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\begin{document}

\title{A comparison of central-tendency and interconnectivity approaches to
clustering multivariate data with irregular structure}



\author[1]{Mark Tozer}%
\author[1]{David Keith}%
\affil[1]{University of New South Wales}%


\vspace{-1em}



  
  \date{May 26, 2024}


\begingroup
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\selectlanguage{english}
\begin{abstract}
Abstract Questions: Most clustering methods assume data are structured
as discrete hyper-spheroidal clusters to be evaluated by measures of
central-tendency. If vegetation data do not conform to this model, then
vegetation data may be clustered incorrectly. What are the implications
for cluster stability and evaluation if clusters are of irregular shape
or density? Location: Southeast Australia Methods: We define
misplacement as the placement of a sample in a cluster other than
(distinct from) its nearest neighbour and hypothesise that optimising
homogeneity incurs the cost of higher rates of misplacement. The
Chameleon algorithm emphasises interconnectivity and thus is sensitive
to the shape and distribution of clusters. We contrasted its solutions
with those of traditional non-hierarchical and hierarchical
(agglomerative and divisive) approaches. Results: Chameleon-derived
solutions had lower rates of misplacement and only marginally higher
heterogeneity than those of k-means in the range 15--60 clusters, but
their metrics converged with larger numbers of clusters. Solutions
derived by agglomerative clustering had the best metrics (and divisive
clustering the worst) but both produced inferior high-level solutions
clusters to those of Chameleon by merging distantly-related clusters.
Conclusions: Our results suggest that Chameleon may have an advantage
over traditional algorithms at when data exhibit discontinuities and
variable structure, potentially producing more stable solutions (due to
lower rates of misplacement), but scoring lower on traditional metrics
of central-tendency. Chameleon's advantages are less obvious in the
partitioning of data from continuous gradients, however its graph-based
partitioning protocol facilitates hierarchical integration of solutions.%
\end{abstract}%



\sloppy


\textbf{A comparison of central-tendency and interconnectivity
approaches to clustering multivariate data with irregular structure}

\textsuperscript{\textbf{1}}\textbf{Mark Tozer and David Keith}

Centre for Ecosystem Science, School of Biological, Earth and
Environmental Science, University of NSW, Sydney, NSW 2052, Australia

\textsuperscript{1} Corresponding author (Email:
m.tozer@student.unsw.edu.au )

\textbf{Suggested running title:}

Dealing with irregular data structure

\textbf{Keywords}

Chameleon, classification, misclassification, cluster-optimisation,
clustering, CLUTO, vegetation databases, cluster metrics, homogeneity

\textbf{Abstract}

\textbf{Questions:} Traditional clustering methods generally assume data
are structured as discrete hyper-spheroidal clusters to be evaluated by
measures of central-tendency. If vegetation data do not conform to this
model, then vegetation data may be clustered incorrectly. What are the
implications for cluster stability and evaluation if clusters are of
irregular shape or density?

\textbf{Location:} Southeast Australia

\textbf{Methods:} We define mis-classification as the placement of a
sample in a cluster other than its nearest neighbour and hypothesise
that: i) optimising homogeneity incurs the cost of higher rates of
mis-classification; and ii) misclassification varies with thematic
scale. We comparied the performance of an algorithm (Chameleon) which
operates on interconnectivity and thus is sensitive to the shape and
distribution of clusters with that of three traditional algorithms over
varying scales.

\textbf{Results:} Chameleon-derived solutions had lower rates of
misclassification and only marginally higher heterogeneity than those of
k-means in the range 15--60 clusters, but their metrics converged at
finer thematic scales. Solutions derived by agglomerative clustering had
the best metrics (and divisive clustering the worst) but both produced
inferior high-level solutions clusters to those of Chameleon by merging
distantly-related clusters.

\textbf{Conclusions:} Our results suggest that Chameleon may have an
advantage over traditional algorithms at thematic scales at which data
exhibit discontinuities and variable structure, potentially producing
more stable solutions (due to lower rates of mis-classification), but
scoring lower on traditional metrics of central-tendency. Chameleon's
advantages are less obvious in the partitioning of continuous data,
however its graph-based partitioning protocol facilitates hierarchical
integration of solutions.



\textbf{Introduction}

Vegetation classification is the process of delimiting types of
vegetation on the basis of their relative homogeneity and distinctness
from other types (van der Maarel \& Franklin 2013). Classification
facilitates not only the description of vegetation, but also the study
of its relationships with the environment and attendant interacting,
co-dependant organisms. Vegetation classification is thus a first step
to the classification of ecosystems (\emph{sensu} Tansley 1935), and
vegetation typologies have come to underpin a wide variety of
conservation and natural resource management applications including the
selection of protected areas, ecosystem risk assessment and market-based
mechanisms such as biodiversity offsets (Bland \emph{et al.} 2019).
Despite a relatively short history, the science has spawned a wide range
of schools (Whittaker 1978, van der Maarel \& Franklin 2013).
Increasingly however, vegetation classification centres on the
clustering of quantitative plot samples (De C\selectlanguage{ngerman}áceres \emph{et al} .,
2015; 2018). When recorded with systematic procedures, plot samples have
the advantage of allowing observations from different sources to be
consolidated over time, while computer-generated clustering solutions
confer a degree of objectivity in the elucidation of patterns.

The utility of clustering in the development of vegetation
classifications is beyond question, although it is complicated by three
inter-related problems. First, excepting simulated datasets, there is no
agreed external point of reference with which clustering solutions can
be compared. Instead, solutions based on field data must be evaluated on
internal criteria (Aho \emph{et al} . 2008), either geometric (eg
cluster homogeneity) or non-geometric (eg species/cluster fidelity).
Since these vary in the way they weight particular characteristics of
the solution, the best clustering solution may depend on its
application. Second, the hyper-spatial structure of vegetation data is
generally unknown. The choice of both clustering algorithm and
evaluation metrics therefore requires a user-supplied model which
usually (but not invariably (Aho\emph{et al} . 2008)) means clusters are
assumed to be spheroidal, if only because the majority of operators
default to a few well-tested algorithms (Kent 2011). This is
problematic, because algorithms which seek to optimise central tendency
can generate sub-optimal solutions when applied to data with irregular
structure, and internal metrics which assume a spheroidal model may not
be appropriate measures of cluster quality. Third, biases in the both
the geographic and environmental distribution of samples means that
cluster metrics are often optimised for data which sample the range of
floristic variation either unevenly or incompletely. That is, biases may
induce irregularities in data structure even if assemblages in the field
form a continuum. It is not surprising then, that clustering solutions
are notoriously idiosyncratic and highly sensitive to data structure,
transformations, choices of algorithm and resemblance measures
(Tichy\emph{et al} ., 2014).

The potential limitations of assuming a spheroidal model to data of
irregular structure are illustrated in Figure 1. The data are points on
a cartesian plain, normally and randomly distributed around each of six
pre-defined centroids. The k-means algorithm fails to retrieve the
underlying data structure; in i) incorrectly splitting cluster C while
merging clusters D and F; and ii) incorrectly splitting clusters C and F
to partially merge with clusters A and D, respectively. The resulting
solutions appear what Barton \emph{et al} . (2019) termed `unnatural',
although they conceded the vagueness (\emph{sensu} Regan et al, 2002) of
the assignation, relying as it does on an appeal to the human eye. Less
subjectively, the solution is `incorrect', for example in Figure 1(ii)
in assigning samples that are co-located in space in the region of
centroid C to different groups, while drawing in remotely-located
samples from the region of centroid A. The implication is there is a
high likelihood of alternative solutions arising as further data are
added, or if the clustering algorithm is changed or supplied different
parameters.

The problem illustrated in Figure 1 arises primarily to the
insensitivity of the algorithm to variations in the density of points,
however a failure to recover `natural' or `correct' clusters of
irregular shape has similarly been documented in a wide range of
algorithms operating on assumptions of central tendency (Karypis 1999,
Zhao \& Karypis 2005, Han \emph{et al} . 2012, Barton \emph{et al} .
2019). The core principle underpinning algorithms which seek to retrieve
clusters of irregular shape and/or density is sample inter-connectivity.
That is, cluster membership depends on interconnections among sample
(based on pairwise similarity) rather than shared proximity to an
artificial centroid or medoid. Schmidtlein \emph{et al} . (2010), for
example, noted two vegetation samples with no species in common could
nevertheless share cluster membership provided they were connected in a
chain of close neighbours. This implies clusters generated by an
algorithm sensitive to irregular data structure are likely to be more
heterogeneous than those derived with reference to a spheroidal model,
particularly at thematic scale where discontinuities and variation in
sample density exist.

