Comparative Analysis of Clustering Algorithms on Synthetic Circular Patters Data

Comparative Analysis of Clustering
Algorithms on Synthetic Circular Patters
Data
Hardev Ranglani
Clustering algorithms play a pivotal role in discovering hidden patterns in
unlabeled data, but their performance varies significantly across datasets
with complex geometries. This paper explores the performance of various
clustering techniques in identifying distinct circular clusters within the
Synthetic Circle Data Set, a benchmark dataset designed to test algorithms
on non-linear structures. We evaluate popular clustering methods, includ-
ing k-means, DBSCAN, Gaussian Mixture Models, hierarchical clustering,
and emerging techniques like Self Organizing Maps, Mean Shift Clustering
and Spectral Clustering. Using metrics such as Adjusted Rand Index (ARI),
Normalized Mutual Information (NMI), and Silhouette Score, along with
detailed visualizations, we systematically compare the algorithms’ ability
to recover the true circle-based clusters without prior labels. Our find-
ings highlight the strengths and limitations of each method, revealing that
density- and graph-based algorithms consistently outperform traditional
techniques like k-means in handling circular patterns.
Keywords: Clustering, K-Means algorithm, Non-linear patterns, Density-Based
Clustering, Hierarchical Clustering, Gaussian Mixture Models, Adjusted Rand Index,
Spectral Clustering
1 Introduction
Clustering, an essential task in unsupervised machine learning, is widely used to
discover underlying patterns and structures in unlabeled data. Despite its prevalence,
clustering algorithms often face significant challenges when applied to datasets with
complex geometries, such as non-linear or concentric patterns. Traditional algorithms,
like k-means, work well at clustering linearly separable data but frequently struggle to
identify non-linear relationships. This limitation becomes a bigger issue in datasets
with overlapping, circular, or nonconvex structures, where clustering boundaries are
inherently non-Euclidean. Addressing these challenges requires evaluating advanced
clustering algorithms that can adapt to such complexities.
To analyze clustering performance on data with non-linear patterns, this study
utilizes the Synthetic Circle Data Set from the UCI Machine Learning Repository—a
EXL Service Inc.
Machine Learning and Applications: An International Journal (MLAIJ) Vol.12, No.1, March 2025
DOI:10.5121/mlaij.2025.12103 33

benchmark dataset consisting of two-dimensional points arranged into multiple circu-
lar clusters (Synthetic Circle Data Set, 2024) The simplicity of this data set makes it
ideal for clustering evaluations, as its two-dimensional structure facilitates easy visual-
ization and interpretation of the results. Each observation is already associated with a
ground-truth label identifying the circle it belongs to, enabling rigorous comparisons
between predicted clusters and true clusters. The overall goal is to assess whether
clustering algorithms can identify the individual circles without access to the true
labels during the clustering process.
Clustering algorithms differ significantly in their ability to adapt to such data com-
plexities. Density-based methods like DBSCAN (Ester et al., 1996) are known for
their ability to handle irregular cluster shapes and noise, while spectral clustering
approaches leverage graph-based representations to uncover nonlinear patterns (Ng
et al., 2001). Gaussian Mixture Models (Reynolds, 2009) and hierarchical clustering
methods (Johnson, 1967; Murtagh & Contreras, 2012) offer flexibility in modeling
and structuring clusters, but their performance can depend heavily on parameter
tuning. Meanwhile, recent advances such as HDBSCAN (Campello et al., 2015) aim to
extend traditional density-based approaches by dynamically determining the number
of clusters and addressing varying densities. These methods, along with founda-
tional algorithms like k-means, form a robust foundation for evaluating clustering
performance on the Synthetic Circle Data Set.
The Synthetic Circle Data Set provides several advantages for this analysis. It has
only 2 features- the X and Y co-ordinates of the data point and the target variable is the
"class" which is basically a label for which circle the data point belongs to. So overall,
the data has only 3 columns. This allows for easier visualizations that clearly illustrate
the success or failure of different clustering methods. The circle label in the dataset
facilitate quantitative evaluations using metrics like Adjusted Rand Index (Hubert &
Arabie, 1985) and Silhouette Score (Rousseeuw, 1987), enabling objective comparisons
of the quality of the clustering. By systematically analyzing and comparing clustering
algorithms, this study seeks to identify methods that work well in recovering circular
clusters while highlighting the limitations of others. Circular or arbitrarily shaped
clusters are commonly encountered in fields such as biology, social networks, and
geospatial analysis, and understanding which algorithms are best suited to these
structures can enhance the effectiveness of clustering algorithms.
1.1 Overall Research Goal and Novelty of this work
The overall objective of this study is to evaluate and compare the performance of
various clustering algorithms on the Synthetic Circle Data Set, which is a dataset
with non-linear, circular cluster structures. The goal here is to determine how well
algorithms can recover true clusters (circles) without access to ground-truth labels,
using metrics like ARI, NMI, and Silhouette Score. This study addresses a gap in
clustering algorithms research by focusing on datasets with circular geometries, unlike
traditional convex datasets like the Iris dataset. The Synthetic Circle Data Set from
the UCI ML repository serves as a novel benchmark, enabling precise evaluation of
clustering methods on non-linear patterns. The results highlight the effectiveness of
density-based and hierarchical methods for circular clusters and provide a practical
framework for evaluating algorithms on non-linear geometries. This study offers
actionable insights for real-world applications in biology, geospatial analysis, and social
networks, addressing clustering challenges often overlooked in traditional benchmarks.
Thus, the overall question answered by this analysis can be summarized as- How
34

