A New Cluster Validation Index Based on Stability Analysis

Adane Nega Tarekegn

, Bjørnar Tessem

and Fazle Rabbi

Department of Information Science and Media Studies, University of Bergen, Bergen, Norway

Keywords: Cluster Validation

, Stability Analysis, Clustering Algorithm, Text Clustering, CSAI.

Abstract: Clustering is a frequently employed technique across various domains, including anomaly detection,

recommender systems, video analysis, and natural language processing. Despite its broad application,

validating clustering results has become one of the main challenges in cluster analysis. This can be due to

factors such as the subjective nature of clustering evaluation, lack of ground truth in many real-world datasets,

and sensitivity of evaluation metrics to different cluster shapes and algorithms. While there is extensive

literature work in this area, developing an evaluation method that is both objective and quantitative is still

challenging task requiring more effort. In this study, we proposed a new Clustering Stability Assessment

Index (CSAI) that can provide a unified and quantitative approach to measure the quality and consistency of

clustering solutions. The proposed CSAI validation index leverages a data resampling approach and prediction

analysis to assess clustering stability by using multiple features associated within clusters, rather than the

traditional centroid-based method. This approach enables reproducibility in data clustering and operates

independently of the clustering algorithms used, which makes it adaptable to various methods and

applications. To evaluate the effectiveness and generality of the CSAI, we have carried out an extensive

experimental analysis using various clustering algorithms and benchmark datasets. The obtained results show

that CSAI demonstrates competitive performance compared to existing cluster validation indices and

effectively measures the quality and robustness of clustering results across multiple samples.

1 INTRODUCTION

Clustering is one of the fundamental tasks in

unsupervised learning, where the input is a set of

unlabeled samples, each described by a vector of

feature values. The goal is to partition a set of

observations into distinct groups such that

observations within each cluster exhibit substantial

similarity, while those in different clusters should be

dissimilar (Duan et al., 2023). The cohesion of these

groups may stem from various factors, including

proximity in the distance, inherent trends, patterns,

and interrelationships present within the dataset.

Clustering is often employed across different tasks

(Oyewole & Thopil, 2023; Halim et al., 2015), such

as anomaly and event detection, recommendation

systems, graph analysis, customer segmentation, and

natural language processing. It is frequently applied

to get an intuition about the underlying structure in

https://orcid.org/0000-0002-4005-1238

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https://orcid.org/0000-0001-5626-0598

the data, find meaningful groups, feature extraction,

topic modelling, and summarization (Nakshatri et

al., 2023). Clustering can also serve as a method for

representation learning in the absence of reference

data for supervised learning tasks (Li et al., 2022;

Tarekegn et al., 2021), and it can improve

classification performance. Clusters can be

generated using either hard clustering or soft

clustering based on the level of membership

assignment to each data point. Hard clustering

assigns each data point exclusively to one cluster,

while soft clustering assigns degrees of membership

to multiple clusters, indicating the likelihood

or probability of each data point belonging to each

cluster (Murtagh, 2015). Depending on the specific

nature of the problem, various algorithms have been

developed, which can be categorized into various

groups, including partitioning-based, hierarchical-

based, density-based, and model-based (Xu &

Wunsch, 2005; Ezugwu et al., 2022).

Tarekegn, A. N., Tessem, B. and Rabbi, F.

A New Cluster Validation Index Based on Stability Analysis.

DOI: 10.5220/0013309100003905

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 14th International Conference on Pattern Recognition Applications and Methods (ICPRAM 2025), pages 377-384

ISBN: 978-989-758-730-6; ISSN: 2184-4313

377

Despite the proven efficiency of various

clustering algorithms in data grouping, a crucial

aspect of clustering analysis is evaluating algorithm

performance (W. Wu et al., 2020; Tarekegn et al.,

2020). This involves determining not only the

consistency of clusters but also assessing the validity

of results that best fit the underlying structure of the

data (Kim & Ramakrishna, 2005). Different methods

often lead to different clusters, and even for the same

algorithm, variations in parameter selection or the

sequence in which input patterns are presented may

affect the results. Thus, effective evaluation

standards and criteria are essential to provide users

with a degree of confidence in the clustering results

derived from the employed algorithms. However,

cluster validation poses significant challenges due to

factors such as the subjective nature of evaluating

clustering results, the absence of predefined labels

or ground truth in many real-world datasets, and the

sensitivity of evaluation metrics to different cluster

shapes and densities (Xu & Wunsch, 2005).