Potential irregularities in data structure are rarely accounted for in
vegetation classification. Schmidtlein \emph{et al} . (2010) documented
a promising approach, however our investigations of their ISOMAP
algorithm suggested its ``brute force'' approach is too computationally
demanding for a dataset many thousands of samples (Schmidtlein \emph{et
al} . (2010) investigated datasets ranging in size up to 305 samples and
warned users of ISOMAP that the algorithm is slow, and not to
complain!). Chameleon (Karypis et al., 1999, see methods for a detailed
description) is one of several alternative algorithms designed to
recover clusters of variable shape which may, therefore, reproduce
landscape scale relationships more faithfully than traditional
clustering techniques (Han \emph{et al} ., 2012). Chameleon assesses
both interconnectivity and closeness of objects as a basis for
determining merging decisions, an approach which results in fewer
``wrong'' decisions than algorithms that consider only one or the other
(Karypis\emph{et al} ., 1999). Focussing on interconnectivity allows the
algorithm to adapt automatically to the characteristics of the clusters
(density and hyperspatial distribution), rather than relying on a static
model (eg discrete spherical clusters). Therefore, provided they are
strongly interconnected, samples spanning a compositional continuum can
be retrieved as a single cluster even if the distribution of samples
along the continuum is uneven, because Chameleon is relatively
insensitive to variations in hyperspatial density (Han \emph{et al} .,
2012).

We suggest that a failure to take account of the underlying structure of
vegetation data is likely to be one factor contributing to
idiosyncrasies among clustering solutions, however the affect is likely
to be dependent on the expression and nature of discontinuities in the
data structure. We postulate that accounting for data structure is more
likely to be important at broad thematic scales (as represented by the
points in Figure 1 collectively) because discontinuities are likely to
arise both naturally (eg between regions which share few species), due
to variable data coverage (De Cáceres \emph{et al} ., 2018,
Gellie\emph{et al} . 2018) or because environmental gradients are
discontinuous in geographic space (Austin 2013). Conversely, there may
be no disadvantage in assuming a spheroidal model where clustering
essentially amounts to partitioning a continuum (ie partitioning the
individual clusters in Figure 1). In this paper, we investigate two
hypotheses: i) that an algorithm sensitive to hyperspatial
irregularities in the density and arrangement of samples will produce
clusters which are likely to be more `correct' (in the sense that
samples are co-located with their close neighbours), but at the cost of
poorer internal metrics relative to algorithms that seek to optimise
around central tendency; and ii) Differences between the respective
algorithms will decline with decreasing thematic scale of cluster
solutions. To test these hypotheses, we used a large regional data set
of 7541 plot samples to compare the performance of traditional
clustering algorithms (k-means, hierarchical agglomerative and divisive)
with the Chameleon algorithm using both internal metrics (homogeneity,
indicator species) and the concept of `correctness' which we apply as
the misclassification rate: the proportion of samples which do not
cluster with their nearest neighbour.

\textbf{Methods}

\emph{The Chameleon algorithm}

Chameleon models the feature space as a k-nearest neighbour graph
(sparse graph) with samples forming vertices connected by links that are
proportional to pairwise similarity between samples (Figure 2). The user
specifies the number of links between samples (neighbourhood range) and
then in the first phase, links are progressively dissolved (in order of
increasing similarity) until a user-specified number of sub-partitions
has formed. In this partitioning phase the algorithm seeks to minimise
the summed length of all links hence minimising the affinity between
samples in different sub-partitions (Karypis, 1999). Sub-partitions are
then (optionally) merged using a hierarchical agglomerative clustering
algorithm to resolve the number of groups required of the solution. An
advantage of this approach is that it encapsulates the concept of
environmental /compositional continua by weighting cluster
interconnectivity over homogeneity. That is, samples that are distantly
related may still share a cluster if they are linked by strongly
interconnected neighbours. One of the key features of the Chameleon
algorithm is that the structure of inter-sample relationships is
preserved through the partitioning phase because co-membership of
sub-partitions is dependent on pairwise inter-sample connectivity. In
contrast, traditional algorithms merge or split samples progressively
and the outcome at each step depends on comparing samples with
intermediate clusters, the compositional characteristics of which are
artificial and reflect the range of the samples merged (Han \emph{et al}
., 2012).

\emph{Study Area}

The study area encompassed the South East Highlands and Australian Alps
Bioregions of the state of NSW, Australia (Thackway \& Creswell, 1995),
an area of 96,089 km\textsuperscript{2} encompassing mountains and
plateaus of the Great Dividing Range. Average annual precipitation
ranges from 460 ---- 2,344 mm and mean annual temperatures are 3 --
16\textsuperscript{o} C. The area is underlain by a complex series of
heavily folded metamorphosed sedimentary rocks deposited from the
Ordovician to Devonian periods and interspersed with numerous granite
intrusions and, to a much lesser extent, basalts deposited in the
Paleogene.

Primary factors influencing the distribution of vegetation formations in
our study area include temperature, rainfall, topography, soils and
drainage (Jenny, 1983; Costin, 1954; Beadle, 1981; Keith, 2004). Alpine
assemblages are restricted to elevations more than 1830 m above sea
level where winter temperature minima fall below the physiological
tolerance of trees (Keith, 2004). Tree cover progressively increases
with decreasing elevations severity of winter conditions declines.
Sub-alpine grassy woodlands occupy the sub-alpine tracts,
characteristically with short gnarled trees and a large compliment of
cold-tolerant species also found in the alpine zone. On the southwest
flank of the Great Divide, sub-alpine woodlands grade into tall wet
sclerophyll forests, sustained by high orographic rainfall originating
in south-westerly air flows. To the east, depending on soils lithology
texture and fertility, sub-alpine woodlands grade into either Dry
Sclerophyll Forest or Grassy Woodlands as annual rainfall declines in
the shadow of the Divide. Grasslands replace Grassy Woodlands in frost
hollows, the heaviest-texture soils and the most moisture-limited sites
(Costin, 1954). Further east of the tablelands, grasslands and grassy
woodlands are replaced by mosaics of wet and dry sclerophyll forests on
the escarpment ranges as rainfall increases with increasing elevation
and exposure to oceanic weather systems (Keith, 2004). Wetlands occur
throughout the bioregions in areas of impeded drainage, while heathlands
are among the local expressions of edaphic and topographic variation.

\emph{Compilation of Floristic Data}

We sourced a total of 7541 floristic plot samples from a database
compiled and administered by the Department of Planning, Industry and
Environment (DPIE, 2019). These comprised all survey data collected in
(or within 25 km of) the South East Highlands and Australian Alps
Bioregions which met the following criteria: i) the sample location was
recorded with an accuracy of \emph{\textless{}} c.100m; ii) the sample
area was 0.04 ha; and iii) all vascular plant species were recorded.
Individual species records were reviewed and modified to resolve
inconsistencies in taxonomy (see Methods in Tozer \emph{et al} ., 2010).
Taxa identified only at the generic level were removed along with
records of naturalised species. Cover-abundance scores were transformed
to presence-absence to eliminate the possible effects of bias in
cover-abundance estimates by different observers. This transformation
was considered an appropriate strategy to achieve a balance between
information-loss and maximising the pool of available data in
circumstances where the data set is both large and likely to be
heterogeneous (Goodall, 1978).

\emph{Chameleon performance evaluation}

We performed all Chameleon analyses on pairwise Bray-Curtis
compositional similarities between samples (Clarke, 1993) using the
scluster function in CLUTO software version 2.1.2 (Karypis, 1999).
First, since we found little information in the literature to guide
parameter-setting, we assessed solutions of 15 clusters over a range of
neighbourhood sizes (15 -- 1000 neighbours), degrees of sub-partitioning
(up to 500 sub-partitions or agglomerative phase omitted) and linkage
functions (single or complete) (Table 1). We focused carried out our
initial trials using the cluster-weighted single-link criterion function
in the agglomerative phase, as recommended for non-spherical clusters
(Karypis, 1999). For each solution we calculated average pairwise
within-cluster association (homogeneity) and the proportion of samples
located in clusters other than that of their nearest neighbour
(misclassification rate). We found using the single linkage function
caused chaining (\emph{sensu} Peet \& Roberts 2013) when the number of
sub-partitions specified was larger than 30. We repeated the relevant
trials using an option forcing Chameleon to prioritise large clusters
over small in the partitioning phase. On the basis of the preliminary
results, we undertook subsequent analyses using the complete linkage
function and assessed performance over a range of thematic scales (15 --
250 clusters) and degrees of sub-partitioning (30 -- 500 sub-partitions)
with neighbourhood size fixed at either 30 or 1000 (Table 1).

