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 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 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 (sensuPeet & 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 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.