Fuzzy and Evidential Contribution to Multilevel Clustering
Martin Cabotte, Pierre-Alexandre H
ebert and
Emilie Poisson-Caillault
Univ. Littoral C
ote d’Opale, LISIC - UR 4491,
Laboratoire d’Informatique Signal et Image de la C
ote d’Opale, F-62100 Calais, France
Multilevel Clustering, Cmeans, Ecm, Split Criterion, Fuzzy Silhouette, Credal Partition, Spectral Clustering.
Clustering algorithms based on split-and-merge concept, divisive or agglomerative process are widely devel-
oped to extract patterns with different shapes, sizes and densities. Here a multilevel approach is considered in
order to characterise general patterns up to finer shapes. This paper focus on the contribution of both fuzzy
and evidential models to build a relevant divisive clustering. Algorithms and both a priori and a posteriori
split criteria are discussed and evaluated. Basic crisp/fuzzy/evidential algorithms are compared to cluster four
datasets within a multilevel approach. Finally, same framework is also applied in embedded spectral space in
order to give an overall comparison.
Extracting general behaviours or particular patterns in
data is a common task in various industrial, medical or
environmental applications. K-means and its deriva-
tive algorithms are appreciated for their understand-
ability and explicability of the resulting clusters. They
also are low-cost algorithms on several aspects in-
cluding computation time, development time, energy.
However, in real-life datasets, clusters can have dif-
ferent sizes, densities, shapes (non-convex, related or
thread-like) and ambiguous boundaries. Algorithms
such as K-means don’t perform well on such diverse
datasets, but other clustering approaches such as spec-
tral or multilevel ones suit more. In the past decades,
several multilevel methods were developed in order
to work at different scales depending on the cluster’s
To deal with non-linearly separable clusters, spec-
tral approaches were proposed, with the aim to trans-
fer data into an embedded spectral space, where the
boundaries between clusters become linear. Numer-
ous clustering problems are based on multiscale prob-
lems, leading to the emergence of multilevel cluster-
ing such as recursive biparted spectral clustering al-
gorithm (Shi and Malik, 2000), spectral hierarchical
clustering HSC (Sanchez-Garcia et al., 2014) or MSC
(Grassi et al., 2019). But computing spectral embed-
ded spaces is time-consuming when applied on nor-
mal/large datasets and can result in not perfectly accu-
rate clustering results when noise and ambiguity ap-
Fuzzy/evidential partition could help to deal with
such noisy datasets. We propose to compare a large
diversity of clustering approaches , on a selection of
possibly noisy datasets: crisp vs fuzzy or evidential,
direct vs multilevel/hierarchical, working in the raw
initial features space or in a spectral embedded space.
This paper is organised as follows: Section 2 pro-
vides multilevel clustering background and a presen-
tation of their split criteria . Section 3 gives the gen-
eral protocol of the methods evaluation and Section
4 analyses the results to highlight benefits and draw-
backs of each method.
2.1 Multilevel Approach
A divisive hierarchical clustering approach is used to
build a multilevel structuring of datasets: from global
shapes to finer parts leading to a more precise analysis
of datasets. An illustration of a three-level clustering
is given by Figure 1 (Grassi et al., 2019).
This process can be carried out either from the fea-
tures data or from an embedded space. In both cases,
data would may be first normalised if necessary (min-
max scaling for instance).
Cabotte, M., Hébert, P. and Poisson-Caillault, É.
Fuzzy and Evidential Contribution to Multilevel Clustering.
DOI: 10.5220/0011550800003332
In Proceedings of the 14th International Joint Conference on Computational Intelligence (IJCCI 2022), pages 217-224
ISBN: 978-989-758-611-8; ISSN: 2184-2825
2022 by SCITEPRESS Science and Technology Publications, Lda. All rights reserved
Figure 1: Multilevel clustering approach.
2.1.1 Multilevel Clustering Framework
Algorithm 1 describes the general MultiLevel Clus-
tering framework (MLC). At each level, for each clus-
tering, the number of clusters K is a posteriori se-
lected by maximising the silhouette criterion of the
K-means algorithm. Then, the sub-dataset is clustered
into K clusters. For each cluster, a split criterion is
computed to assess its quality/compactness (cf. Sec-
tion 2.2). If the criterion doesn’t reach a given thresh-
old, the algorithm restarts the same clustering proce-
dure on it. The whole process stops either when the
maximum level is reached, or when all clusters verify
the split-criterion’s threshold.