A clustering book by Jain and Dubes (Jain & Dubes,

1988) stated that “the validation of clustering

structures is the most difficult and frustrating part of

cluster analysis. Without a strong effort in this

direction, cluster analysis will remain a black art

accessible only to those true believers who have

experience and great courage ". This observation

has persisted for several years despite substantial

advancements in the field. Numerous evaluation

methods exist in the literature, with a common

approach being the use of internal validity measures

(Jegatha Deborah et al., 2010) that rely solely on

metrics such as compactness, cluster separation, and

roundness to assess clustering quality. However,

these methods are designed for evaluating convex-

shaped cluster, making them less effective for non-

convex clusters and unable to directly assess cluster

stability (Liu et al., 2022) across different runs or

perturbations.

In this study, we propose a new Clustering

Stability Assessment Index (CSAI) for measuring

the validity and stability of the clustering solutions

based on multiple feature information associated

with clusters. The proposed method employs the

data resampling approach, where the training data is

grouped into several partitions for cluster

construction, from which the prediction of cluster

memberships for test data points is made. CSAI was

motivated by the news event detection task in which we

would like to assess the quality of news text clusters.

The main contributions of this study include:

1. We propose stability-based cluster validation

index (CSAI) that leverages information from

multiple features associated with clusters.

2. The proposed method is evaluated using

benchmark textual datasets, each containing

multiple partitions for cluster construction and

a separate validation set for assessing the

quality of results.

3. The effectiveness of the proposed validation

index is compared against the widely used

clustering validation indices using various

clustering algorithms.

4. CSAI employs the structure of features within

clusters instead of centroids and adopts the

mean squared error expression as a distance

measure.

2 PROPOSED METHOD

In this paper, we propose CSAI to measure the quality

and consistency of clustering solutions. It employs

stability analysis to examine the robustness of

clustering algorithms across various samples, where

each cluster is characterized by aggregation of multiple

features. CSAI score is computed based on partitioning

the training data into several samples for cluster

generation and holding separate validation data for

evaluation of clustering results. The data partitioning

process shares some similarities with the conventional

cross-validation approach in supervised learning, but

here, the training data is resampled into several parts

without reserving any portion for validation purposes.

With this validation data, the clustering model applied

to the training data can be used to predict the cluster

membership of new data points. This strategy is in line

with (Tibshirani & Walther, 2005; Xu & Wunsch,

2005) that an effective clustering model should handle

new data points without relearning. The degree of

clustering stability is then measured by applying a

normalized distance measure between the clusters on

the training data and the validation set. This distance

quantifies how close or similar the features of the

validation data are to those of the training data within a

particular cluster.

In CSAI, the distance or similarity is calculated

individually for each cluster and then averaged across

all clusters in each of the partitions, which provides a

measure of the average discrepancy between the

similarity scores of features in the training and

validation data. By ‘features’, we refer to the

transformed representation of the original textual data

obtained through embeddings and manifold learning

techniques or other transformations. By leveraging

ICPRAM 2025 - 14th International Conference on Pattern Recognition Applications and Methods

378

these aggregated properties of cluster-associated data,

the CSAI index can be computed independent of the

clustering method used and its mode of operation.

Moreover, the distance measure employs a more robust

quality estimator, utilizing the mean square error

measure instead of relying solely on the Euclidean

distance.

