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Plant Ecology 138: 77–87, 1998.
© 1998 Kluwer Academic Publishers. Printed in the Netherlands.
77
Analysis of vegetation structural diversity by Burnaby’s similarity index
L. Carranza, E. Feoli & P. Ganis
Department of Biology, University of Trieste, I-34127 Trieste, Italy
Received 12 December 1997; accepted in revised form 6 May 1998
Key words: Character state, Classification, Eigenvectors, Fuzzy ordination, Mixed data, Probability, Weight
Abstract
The application of Burnaby’s similarity index is discussed by using structural data from mediterranean vegetation.
The index, suggested to compare objects described by characters measured on different scales (mixed data), was
applied in a fuzzy theory context. Ordinations of vegetation relevés and structural characters by joint plots have
been obtained. These are very useful to map vegetation structural diversity in multidimensional spaces and to test
the efficiency of an intuitive classification based on qualitative assessment.
Introduction
Vegetation is a complex system whose components
(plant species) are difficult to measure. In vegetation
science data are collected in different ways accord-
ing to the aim of the research (Mueller Dombois &
Ellenberg 1974), in the majority of the cases only
nominal and ordinal scales are used. Complexity has
several definitions and implications in vegetation sci-
ences (Anand 1994; Anand & Orlóci 1996). However,
Feoli & Zuccarello (1994) prefer to associate the con-
cept of complexity with the difficulty to measure the
vegetation states: a sytem is complex when at least
some of its state variables are practically unmeasur-
able by any kind of instrument or, if measures are
done, they are very imprecise. As a consequence of
vegetation complexity the description of vegetation
structure is often only qualitative. Quantitative struc-
tural description of vegetation can be done indirectly,
as was suggested by Feoli (1984), on the basis of
multiplication of the matrix relevés-species by the
matrix species-structural characters (or textural char-
acters accordingBarkman1979, 1988). In this case the
resulting matrix (relevés-structural characters) gives
for each character and relevé a numerical weight. This
is the number of species, in case of binary data, or the
total cover of species, in case of cover data presenting
the character in each relevé. Feoli et al. (1985) and
De Patta Pillar & Orlóci (1991, 1993) show how the
structural characters can be arranged in hierarchical
order and how vegetation entities (relevés or synthetic
phytosociologicaltables) can be numerically analysed
in a hierarchicalprocess. The disadvantageofthis kind
of descriptionis that it does not take into consideration
the size of the plants and their vertical arrangement in
the strata. The proposal of Orlóci & Orlóci (1985) to
describe directly the vegetation structure by character
set types may have the same disadvantage and it looks
more difficult to be carried out in the field. Arrigoni
(1996a) tries to offer a standard that can be more eas-
ily followed by phytosociologists and plant ecologists
interested to collect data for mapping physiognomic
types. The method suggested by Arrigoni will not be
discussed here. What is relevant for this paper is the
type of data that it generates. This type of data is very
common when dealing with complex systems, namely
data originated by measuring the different characters
by different scales (nominal, ordinal, continuous that
is ‘mixed data’). In Arrigoni’s approach each relevé
is a vector describing the vegetation by strata. Each
stratum is described by its percentage cover value, its
average height in meters, and by the presence of the
structural (textural) character characterising it.
There are many similarity indices that can be
used to compare structural descriptions of vegetation
(Sneath & Sokal 1973; Orlóci 1978; Dale 1988; Po-
dani 1995), however only few have been proposed for
mixed data. These may be grouped into two types:
78
those that keep into consideration the association be-
tween characters and those that do not. The similarity
indices of Goodall (1964, 1966, 1993) and Gower
(1971) are of the second type. The most widely used
indexthat keeps into consideration the association pat-
tern between characters is the Mahalanobis distance
(Orlóci 1978), however it applies directly to continu-
ous data only. One similarity index accepts mixed data
is COCHIS (Feoli & Lagonegro, 1983). According
to COCHIS, variables are divided into two sets, one
which contributes to similarity and the other which
contributes to dissimilarity. For each set an index (S
1
and S
2
) is constructed by summing the probabilities
of character association. This is calculated by chi
square test or correlation coefficient depending on the
variable scale. By these two indices (S
1
,S
2
)asim-
ilarity between two relevés (S
jk
) may be computed
in many ways, COCHIS uses S
jk
= S
1
/(S
1
+ S
2
).