effectively can clustering algorithms identify true clusters in data with non-linear,
circular geometries without access to ground-truth labels?
This paper contributes to the literature by providing a performance evaluation of
clustering algorithms on data with non-linear structures, emphasizing their suitability
for separating circular patterns. The insights gained from this analysis can guide
practitioners in selecting the most effective methods for clustering on complex datasets,
such as those encountered in biological, geographical, and social network analyses.
Furthermore, the findings highlight the importance of aligning algorithm choice with
the inherent geometry of the data, a consideration often overlooked in clustering ap-
plications. The rest of the paper is structured as follows: The Literature review section
discusses the previous related work on the subject, the methodology section describes
the dataset briefly describes each clustering algorithm, along with the evaluation
metrics used to assess the performance of clustering algorithms, the results section
describes in detail the performance of each of the algorithms, and the Conclusion and
Future work section mentions how the findings mentioned in this paper can be used
for future analysis.
2 Literature Review
Clustering, as a fundamental unsupervised learning task, has been extensively studied,
with significant progress seen in developing algorithms tailored to various data struc-
tures and application domains. However, challenges still exist in effectively clustering
data with complex geometries, such as circular or concentric patterns. This section
reviews key advancements in clustering algorithms, their application to non-linear
and geometrically complex datasets, and how the Synthetic Circle Data Set provides a
unique benchmark for comparing these methods.
2.1 K-Means and its Limitations
K-means clustering (MacQueen, 1967) remains one of the most popular clustering
algorithms due to its simplicity and computational efficiency. However, it relies
on Euclidean distance that makes it often unsuitable for non-convex or non-linear
cluster shapes (Kanungo et al., 2002). Various extensions, such as kernel k-means
(Schölkopf et al., 1998), try to overcome this limitation by mapping the data into
a higher-dimensional feature space so that clusters may become linearly separable.
Despite these advances, k-means is sensitive to initialization and the need to specify
the number of clusters remain significant challenges.
2.2 Density-Based Clustering
DBSCAN (Density-Based Spatial Clustering of Applications with Noise) (Ester et
al., 1996) performs clustering by identifying clusters as dense regions of data points.
Its ability to handle noise and detect arbitrarily shaped clusters makes it particularly
effective for non-linear patterns. However, DBSCAN’s performance is highly sensitive
to parameters like eps(the maximum distance between two samples for one to be
considered as in the neighborhood of the other) and minsamples(the number of points,
including the core point itself that must exist within an eps neighborhood for a point to
be considered a core point) , and it struggles with datasets featuring varying densities.
HDBSCAN (Campello et al., 2015) tries to addresses these limitations by dynamically
35