2.1 Theoretical Framework for CSAI

The proposed CSAI method can be expressed formally

as follows. Assume a dataset X consist of 𝑝 data points

and A be a clustering algorithm. Each data point 𝑥



for

𝑟=1,2,…,𝑝, is represented as a vector of 𝑞 real-

valued features: 𝒙

𝒓

=𝑥



,𝑥



,…,𝑥



. Consider the

dataset X is randomly divided into 𝑚+1 subsets or

partitions: 𝑋



,𝑋



,…,𝑋



,𝑋



, and let 𝑋



= V is

used as the validation set, while all the remaining

partitions constitute the training set, denoted as

⋃

𝑋







Our next step is to take each partition in

𝑇: 𝑋



,𝑋



,…,𝑋



, and run an identical clustering

method A resulting in 𝑛 clusters each, the clusters

being named 𝐶



, 𝑖=1,2,…,𝑚 and j=1,2,…,𝑛. The

cluster membership of the data points in V is

subsequently determined based on their similarity to

the data points in 𝑇.

Compute the representative points for each cluster

𝐶



in T and 𝑉 by aggregating all features that are

associated with each cluster of 𝑇 and 𝑉. This process

transforms the values of T and V, each containing 𝑛

points in s-dimensional feature space. These points are

represented as 𝑥



and 𝑣



, respectively, where the

values can be the mean, sum, or maximum of all

features in each T and 𝑉. The distance between the

representative points 𝑥



and 𝑣



is indicative of both

cluster validity and stability. In both cases, less distance

implies higher quality clustering, reflecting accurate

(valid) and consistent (stable) clustering results. We

use a distance based on the mean square distance

expression (Eq. (1)), with division by 𝑞 to reflect the

average magnitude of the distance between 𝑥



and 𝑣



𝑑𝑥



,𝑣



=







∑

𝑥



−𝑣











(1)

where 𝑠<𝑞 is the total number of features,𝑥



and

𝑣



represent the k



feature obtained in the j



cluster

of the i



partition in T and 𝑉, respectively. To make

comparability of the distance scores across datasets and

algorithms, it is necessary to divide all scores by the

size of the largest value range among all features in the

representative points in T. This range value is

calculated in Eq. (2) for each partition.

w=max



(



max



(



max



𝑥





−min



𝑥





)) (2)

The CSAI score on each partition, obtained using

clustering algorithm A, represents the average distance

(𝑑) for all clusters in T



, normalized by 𝑤, as shown in

Eq. (3).

CSAI

(

𝑇



,𝐴

)

𝑤.𝑛





𝑠

𝑥



−𝑣















(3)

Finally, the global CSAI score is computed across all

the training partitions in 𝑇 using Eq. (4), which is

obtained by adding Eq. (3) for multiple partitions.

CSAI

(

T,A

)



𝑤.𝑛





𝑠

𝑥



−𝑣





















(4)

where 𝑚 represents the total number of partitions in the

training data, and 𝑛 is the total number of clusters

generated from each partition in the training data. 𝑤,

denote the difference between the maximum and

minimum values of the features in each partition of T.

The specific characteristic of CSAI is that it is based

on the exploitation of aggregated features attached to

clusters and the concept of stability. The aggregated

features can be helpful for discovering non-linear

relationships within the data and assessing the

robustness of clustering results, which might not be

feasible in the original data space. The similarity or

distance between these features is computed for each

cluster, which allows CSAI to assess both the validity

and the robustness of clustering solutions. The concept

of stability, on the other hand, is important for

determining the reliability of the clustering results and

can enhance the practicalability of results in real-world

applications. The other important feature of CSAI is

that a separately stored validation dataset is used for

quality assessment. The validation data, whether

sampled from the original dataset or collected

independently, should be excluded from the clustering

generation process, and solely used for result

evaluation. CSAI offers a unified approach to

validating clustering quality, aiding in the assessment

of clustering stability and generalizability while also

addressing the reproducibility issues in the field of

unsupervised learning (Hutson, 2018). Moreover,

CSAI can operate independently of the clustering

algorithm used.

