address

this problem in the paper. Following suggestions

included in various publications on the subject (e.g.,

Everitt and Hand, 1981), we assumed the number of

components based on the analysis of the problem and

on the purpose of the research. Since we analyzed only

two natural classes of trees in the stand (dominant and

suppressed stand, Fig. 3), we assumed that the number

of components in a mixture was equal to two. This

does not preclude the fact that analyzed classes can

comprise sub-classes. Obviously, in the case of

M. Zasada, C.J. Cieszewski / Forest Ecology and Management 204 (2005) 145–158150

Fig. 2. Karl Pearson’s crab data (1822) fitted using (a) two normal (left) and (b) one Weibull (right) distributions (MacDonald, 1988,

reproduction permitted by the Author).

Fig. 3. Sample results of fitting the mixture to the data from BD150

sample plot using the two-stage method with adjustment of propor-

tions. The criterion of fitting was the best estimation of the overall

diameter distribution.

analyzing any other division of the stand (such as e.g.,

detailed Kraft classes), we would have to change this

assumption accordingly.

We performed all analyses using ‘‘R’’ software, a

free of charge language and environment for statistical

computing and graphical data presentation. It was

created by Ihaka and Gentleman (1996) and developed

as an Open Source project under the GNU General

Public License. For estimation of the mixture

parameters we used functions and procedures included

in the Rmix package. The package is an implementa-

tion of Peter Macdonald’s MIX software (Macdonald

and Pitcher, 1979) ported by the author and his

collaborators from Fortran to ‘‘R’’. Rmix fits mixture

distributions to grouped data by the maximum

likelihood method using EM and quasi-Newton

algorithms. The estimated parameters can be con-

strained in many different ways to help with fitting

when component distributions strongly overlap.

For each analyzed plot and scenario we calculated

the parameters of constituent distributions (means

and standard deviations), mixing proportions and

their standard errors as well as a goodness-of-fit

(x2-test) of the overall distribution, and compared the

estimated parameters to their real values calculated

from the available data. We took into account three

theoretical distributions, normal, beta and gamma, as

components of the mixture. We also used different

sets of starting parameters for the dominant and sup-

pressed stands. The means and standard deviations of

M. Zasada, C.J. Cieszewski / Forest Ecology and Management 204 (2005) 145–158 151

constituent distributions were in one case assumed as

equal to their values in the overall diameter

distribution, and in the other case calculated by

multiplying the parameters of the total observed

distribution by average ratios of means and standard

deviations in a given class derived from the source

data. Because this approach requires having additional

information/relationships available from data, we also

examined starting values obtained without using

relationships derived from the data, following the

method described by Seidel and Sevcıkova (2002).

The authors used in their studies starting values of

means calculated from the mean value of the overall

distribution multiplied by the arbitrarily chosen values

of 0.5 and 1.5 (0:5x and 1:5x). In our case, fitting

results were the same as when using relationships

derived from the data. If the starting values for

component proportions were required, we calculated

their initial values using the simple nonlinear model

relating the proportion of dominant and codominant

trees in the stand to its average breast height diameter.

Parameters of the chosen power function were

calculated using a nonlinear least square function

from the ‘‘nls’’ library of the ‘‘R’’ software. We

present the summary of formulas for starting values

calculations in Table 2.

First we examined an unconstrained method of

fitting. Then we examined other methods of fitting

with some of the parameters set as fixed. We had to

keep certain parameters fixed to assure feasibility of

fitting and to check the best way of constraining

particular parameters during fitting. The analyzed

variants included keeping the same coefficient of

variation of dbh in both constituent distributions, and

fitting the mixture in two steps. We used two variants

of the two-stage fitting. In the first variant we initially

estimated the proportions by fitting the mixture with

both means and standard deviations fixed, and later we

Table 2

The formulas for calculation of starting values of means, standard deviat

Variable

Mean of the first component (suppressed stand)

Mean of the second component (dominant stand)

Standard deviation of the first component (suppressed stand)

Standard deviation of the second component (dominant stand)

Proportion of the first component (suppressed stand)

Proportion of the second component (dominant stand)

adjusted means and standard deviations by fitting the

mixture with fixed values of proportions obtained in

the first phase. In the second variant first we estimated

the means and standard deviations by fitting the

mixture with fixed proportions, and then we adjusted

proportions by fitting the mixture with fixed values of

means and standard deviations that were obtained in

the first phase.