\emph{Comparison of algorithms}

We compared Chameleon cluster member-sets with those derived using: i)
k-means clustering; ii) flexible unweighted pair-group averaging with
arithmetic mean (Belbin \emph{et al} . 1992); and iii)
polythetic-division (MacNaughton-Smith \emph{et al} ., 1965;
Belbin\emph{et al} ., 1984). We transformed the adjacency matrix
supplied to scluster to dissimilarity (1-simmilarity) and used each
algorithm to compute solutions ranging from 15 -- 250 clusters (Table
1.) We characterised each solution in terms of homogeneity and
misclassification rate (as described above), the number of species
occurring at higher frequencies within each cluster than in the dataset
as a whole (cumulative hypergeometric probability \textgreater{}0.999)
and the number of species with standardised phi \textgreater{} 0.35
(Tichy~\selectlanguage{english} \selectlanguage{english}\& Chytry~ \selectlanguage{english}2006).

\emph{Comparing clustering solutions with a reference classification}

We assessed the extent to which clustering solutions (15 classes)
produced by each algorithm retrieved species-sets characterising the
units of an established subcontinental-scale vegetation classification
that covers 800,000 km\textsuperscript{2} in southeastern Australia
(Keith 2004), including the study area (c. 11\% of total area). The
reference classification was developed from the top-down based on an
extensive review of vegetation studies, field reconnaissance and
qualitative synthesis of vegetation maps available at the time (Keith
2004). Its highest level of classification (vegetation formation) is
based on structural/physiognomic features. Formations are subdivided
into vegetation classes based on geographically distinct expressions of
structural and compositional features. Fifteen of 99 vegetation classes
recognised in the reference classification are mapped within the study
area and are described with lists of indicative species (Keith, 2004).
For each clustering solution, we identified the species diagnostic of
each cluster as those with a frequency of occurrence statistically
higher within the cluster samples than across the dataset as a whole
(cumulative hypergeometric probability \textgreater{} 0.999). We
compared these with the species identified as diagnostic of the
reference classes, compiling a confusion matrix with the units of the
respective classifications as rows and columns and cell values
calculated as the proportion of reference class species that were
identified as diagnostic of each cluster class.

\textbf{Results}

A summary of the analytical trials performed and a brief synopsis of the
results is contained in Table 1. A detailed description of the results
follows:

\emph{Performance of Chameleon under combinations of varying parameters-
single linkage}

Trends in the mis-classification rate and average within cluster
homogeneity in Chameleon cluster solutions generated using the weighted
single-link functions are summarised in Figure 3. The misclassification
rate rose with increasing neighbourhood size (Figure 3A). This result
may reflect aberrations caused by forcing members of small clusters to
forge links with samples in other clusters as illustrated by Chameleon's
attribution of the simulated data we presented in Figure 1 given
neighbourhoods of different sizes (Figure 4). Solutions derived by
agglomeration from 30 sub-partitions had consistently lower rates of
misclassification, but beyond 30 sub-partitions solutions became
increasing uneven (chaining) and mis-classification rates became
meaningless because a high proportion of samples were concentrated in
few clusters. The problem of chaining was not corrected by directing the
algorithm to prioritise large clusters over small in the partitioning
phase however more even clusters were produced when the cluster-weighted
complete link function was employed in the agglomerative phase of the
algorithm and subsequent analyses were performed using this option, as
described in the next section. There was no clear trend in
within-cluster homogeneity with increasing neighbourhood size when the
agglomeration phase was omitted (Figure 3B). Solutions derived by
agglomeration from 30 sub-partitions had highest homogeneity with a
neighbourhood size of 100. Beyond 30 sub-partitions the data showed no
clear trend and varied erratically depending on the uneven-ness of the
solutions.

Clusters of 15 solutions generated using the cluster-weighted complete
link function exhibited higher rates of mis-classification and lower
within-cluster homogeneity when either neighbourhood size (n) or the
number of sub-partitions (a) in the agglomerative phase were increased,
although increasing n disproportionately affected the mis-classification
rate while increasing a disproportionately affected cluster homogeneity
(Figure 5).

Both the rate of mis-classification and within-cluster homogeneity
increased with increasing thematic resolution (Figure 6). Chameleon
solutions derived using small neighbour sizes and either: modest numbers
of sub-partitions (twice the number of classes in the solution); or with
the agglomeration phase omitted, were better (lower rates of
misclassification and higher homogeneity) than those derived with the
divisive algorithm, but worse than those derived with the agglomerative
algorithm (Figure 6). However, 15- class solutions derived by Chameleon
were more even than those produced by either the agglomerative or
divisive algorithms (Figure 7). Chameleon solutions were better than
those of k-means at broad thematic scales (15 -- 60 classes) but
equivalent at finer scales (90 -- 250 classes). Chameleon produced more
even 15-class solutions than k- means (Figure 7).

Clusters derived by Chameleon solutions were generally characterised by
fewer diagnostic species than those derived using the traditional
algorithms (Table 2), however species diagnostic of Chameleon clusters
corresponded more with those characterising units of a reference
classification for our study area than those diagnostic of cluster
derived by agglomerative or divisive algorithms, both in the range of
units represented and with less overlap between unrelated units (Table
3a, 3b, 3c). Clusters derived by k-means retrieved units of the
reference classification with efficiency similar to Chameleon (Table
3d).

\textbf{Discussion}

\emph{Is there evidence of irregular structure in vegetation data
reflected in the performance of different algorithms?}

We hypothesised that in cases where the structure of vegetation data is
variable (irregular shaped clusters or variable density), an algorithm
sensitive to such variability would perform better (lower rates of
mis-classification) than one that optimises central-tendency (more
homogenous clusters). While the structure of our vegetation data is
unknown, it is unlikely to be regular, neither continuous along
environmental gradients nor arranged in discrete clusters. Theory and
empirical evidence suggest that assemblages of species form
multi-dimensional continua (Whittaker, 1975; Goodall, 1978; Kent, 2011).
However, discontinuities may arise where environmental gradients are
either discontinuous in geographic space, or parts of the environmental
spectrum are not represented (Austin, 2013). Discontinuities are also
likely to arise in our data at broader thematic scales due to biases in
the distribution in sample (Gellie \emph{et al} . 2018) and they
patently exist at continental scales between climatically similar
sub-continental regions which are separated by water or large areas with
unsuitable climate and so share few species (Tozer \emph{et al} . 2017).
We further hypothesised, therefore, that Chameleon's primary advantage
was likely to be in the elucidation of upper-hierarchal clusters.

Overall, the results of our analyses support both hypotheses, although
it is clear that: i) the utility of the different clustering methods
cannot be encapsulated solely in terms of cluster homogeneity and rates
of misclassification, ii) internal evaluators can be misleading in terms
of cluster quality; and iii) the superior performance of Chameleon in
elucidating upper-hierarchical clusters is entirely dependent on
selecting appropriate parameters from an infinite range of combinations.
The clearest support for our hypotheses was evident in the comparison
between solutions derived using Chameleon clusters with those derived by
k-means over the range from 15 -- 250 clusters. Chameleon's best 15 and
30 cluster solutions exhibited significantly lower rates of
mis-classification than those of k-means at the cost of an increase in
heterogeneity (Figure C), while at progressively higher levels of
thematic detail (60 -- 250 clusters) there was a convergence in the
respective metrics. We speculate that increasing rates of
misclassification at finer thematic scale is indicative of the
partitioning of a continuum. That is, at fine thematic scales
communities increasingly intergrade such that the proportion of their
(ever decreasing) member-sets which most closely resemble samples in
adjacent clusters increases. If there was indeed variability in the
structure of our data at broad thematic scales, then the algorithms
performed as hypothesised. We conclude there was a clear advantage in
using Chameleon over k-means to elucidate our upper-hierarchical
clusters (and relatively little cost), but no apparent advantage at
finer thematic scales in terms of cluster metrics. However, since
Chameleon solutions of progressive finer scale can be produced by
continually partitioning the sparse graph, the algorithm potential
offers a straightforward method of integrating plot-based
classifications at multiple hierarchical scales.