A New Cluster Validation Index Based on Stability Analysis

379

2.2 Algorithm for CSAI

The proposed algorithm for the CSAI method is

presented in Algorithm 1. By analysing the theoretical

framework and algorithmic procedure, we can

determine the time complexity of the CSAI module. In

Algorithm 1, lines 1 and 2 involve iterating over the

dataset to split it into multiple subsamples. Step 3 of

Algorithm 1 iterates over the number of partitions in

the training data to construct clusters on each partition.

The subsequent steps include essential aspects of

CSAI, such as determining the number of partitions

and clusters, assigning data points to clusters, and

aggregating features within clusters. Feature

aggregation transformation involves statistical

calculations such as mean, median, mode, standard

deviation, etc., Then, from step 9 onwards, the error

between the training and validation datasets is

computed within each cluster and over partitions.

Overall, let M be the number of partitions in the

training dataset, N be the number of clusters in each

partition, and F be the number of features in the latent

space. Therefore, the time complexity of the CSAI

index can be approximated as O(M⋅N

⋅F), reflecting

the computational cost of the CSAI score calculations.

Algorithm 1: Proposed Clustering Stability Assessment

Index (CSAI).

https://www.gdeltproject.org/

3 EXPERIMENTS AND RESULTS

3.1 Datasets and Algorithms

Three publicly available datasets: 20 newsgroups

(20NG) (Rennie, 2003), MN-DS (Petukhova &

Fachada, 2023) and arXiv (Clement et al., 2019)

datasets, and one curated news dataset from the Global

Database of Events, Language, and Tone(GDELT)

were used to evaluate the effectiveness of our proposed

validation index. The 20NG dataset is a widely

recognized benchmark in machine learning and text

analysis, containing a total of 18,846 newsgroup posts

organized into 20 different categories. The MN-DS

dataset is based on the NELA-GT-2019 dataset

(Gruppi et al., 2020), which is a collection of 10,917

news articles, with most of them sourced from

mainstream outlets, such as ABC News, BBC, and The

Guardian. The ArXiv dataset is a collection of research

papers submitted to arXiv.org, a preprint repository

where scholars from various fields share their scientific

papers. It typically includes metadata such as title,

authors, abstract, publication date, and categories. The

fourth dataset is a collection of news articles from

GDLET, which is a comprehensive repository of

global news events and coverage. GDELT processes

news articles, television broadcasts, online news

sources, and other media to extract valuable insights.

For each textual dataset, we utilized the Sentence-

BERT (SBERT) (Reimers & Gurevych, 2019) for text

embedding, which produces 768-dimensional output

representation. Subsequently, we applied Uniform

Manifold Approximation and Projection (UMAP)

(McInnes et al., 2018) for generating low-dimensional

representations and visualizations (2D for visualization

and 10D for clustering).

In terms of clustering algorithms, four different

clustering algorithms: k-means and k-medoids,

agglomerative clustering, and Gaussian mixture model

(model-based clustering) were used for the evaluation

of CSAI. K-means clustering is among the most

popular clustering algorithms, which works by

partitioning a set of data points into distinct, non-

overlapping clusters (Dharmarajan & Velmurugan,

2013). K-medoid is an extension of K-means to

discover clusters, while it adopts “modes” instead of

cluster centres (Kaufman, 1990). One limitation of both

K-means and K-medoids is that they struggle to handle

noise and outliers and are not well-suited for detecting

clusters with non-convex shapes. Hierarchical

clustering, on the other hand, constructs a hierarchy of

clusters, visualized as a tree-like structure called a

ICPRAM 2025 - 14th International Conference on Pattern Recognition Applications and Methods

380

dendrogram. A Gaussian Mixture Model (GMM) is a

probabilistic approach for clustering and density

estimation (Lin et al., 2019). It assumes that data is

generated from a mixture of several Gaussian

distributions, each representing a distinct cluster.

3.2 Analysis of Results

In this study, multiple experiments were conducted to

assess the effectiveness of the proposed CSAI index.