Algorithm 1: Multilevel clustering framework.
Require: X data, 1 chosenT hreshold 1
Require: MaxLevel 1, MaxK 1
clToSplit 1 ; NextclToSplit {}
treeLabelled NU LL
level 1
while clToSplit! = {} and level levelMax do
for X clToSplit do
silhouette(clustering(X, k))
clustering(X , K
treeLabelled appendTree(clust
for Y clust
criteriaCluster ComputeSplitCriterion
if criteriaCluster chosenT hreshold then
NextclToSplit Y NextclToSplit
end if
end for
end for
level level + 1
clToSplit NextclToSplit; NextclToSplit {}
end while
return treeLabelled
2.1.2 Embedded Space Variant
To relax data shape requirements and to avoid the se-
lection of one suitable partitioning algorithm, each
clustering may be applied in the embedded space
generated by Algorithm 2 (# means cardinal num-
ber). This space consists in the K first eigenvectors
of the NJW Laplacian L (Ng and Weiss, 2001), built
from the local kernel gaussian ZP-similarity matrix W
(Zelnik-manor and Perona, 2004). At each level, K is
set to the number of the largest Principal EigenValues
PEV of L.
Algorithm 2: Spectral clustering with K estimation.
Require: X, neighbour, PEV threshold
W ZP.similarity.matrix(X ,neighbour)
W check.gram.similarityMatrix(W )
L compute.laplacian.NJW (W )
K #(W $eigenValues > PEVthreshold)
dataSpec W$eigenVectors[, 1 : K]
clustering(dataSpec, K)
return clust
2.2 Stopping Split Criteria
In multilevel clustering, split criterion is an important
feature. This criterion can direct the clustering to-
wards geometry-based clustering, density-based clus-
tering, etc.
The first 3 split criteria studied are a priori crite-
ria: they estimate the necessity of a subdivision (with-
out doing it). They assess the homogeneity of the
clusters: low values indicate that they should be sub-
divided. They are:
The number of wrongly-clustered points named
CardSil, with CardSil = #(silhouette
< 0) <
CardT hreshold, i cluster C
, as implemented
in the R-package sClust. A threshold set to 0
requires that all points are closer to its cluster’s
neighbours than to points of other clusters.
The average silhouette of a cluster named Sil =
) > SilT hreshold, i [1;C
]. It
evaluates the overall geometry of a given cluster.
By heuristics, SilThreshold is set to 0.7.
A fuzzy generalisation of the silhouette criterion:
the Fuzzy Silhouette (Campello and Hruschka,
2006). This fuzzy silhouette (FS) can be in-
terpreted as the silhouette criterion of the clus-
ter cores: it decreases the impact of ambiguous
All the above criteria are a priori split criteria.
If the value of the criterion doesn’t reach a certain
threshold, the clustering of the sub-dataset is com-
puted. We also propose an a posteriori criterion to
deal with fuzzy and evidential approaches:
The Mass criterion is based on fuzzy or eviden-
tial membership functions. This criterion evalu-
FCTA 2022 - 14th International Conference on Fuzzy Computation Theory and Applications
ates the quality of a level clustering by assessing
the non-ambiguity between clusters:
Mass =
Indeed, for each point i of the obtained defuzzified
cluster C
, m
) denotes the membership degree
of i to C
: the higher the value, the lower the am-
biguity of its cluster assignment. So, a high Mass
value means that the obtained current clustering is
coherent: it should be kept, and a sub-clustering
level may then be considered. Conversely, a low
value means that the obtained clustering should
not be kept, because of its high ambiguity. In
section 4, 2 variants of this criterion are going to
be compared Mass25 and Mass100: Mass100 de-
notes the original Mass criterion, whereas Mass25
is a more selective variant which averages the
25% lowest membership degrees of each cluster.
In order to compare approaches, a protocol has been
set up. Considering the variety of algorithms, several
protocols are explained in this section. Furthermore,
parameters settings is also described. Then, quality
criteria are listed followed by there explanation. Fi-
nally, datasets are shown.