For comparison, we ranked each variant of fitting

based on the overall distribution fit and for results of

means, standard deviations, and the mixing propor-

tions. Ranking of the distribution fitting was based on

values of x2-statistic and ranking of the parameter fit

was based on differences between estimated and

observed values of parameters. We assigned ‘‘1’’ for

the case with the best fit/lowest difference, ‘‘2’’ for the

second, and ‘‘3’’ for the last one. We also calculated a

bias and the root mean square error (RMSE) of the

fitting, following the approach used by Maltamo et al.

(1995), Liu et al. (2002), and Zhang et al. (2001) in

similar studies. The variant with the lowest sum of

ranks, lowest bias, and lowest RMSE was considered

as the most suitable for potential practical use.

4. Results

We started fitting parameters of mixtures using

means and standard deviations of the overall distribu-

tion as initial values. However, the performance of this

approach was in many cases unsatisfactory due to

problems with convergence of the estimation process.

To improve the fitting process, we replaced starting

parameter values with those closer to their real values,

as described above. Results of unconstrained fitting for

normal, lognormal and gamma mixtures are presented

in Table 3a. The hypothesis that the compound

distribution is the mixture of distributions with fitted

ions and mixture proportions

Symbol Formula

musup mu 0.75 and mu 0.5

mudom mu 1.1 and mu 1.5

Ssup S 0.6

Sdom S 0.85

pisup 1 � pidom

pidom 0.18079 mu0.26647

M. Zasada, C.J. Cieszewski / Forest Ecology and Management 204 (2005) 145–158152

Table 3

Results of the (a) unconstrained fit, (b) fit with the same coefficients of variation of dbh in both constituent distributions, (c) two-stage fit with the

adjustment of means and standard deviations, and (d) two-stage fit with the adjustment of proportions

Overall fit Proportions mudom musup Sdom Ssup

FitP

Rank FitP

Rank FitP

Rank FitP

Rank FitP

Rank FitP

Rank

(a) Unconstrained fit

Normal 38 113 29 82 29 85 25 77 25 104 27 78

Lognormal 38 83 29 83 30 75 28 86 36 72 27 93

Gamma 40 62 28 83 29 82 28 82 32 75 28 85

(b) The same coefficients of variation

Normal 37 109 21 102 31 93 22 93 27 73

Lognormal 38 80 25 62 29 58 30 62 29 71

Gamma 39 69 29 73 32 76 30 73 27 70

(c) Two-stage fit with the adjustment of means and standard deviations

Normal 37 108 18 85 18 90 33 75 22 102 34 76

Lognormal 37 86 19 78 25 62 39 74 39 52 39 69

Gamma 39 64 17 62 23 76 37 79 35 62 38 65

(d) Two-stage fit with the adjustment of proportions

Normal 42 109 33 64 30 79 32 82 25 97 36 62

Lognormal 41 85 31 60 34 65 37 68 41 53 35 89

Gamma 42 64 32 62 35 70 37 75 40 59 38 68

‘‘Fit’’ denotes number of cases when the null hypothesis on the fitted distribution or parameter correspondence to their empirical counterparts

was not rejected, and ‘‘P

Rank’’, sum of ranks for each plot. Initial parameters were calculated according to formulas from Table 2. Goodness-

of-fit was tested using x2-test. Remaining parameters were tested using t-test. All tests were performed for a = 0.05 significance level. Note lack

of results for standard deviation of the dominant stand (Sdom) while fitting distributions with the same coefficient of variation (b); this is caused by

an assumption that value of Sdom is in this case held as fixed.

parameters could not be rejected 38 times in the cases of

normal and lognormal, and 40 times for the gamma

distribution. Because of the heavily overlapped

components, in some cases the EM algorithm used

for fitting failed. Also, fitted parameters (component

proportions, means, and standard deviations) were in

many cases significantly different from their empirical

values (see relatively low ‘‘Fit’’values for parameters in

Table 3a). The parameters matched up at a maximum of

70% of analyzed plots. Keeping equal dbh coefficients

of variation in both sub-distributions did not provide

much improvement over the unconstrained fitting

(Table 3b). Results of the two-stage scenarios are

presented in Table 3c and d. Detailed values of bias and

RMSE for all analyzed distributions and fitting variants

are presented in Table 4. Constraining parameters did

not help much in obtaining better fit statistics (see

values of RMSE and x2 in Table 4), but in many cases

improved the ability of the mixture to fit particular

parameters (Table 3), especially when the two-stage

fitting was used. We present a sample result of such

fitting in Fig. 3.