Accounting for the performance of agglomerative and divisive clustering
algorithms is more complicated. First, on the basis of cluster
homogeneity and rates of mis-classification, our agglomerative algorithm
performed better than either Chameleon or k-means, scoring higher on
both metrics at all levels of thematic detail, while our divisive
algorithm scored worse (Figure C). Both, however, produced 15-cluster
solutions of much greater unevenness in membership numbers than k-means
or Chameleon (Figure D) which, if evidence of chaining (\emph{sensu}Peet
\& Roberts 2013), could suggest that both solutions were less
informative in relation to the nature of upper-hierarchical groupings.
Conversely, our three traditional algorithms scored equally highly in
terms of the number of diagnostic species and clearly higher than the
best Chameleon solutions, suggesting that unevenness in cluster
membership numbers could, in fact, be symptomatic of biases in the
distribution of samples among `natural' clusters, and that the three
traditional algorithms performed better in detecting these uneven
clusters (as evidenced by higher numbers of diagnostic species).

Comparisons with a reference classification suggest unevenness in the
cluster size is more likely to be indicative of chaining, because
indicator species of the largest clusters tended to represent large
numbers of known classes, some of which are relatively distantly
related, a phenomenon most strongly evident in the agglomerative and
divisive solutions (Tables 3a-c). This reflects a well-known weakness of
agglomerative or divisive methods which incorporate merge or split
decisions based on the aggregate properties of clusters. Such methods
require either unrealistic assumptions concerning the structure of the
data and/or sequential merge/split decisions which cannot be reversed,
and which are necessarily sensitive to the composition of the dataset
(Han \emph{et al} ., 2012). While we did not evaluate the quality of
solutions of greater than 15 classes, we suggest our agglomerative
algorithm outperformed all others in producing 250-cluster solutions
with low rates of misclassification and high homogeneity, but that
subsequent, upper-hierarchical groupings because progressively less
meaningful because of poor merging decisions. We conclude that Chameleon
and k-means generated the most informative solutions of 15 clusters with
the former perhaps better representing the natural structure of the data
while the latte produced more homogeneous groupings.

\emph{Are `natural' clusters necessarily less homogeneous?}

Although a degradation of cluster homogeneity is implicit in our model,
the degree to which this is realised is likely to be highly dependent on
the structure of individual datasets. In our case study, the
mis-classification rate achieved by Chameleon was half that of k-means
at the cost of a 10\% reduction in cluster homogeneity. We speculate
that if the clusters Chameleon retrieved in our dataset are indeed
irregular shapes, then our results suggest they are unlikely to be
highly elongated, and variability in our data structure tends toward
uneven density rather than irregular shape.

The question of whether `natural' clusters necessarily have fewer
diagnostic species is more difficult to resolve based on our
analyses.\emph{A priori} , we inclined to the notion that more
heterogenous clusters would mean fewer diagnostic species, the pattern
reflected in our results, however Schmidtlein et al\emph{.} (2010)
demonstrated that Isopam, an algorithm that adapts to irregular cluster
shapes, consistently out-performed other algorithms in terms of the
number of indicator species (\emph{sensu} Dufr\selectlanguage{ngerman}êne \& Legendre 1997) and
was also highly ranked in terms of the number of species with
standardized phi \textgreater{}0.35 (Tichy~\selectlanguage{english} \selectlanguage{english}\& Chytry~ \selectlanguage{english}2006). Higher
numbers of diagnostic species could reflect the sampling of a wider
species pool, since samples sharing no species can occupy the same
cluster if comprise an interconnected neighbourhood (Schmidtlein et
al\emph{.} 2010). However, it is not clear that higher numbers of
diagnostic species is not an artefact of Isopam's partitioning of the
ordinations space by medoids, notwithstanding the fact the ordination
axes are adjusted to accommodate non-linearities (and hence
irregularities).

On the evidence of our results, we conclude that our original contention
is supported, that cluster solution derived by algorithms sensitive to
data structure are unlikely to be as compact or homogenous as those
derived by optimising central tendency, although the differences may not
always be pronounced, depending on the characteristics of individual
datasets and thematic scale of investigation. In that case, we suggest
that further research is required into metrics which give insights into
how well cluster solutions model the structure of vegetation data (eg
within-cluster inter-connectedness, mis-classification rates) to better
understand the potential trade-offs involved in maximising homogeneity
or indicator values.

\emph{Are natural clusters likely to be more stable/robust to new data?}

Clustering solutions are notoriously sensitive to classification
protocols, and it has generally proven difficult to retrieve the units
of individual CCSs via meta-analysis of combined data (Tich\selectlanguage{ngerman}ý \emph{et
al} . 2011, 2014). Wiser \& De Cáceres (2013) and Tichý \emph{et al} .
(2014) characterised this problem in terms of the need to preserve units
of one or more CCSs while allowing for previously unrecognised units to
be identified following the acquisition of new plot data. Their
respective solutions comprise alternative forms of semi-supervised
clustering, promising approaches that allow for units to be ``fixed'' by
specifying their plot membership a priori while allowing for
unattributed plots to form new clusters. The question of when units
should be ``fixed'' must still be addressed. If the problem arises
either because algorithms cluster irregular data in idiosyncratic ways
or there are biases in the distribution of samples in compositional then
some understanding of the underlying data structure is likely to be
informative.

In theory, algorithms sensitive to data structure may reduce the extent
of this problem, at least at some thematic scales. Tozer et al. (2022)
concluded Chameleon's novel approach to modelling inter-sample
relationships greatly facilitated the revision of an earlier
broad-thematic classification of forested wetlands based on
substantially fewer plot samples (Keith \& Scott 2005). Unlike many
traditional methods which incorporate merge or split decisions based on
the aggregate properties of clusters, Chameleon operates on
inter-connected neighbourhood sets structured, in Tozer \emph{et al} 's
(2022) case, on the same similarity metric used in the original
analysis. They considered these features pivotal, because the algorithm
could potentially minimise the impact of incremental additions of new
data by retaining connections between samples from the original set
(Tozer \emph{et al} . 2022) (although they speculated that this could
best achieved by specifying a large neighbourhood which, on the basis of
our results, we suggest is not appropriate for minimising the rate of
misclassification). Tozer \emph{et al} . (2022) reasoned that since
Chameleon dissolves connections between relatively weakly-connected
samples in the partitioning phases, strong pairwise relationships
between samples underpinning clusters in the original analysis were
potentially preserved (and reflected more faithfully) in their new
Chameleon-derived clusters (provided they were not displaced by a
sufficiently large number of more strongly inter-connected samples).
This interesting feature requires further study.

\emph{Conclusion}

Our study demonstrates that scale-dependent irregularities in vegetation
data can exist and potentially affect the utility and stability of
clustering solutions underlying vegetation classification schemes. The
existence of clusters of irregular shape and density implies that novel
metrics are required in their evaluation because clusters which are
`natural' in the sense they reflect human responses to visual cues
(Barton \emph{et al} . 2019) are unlikely to score well on traditional
metrics that assume a spheroidal model (Aho \emph{et al} . 2008).
Evaluating the utility of such cluster solutions requires metrics which
assess inter-connectivity rather than central tendency.

While the results presented here demonstrate the potential utility of
the Chameleon algorithm in vegetation science, there exist several
issues requiring further investigation. Although Chameleon produced
informative solutions at broad thematic scales, solutions derived with
different parameters varied markedly, and some were clearly inferior to
those of the traditional algorithms we evaluated. While its theoretical
advantages are widely cited, we found very few examples in the
literature to guide how Chameleon should be parameterised, and none
pertaining to analysis of vegetation data. Although the algorithm can be
implemented on wide range of distance metrics, we opted to import a
distance matrix which underpins Consistent Classification Sections
(CCS,\emph{sensu} De Cáceres \emph{et al} . 2015) within our study area
in order to maximise the potential for integrating our results with
those CCSs. On the basis of our trials, we recommend the use of small
neighbourhood sizes over large and either omitting the agglomerative
phase or restricting the number of partitions to no more than twice the
number of samples in the desired solution. Although Karypis (2003)
recommended using a cluster-weighted single linkage function in the
implementation of Chameleon, we found this induced chaining in our
solutions, while the cluster-weighted complete linkage function produced
satisfactory results and we recommend this function if an agglomerative
step is employed. Further experimentation with each of these parameters
using other datasets is clearly required.