3.1 Clustering Algorithms Compared
In order to evaluate fuzzy and evidential contribu-
tion to multilevel clustering, 3 types of algorithms are
Direct Algorithms such as K-means, cmeans
(e1071 :: cmeans) as fuzzy algorithm and ecm
(evclust :: ecm) algorithm (Masson and Denœux,
2007) for credal clustering as witness values.
Agglomerative Algorithms such as high-density
based scanning (dbscan :: hdbscan), agglomera-
tive Ward.d2 clustering (stats :: hclust) and HSC
(sClust :: HierarchicalSC) to compare how well
the fuzzy and evidential multilevel algorithms per-
form well on dense datasets.
Multilevel Algorithms based on the previous di-
rect algorithms in the initial and spectral cluster-
ing space.
3.2 Overview of the General Protocol
During the process, it is necessary to ensure a good
convergence of all elementary clustering. Thus, each
clustering is computed 10 times, and the best result is
kept, according to its own optimization’s criteria.
Moreover, global clustering approaches are iter-
ated 10 times to make the assessment more robust.
The mean value of each quality criterion is computed
as the final result (standard deviation near zero, not
3.3 Parameter Settings
Let K
be the ground truth number of classes. For di-
rect approach and agglomerative clustering based on
Ward, the K input is set to K
. For others methods, K
is estimated in order to be close to the ground truth.
3.3.1 K Estimation: A ”Fair” Estimation
At each level of multilevel approaches, a number of
clusters K has to be automatically determined. Two
methods are used, depending on the working space
In the initial space, K is determined using silhouette
criterion. Silhouette criterion is computed for each
i J2, 10K and the i value maximizing the silhouette
criteria is chosen as the optimal K value.
In spectral embedded space, K is set as the num-
ber of prime eigenvalues of Laplacian matrix, which
are higher than a PE V treshold (0.999 by default, i.e.
close to 1-value for numerical error computation)
3.3.2 Split Threshold Estimation: A Supervised
Split threshold has a huge impact in multilevel ap-
proaches that could stop clustering to its first level
or result in an over-clustering. In order to avoid an
a priori threshold leading to this kind of aberrant fi-
nal number of clusters, it is rather computed follow-
ing Algorithm 3. The threshold is iteratively tuned,
either by incrementation or decrementation depend-
ing on the split criterion, until the number of clusters
reaches the true number of classes.
3.4 Agglomerative Algorithms: A
Specific Evaluation Protocol
Two agglomerative methods compared in this paper
have specific architectures, which require specific pa-
rameter settings.
- HC -Ward D2 clustering (stats::hclust ) builds a
clustering tree according to within-cluster variance
minimum. Then, dendrogram is cut (stats::cutree) ac-
cording to a K
Fuzzy and Evidential Contribution to Multilevel Clustering
- dbscan::hdbscan agglomerates points according to
a core density with a minimum number of points per
cluster (minPts). Two different values of minPts pa-
rameter are considered to reach the K
ground truth:
the closest lower and upper K value.
Algorithm 3: Threshold determination.
Require: data, SplitCriterion, K
if SplitCriterion == mass then
threshold 1
threshold 1
end if
NbFinalCluster 0
while NbFinalCluster < K
if SplitCriterion == mass then
threshold threshold 0.05
threshold threshold + 0.05
end if
cluster MLclustering(data, threshold)
NbFinalCluster #unique(cluster)
end while
return threshold
3.5 Quality Criteria
To evaluate and compare clustering algorithms, sev-
eral unsupervised quality criteria are computed:
The Silhouette Score, cluster :: silhouette
achler et al., 2012) measures compactness of
a cluster compared to the minimum inter-cluster
distance. A silhouette score of 1 means that each
clusters is compact and distant whereas a negative
score means that inter-cluster distance are smaller
than intra-cluster distance.
The Adjusted Rand Index, pd fCluster ::
ad j.rand.index (Azzalini and Menardi, 2013) is
a corrected-for-chance Rand Index. It measures
the ratio of the agreement between the true and
the predicted partitions over the total number of
The non-Overlap, corresponds to a part of the
Rand Index. The non-overlap index measures the
ratio of non-overlapping pairs over the total num-
ber of pairs. A value of 1 means that all points
are affected to non-overlapping clusters (a cluster
included in a class) and that real classes could be
After assigning clusters to classes using major-
ity vote, two adapted supervised quality criteria are
added to catch non representation of one ground truth
class in the obtained clusters and to detect small
(T P
+ FP
(T P
+ FN
With K
the number of classes, the numbers T P: True
positive, FP: False Positive, FN: False Negative and
= 1 if no cluster is assigned in class i, 0 otherwise.