In all fitting variants we obtained the best overall fit

using the gamma distribution. The gamma distribution

was the best on average three times more frequently

than a normal distribution, and also more frequently

than lognormal (Table 3). Values of RMSE for this

distribution are also lower than for normal and

lognormal ones with comparable low, insignificant

bias (Table 4). The fit of particular parameters

(proportions, means and standard deviations of

components) was usually better when using the

lognormal distribution (Table 3). However, the gamma

distribution also performed fairly well.

Results of particular parameters and distribution

fitting are shown in Table 5. To check which analyzed

theoretical distribution fits the data best, we calculated

a goodness-of-fit of the overall distributions with all

parameters fixed at their true values. The normal

distribution was able to fit the overall distribution in 32

cases, lognormal in 39, and gamma in 40 out of 43

analyzed plots. The best performance was shown by

the gamma distribution, which was superior to the

other functions in 24 out of 43 cases. All proportions,

M. Zasada, C.J. Cieszewski / Forest Ecology and Management 204 (2005) 145–158 153

Table 4

Detailed values of bias, RMSE, and ratio-likelihood x2 statistics for each analyzed plot and case: (a) unconstrained fit, (b) fit with the same

coefficients of variation of dbh in both constituent distributions, (c) two-stage fit with the adjustment of means and standard deviations, and (d)

two-stage fit with the adjustment of proportions for the mixture of (A) normal, (B) lognormal, and (C) gamma distributions

Value (a) (b) (c) (d)

Bias RMSE x2 Bias RMSE x2 Bias RMSE x2 Bias RMSE x2

(A) Mixture of normal distributions

Minimum �0.10 1.79 9.60 �0.07 1.97 9.84 �0.10 1.84 11.13 �0.09 1.83 10.93

Maximum 0.15 5.89 40.76 0.09 6.24 115.9 0.14 6.10 45.44 0.11 5.95 45.49

Mean 0.02 3.40 22.71 0.02 3.67 27.35 0.02 3.51 24.87 0.02 3.59 24.95

(B) Mixture of lognormal distributions

Minimum �0.10 1.55 4.34 �0.09 1.81 4.72 �0.10 1.77 4.68 �0.18 1.83 4.43

Maximum 0.12 5.77 46.03 0.09 6.97 46.42 0.14 6.50 49.03 0.15 5.77 47.41

Mean 0.01 3.31 19.89 0.00 3.42 21.58 0.00 3.49 22.20 0.01 3.42 21.95

(C) Mixture of gamma distributions

Minimum �0.10 1.81 4.31 �0.12 1.79 4.44 �0.10 1.80 4.39 �0.12 1.85 4.60

Maximum 0.12 5.71 44.16 0.11 5.48 46.99 0.14 5.67 45.81 0.17 5.67 45.78

Mean 0.02 3.25 19.52 0.01 3.32 21.30 0.02 3.35 21.51 0.01 3.32 21.46

means and standard deviations of dominant and

suppressed stands matched up their true values for

almost all analyzed plots. The only exceptions were

the standard deviation of a dominant stand fitted using

the normal distribution, and values of the standard

deviation of the suppressed stand. Even though we

obtained a satisfactory fit of particular parameters, in

some cases we were not able to fit the overall

distributions with such values of parameters, espe-

cially when we used the normal distribution. As for the

analysis of the overall fit, the gamma distribution was

superior to other analyzed distributions in fitting

parameters (Table 5).

5. Discussion

Analyses of the mixture distributions and estima-

tion of their parameters are always performed under

Table 5

Results of particular parameters fitting for various distributions with rem

Distribution Proportions mudom

Dist. Best Param. Dist. Param. Dis

Normal 32 3 43 32 43 32

Lognormal 39 16 43 38 43 39

Gamma 40 24 43 39 43 40

‘‘Dist.’’ means number of plots where the fit of the overall distribution was

‘‘Best’’ denotes number of cases with the best fit; ‘‘Param.’’ denotes numbe

a = 0.05 significance level.

the assumption of a certain number of components.