Finally, there is some uncertainty in relation to how the algorithm can
be implemented. We employed the Cluto clustering package (Karypis 2003)
distributed by Chameleon's authors, however we noted some
inconsistencies in relation to the parameters offered compared to the
description of the algorthim (Karypis \emph{et al} . 1999). Furthermore,
Barton \emph{et al.} (2019) have suggested Cluto's implementation does
not entirely embody the Chameleon concept. Barton \emph{et al.} (2019)
offer an alternative implementation which deserves evaluation, although
it relies on an different partitioning algorithm because the original is
proprietary protected.

In summary, while our results support the notion the Chameleon algorithm
is theoretically better suited to the task of elucidating vegetation
classes, the characteristics of its solutions, and the ways in which
these improve upon those retrieved by traditional clustering approaches
requires further quantification. We suggest this is a worthwhile
endeavour because Chameleon offers a conceptually simple model, can
process very large datasets quickly and potentially presents a solution
to the problem of integrating plot-based classifications across
hierarchical levels.

\textbf{Data availability statement}

CLUTO software modules are available for download from Karypis Lab
website (http://glaros.dtc.umn.edu/gkhome/cluto/cluto/download ). Plot
data used in our analyses are available at:
https://www.environment.nsw.gov.au/research/Vegetationinformationsystem.htm
(NSW DPIE 2020, accessed 2\textsuperscript{nd} August 2016). All
analyses were performed on an adjacency matrix of similarity
(1-Bray-Curtis dissimilarity) between the objects to be clustered. Data
were imported in a plain text file with \emph{n} + 1 lines, the first
line containing the number of rows and the remaining \emph{n} lines
containing adjacency values for each row (Karypis 2003).

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\textbf{Tables}

Table 1: Summary of analytical trials undertaken, their purpose and
results. Colours in column match indicative results plotted in Figures
3-6.\selectlanguage{english}
\begin{longtable}[]{@{}llllllll@{}}
\toprule
Algorithm & Number of clusters & Neighbourhood size (Chameleon only) &
Number of sub-partitions (Chameleon only) & link & purpose & Examples
results in figure: & result\tabularnewline
\midrule
\endhead
Chameleon & 15 & 15 - 1000 & 30, 60, 90, 150 & single & Assess
performance under different combinations of neighbourhood size and
number of sub-partitions & 3 & chaining increased with increasing number
of sub-partitions over 30, mis-classification rate increased with
neighbourhood size, no clear patterns in within-cluster homogeneity with
either neighbourhood size or number of sub-partitions\tabularnewline
Chameleon & 15 & 15 - 1000 & agglomerative phase omitted & NA & Assess
performance under different neighbourhood size with no agglomerative
phase & 4 & mis-classification rate decreased strongly,within-cluster
homogeneity decreased weakly with increasing neighbourhood
size\tabularnewline
Chameleon & 15 & 30 & 30, 60, 120, 180, 240, 300, 400, 500 & complete &
Assess effect of increasing number of sub-partitions on a 15-cluster
solution with neighbourhood size of 30 samples & 4 & mis-classification
rate decreased weakly, within-cluster homogeneity decreased strongly
with increasing neighbourhood size\tabularnewline
Chameleon & 15, 30, 60, 90, 120, 150 200, 250 & 30 & agglomerative phase
omitted & NA & Assess performance on cluster solutions of different
thematic scale, neighbour size fixed, agglomerative phase omitted &
5,6,7 & mis-classification rate and within-cluster homogeneity increased
with increasing thematic detail, cluster solutions relatively even in
size\tabularnewline
Chameleon & 15, 30, 60, 90, 120, 150 200, 250 & 30 & 15, 30, 60, 120,
180, 240, 300, 400, 500 & complete & Assess performance on cluster
solutions of different thematic scale, neighbour size fixed, number of
sub-partitions proportional to number of final clusters & 5,6,7 &
mis-classification rate and within-cluster homogeneity increased with
increasing thematic detail, cluster solutions relatively even in
size\tabularnewline
Chameleon & 15, 30, 60, 90, 120, 150 200, 250 & 1000 & 30, 60, 120, 180,
240, 300, 400, 500 & complete & Assess performance on cluster solutions
of different thematic scale, neighbour size fixed, number of
sub-partitions proportional to number of final clusters & 5,6,7 &
mis-classification rate and within-cluster homogeneity increased with
increasing thematic detail, cluster solutions relatively even in
size\tabularnewline
k-means & 15, 30, 60, 90, 120, 150 200, 250 & NA & NA & NA & Assess
performance of k-means algorithm over cluster solutions of different
thematic scale & 5,6,7 & mis-classification rate and within-cluster
homogeneity increased with increasing thematic detail, cluster solutions
relatively even in size\tabularnewline
flexible unweighted pair-group averaging with arithmetic mean (Belbin
\emph{et al}. 1992) & 15, 30, 60, 90, 120, 150 200, 250 & NA & NA &
complete & Assess performance of agglomerative algorithm over cluster
solutions of different thematic scale & 5,6,7 & mis-classification rate
and within-cluster homogeneity increased with increasing thematic
detail, cluster solutions relatively uneven in size\tabularnewline
polythetic-division (MacNaughton-Smith \emph{et al}., 1965; Belbin
\emph{et al}., 1984) & 15, 30, 60, 90, 120, 150 200, 250 & NA & NA &
complete & Assess performance of divisive algorithm over cluster
solutions of different thematic scale & 5,6,7 & mis-classification rate
and within-cluster homogeneity increased with increasing thematic
detail, cluster solutions relatively uneven in size\tabularnewline
\bottomrule
\end{longtable}

Table 7: Total number of diagnostic species across all classes as
determined by frequency (statistically higher than background frequency)
or standardised phi (Tichy~ \selectlanguage{english}\& Chytry~ \selectlanguage{english}2006).\selectlanguage{english}
\begin{longtable}[]{@{}lllllll@{}}
\toprule
Algorithm & total number of species with class hypergeometric
probability \textgreater{} 0.999 & total number of species with class
hypergeometric probability \textgreater{} 0.999 AND class frequency
\textgreater{}= 0.3 & total number of species with class hypergeometric
probability \textgreater{} 0.999 AND class frequency \textgreater{}= 0.3
(median frequency) & total number of species with PHI \textgreater{}=
0.35 & median & total number of species with PHI \textgreater{}= 0.35
A1:G1AND class frequency \textgreater{}= 0.3 (median
frequency)\tabularnewline
\midrule
\endhead
k-means & 3615 & 303 & 303 (0.42) & 203 & 0.44 & 121
(121)\tabularnewline
Agglomerative & 3478 & 282 & 282 (0.42) & 227 & 0.43 & 125
(125)\tabularnewline
Divisive & 3118 & 278 & 278 (0.45) & 222 & 0.44 & 144
(144)\tabularnewline
Chameleon (n=10, a = 15) & 3569 & 252 & 252 (0.42) & 123 & 0.41 & 66
(66)\tabularnewline
Chameleon (n=40, a = 15) & 3572 & 268 & 268 (0.41) & 123 & 0.42 & 66
(66)\tabularnewline
Chameleon (n=100, a = 15) & 3416 & 269 & 269 (0.44) & 55 & 0.39 & 31
(31)\tabularnewline
Chameleon (n=1000, a = 15) & 3416 & 269 & 269 (0.44) & 60 & 0.4 & 34
(34)\tabularnewline
Chameleon (n=1000, a = 60) & 4646 & 549 & 549 (0.44) & 173 & 0.41 & 115
(115)\tabularnewline
\bottomrule
\end{longtable}