In updating COCHIS, to make it more efficient, we
reviewed again the literature in Gower (1971). There
we discovered that an index based on the same idea
of COCHIS has been already proposed by Burnaby
(1970) in a geological context. However, in Burnaby’s
index each character is weighted not only on the basis
of its association pattern with the other characters, as
in COCHIS, but also on its information content. In
the present paper Burnaby’s index is discussed as an
idea to base comparative analysis of vegetation struc-
ture rather than as a strict technical tool. The index is
used to produce ordination diagrams to map the struc-
tural diversity of vegetation based on fuzzy set theory
(Zimmerman 1984; Feoli & Zuccarello 1986; Roberts
1986; Marsili-Libelli 1989). The performance of the
index is discussed on the basis of its application to
data from Arrigoni (1996a). We use Arrigoni’s data
since they give a clear example of a structural descrip-
tion of relevés based on a critical revision of plant
growth form classifications (Arrigoni 1996b). A com-
puter program is offered to apply the Burnaby index
(P. Ganis, BURNY-COCHIS).
Burnaby’s similarity index
As COCHIS, Burnaby’s idea opens many alternatives.
We focus only on two, the one suggested by Burnaby
(1970), giving weight to independent characters and
the other implemented by COCHIS giving weight to
associated characters.
Given N objects described by M characters mea-
sured on different scales, Burnaby’s similarity index
between two objects j and k is defined by the following
formula:
K
v
jk
=
M
P
i=1
w
i
v
i
jk
I
i
jk
M
P
i=1
w
i
I
i
jk
, (1)
where w
i
is the weight assigned to the ith character
based on its independencewith all the other characters.
It is defined as:
w
i
=
N
2
N
2
+
M
P
h6=i
(χ
2
hi
)
2
, (2a)
where χ
2
hi
is the chi square statistic computed for the
M-1 pairs of characters. w
i
ranges between 1/M for
complete association to 1 for complete independence.
For weighting association the following formula is
used:
W
i
=
1
Mw
i
, (2b)
v
i
jk
assumes differentvalues accordingto the scale
of the ith examined character. For nominal scale v
i
jk
is
1 if the character states agree, 0 if not. For ordinal and
continuous scale, v
i
jk
is calculated according to:
v
i
jk
= 1 −
x
0
ij
− x
0
ik
x
0
max
− x
0
min
!
2
, (3)
where x
0
ij
, x
0
ik
, x
0
max
and x
0
min
are quantile class marks
(1, 2, 3, 4, 5 if quintiles are used, as suggested by
Burnaby), all calculated using ranks of the character
values.
I
i
jk
is the information weight of the states of the ith
character in the jth and kth relevé calculated according
to the following formula:
I
i
jk
=−
logp(x
ij
) + logp(x
ik
)
, (4)
where p(x
ij
) and p(x
ik
) are respectively the proba-
bilities to find the character i in the states of j and
k relevés. I
i
jk
is the information carried out by the
probability of finding the two states of character i in
the couple of relevés under comparison. For ordinal
and continuous variables there is no weight since the
probabilities of quantiles are equal.
Burnaby suggests to compute w
x
i
always on 2 ×
2 contingency tables independently of the character
scale of measure. To achieve this, the qualitative
79
characters with m states are transformed into m di-
chotomouscharactersdescribingthe presence-absence
of each character state. Ordinal and continuous scales
are transformed in quintiles and the chi square statis-
tic is computed on 2 × 2 contingency tables obtained
by removing the median classes and by grouping the
remaining cells by tetrads. This suggestion has been
strongly criticised by Gower (1970, 1971). However,
several trials carried out by us have demonstrated that,
when compared to other ways of obtaining contin-
gencytables from continuousvariables, it gives results
closest to the product moment correlation coefficient.
The use of quantiles instead of other types of classes
(i.e., the classical equal rangefrequency classes) guar-
anteesthe equidistributionof the objectsin the classes,
avoiding empty classes (i.e., in the case of bi- or multi-
modal distributions) or classes with few elements (i.e.,
in the tails of the normal distributions).