The issue of determining the number of components in

a mixture is still not satisfactorily resolved, especially

in the case of unimodal distributions. Various authors

have tried to use formal tests to address this problem

by comparing fits obtained with various numbers of

components (e.g., Wolfe, 1970, 1978). The fact of

unimodality, observed in this study, does not imply

lack of the mixture. Furthermore, in many cases one

flexible distribution fits a unimodal distribution as well

as a mixture (e.g., Pearson, 1894 versus Macdonald,

1988, see Fig. 2). This means that an appropriate fit

can be obtained using various numbers of components,

as shown by Basford et al. (1997), that in many cases

have no practical meaning. If there is no biological or

theoretical evidence that the population is a mixture,

the fact that a mixture of two distributions fits well

does not prove that there are two components in the

distribution. That is why in most cases it is

aining parameters held fixed

musup Sdom Ssup

t. Param. Dist. Param. Dist Param. Dist.

43 34 41 35 37 35

43 38 43 40 42 40

43 39 43 40 40 40

attainable with given set of parameters at a = 0.05 significance level;

r of plots where the parameter was not different from its true value at

M. Zasada, C.J. Cieszewski / Forest Ecology and Management 204 (2005) 145–158154

recommended to assume the number of components

based on some other analysis before fitting the

parameters of the mixture (Everitt and Hand, 1981,

p. 22) as we did.

For practical application, the simplest case would

occur where the parameters of the mixture distribu-

tions are fitted without any constraints. However, in

many cases this approach is not only non-optimal, but

also is often unable to converge the fit and give any

reasonable results. The real problems arise when the

components of the distribution strongly overlap, as in

all analyzed cases. Parameter estimation becomes a

difficult task even when the number of sub-distribu-

tions is assumed known. The fact of overlapping

falsifies information about individual components. In

such a case using better start information helps to fit

the mixture. Nevertheless, in many cases even the

starting values of parameters set close to their true

values do not provide the appropriate fit of the overall

distribution or mixture components. In such cases, the

most frequently used solution is to hold some of the

parameters fixed or constrained in a certain way (e.g.,

by having the same coefficient of variation, see

Macdonald and Green, 1988).

An illustration of difficulties in fitting using our

initial assumptions can be illustrated using an example

of the Weibull distribution (Weibull, 1951). We were

not able to use it even though it is one of the most

widely used functions for diameter distribution

modeling (e.g., Bailey and Dell, 1973; Garcıa, 1981;

Shiver, 1988; Borders and Patterson, 1990). If the

component distributions strongly overlap, serious

problems exist with using the EM algorithm and other

methods for fitting a mixture (e.g., Wolfe, 1967). The

standard two-parameter Weibull distribution does not

apply in a case of modeling diameter distributions. Its

location parameter (that is, the minimal value of the

analyzed variable) equals zero, and we can expect that

the minimal dbh value in most stands (especially for the

dominant stand) will be significantly greater than zero.

This is why we tried to use a three-parameter Weibull

function as a mixture compound. Unfortunately, there

are problems in estimating the three-parameter Weibull

distribution mixture using means and standard devia-

tions of components instead of location, shape and scale

parameters. When the components strongly overlap,

information about the shape is lost, so the estimation

process may fail (Peter Macdonald, personal commu-

nication). Liu et al. (2002) and Zhang et al. (2001)

successfully used a mixture of three-parameter Weibull

distributions in their studies, but they fit the mixture

using the classic approach (i.e., using location, shape

and scale parameters). We drop the use of Weibull for

the sake of technical compatibility with remaining

analyzed cases.

An additional source of uncertainty also exists

when analyzing the diameter distributions of any

classes distinguished in the stand, such as natural

social classes. This is because of the subjective nature

of these classes and common errors in classification

(Assmann, 1961). Liu et al. (2002) stated that in the

finite mixture distribution approach for the multi-

modal diameter distribution it is not necessary to

provide any additional information besides the

number of classes determined using the silvicultural

perspective. In case of unimodal distributions we must

provide not only additional information, such as

approximate mixture proportions or tree group means,

but also – in order to assure convergence of the fit – we

have to keep certain parameters fixed, or better – use a

multi-stage fitting.