Sometimes, majority vote doesn’t perform well and
small classes are affected to bigger ones. So α term
penalises non represented classes in the standard pre-
cision and recall formula.
3.6 Data Presentation
Considering the diversity of clustering algorithms,
different types of datasets are compared based on their
specificity. Some datasets favour some algorithms
(e.g, hierarchical algorithms are designed for dense
datasets, fuzzy and evidential algorithms are made to
characterise ambiguity...). Thus, 3 types of datasets
are used in this comparison: two are non ambiguous
and dense datasets, and the last one has ambiguity and
uncertainty. A representation of the 3 datasets can be
seen figure 2.
(A) Aggregation (Gionis et al., 2007) is a well-
segmented dataset composed of 7 clusters. In this
seven clusters, two pairs of clusters are connected
by few points, which usually disturbs the clas-
sical hierarchical clustering, such as HDBSCAN
(McInnes et al., 2017) or stats::hclust.
(B) Coumpound (Zahn, 1971) is a 2-level dataset
composed of 3 clusters that can be divided into 2
clusters. A pair of clusters has the same specificity
as aggregation dataset (they are connected by few
points). Another pair of clusters is made in a way
that one is included into the other. These two clus-
ters are merged in order to not penalise some clus-
tering algorithms in raw feature space; they can be
easily separated using spectral approaches. The
last pair is a cluster surrounded by a noise cluster.
(C) 6-Bananas is a dataset built using the function
evclust::bananas (Denœux, 2021). This dataset is
designed for multilevel approaches. Three sets of
bananas are positioned the same way as Coum-
pound dataset. This dataset is made for fuzzy and
evidential clustering according to its non-linear
cut between bananas.
FCTA 2022 - 14th International Conference on Fuzzy Computation Theory and Applications
Figure 2: Datasets used with ground truth classes, one color per class - (A) Aggregation - (B) Coumpound - (C) 6-Bananas.
In order to understand how well fuzzy and eviden-
tial algorithm perform, the analysis will begin with
datasets where current multilevel approaches perform
well. Finally, a focus on Bananas dataset will be pre-
sented where fuzzy and evidential methods are more
4.1 Aggregation and Coumpound
Dataset Results
Aggregation and Coumpound are first clustered in
spectral embedded space (cf. the upper part of Ta-
ble 1). The main quality criterion analysed is ARI,
well-suited to the unsupervised approach.
Compound results of multilevel C-means
(Mass25/Mass100) as well as multilevel K-means
(CardSil) have high ARI scores. A deeper under-
standing of theses values is given in the following
rows. Precision* and Recall* scores show that
multilevel K-means isn’t as efficient as multilevel
C-means. Those lower scores can be primarily
explained by a class disappearance induced by a
lower number of resulting clusters.
This low number of final clusters obtained by the
ML K-mean (CardSil) algorithm (cf. fifth row of Ta-
ble 1), can’t be increased because of the architecture
of the split criterion. Indeed, the roughest thresh-
old value is already selected. This highlights a limit
of this split criterion: it may stop sub-clustering too
early, if a cluster appears ”coherent”, despite being
composed of several subclusters.
Aggregation results show that multilevel ECM
(Mass25) perfectly retrieves real classes: its ARI
score almost reaches 1. Multilevel K-means
(Sil), C-means (FS/Mass25/Mass100) and ECM
(FS/Mass100) have a high precision value but over-
cluster the dataset. However, it should be noted that
Mass split-criteria can lead to lower number of clus-
ters than FS/Sil criteria.
In order to challenge a bit more algorithms, clus-
tering is also performed in the initial features space.
According to ARI, Aggregation is the most difficult
dataset for fuzzy and evidential methods. Hierarchi-
cal clustering performs well on this dataset; and mul-
tilevel K-means is the best multilevel algorithm with
only few overlapping pairs (less than 2%) and a good
precision. Coumpound conclusions are slightly dif-
ferent. Multilevel K-means (Sil) results are equal to
C-means (FS) with good precision and non-overlap
values but result in a total of 23 clusters whereas mul-
tilevel C-means (Mass100) and ECM (Mass100) have
better ARI and non-Overlapping scores with half fi-
nal clusters: on this dataset, the resulting clustering is
better using fuzzy and evidential approaches.