Table 3a:: Proportion of characteristic species for each reference class
(rows) shared with clusters from the Chameleon algorithm (15 clusters
based on neighbourhood range of 1000 samples agglomerated from 30
sub-partitions). Dark grey indicates proportions\textgreater{}0.7, pale
grey proportions \textgreater{}0.5. Cells with the same shading in
column one comprise members of the same formation.\selectlanguage{english}
\begin{longtable}[]{@{}lllllllllllllllll@{}}
\toprule
Cluster (15) & 6 & 7 & 9 & 0 & 4 & 5 & 8 & 14 & 3 & 15 & 1 & 12 & 2 & 10
& 11 & 13\tabularnewline
\midrule
\endhead
Alpine Herbfields & 0.88 & 0.00 & 0.00 & 0.12 & 0.19 & 0.19 & 0.19 &
0.00 & 0.00 & 0.00 & 0.00 & 0.19 & 0.08 & 0.00 & 0.00 &
0.04\tabularnewline
Alpine Bogs and Fens & 1.00 & 0.00 & 0.00 & 0.04 & 0.04 & 0.07 & 0.04 &
0.11 & 0.04 & 0.00 & 0.00 & 0.21 & 0.04 & 0.04 & 0.00 &
0.04\tabularnewline
Alpine Heaths & 0.93 & 0.00 & 0.07 & 0.07 & 0.07 & 0.00 & 0.00 & 0.04 &
0.04 & 0.00 & 0.00 & 0.37 & 0.04 & 0.00 & 0.04 & 0.00\tabularnewline
Alpine Fjaeldmarks & 1.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.06 & 0.00 &
0.06 & 0.06 & 0.00 & 0.00 & 0.06 & 0.00 & 0.00 & 0.00 &
0.00\tabularnewline
Southern Tablelands DSF & 0.03 & 0.80 & 0.14 & 0.40 & 0.43 & 0.31 & 0.09
& 0.06 & 0.06 & 0.09 & 0.23 & 0.00 & 0.26 & 0.06 & 0.31 &
0.09\tabularnewline
Southern Escarpment WSF & 0.00 & 0.03 & 0.88 & 0.38 & 0.16 & 0.00 & 0.00
& 0.00 & 0.00 & 0.16 & 0.00 & 0.03 & 0.16 & 0.31 & 0.06 &
0.00\tabularnewline
Montane Wet Sclerophyll Forests & 0.11 & 0.09 & 0.49 & 0.66 & 0.20 &
0.06 & 0.06 & 0.06 & 0.06 & 0.14 & 0.06 & 0.14 & 0.11 & 0.03 & 0.03 &
0.00\tabularnewline
Southern Tableland WSF & 0.02 & 0.17 & 0.41 & 0.68 & 0.66 & 0.27 & 0.15
& 0.00 & 0.02 & 0.12 & 0.05 & 0.07 & 0.44 & 0.20 & 0.39 &
0.15\tabularnewline
Subalpine woodlands & 0.19 & 0.24 & 0.19 & 0.62 & 0.57 & 0.22 & 0.14 &
0.03 & 0.05 & 0.08 & 0.08 & 0.11 & 0.30 & 0.05 & 0.22 &
0.05\tabularnewline
Tableland Clay GW & 0.08 & 0.05 & 0.14 & 0.41 & 0.65 & 0.46 & 0.46 &
0.00 & 0.03 & 0.03 & 0.00 & 0.14 & 0.38 & 0.14 & 0.41 &
0.14\tabularnewline
Southern Tablelands GW & 0.00 & 0.26 & 0.14 & 0.42 & 0.79 & 0.74 & 0.49
& 0.02 & 0.00 & 0.05 & 0.02 & 0.02 & 0.47 & 0.14 & 0.51 &
0.26\tabularnewline
Temperate\_Montane\_grasslands & 0.07 & 0.04 & 0.00 & 0.19 & 0.48 & 0.70
& 0.85 & 0.00 & 0.00 & 0.00 & 0.00 & 0.22 & 0.44 & 0.00 & 0.22 &
0.11\tabularnewline
Southern Montane Heaths & 0.03 & 0.24 & 0.07 & 0.10 & 0.14 & 0.07 & 0.03
& 0.59 & 0.38 & 0.21 & 0.28 & 0.07 & 0.03 & 0.07 & 0.00 &
0.00\tabularnewline
Sydney Montane Heaths & 0.00 & 0.08 & 0.00 & 0.04 & 0.04 & 0.04 & 0.00 &
0.42 & 0.92 & 0.50 & 0.25 & 0.00 & 0.00 & 0.04 & 0.04 &
0.00\tabularnewline
Sydney Montane DSF & 0.00 & 0.20 & 0.10 & 0.07 & 0.10 & 0.00 & 0.00 &
0.13 & 0.63 & 0.80 & 0.43 & 0.00 & 0.10 & 0.00 & 0.07 &
0.00\tabularnewline
South East DSF & 0.04 & 0.37 & 0.33 & 0.11 & 0.11 & 0.02 & 0.00 & 0.20 &
0.43 & 0.65 & 0.50 & 0.00 & 0.17 & 0.15 & 0.09 & 0.00\tabularnewline
Montane Bogs and Fens & 0.49 & 0.04 & 0.23 & 0.05 & 0.15 & 0.04 & 0.09 &
0.32 & 0.21 & 0.06 & 0.02 & 0.72 & 0.04 & 0.00 & 0.02 &
0.06\tabularnewline
Montane Lakes & 0.23 & 0.00 & 0.00 & 0.05 & 0.09 & 0.05 & 0.09 & 0.00 &
0.00 & 0.00 & 0.00 & 1.00 & 0.00 & 0.00 & 0.00 & 0.00\tabularnewline
Central Gorge DSF & 0.00 & 0.20 & 0.04 & 0.15 & 0.14 & 0.16 & 0.11 &
0.20 & 0.02 & 0.18 & 0.23 & 0.00 & 0.75 & 0.68 & 0.27 &
0.14\tabularnewline
\bottomrule
\end{longtable}

\textbf{Table 3b:} Proportion of species characteristic of each
structural/physiognomic class that are diagnostic of units of a
15-cluster solution derived using polythetic division. ). Dark grey
indicates proportions\textgreater{}0.7, pale grey proportions
\textgreater{}0.5. Cells with the same shading in column one comprise
members of the same formation.\selectlanguage{english}
\begin{longtable}[]{@{}llllllllllllllll@{}}
\toprule
Cluster & 2 & 1 & 11 & 8 & 7 & 5 & 9 & 3 & 4 & 6 & 10 & 12 & 13 & 14 &
15\tabularnewline
\midrule
\endhead
Alpine Herbfields & 0.73 & 0.00 & 0.00 & 0.15 & 0.23 & 0.00 & 0.00 &
0.35 & 0.27 & 0.00 & 0.00 & 0.04 & 0.00 & 0.00 & 0.00\tabularnewline
Alpine Bogs and Fens & 0.57 & 0.00 & 0.00 & 0.07 & 0.04 & 0.07 & 0.00 &
0.79 & 0.25 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00\tabularnewline
Alpine Heaths & 0.89 & 0.00 & 0.07 & 0.26 & 0.00 & 0.04 & 0.00 & 0.26 &
0.11 & 0.04 & 0.04 & 0.07 & 0.04 & 0.07 & 0.07\tabularnewline
Alpine Fjaeldmarks & 0.88 & 0.00 & 0.00 & 0.00 & 0.06 & 0.06 & 0.00 &
0.31 & 0.06 & 0.00 & 0.00 & 0.06 & 0.00 & 0.00 & 0.00\tabularnewline
Southern Tablelands DSF & 0.00 & 0.80 & 0.14 & 0.43 & 0.26 & 0.00 & 0.03
& 0.00 & 0.00 & 0.14 & 0.03 & 0.03 & 0.03 & 0.06 & 0.06\tabularnewline
Southern Escarpment WSF & 0.03 & 0.06 & 0.91 & 0.59 & 0.00 & 0.00 & 0.44
& 0.00 & 0.03 & 0.41 & 0.03 & 0.13 & 0.22 & 0.13 & 0.22\tabularnewline
Montane Wet Sclerophyll Forests & 0.14 & 0.14 & 0.40 & 0.83 & 0.06 &
0.03 & 0.14 & 0.00 & 0.09 & 0.20 & 0.03 & 0.03 & 0.11 & 0.09 &
0.11\tabularnewline
Southern Tableland WSF & 0.02 & 0.20 & 0.34 & 0.83 & 0.44 & 0.00 & 0.24
& 0.00 & 0.15 & 0.54 & 0.07 & 0.17 & 0.29 & 0.27 & 0.32\tabularnewline
Subalpine woodlands & 0.22 & 0.24 & 0.14 & 0.78 & 0.24 & 0.00 & 0.05 &
0.00 & 0.08 & 0.27 & 0.05 & 0.08 & 0.11 & 0.16 & 0.16\tabularnewline
Tableland Clay GW & 0.08 & 0.08 & 0.14 & 0.57 & 0.59 & 0.03 & 0.08 &
0.03 & 0.14 & 0.27 & 0.00 & 0.05 & 0.16 & 0.16 & 0.11\tabularnewline
Southern Tablelands GW & 0.00 & 0.21 & 0.12 & 0.49 & 0.79 & 0.00 & 0.12
& 0.00 & 0.05 & 0.28 & 0.05 & 0.09 & 0.21 & 0.21 & 0.16\tabularnewline
Temperate\_Montane\_grasslands & 0.04 & 0.04 & 0.00 & 0.30 & 0.89 & 0.00
& 0.00 & 0.04 & 0.19 & 0.11 & 0.00 & 0.00 & 0.00 & 0.07 &
0.00\tabularnewline
Southern Montane Heaths & 0.00 & 0.76 & 0.00 & 0.14 & 0.07 & 0.21 & 0.07
& 0.00 & 0.03 & 0.03 & 0.00 & 0.00 & 0.03 & 0.00 & 0.00\tabularnewline
Sydney Montane Heaths & 0.00 & 0.67 & 0.04 & 0.04 & 0.04 & 0.92 & 0.00 &
0.04 & 0.00 & 0.04 & 0.04 & 0.04 & 0.04 & 0.00 & 0.00\tabularnewline
Sydney Montane DSF & 0.00 & 0.97 & 0.07 & 0.10 & 0.03 & 0.33 & 0.03 &
0.00 & 0.00 & 0.10 & 0.00 & 0.03 & 0.07 & 0.00 & 0.00\tabularnewline
South East DSF & 0.02 & 0.85 & 0.28 & 0.13 & 0.07 & 0.26 & 0.15 & 0.00 &
0.00 & 0.24 & 0.02 & 0.09 & 0.20 & 0.17 & 0.11\tabularnewline
Montane Bogs and Fens & 0.13 & 0.04 & 0.00 & 0.32 & 0.13 & 0.19 & 0.00 &
0.64 & 0.74 & 0.06 & 0.00 & 0.04 & 0.00 & 0.02 & 0.02\tabularnewline
Montane Lakes & 0.05 & 0.00 & 0.00 & 0.05 & 0.14 & 0.00 & 0.00 & 0.14 &
1.00 & 0.00 & 0.00 & 0.05 & 0.00 & 0.05 & 0.00\tabularnewline
Central Gorge DSF & 0.00 & 0.27 & 0.07 & 0.07 & 0.59 & 0.00 & 0.50 &
0.00 & 0.00 & 0.43 & 0.05 & 0.09 & 0.39 & 0.27 & 0.11\tabularnewline
\bottomrule
\end{longtable}