adjusting density thresholds, making it well-suited for datasets with varying cluster
densities.
2.3 Hierarchical Clustering
Hierarchical clustering techniques, including agglomerative (Johnson, 1967) and
divisive approaches (Murtagh & Contreras, 2012), build tree-like structures (den-
drograms) to represent data groupings at multiple levels of granularity. While these
methods provide flexibility in cluster formation, they often rely on distance metrics that
do not work well with non-linear patterns. Advances such as dynamic dendrogram
cutting (Langfelder et al., 2008) aim to improve their utility for complex data.
2.4 Gaussian Mixture Models
Gaussian Mixture Models (GMMs) (Reynolds, 2009) offer a probabilistic approach
to clustering, modeling data as a mixture of Gaussian distributions. GMMs excel at
handling overlapping clusters and capturing soft memberships, but they assume that
clusters follow Gaussian shapes, which may not hold for non-linear geometries like
circles. Extensions, such as variational Bayesian GMMs (Bishop, 2006), attempt to
address these limitations by introducing more flexible priors.
2.5 Spectral Clustering
Spectral clustering (Ng et al., 2001) uses graph-based representations of data, using
eigenvectors of the graph Laplacian to partition data into clusters. Its ability to handle
non-linear and non-convex patterns makes it a strong candidate for datasets like the
Synthetic Circle Data Set. Despite its strengths, spectral clustering requires careful
selection of similarity measures and parameters.
2.6 Self-Organizing Maps
Self-Organizing Maps (SOMs), introduced by Kohonen (1982), are unsupervised neu-
ral networks that project high-dimensional data onto a lower-dimensional grid while
preserving topological relationships. SOMs have been widely used in clustering and
visualization tasks across fields such as biology and healthcare (Vesanto & Alhoniemi,
2000).However, SOMs struggle with capturing highly non-linear or complex patterns
due to their fixed grid topology, which may oversimplify relationships in intricate
datasets.
2.7 MeanShift Clustering
MeanShift, a density-based clustering algorithm, identifies clusters by shifting data
points toward regions of higher density (Fukunaga & Hostetler, 1975). Unlike K-
means, it does not require the number of clusters to be predefined. However, despite
its flexibility, MeanShift may perform poorly with non-linear or overlapping patterns,
as it relies on the kernel bandwidth, which can fail to adapt dynamically to complex
density distributions.
36

2.8 Evaluation of Clustering Algorithms
Several benchmark datasets, such as the Iris dataset (Fisher, 1936) and synthetic
datasets (Blobs, Moons), have been used to evaluate clustering algorithms. However,
these datasets often do not represent the geometric complexity of real-world data. The
Synthetic Circle Data Set, by contrast, introduces a controlled environment where the
true cluster shapes are circular, making it ideal for evaluating the ability of clustering
algorithms to handle non-linear geometries.
Metrics such as Adjusted Rand Index (ARI) (Hubert & Arabie, 1985), Normalized
Mutual Information (NMI) (Vinh et al., 2010), and Silhouette Score (Rousseeuw, 1987)
are widely used to quantify clustering performance. These metrics allow researchers
to compare algorithms objectively, even across datasets with varying complexities.
Existing benchmarks often fail to capture the intricacies of such patterns, leaving a
gap in the evaluation of clustering methods tailored for non-linear data. This study
addresses this gap by:
1. Using the Synthetic Circle Data Set as a benchmark to evaluate clustering algo-
rithms on non-linear geometries.
2. Systematically comparing algorithms across multiple dimensions, including com-
putational efficiency, clustering accuracy (ARI, NMI), and cluster separability
(Silhouette Score), along with detailed visaulizations
3. Providing actionable insights into the strengths and limitations of each method,
helping practitioners choose appropriate algorithms for real-world tasks involv-
ing complex data structures.
2.9 Differences from current State of the Art
The differences for the current analysis as compared to the existing literature can be
summarized as:
1. This analysis uses a non-linear dataset with predefined circular clusters, which
are found to be rarely addressed in clustering evaluations.
2. This analysis systematically compares a diverse range of algorithms (density-
based, hierarchical, probabilistic, graph-based, and neural-inspired) in a single
framework
3. It also emphasizes actionable insights for practitioners dealing with similar
non-linear structures.
3 Methodology
This section outlines the methodology adopted for evaluating clustering algorithms
on the Synthetic Circle Data Set. It includes details on the data set, the application
of clustering algorithms, the evaluation metrics used, and the overall experimental
setup.
37