To sum up, spectral clustering is improved on
certain datasets using fuzzy/evidential clustering and
is equal on others. The reason is that compact,
dense and distant clusters will still remain in spec-
tral embedded space. But if ambiguity is kept or cre-
Fuzzy and Evidential Contribution to Multilevel Clustering
Table 1: Clustering results obtained in embedded spectral space (top section) and raw feature space (bottom section) by:
K-means (KM), C-means (CM), Evidential C-means (ECM), agglomerative Ward.d2 clustering (HC), High Density Based
Scanning (HDBSCAN) with K estimation (lower and upper closest values of K) and all Multilevel variants (ML).
Direct (1) Agglomerative (2) ML KM (3) ML CM (4) ML ECM (5)
KM CM ECM HC HDBSCAN CardSil Sil FS Mass25 Mass100 FS Mass25 Mass100
Embedded spectral space
Coumpound K*=6
ARI 0.49 0.43 0.43 0.51 0.86-0.45 0.81 0.36 0.26 0.85 0.85 0.26 0.58 0.58
NonOverlap 0.92 0.91 0.91 0.92 0.94 0.92 1 1 0.94 0.94 1 0.99 0.94
Precision* 0.7 0.52 0.52 0.7 0.92 0.7 0.99 1 0.94 0.94 0.99 0.97 0.94
Recall* 0.67 0.5 0.5 0.67 0.79-0.78 0.67 0.99 1 0.83 0.83 0.99 0.93 0.83
NbClusters 6* 6* 6* 6* 5-7 4 17 28 7 7 21 14 13
Aggregation K*=7
ARI 0.96 0.95 0.77 0.99 0.99-0.44 0.81 0.33 0.29 0.85 0.29 0.29 0.96 0.45
NonOverlap 1 1 0.99 1 1-0.97 0.93 1 1 0.97 1 1 1 1
Precision* 0.96 0.94 0.77 0.99 0.99-0.96 0.64 0.95 1 0.84 1 1 1 1
Recall* 0.99 0.98 0.85 0.99 1-0.89 0.71 0.99 0.99 0.85 0.99 0.99 0.99 0.99
NbClusters 7* 7* 7* 7* 7-20 5 21 38 14 37 38 8 26
6-Bananas K*=6
ARI 0.65 0.63 0.64 0.66 0.59-0.57 0.57 0.35 0.32 0.55 0.49 0.32 0.41 0.49
NonOverlap 0.95 0.95 0.95 0.95 0.93-0.93 0.83 0.99 0.99 0.88 0.93 0.99 0.98 0.93
Precision* 0.82 0.81 0.82 0.84 0.63- 0.63 0.25 0.92 0.93 0.46 0.68 0.92 0.87 0.67
Recall* 0.82 0.81 0.82 0.83 0.72-0.71 0.5 0.91 0.92 0.61 0.74 0.91 0.85 0.74
NbClusters 6* 6* 6* 6* 6-8 3 23 24 7 13 24 18 13
Feature space
Coumpound with class fusion K*=5
ARI 0.57 0.51 0.48 0.59 0.76 - 0.84 0.5 0.28 0.28 0.45 0.8 0.35 0.47 0.83
NonOverlap 0.94 0.95 0.93 0.94 0.94-0.98 0.74 0.94 0.94 0.79 0.97 0.94 0.79 0.94
Precision* 0.84 0.64 0.63 0.91 0.89-0.94 0.47 0.93 0.93 0.49 0.67 0.92 0.44 0.92
Recall* 0.74 0.6 0.59 0.79 0.76-0.9 0.4 0.8 0.8 0.5 0.7 0.8 0.48 0.8
NbClusters 5* 5* 5* 5* 6-9 2 23 24 6 7 12 6 11
Aggregation K*=7
ARI 0.76 0.74 0.55 0.81 0.81-0.67 0.66 0.56 0.52 0.63 0.59 0.55 0.52 0.52
NonOverlap 0.99 0.99 0.92 1 0.93-0.93 0.98 0.99 0.97 0.93 0.94 0.97 0.95 0.95
Precision* 0.76 0.76 0.47 0.79 0.64-0.64 0.95 0.97 0.79 0.65 0.66 0.76 0.67 0.67
Recall* 0.83 0.83 0.54 0.86 0.71-0.71 0.89 0.93 0.83 0.61 0.7 0.82 0.66 0.66
NbClusters 7* 7* 7* 7* 5-55 14 18 15 13 17 25 14 14
6-Bananas K*=6
ARI 0.57 0.59 0.57 0.67 0.57-0.03 0.57 0.37 0.37 0.54 0.54 0.38 0.49 0.51
NonOverlap 0.94 0.94 0.93 0.94 0.83-0.98 0.83 0.94 0.94 0.96 0.96 0.94 0.96 0.92
Precision* 0.76 0.78 0.79 0.86 0.25-0.92 0.25 0.73 0.73 0.84 0.84 0.72 0.8 0.63
Recall* 0.76 0.78 0.75 0.82 0.5-0.87 0.5 0.81 0.81 0.84 0.84 0.8 0.79 0.72
NbClusters 6* 6* 6* 6* 3-218 3 75 64 10 10 50 14 14
ated while transferring data into spectral embedded
space, fuzzy/evidential methods will improve cluster-
ing thanks to their membership characterisation.