Figure 1 depicts the CSAI score generated by four

different clustering algorithms (k-means, k-medoid,

agglomerative, and GMM) across three datasets. From

the figure, lower CSAI scores were observed on the

MN-DS dataset with k-means and k-medoid

algorithms. On the arXiv dataset, the better scores of

CSAI were scored with agglomerative and GMM,

while the worst was observed on k-medoid. Notably,

when comparing clustering algorithms based on CSAI

values, the agglomerative clustering algorithm

consistently outperformed others across all three

datasets (MN-DS, GDELT, and arXiv).

Figure 1: CSAI scores on different clustering algorithms.

In addition to evaluating each clustering algorithm

using the overall CSAI, we have also computed the

CSAI score across different partitions of each dataset

to gauge the stability of each algorithm. Understanding

the variability of CSAI can aid in selecting alternative

models; for instance, models with less variability in

CSAI score may be preferred over those with higher

variability. Additionally, CSAI variation can assist in

choosing the final clustering solution. The variation of

each model's CSAI score across the five samples in

each partition of the training data is shown in Figure 2

for the GDELT dataset and Figure 3 for the MN-DS

dataset. To quantify the stability assessment more

precisely, we computed the standard deviation (SD) of

the CSAI values across different subsamples. On

GDELT, it is evident that the highest instability was

observed with the k-medoid algorithm (SD:0.3664),

Figure 2: CSAI scores of four models across five samples

on the GDELT dataset.

showing significant differences in CSAI values across

the five samples. Furthermore, a slight variation in

CSAI scores was noted with AGM (SD:0.1860), where

almost identical CSAI values were scored on samples

2, 3, and 5, as well as the similarity between sample 1

and sample 4. The most plausible clustering solution

was observed in sample 2, with a better score of CSAI

on AGM, which produced nine reasonable clusters. On

MN-DS, all the clustering algorithms have shown

stability across the 5 samples, except GMM

(SD:0.3818), which has quite different values across

the five samples. On 20NG and arXiv datasets, the k-

means algorithm produced the most stable CSAI scores

with SD of 0.0780 and 0.1962, respectively.

Figure 3: CSAI scores of four models across five samples

on the MN-DS dataset.

3.3 CSAI Results with Different Indices

In addition to evaluating clustering results using CSAI,

we also considered other well-known clustering

validation indices, including silhouette index (SS)

(Rousseeuw, 1987), Davies-Bouldin (DB) (Davies &

Bouldin, 1979), Calinski-Harabasz (CH)

(Bandyopadhyay & Saha, 2008), Dunn index (DI)

(Dunn, 1974), Ray-Turi index (RT) (Ray & Turi,

1999), S_Dbw (Halkidi & Vazirgiannis, 2001), and

Xie-Beni (XB) (Tang et al., 2005). While these indices

are commonly used, they focus on a single aspect of

clustering quality (e.g. cluster validity). In contrast,

CSAI integrates both the stability and validity of

A New Cluster Validation Index Based on Stability Analysis

381

Table 3: Clustering results in terms of CSAI score and other indices on the four different datasets and algorithms.