3.1 The Synthetic Circle Dataset
This dataset comprises 10000 two-dimensional points arranged into 100 circles, each
containing 100 points, and it is available on the UCI Machine Learning Repository. It
was designed to evaluate clustering algorithms, by providing a clear and structured
clustering challenge. Figure 1 shows a sample of 5 records of the data, which contain
only 3 features- the x-coordinate and the y-coordinate of the data point and the ’class’
label indicating which circle the data point belongs to, which can be between 0 to 99.
Figure 2 shows a scatter plot of the data in 2 dimensions, clearly indicating the label of
which data point belongs to which circle. These labels are used solely for evaluation
purposes and are not provided as input to the clustering algorithms. The challenge
for the algorithms is to identify each of these 100 circles as 100 separate clusters
based purely on the x and y coordinates of the points.
Figure 1: Sample of 5 records of the Synthetic Circle Dataset with 3 features
3.2 Clustering Algorithms
A variety of clustering algorithms are applied to the dataset, chosen for their different
approaches to handling non-linear and geometrically complex data:
1. K-Means: A centroid-based algorithm that partitions data into k clusters using
Euclidean distance. It partitions data into k clusters by iteratively minimizing
the within-cluster sum of squares. The algorithm alternates between assigning
each data point to the nearest centroid and updating the centroids based on the
mean of the assigned points. The optimization goal is to minimize:
J =
k
∑
i=1
∑
x∈Ci
||x − µi||2
,
where Ci is the ith cluster and µi is its centroid.
2. DBSCAN: A density-based algorithm that clusters points by identifying dense
regions based on two parameters: ϵ(the radius of the neighborhood) and minPts
(minimum number of points required for a dense region). Points are classified as
core, border, or noise. The algorithm grows clusters from core points by including
points within ϵ that are directly or indirectly density-reachable. Mathematically,
38

Figure 2: Scatter plot of the Synthetic Circle Dataset, representing 100 circles
for a point p, the neighborhood is defined as
N(p) = {q : dist(p, q) ≤ ϵ}
and p becomes a core point if |N(p)| ≥ minPts.
3. Hierarchical Clustering: It organizes data into a dendrogram that represents
nested groupings based on their similarity. It can be agglomerative, starting
with each data point as its cluster, or divisive, starting with one large cluster.
Clusters are merged or divided based on linkage criteria such as single-linkage
(minimum distance between clusters), complete-linkage (maximum distance),
or average-linkage (mean distance). Using Ward’s method, the distance between
clusters u and v after merging with cluster s is updated as:
d(u, v) =
r
|v| + |s|
T
d(v, s)2 +
|u| + |s|
T
d(u, s)2 −
|s|
T
d(u, v)2
where
T = |u| + |v| + |s|
4. Gaussian Mixture Model (GMM): It is a clustering algorithm based on the as-
sumption that data is generated from a mixture of several Gaussian distributions
with unknown parameters. The Expectation-Maximization (EM) algorithm
estimates the parameters iteratively. The probability density function of the data
is:
p(x) =
K
∑
k=1
πkN (x|µk, Σk)
39

, where πk is the weight, µk is the mean, and Σk is the covariance matrix. GMM
assigns points to clusters probabilistically, making it more flexible than hard
clustering methods like k-means.
5. Self-Organizing Maps (SOM) are a type of neural network used for clustering
and dimensionality reduction. They project high-dimensional data onto a low-
dimensional (usually 2D) grid, preserving topological relationships. During
training, data points adjust the weights of the winning neuron and its neighbors
using:
wi(t + 1) = wi(t) + η(t)hci(t)[x(t) − wi(t)]
where hci(t) is the neighborhood function, and η(t) is the learning rate.
6. Spectral Clustering algorithm uses the eigenvalues of a graph Laplacian ma-
trix derived from the data to form clusters. It embeds the data into a lower-
dimensional space, capturing the structure of the data graph, and applies a
standard clustering algorithm like k-means. The normalized graph Laplacian is
computed as:
Lsym = D−1/2
LD−1/2
, where D is the degree matrix, and L = D − W is the unnormalized Laplacian
with W as the adjacency matrix. This method is effective for capturing non-linear
cluster structures.
7. Mean Shift Clustering algorithm: It identifies clusters by locating areas of high
density in the feature space. Starting with random initial points, it iteratively
shifts them toward the mean of their neighborhood defined by a kernel function,
such as Gaussian. The update step for each point xi is given by:
xt+1
i =
∑j K(xt
i − xj)xj
∑j K(xt
i − xj)
where K is the kernel function.
Each algorithm is configured with parameters optimized for the data set, ensuring
a fair comparison. The primary objective of this study is to evaluate how effectively
clustering algorithms can recover the true circular clusters. Specifically, the algorithms
are evaluated on the basis of grouping observations into clusters that correspond to
the underlying circles in the dataset, and achieving this clustering without access to
the ground-truth labels (circle_id), which are used only for evaluation.
3.3 Evaluation Metrics
To objectively compare the performance of the clustering algorithms, the following
metrics are used:
1. Adjusted Rand Index (ARI): Measures the similarity between the predicted
clusters and the true labels, adjusted for chance. Values range from -1 (poor
agreement) to 1 (perfect agreement).
2. Normalized Mutual Information (NMI): Captures the shared information be-
tween the predicted and true clusters. NMI values range from 0 (no shared
information) to 1 (perfect match).
40