\textbf{Table 3c:} Proportion of species characteristic of each
structural/physiognomic class that are diagnostic of units of a
15-cluster solution derived using pairwise unweighted group-averaging.
). Dark grey indicates proportions\textgreater{}0.7, pale grey
proportions \textgreater{}0.5. Cells with the same shading in column one
comprise members of the same formation.\selectlanguage{english}
\begin{longtable}[]{@{}llllllllllllllll@{}}
\toprule
Cluster & 14 & 13 & 15 & 7 & 4 & 5 & 11 & 1 & 2 & 8 & 10 & 6 & 3 & 9 &
12\tabularnewline
\midrule
\endhead
Alpine Herbfields & 0.58 & 0.46 & 0.46 & 0.12 & 0.00 & 0.15 & 0.00 &
0.19 & 0.15 & 0.04 & 0.00 & 0.00 & 0.04 & 0.04 & 0.00\tabularnewline
Alpine Bogs and Fens & 0.50 & 0.79 & 0.43 & 0.07 & 0.00 & 0.18 & 0.00 &
0.07 & 0.00 & 0.00 & 0.11 & 0.00 & 0.04 & 0.04 & 0.00\tabularnewline
Alpine Heaths & 0.85 & 0.30 & 0.56 & 0.07 & 0.07 & 0.41 & 0.07 & 0.07 &
0.00 & 0.04 & 0.07 & 0.00 & 0.00 & 0.00 & 0.04\tabularnewline
Alpine Fjaeldmarks & 0.25 & 0.31 & 0.88 & 0.00 & 0.00 & 0.00 & 0.00 &
0.00 & 0.06 & 0.00 & 0.06 & 0.00 & 0.00 & 0.00 & 0.00\tabularnewline
Southern Tablelands DSF & 0.40 & 0.49 & 0.03 & 0.74 & 0.23 & 0.40 & 0.63
& 0.31 & 0.14 & 0.34 & 0.43 & 0.43 & 0.29 & 0.26 & 0.17\tabularnewline
Southern Escarpment WSF & 0.22 & 0.34 & 0.06 & 0.22 & 0.97 & 0.44 & 0.34
& 0.13 & 0.00 & 0.09 & 0.25 & 0.06 & 0.09 & 0.06 & 0.50\tabularnewline
Montane Wet Sclerophyll Forests & 0.34 & 0.37 & 0.17 & 0.31 & 0.54 &
0.80 & 0.43 & 0.23 & 0.06 & 0.20 & 0.29 & 0.09 & 0.03 & 0.00 &
0.23\tabularnewline
Southern Tableland WSF & 0.46 & 0.71 & 0.05 & 0.39 & 0.49 & 0.66 & 0.68
& 0.66 & 0.17 & 0.24 & 0.44 & 0.20 & 0.32 & 0.20 & 0.34\tabularnewline
Subalpine woodlands & 0.70 & 0.54 & 0.14 & 0.43 & 0.32 & 0.78 & 0.51 &
0.46 & 0.14 & 0.32 & 0.43 & 0.19 & 0.19 & 0.14 & 0.30\tabularnewline
Tableland Clay GW & 0.35 & 0.46 & 0.05 & 0.22 & 0.24 & 0.43 & 0.38 &
0.68 & 0.43 & 0.14 & 0.27 & 0.08 & 0.38 & 0.14 & 0.16\tabularnewline
Southern Tablelands GW & 0.51 & 0.53 & 0.02 & 0.44 & 0.23 & 0.33 & 0.51
& 0.72 & 0.56 & 0.21 & 0.33 & 0.14 & 0.42 & 0.28 & 0.16\tabularnewline
Temperate\_Montane\_grasslands & 0.33 & 0.48 & 0.00 & 0.19 & 0.04 & 0.22
& 0.26 & 0.59 & 0.78 & 0.11 & 0.19 & 0.04 & 0.33 & 0.15 &
0.11\tabularnewline
Southern Montane Heaths & 0.28 & 0.17 & 0.03 & 0.38 & 0.14 & 0.10 & 0.45
& 0.10 & 0.03 & 0.66 & 0.38 & 0.34 & 0.10 & 0.07 & 0.03\tabularnewline
Sydney Montane Heaths & 0.17 & 0.13 & 0.00 & 0.13 & 0.04 & 0.04 & 0.67 &
0.04 & 0.04 & 0.25 & 0.92 & 0.29 & 0.04 & 0.00 & 0.04\tabularnewline
Sydney Montane DSF & 0.17 & 0.17 & 0.00 & 0.23 & 0.10 & 0.07 & 1.00 &
0.07 & 0.00 & 0.27 & 0.60 & 0.60 & 0.10 & 0.03 & 0.13\tabularnewline
South East DSF & 0.24 & 0.30 & 0.02 & 0.35 & 0.37 & 0.13 & 0.72 & 0.09 &
0.02 & 0.26 & 0.50 & 0.74 & 0.24 & 0.07 & 0.24\tabularnewline
Montane Bogs and Fens & 0.23 & 0.72 & 0.09 & 0.11 & 0.04 & 0.30 & 0.13 &
0.32 & 0.04 & 0.06 & 0.43 & 0.02 & 0.04 & 0.04 & 0.04\tabularnewline
Montane Lakes & 0.09 & 0.36 & 0.00 & 0.05 & 0.00 & 0.05 & 0.00 & 0.77 &
0.05 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00\tabularnewline
Central Gorge DSF & 0.20 & 0.14 & 0.00 & 0.32 & 0.18 & 0.05 & 0.41 &
0.09 & 0.16 & 0.05 & 0.18 & 0.41 & 0.89 & 0.11 & 0.34\tabularnewline
\bottomrule
\end{longtable}