Also, CardSil split criterion shows its limits in raw
features space, because the lowest possible estimated
K is often lower than the ground-truth value. Thus,
finer shapes can’t be retrieved, restraining multilevel
4.2 A More Difficult Dataset:
Bananas dataset combines several difficulties: cluster
boundaries are non-linear, very close to each other,
ambiguous, and the cluster densities are weak (each
banana has 125 points, ambiguity included). There-
fore, clustering is particularly hard.
Clustering in Spectral Space. The too high con-
nexity between bananas does not allow building a rel-
evant spectral space. But this spectral space causes in
most cases lower final number of clusters and homo-
geneous clusters (non-overlap scores are almost equal
to 1).
Clustering in Raw Features Space. Direct meth-
ods with true value of K achieve quite good ARI re-
sults. However, they do not clearly exceed 0.57, the
ARI score obtained by the 2 methods resulting in
K = 3 clusters (pairs of bananas are identified, but
without a finer division). HC with the true value of K
gives the best clustering with an ARI equal to 0.67 and
only 6% of overlapping pairs of points. HDBSCAN is
not particularly efficient on this dataset. With 3 min-
imum points per cluster (minPts = 3), the algorithm
gives only 3 final clusters, and when the minimum
number of points per cluster is decreased to 2 points,
FCTA 2022 - 14th International Conference on Fuzzy Computation Theory and Applications
the algorithm gives a total of 218 clusters resulting in
a disastrous ARI.
Multilevel K-means (CardSil), as mentioned
above, stops at the first level and only detects the pairs
of bananas.
Then two groups can be identified:
the Sil/FS split criterion, which builds more than
50 final clusters. This overclustering leads to
weak ARI scores. The crisp multilevel FS K-
means is the worst, with 25 more clusters than
ECM but the same quality criterion values.
the Mass25 and Mass100 group determines be-
tween 10 and 14 final clusters. These split criteria
have higher ARI (0.54 with MC CM) and almost
equal non-overlap scores with less clusters, mean-
ing that they perform better than the Sil/FS ones.
To summarise, fuzzy and evidential approach can
improve multilevel clustering results, in spite of noisy
non-linearly separable clusters. However, on this
dataset, ML variants do not achieve the best ARI
(HC’s score), which can be explained by their higher
number of final clusters. But those clusters are very
homogenous (high non-overlap scores).
In this paper, we have mainly proposed a compari-
son between clustering methods: direct vs multilevel,
then crisp vs fuzzy/evidential. To enhance the fuz-
zy/evidential multilevel algorithms, a new split crite-
rion has also been proposed (Mass a posteriori split
Several conclusions were obtained. First, direct
methods may result in a bad structure recognition
due to particular geometry shapes like nested or close
clusters. Agglomerative methods may also be dis-
turbed by connected noisy clusters. This problem
also affects HDBSCAN, despite its ability to cluster
noise. Moreover, the number of clusters obtained by
HDBSCAN can be extremly sensitive to its parameter
minPts on this type of datasets.
An other shortcoming of agglomerative methods,
and spectral clustering as well, is their complexity:
they are not suitable for large datasets because of too
high computing times.