Data and methods
The data from Arrigoni (1996a) are given in Table 1.
They correspond to 23 relevés described by 40 char-
acters, 26 of them are nominal with two states (binary
characters), 4 are nominal with more than 2 states and
10 are continuous (height of plants and percentage of
cover of the strata). The data have been collected on
the calcareous mountain of Sardinia between the sea
level to 1400 m above sea level.
The 23 relevés are classified by Arrigoni into 4
main plant formations of Mediterranean vegetation,
namely evergreen forest (relevés 1 to 6), evergreen
maquis (relevés 7 to 11), garrigue (relevés 12 to 18)
and a mixed formation on rocks (relevés 19 to 23).
Burnaby’s index is compared to Gower’s and
Goodall’s index. Two options are considered: one
giving weight to character association and the other
giving weight to character independence. Ordination
axes have been obtained by averaging the similar-
ity values of relevés within the four plant formations
of the Arrigoni’s classification. According to Feoli
& Zuccarello (1986) and Zhao (1986) (see Marsili-
Libelli 1989) these average values are the degrees
of belonging of the relevés to the fuzzy sets corre-
sponding to the plant formations. In this case the axes
represent independent fuzzy sets and not a fuzzy par-
tition. Ordination axes have been also obtained by the
eigenvectors of the similarity matrices calculated by
Burnaby’s, Gower’s and Goodall’s indices. The ordi-
nations are non-centred (Noy-Meir 1973; Feoli 1977)
since they use similarity matrices with values ranging
between0 and 1. The first eigenvectoris alwaysunipo-
lar, other unipolar eigenvectors are obtained if the
data matrix presents disjoint submatrices. Noy-Meir
(1973) and Feoli (1977) show the advantage of us-
ing non-centred ordination for interpreting clusters by
eigenvectors. According to an algebraic theorem, al-
ready presented in ecological context by Feoli (1977),
each disjoint submatrix of a similarity matrix has its
independent set of eigenvalues and eigenvectors. The
magnitude of eigenvaluesdepends on the combination
between the size of the submatrix and its average sim-
ilarity. From this it follows that if a similarity matrix
has submatrices ‘tendencially’ disjoint the elements of
the submatrices with highest dimension and/or highest
average similarity, have higher scores in the corre-
sponding eigenvectors. This allows to interpret the
eigenvectors in terms of clusters of elements corre-
sponding to the submatrices, i.e. to characterise the
eigenvectors by clusters of elements. Thus, the co-
efficient of correlation calculated between the fuzzy
axes and the eigenvectors of the similarity matrices
gives a measure of the correspondence between the
eigenvectors and the clusters of elements.
The Jancey’s relocation method (Anderberg1973),
by using the 4 fuzzy axes and the first eigenvectors of
the similarity matrices capable to represent the clusters
of relevés, was applied to test the best classification in
terms of separation between the clusters suggested by
Arrigoni’s classification. The within sum of squares is
used as the optimality criterion: the lower the sum of
squares the higher the separation between clusters.
Joint plots have been obtained by binarising all the
qualitative data and by the multiplication of the matrix
so obtained by the matrix of fuzzy sets and by the ma-
trix of eigenvectors. The matrix multiplication is done
in such a way that the scores of each character state
and/or each continuous variable in the ordination axes
are the weighted averages according to the following
formula:
S
ih
=
N
P
j=1
x
ij
b
jh
N
P
j=1
x
ij
, (5)
where x
ij
is the score in the data matrix, b
jh
is the
score of the jth relevé in the h fuzzy set or eigenvector.
This multiplication is done in analogywith the method
of reciprocal averaging (Orlóci 1978; ter Braak 1995).
However the use of one of the similarity indices for
80
Table 1. Table of 23 relev
´
es described by 40 structural characters from Arrigoni 1996a. Symbols: Char. type = character type: B.binary,Q. qualitative, C. continuous. Codes for strata:
S1 (0–0.5 m), S2 (0.5–2 m), S3 (2–5 m), S4 (5–12 m), S5 (12–25 m). State of qualitative characters: Herbaceous: 1. Polymorphous 2. Tuberous 3. Graminoid 4. Succulent (crassulent) 5.