\textbf{Table 3d:} Proportion of species characteristic of each
structural/physiognomic class that are diagnostic of units of a
15-cluster solution derived using k-means. ). Dark grey indicates
proportions\textgreater{}0.7, pale grey proportions \textgreater{}0.5.
Cells with the same shading in column one comprise members of the same
formation.\selectlanguage{english}
\begin{longtable}[]{@{}llllllllllllllll@{}}
\toprule
Cluster & 14 & 13 & 15 & 7 & 4 & 5 & 11 & 1 & 2 & 8 & 10 & 6 & 3 & 9 &
12\tabularnewline
\midrule
\endhead
Alpine Herbfields & 0.88 & 0.62 & 0 & 0 & 0.12 & 0.04 & 0.04 & 0.12 & 0
& 0.04 & 0.31 & 0.04 & 0.04 & 0 & 0.04\tabularnewline
Alpine Bogs and Fens & 0.43 & 0.43 & 0 & 0.04 & 0.21 & 0.11 & 0.07 & 0 &
0 & 0.04 & 0.18 & 0 & 0.07 & 0.04 & 0\tabularnewline
Alpine Heaths & 0.3 & 0.89 & 0 & 0.11 & 0.44 & 0.07 & 0 & 0.04 & 0 & 0 &
0.11 & 0 & 0 & 0 & 0\tabularnewline
Alpine Fjaeldmarks & 0.63 & 0.88 & 0 & 0 & 0 & 0 & 0.06 & 0.06 & 0 & 0 &
0.06 & 0 & 0 & 0 & 0\tabularnewline
Southern Tablelands DSF & 0 & 0 & 0.74 & 0.26 & 0.4 & 0.31 & 0.09 & 0 &
0.09 & 0.26 & 0.03 & 0 & 0.57 & 0.4 & 0.2\tabularnewline
Southern Escarpment WSF & 0 & 0 & 0.03 & 0.97 & 0.41 & 0.19 & 0 & 0 &
0.09 & 0.09 & 0.03 & 0 & 0 & 0 & 0\tabularnewline
Montane Wet Sclerophyll Forests & 0 & 0.09 & 0.09 & 0.57 & 0.8 & 0.23 &
0.06 & 0.03 & 0.14 & 0.06 & 0.14 & 0 & 0 & 0.06 & 0\tabularnewline
Southern Tableland WSF & 0 & 0 & 0.17 & 0.56 & 0.59 & 0.68 & 0.12 & 0 &
0.12 & 0.29 & 0.15 & 0.05 & 0.24 & 0.07 & 0.15\tabularnewline
Subalpine woodlands & 0 & 0.16 & 0.27 & 0.38 & 0.84 & 0.49 & 0.11 & 0 &
0.08 & 0.16 & 0.14 & 0.03 & 0.19 & 0.11 & 0.08\tabularnewline
Tableland Clay GW & 0.03 & 0.05 & 0.08 & 0.27 & 0.41 & 0.62 & 0.46 &
0.03 & 0.03 & 0.32 & 0.16 & 0.08 & 0.3 & 0.03 & 0.14\tabularnewline
Southern Tablelands GW & 0 & 0 & 0.23 & 0.23 & 0.3 & 0.77 & 0.53 & 0 &
0.05 & 0.33 & 0.05 & 0 & 0.53 & 0.09 & 0.23\tabularnewline
Temperate\_Montane\_grasslands & 0.04 & 0.04 & 0.04 & 0.04 & 0.07 & 0.52
& 0.85 & 0 & 0 & 0.22 & 0.22 & 0.19 & 0.22 & 0 & 0.15\tabularnewline
Southern Montane Heaths & 0 & 0 & 0.52 & 0.07 & 0.14 & 0.1 & 0.03 & 0.34
& 0.24 & 0.07 & 0.07 & 0 & 0.21 & 0.28 & 0.07\tabularnewline
Sydney Montane Heaths & 0 & 0 & 0.13 & 0.04 & 0.04 & 0.04 & 0 & 0.88 &
0.67 & 0.04 & 0 & 0 & 0.04 & 0.25 & 0\tabularnewline
Sydney Montane DSF & 0 & 0 & 0.2 & 0.1 & 0.07 & 0.07 & 0 & 0.37 & 0.8 &
0.07 & 0 & 0 & 0.1 & 0.43 & 0.03\tabularnewline
South East DSF & 0 & 0.02 & 0.28 & 0.33 & 0.15 & 0.11 & 0 & 0.22 & 0.65
& 0.24 & 0.02 & 0 & 0.09 & 0.54 & 0.04\tabularnewline
Central Gorge DSF & 0 & 0 & 0.09 & 0.16 & 0.02 & 0.09 & 0.09 & 0 & 0.18
& 0.86 & 0 & 0 & 0.23 & 0.32 & 0.11\tabularnewline
Montane Bogs and Fens & 0.4 & 0.11 & 0.04 & 0.04 & 0.23 & 0.21 & 0.06 &
0.32 & 0.09 & 0.02 & 0.81 & 0.15 & 0.04 & 0 & 0.04\tabularnewline
Montane Lakes & 0.23 & 0.05 & 0 & 0 & 0.05 & 0.23 & 0.09 & 0 & 0 & 0 &
0.73 & 0.73 & 0.05 & 0 & 0\tabularnewline
\bottomrule
\end{longtable}

Figure 1: Sample clusters (A-F) Simulated data created by supplying
cartesian coordinates for six centroids and generating random
coordinates normally distributed around each centroid with sample sizes
i) n= 30, 50, 500, 50, 70, 300), and ii) n = 20, 100, 500, 20, 100, 500)
with standard deviation =1. The boundaries of each cluster are
approximated by circles, colours indicate cluster membership as
determined by k-means operating on a matrix of Euclidean distances.

\textbf{Hosted file}

\verb`image1.emf`  available at \url{https://authorea.com/users/505959/articles/584876-a-comparison-of-central-tendency-and-interconnectivity-approaches-to-clustering-multivariate-data-with-irregular-structure}

Figure 2: Graphical representation of Chameleon's two-phase algorithm
(reproduced form Karypis \emph{et al} . 1999)\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image2/image2}

\end{center}
\end{figure}

Figure 3: Mis-classification rate (A) and average similarity among
objects within clusters (B) function of neighbourhood size as a function
of neighbourhood size. Results for solutions obtained with more than 30
sub-partitions are not shown in (A) because samples were frequently
concentrated in a single cluster (chaining). Trials incorporating an
agglomeration phase (a \textgreater{}15) were performed using a weighted
single linkage function.

\textbf{Hosted file}

\verb`image3.emf`  available at \url{https://authorea.com/users/505959/articles/584876-a-comparison-of-central-tendency-and-interconnectivity-approaches-to-clustering-multivariate-data-with-irregular-structure}

Figure 4: Clustering of simulated data (Figure 1) by Chameleon
illustrating increasing rates of mis-classification with increasing
neighbourhood size. Samples with the same colour were placed in the same
cluster.

\textbf{Hosted file}

\verb`image4.emf`  available at \url{https://authorea.com/users/505959/articles/584876-a-comparison-of-central-tendency-and-interconnectivity-approaches-to-clustering-multivariate-data-with-irregular-structure}

Figure 5: Trends in mis-classification rate and within-cluster
homogeneity with increasing neighbourhood size or increasing number of
sub-partitions in the agglomeration phase. The effect of increasing
sub-partitions using the single-linkage function is not shown due to
chaining as described above). Trendlines are least-squares linear
regressions. Data describing the respective 15-cluster solutions derived
by k-means, agglomerative and divisive algorithms are plotted for
comparison (see figure 6).

\textbf{Hosted file}

\verb`image5.emf`  available at \url{https://authorea.com/users/505959/articles/584876-a-comparison-of-central-tendency-and-interconnectivity-approaches-to-clustering-multivariate-data-with-irregular-structure}

Figure 6: Trends in mis-classification rate and within-cluster
homogeneity with increasing thematic detail (15 -- 250 clusters).

\textbf{Hosted file}

\verb`image6.emf`  available at \url{https://authorea.com/users/505959/articles/584876-a-comparison-of-central-tendency-and-interconnectivity-approaches-to-clustering-multivariate-data-with-irregular-structure}

Figure 7: Cluster sizes ranked in order of increasing size and plotted
as a proportion of the number of samples in the largest cluster.

\textbf{Hosted file}

\verb`image7.emf`  available at \url{https://authorea.com/users/505959/articles/584876-a-comparison-of-central-tendency-and-interconnectivity-approaches-to-clustering-multivariate-data-with-irregular-structure}

\selectlanguage{english}
\FloatBarrier
\end{document}