Multilevel approaches like multilevel C-means or
ECM can help to recognize noisy/ambiguous clusters,
what’s more, in a reasonable computing time. On
some datasets, the final clustering appears a lot better
than those obtained with direct methods: the ambigu-
ity between clusters may be better processed working
with several levels.
Then the comparison of split-criteria leads to the
conclusion that those based on soft membership de-
grees can limit the over-clustering, with a K-number
closer to the ground truth.
Nevertheless, multilevel approaches based on K-
means and its fuzzy/evidential extensions are clearly
not perfect. In particular, they are based on a delicate
task, the automatic estimation of the cluster number,
which is repeated frequently.Their parameters are es-
timated in order to obtain a final number of clusters
close to the known number of classes. Another reason
which may disadvantage multilevel methods, is the
difficulty to obtain a fair comparison with other clus-
tering methods, when the final number of clusters dif-
fers. Split criteria thresholds were chosen to make this
number closer to the ground-truth, but it often failed:
they tend to overcluster. But, the fuzzy/evidential
approach provides ambiguity information on clusters
that could be used to perform a fusion and retrieve
original classes. The good non-overlapping scores
obtained in the experiments tend to support this idea.
Further works will therefore investigate the char-
acterization of points and clusters ambiguity in fuzzy
and evidential algorithms, in order to improve each
clustering step, and to drive the merger process to
the building of a more coherent final clustering tree.
Moreover, such an approach would reduce the com-
puting time, by making the spectral embedding step
This work is a part of the JERICO-S3 project, funded
by the European Commission’s H2020 Framework
Programme under grant agreement No. 871153.
Project coordinator: Ifremer, France.
Azzalini, A. and Menardi, G. (2013). Clustering Via Non-
parametric Density Estimation: the R Package pdf-
Campello, R. and Hruschka, E. (2006). A fuzzy extension
of the silhouette width criterion for cluster analysis.
Fuzzy Sets and Systems, 157(21):2858–2875.
Denœux, T. (2021). evclust: An R Package for Evidential
Clustering. Bananas dataset.
Gionis, A., Mannila, H., and Tsaparas, P. (2007). Clus-
tering aggregation. ACM Transactions on Knowledge
Discovery from Data, 1(1):4.
Grassi, K., Poisson Caillault, E., and Lefebvre, A. (2019).
Multilevel spectral clustering for extreme event char-
acterization. In OCEANS 2019 - Marseille. IEEE.
Fuzzy and Evidential Contribution to Multilevel Clustering
Masson, M.-H. and Denœux, T. (2007). Algorithme
evidentiel des c-moyennes ecm : Evidential c-means
algorithm. In Rencontres Francophones sur la
Logique Floue et ses Applications (LFA’07), pages
17–24, N
ımes, France, Novembre, 2007.
McInnes, L., Healy, J., and Astels, S. (2017). hdbscan: Hi-
erarchical density based clustering. The Journal of
Open Source Software, 2(11):205.
achler, M., Rousseeuw, P., Struyf, A., Hubert, M., and
Hornik, K. (2012). Cluster: Cluster Analysis Basics
and Extensions, R-CRAN packages.
Ng, A.and Jordan, M. and Weiss, Y. (2001). On Spectral
Clustering: Analysis and an algorithm. In Advances
in Neural Information Processing Systems (NIPS’01),
pages 849–856. MIT Press.
Sanchez-Garcia, J., Fennelly, M., Norris, S., Wright, N.,
Niblo, G., Brodzki, J., and Bialek, J. W. (2014). Hi-
erarchical spectral clustering of power grids. IEEE
Transactions on Power Systems, 29(5):2229–2237.
Shi, J. and Malik, J. (2000). Normalized cuts and image
segmentation. IEEE Trans. Pattern Anal. Mach. In-
tell., 22(8):888–905.
Zahn, C. (1971). Graph-theoretical methods for detecting
and describing gestalt clusters. IEEE Transactions on
Computers, C-20(1):68–86.
Zelnik-manor, L. and Perona, P. (2004). Self-tuning spec-
tral clustering. In Saul, L., Weiss, Y., and Bottou, L.,
editors, Advances in Neural Information Processing
Systems (NIPS’04). MIT Press.
FCTA 2022 - 14th International Conference on Fuzzy Computation Theory and Applications