Caulescent Life cycle: 1. Perennial 2. Annual 3. Annual and perennial Vegetative cycle: 1. Vernal 2. Latevernal 3. Estival 4. Mixed Leaf consistency: 1. Sclerophyllic 2. Laurel-leaf 3.
Other Leaf cycle: 1. Evergreen 2. Deciduous and semideciduous.
81
Figure 1. Ordination of the 23 relev
´
es in Table 1 according to the fuzzy sets (f.s.) corresponding to evergreen forest and vegetation on rocks,
based on Burnaby similarity index weighting character independence.
Table 2. Evaluation of the efficiency of the similarity indices in producing clusters of relev
´
es by means of sum of squares from centroids;Weight
ind.: weight based on character independence, Weight ass.: weight based on character association (see text).
mixed data (Burnaby’s, Gower’s and Goodall’s in-
dices) has the advantage that the continuous variables
may be left untransformed. The structural diversity of
vegetation is evaluated by the number of qualitative
structural characters in the relevés and by the number
of strata.
Results
Table 2 shows the efficiency of the Burnaby’s index
with respect to Gower’sand Goodall’s indices in terms
of sum of squares calculated directly and calculated
by the Jancey’s relocation method. This table shows
that the Burnaby’s index gives rise to more efficient
classifications especially if we consider the reloca-
tion. Table 3(a,b) shows respectively the scores of the
relevés in the ordination axes given by 4 fuzzy sets
(fuzzy axes) and by the first 3 eigenvectors (capa-
ble to represent the clusters of relevés). In this table
the relevés that, according to the Jancey’s method,
are relocated in other clusters are also indicated. The
Burnaby’s option ‘weighting character independence’
is more efficient than the one weighting character
association. In terms of number of relocations the
Burnaby option weighting character independence is
the most efficient. In fact only three relocations are
suggested.
Table 3 is a very useful table since it shows the
degree of belonging of the relevés to the sets corre-
sponding to the four plant formations and their scores
82
Table 3. Fuzzy axes (a) and eigenvectors (b) for the ordinations of the relev
´
es. Symbols: Arrigoni’s Plant formations: A. Evergreen forest, B. Evergreen maquis, C. Garrigue, D. Vegetation
on rocks. In (a) 1, 2, 3, 4 indicate fuzzy sets corresponding to the plant formations, in (b) 1, 2, 3 indicate the eigenvectors corresponding to the plant formations. The shaded cells indicate
the allocation of relev
´
es according to Jancey’s method.
(a) (b)
83
Figure 2. Ordination of some selected character states by fuzzy axes (f.s.) corresponding to those in Figure 1. (a): ordination of the main life
forms of the strata, (b): ordination of the type of leaf cycles of the strata, (c): ordination of the height of the vegetation and cover of the strata.
84
Table 4. Correlation coefficients between the fuzzy sets and the eigenvectors in Table 3. The shaded cells indicate the maximum positive or negative
correlation.
in the first three eigenvectors. The fact that three
eigenvectors are enough to represent the four clusters
of relevés proves the gradual change in the structure
of the considered vegetation. The continuity of the
change is well reflected by the high number of relo-
cated relevés of the two intermediate plant formations,
namely evergreen maquis and garrigue.
Table 4 presents the correlation coefficients be-
tween the fuzzy ordination axes and the eigenvectors
for the three similarity matrices (Burnaby, Gower and
Goodall); it is clear that eigenvector 1 corresponds
to the plant formation on rocks, the negative side
of eigenvector 2 corresponds to the evergreen for-
est and the positive side to garrigue, eigenvector 3
corresponds to evergreen maquis.
All the jointplots of the main ordination axesshow
more or less similar ordination patterns. There is a
clear structural gradient from the evergreen forest to
the vegetation formationon rocks. For this reason only
theordinationbased on the fuzzysets correspondingto
evergreen forest and the rock vegetation is presented
(Figure 1). The evergreen forest shows the highest
structuraldiversity, the rock vegetation the lowest one.