Assmann (1961) and other researchers (e.g.,

Lonnroth, 1925; Wroblewski, 1993) claim that the

diameter distribution of height layers in the stand as

well as social classes can be closely approximated by

the normal distribution. Results obtained here clearly

showed that this can be true in some cases but such an

approach is not optimal. Much better results were

obtained when using distributions exhibiting certain

skewness even if the overall distribution was close to

symmetrical. However, the normal distribution

showed relatively good performance in unconstrained

fitting. This is probably because of the simpler form of

this distribution; fewer parameters/less flexibility

allow us to obtain a good fit even without a need to

constrain parameters.

Relatively big problems arise when coming to

decide which criteria should be used for choosing the

best fit. Good fit of particular parameters does not

imply good fitting of the overall distribution. At the

same time, good fit to the overall distribution does not

guarantee success in fitting single parameters.

Possibly the problem exists because the elasticity of

the used distributions is too low. Maybe overcoming

the problems with fitting a mixture of three-parameter

Weibull distribution would help to solve this problem.

M. Zasada, C.J. Cieszewski / Forest Ecology and Management 204 (2005) 145–158 155

Also, much more attention has to be put on calculation

of starting values of distribution parameters, espe-

cially distribution proportions. Once we have some

parameters of the mixture estimated from the

attributes of the whole stand, we can use them in

the finite mixture approach as fixed components, as for

example the parameter recovery described by Bailey

et al. (1982, 1985) in papers on structure of southern

pine plantations or the approach used by Hessenmoller

and von Gadow (2001). This helps to obtain good

values of remaining parameters. The variant of two-

stage fitting with starting values of proportions

calculated using a nonlinear relationship is an example

of such a solution.

In our study, we obtained the poorest fit for

standard deviations of constituent distributions,

especially that of the suppressed stand. However, it

seems that as long as the results for the overall

distribution and location parameters of constituent

distributions are satisfactory, possible errors coming

from this source should not cause significant problems

in the operational use of this approach.

Another problem we would like to discuss is related

to the sample size. So far, described analyses were based

on relatively big samples, consisting usually of a few

hundred trees.This fact dramatically limitsapplicability

of the method in practice, since in forest inventory

foresters usually deal with much smaller numbers of

measurements per plot. Resolving this issue and other

above-discussed problems could help to incorporate the

presented method into growth and yield models.

6. Summary and conclusion

In this study, we present applicability of the finite

mixture distribution approach for describing diameter

structure of natural tree classes based on the Kraft

classification on an example of pure, even-aged Scots

pine stands in Poland. Our main goal was to propose the

best way of handling the overall diameter distribution as

a sum of natural social classes in terms of both

applicable theoretical distributions and analysis meth-

ods. We performed our study under an assumption that

all starting values for parameter fitting should be

available as a result of a routine forest inventory.

For practical application, the simplest case occurs

where the parameters of the mixture distributions are

fitted without any constraints. However, in many cases

this approach fails when the components of the dist-

ribution strongly overlap, as in all analyzed plots. The

fact of overlapping falsifies information about indivi-

dual components. In such a case, using better start

information helps to fit the mixture. For unimodal

distributions, due to difficulties in fitting, we should also

keep certain parameters fixed and use a multi-stage

fitting. Significant improvement can be also obtained by

using appropriate distributions as the mixture compo-

nents. Such a distribution should exhibit certain

skewness and be relatively flexible (as gamma or

Weibull distributions).

Despite possible problems with fitting parameters to

the strongly overlapping distributions, especially using

small sample sizes, this approach shows much potential

in practical applications. However, high attention has to

be paid to choose good starting values of parameters

based on relationships existing between variables

within stands. This requires additional research effort.

This also calls for deeper research on the relation of the

mixture parameters to the parameters of the entire

stand. Furthermore, described analyses were based on

relatively big samples consisting of a few hundred trees.

This fact limits applicability of the method in practice,

since in forest inventory foresters usually deal with

much smaller numbers of measurements. Resolving the

above-mentioned problems could help to use the

described method in practical forestry applications.

Acknowledgments

The authors are grateful to Professor Peter D.M.

Macdonald from the Department of Mathematics and

Statistics, McMaster University, Canada, for making

the beta version of the Rmix library available for this

study, and for his invaluable support. We also thank the

associate editor and two anonymous reviewers for their

involvement, doubts, and inestimable suggestions.

Finally, thanks are due to Ingvar R. Elle for his help

in improving the English language usage in the paper.

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