3. Silhouette Score: Evaluates cohesion within clusters and separation between
clusters. Values range from -1 (poorly defined clusters) to 1(well-separated
clusters).
4. Visual Assessment: Scatter plots of the clustered data, with each data point
colored according to its cluster, along with centroids of the cluster, are compared
to see if the circles are correctly identified by each cluster. Additionally, a Voronoi
diagram is also overlaid to visualize the partitioning of the feature space, illus-
trating the boundaries between clusters. This visualization allows for a direct
comparison of the algorithm’s clustering results with the expected structure of
the data, particularly highlighting its ability (or inability) to separate the circles.
4 Results
The results demonstrate that density-based methods (DBSCAN, MeanShift) and Hi-
erarchical Clustering are highly effective at identifying non-linear, circular clusters,
outperforming traditional methods like k-means and Gaussian Mixture Models. This
highlights the importance of choosing algorithms tailored to the data’s geometric
complexity. The research also underscores the limitations of Self-Organizing Maps
and Spectral Clustering for such tasks, offering valuable insights into their applicability.
The detailed results for each of the clustering algorithms are highlighted in the next
subsections.
4.1 k-Means algorithm
As seen in the figure 3, the k-means algorithm does a decent job of separating each
circle into its own cluster, but some of the circles are not clearly separated into distinct
clusters. The Adjusted Rand Index (ARI) for the k-Means algorithm is 0.9688, the
Normalized Mutual Information (NMI) is 0.99166 and Silhouette Score is 0.59042,
highlighting the performance of the k-Means algorithm.
4.2 DBSCAN algorithm
As seen in the figure 4, the DBSCAN algorithm does a much better job of separating
each circle into its own cluster, as all of the circles are clearly separated into distinct
clusters. The Adjusted Rand Index (ARI) for the k-Means algorithm is 1.0, the Nor-
malized Mutual Information (NMI) is 1.0 and Silhouette Score is 0.6085, highlighting
the performance of the DBSCAN algorithm.
4.3 Agglomerative Clustering algorithm
As seen in the figure 5, the Agglomerative Clustering algorithm also does a good job
of separating each circle into its own cluster, as all of the circles are clearly separated
into distinct clusters. The Adjusted Rand Index (ARI) for the k-Means algorithm is
1.0, the Normalized Mutual Information (NMI) is 1.0 and Silhouette Score is 0.6085,
highlighting the performance of the DBSCAN algorithm.
4.4 Gaussian Mixture Models algorithm
As seen in the figure 6, the Gaussian Mixture models algorithm is able to separate
most of the circles into its own cluster, while some of the circles are overlapping.
41

Figure 3: Scatter plot of the kmeans algorithm. Most of the circles are separated into a different cluster,
while come circles have overlapping clusters
Figure 4: Scatter plot of the DBSCAN algorithm. All the circles are separated into a different cluster
42

Figure 5: Scatter plot of the DBSCAN algorithm. All the circles are separated into a different cluster
The Adjusted Rand Index (ARI) for the GMM algorithm is 0.94630, the Normalized
Mutual Information (NMI) is 0.98971 and Silhouette Score is 0.5688, highlighting the
performance of the GMM algorithm.
4.5 Spectral Clustering algorithm
As seen in the figure 4, the Spectral Clustering algorithm is unable to separate most of
the circles into its own cluster. The Adjusted Rand Index (ARI) for the GMM algorithm
is 0.2337, the Normalized Mutual Information (NMI) is 0.8195 and Silhouette Score is
-0.14351, highlighting the performance of the Spectral Clustering algorithm.
4.6 Self-Organizing Maps algorithm
As seen in the figure 4, the Self-Organizing Maps algorithm is unable to separate most of
the circles into its own cluster. The Adjusted Rand Index (ARI) for the SOM algorithm
is 0.597084, the Normalized Mutual Information (NMI) is 0.8788 and Silhouette Score
is 0.3216, highlighting the performance of the Spectral Clustering algorithm.
4.7 Mean Shift clustering algorithm
As seen in the figure 4, the Means Shift algorithm is able to perfectly separate all of the
circles into its own cluster. The Adjusted Rand Index (ARI) for the SOM algorithm is
1.0, the Normalized Mutual Information (NMI) is 1.0 and Silhouette Score is 0.6085,
highlighting the performance of the Spectral Clustering algorithm.
43