The average number of qualitative characters in the
relevés and the number of different strata are respec-
tively: evergreen forest 19.5, 4.5; evergreen maquis
18, 3.4; garrigue 15.5, 2; vegetation on rocks 15, 2.2.
The number of strata in the last plant formation is
some time higher than that in garrique since trees can
grow within rocks.
By superimposing the fuzzy ordination of struc-
tural characters (Figure 2a–c) to the relevés ordination
in Figure 1, the correspondence between the structural
characters and the gradient becomes evident. Table 5
presents the degrees of belonging of the structural
charactersto the sets correspondingto Arrigoni classi-
fication calculated according to formula (5) and shows
also their weight in terms of association and indepen-
dence calculated according to formulas (2a) and (2b).
The table shows that the extremeformations(A and D)
are characterised by more characters than the interme-
diate formations (B and C) and that in C and D there
are many characters with similar degree of belonging.
Discussion
According to Gower (1970), Burnaby formulated his
index already in 1965, but he was reluctant to publish
it because he wanted to analyse in more details some
of its properties and because Goodall (1964, 1966)
had anticipated some of his ideas. Probably nobody
applied Burnaby’s index in ecological work because it
was presented in a geological journal, without an ac-
cessible computer program and because Gower (1970)
stressed many points that he was judging weak. We do
not want to discuss here the criticism of Gower, since
the computer program BURNY-COCHIS lets the user
to apply the idea of Burnaby (1970) and of Feoli &
Lagonegro (1983) in a very flexible way. This means
that the variables may be transformed in different
ways and that the contingency tables for association
measures can be obtained also according to Gower’s
suggestions (i.e., beside using only 2 × 2 contingency
tables).
Tests to measure the best performance on the ba-
sis of class separation based on internal or external
criteria are largely available and the Jancey method
is one of them! What we want to stress here is the
idea of weighting the characters by their informa-
tion content and by their association. The context is
pragmatic: weighting characters by their information
content means to give more weight to rare characters,
i.e. to give more similarity to vegetation states that
have rare characters in common. This may be useful
from a conservation point of view. Weighting charac-
85
Table 5. Degree of belonging of the character states to the four fuzzy sets corresponding to the plant formation of Arrigoni based on Burnaby index
weighting character independence (see Table 3 for symbols). The weight of character is also indicated; Weight ind.: weight based on character
independence, Weight ass.: weight based on character association (see text). The shaded cells show the characters that are more linked to the four
plant formations (arbitrary threshold 0.6 has been chosen).
86
ters by their association or independence (in this case
the association is also considered before to compute
the independence) is particularly useful to limit the
negative influence of many redundant non-predictive
characters over few predictive ones. This is in line
with what Intersection Analysis (Feoli & Lagonegro
1979; Feoli et al. 1981) is supposed to do, namely
to give the same importance to groups with many
characters and to character groups with few charac-
ters. As was shown by Feoli et al. (1981), polythetic
classifications, based on the Adansonian principle of
equal weight, may not be free from ecological mis-
classifications. Polythetic classifications may be less
meaningfulthanmonotheticones in the sense that may
be less predictive with respect to chemical physical
factors of ecological relevance. The Burnaby’s idea
of weighting characters may integrate monothetic and
polythetic classifications as supported also by Feoli &
Lagonegro (1983) by COCHIS. The fact that the clas-
sification based on Burnaby’s index is more similar to
the one given by Arrigoni (1996a) suggests that the
logic of Burnaby’s index is more close to the process
of classification worked out by our brain, namely,
given a system to be analysed, the analysis of char-
acter variation is done considering their association
network rather than by considering only one character
at a time. The association pattern between characters
is a very important character itself that in vegetation
system analysis should not be neglected. The possibil-
ity to quantify relationships in a clear fuzzy set theory
context is one advantage of using Burnaby’s index in
vegetation studies.
Acknowledgements
This study was sponsored by Italian C.N.R. and by
Italian M.U.R.S.T. (funds of 40%). We would like to
thank M. Dale, C. Candian, E. Pitacco and one anony-
mous reviewer for useful comments and suggestions.
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