Figure 6: Scatter plot of the GMM algorithm. Most of the circles are separated into a different cluster
Figure 7: Scatter plot of the Spectral clustering algorithm. Most of the circles are not separated into a
different cluster
44

Figure 8: Scatter plot of the SOM algorithm. Most of the circles are not separated into a different cluster
Figure 9: Scatter plot of the SOM algorithm. Most of the circles are not separated into a different cluster
45

4.8 Overall Analysis
Below is the table comparing the results of all algorithms-
Clustering Algorithm Adjusted Rand Index (ARI) Normalized Mutual Information (NMI) Silhouette Score
DBSCAN 1.00 1.00 0.61
Hierarchical 1.00 1.00 0.61
MeanShift 1.00 1.00 0.61
kMeans 0.97 0.99 0.59
Gaussian Mixture Models 0.95 0.99 0.57
Self Organizing Maps 0.60 0.88 0.32
Spectral 0.23 0.82 -0.14
Table 1: Clustering Algorithm Performance Metrics
The results in table 1 show that DBSCAN, Hierarchical Clustering, and MeanShift
are the most effective algorithms for this dataset, primarily due to their ability to
handle non-linear and circular patterns robustly. In contrast, algorithms like k-Means,
GMM, SOMs, and Spectral Clustering are less suited to the dataset’s non-linear struc-
ture, requiring careful parameter tuning or fundamental modifications to achieve
comparable performance.
DBSCAN performs well likely because it excels at detecting arbitrarily shaped
clusters, such as circles, and is robust to noise. Hierarchical Clustering (likely agglom-
erative) performs equally well because its bottom-up approach effectively captures
the nested and non-linear structure of the data. MeanShift also demonstrates strong
performance due to its density-based nature, which aligns well with the clustered
circular geometry of the dataset.
k-Means and Gaussian Mixture Models (GMM) perform slightly worse, with ARI
values of 0.97 and 0.95, respectively. While they capture most of the clusters correctly,
they are not able to perfectly separating overlapping or noisy clusters as they rely on
Euclidean distance and Gaussian assumptions.
Self-Organizing Maps (SOMs) and Spectral Clustering perform poorly compared
to the other methods. SOMs clearly fail to adapt to the exact circular structure, with
an ARI of 0.60 and a relatively low Silhouette Score of 0.32. This shows that while
some clusters are correctly identified, others overlap or are misclassified. Spectral
Clustering exhibits the weakest performance (ARI: 0.23, MI: 0.82, Silhouette: -0.14),
likely because of challenges in configuring the graph similarity matrix or eigenvalue-
based partitioning for this dataset.
4.9 Significance of the Results
The significance of this analysis can highlighted by the fact that the results demonstrate
that density-based methods (DBSCAN, MeanShift) and Hierarchical Clustering are
highly effective at identifying non-linear, circular clusters, outperforming traditional
methods like k-means and Gaussian Mixture Models. This highlights the importance
of choosing algorithms tailored to the data’s geometric complexity. The research also
underscores the limitations of Self-Organizing Maps and Spectral Clustering for such
tasks, offering valuable insights into their applicability.
This analysis thus demonstrates the importance of selecting the right clustering
algorithm for datasets with circular or non-linear geometries. These results can have
practical implications for domains like biology (e.g., detecting circular patterns in
molecular structures), social networks (e.g., circular communities), and geospatial
analysis (e.g., clustering geographic regions with circular features). This analysis
46

shows that practitioners working with non-linear data should prioritize density- or
hierarchical-based clustering approaches. The code used to perform the analysis and
get all the resutls can be found on this github repository
5 Conclusion & Future Work
This study evaluated the performance of several clustering algorithms on the Syn-
thetic Circle Data Set, focusing on their ability to identify circular clusters without
prior knowledge of the true labels. future work could extend this analysis to higher-
dimensional or noisier datasets, where overlapping clusters and real-world complexi-
ties present additional challenges. Automated parameter tuning and enhancements
to existing methods, such as custom distance metrics or graph representations, could
further improve their adaptability. Applying these findings to real-world problems
in biology, geospatial analysis, and social networks would validate their practical
utility.This study provides a foundation for understanding and improving clustering
performance on geometrically intricate datasets.
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