Datasets

Validation

Indices

K-means

Clustering

K-medoid

Clustering

Agglomerative

Clustering

Gaussian

Mixture Model

MN-DS

Dataset

5.7097 1.9122 1.1327 1.0005

RT 0.3466 0.4336 0.3512 0.3389

1.1635 0.6474 0.75604 0.8325

S_Dbw

0.50631 0.5577 0.5672 0.4638

CSAI

0.4523 0.3489 0.2986 0.6269

GDELT Dataset

1.2474 1.5868 1.0859 1.1474

RT 0.4594 1.6694 1.7936 0.4191

2.5290 3.0951 2.4871 0.2862

S_Dbw

0.5377 0.6125 0.47813 0.53000

CSAI 0.4330

0.7327 0.2712 0.6619

ArXiv

Dataset

DB 1.1479

1.3917 1.2371 1.4070

RT 0.2205 1.7617 0.3645 0.2381

XB 0.6965

0.8889 0.8512 0.6040

S_Dbw 0.5415

0.6286 0.5129 0.6847

CSAI 0.6490

0.8025 0.2604 0.4113

20NG

Dataset

DB 0.9374 1.1159 0.9453 1.4071

RT 0.1934 1.9904 0.2259 0.23810

XB 0.9522

1.7111 1.2775 0.6040

S_Dbw 0.4320

0.5574 0.3426 0.6847

CSAI 0.6392 0.5499

0.3099 0.3084

clustering results, offering a more holistic evaluation

approach. CSAI, DB, RT, S_Dbw and XB have values

ranging between zero (0) and positive infinity (+∞),

where the lower values indicate better clustering

results. CH and DI are also bounded by (0, +∞), but

unlike CSAI, higher values indicate better clustering,

and lower values suggest poor clustering. SS score

takes values in the interval [-1, 1]. Negative values

represent the wrong placement of data points, while

positive values indicate better assignments. When

comparing and analysing the different clustering

indices, we observed that CSAI can be better compared

with DB, RT, S_Dbw, and XB as they share the same

value range and interpretation of index values. Table 3

presents the scores of these indices and CSAI involving

four clustering algorithms across four different

datasets, including MN-DS, GDELT, ArXiv, and

20NG datasets. For CSAI to be comparable across

different datasets and clustering structures,

normalization of the index scores was carried out,

following literature guidelines (J. Wu et al.,

2009);(Rezaei & Franti, 2016). On the MN-DS dataset,

agglomerative clustering yields the lowest CSAI score,

indicating that it provides the most stable and valid

clusters. K-means and K-medoids produce higher

CSAI values, suggesting less stability. The GMM, with

a CSAI of 0.6269, performs the worst on this dataset.

Agglomerative clustering consistently outperforms

other algorithms in terms of cluster quality, as indicated

by the lowest CSAI scores. On the same dataset, RT

exhibited slightly better scores in the cases of k- means

and GMM. On GDELT, CSAI achieved the best scores

on k-means and agglomerative hierarchical clustering,

S_Dbw favoured K-medoid, and the XB score was the

best on the Gaussian model.

In this context, the word “best” refers to the

clustering algorithm with the lowest CSAI score. On

the other hand, k-medoid clustering performs better

with the S_Dbw index on GDELT and ArXiv datasets.

Overall, the most notable observation from Table 3 is

that the CSAI index showed the best value on

agglomerative clustering indicating it produces more

stable and quality clustering results compared to other

algorithms across the four datasets. This indicates that

CSAI is best suited to hierarchical clustering while also

showcasing its effectiveness across various clustering

algorithms. Unlike indices such as RT, which favors

compact clusters, CSAI balances compactness and

stability, aligning closely with S_Dbw in terms of score

while diverging from RT and XB in certain cases. This

highlights CSAI's ability to provide a more holistic

evaluation of clustering quality.

ICPRAM 2025 - 14th International Conference on Pattern Recognition Applications and Methods

382

4 CONCLUSIONS

In this paper, we proposed CSAI, a new cluster

evaluation index for dealing with the validity and

robustness of clustering solutions. This approach is

used to assess the quality of clustering solutions and the

reproducibility of data clustering through the concept

of model stability. Building up on some of the

limitations in the literature, CSAI distinguishes itself

by fully leveraging aggregated feature structures

pertaining to clusters rather than focusing on cluster

centroids. For evaluation, we conducted extensive

experiments on four different publicly available

datasets to validate the generality and effectiveness of

the proposed algorithm. In our experiments, we

selected the widely used clustering algorithms,

including k-means, k-medoid, agglomerative

hierarchical clustering and Gaussian mixture model.

Our experimental results demonstrate that CSAI is a

promising unified solution that can effectively evaluate

the quality of clustering solutions and their stability.

More importantly, the CSAI index exhibited the

highest efficacy in agglomerative hierarchical

clustering, surpassing other indices, highlighting its

suitability for hierarchical clustering while also

highlighting its effectiveness on other clustering

algorithms.

As limitation of the study, CSAI’s efficacy has

been evaluated using only textual datasets, despite the

potential for the proposed method to work on other

domains and data modalities such as images and other

datasets. Thus, further research will be towards using

these diverse data types to validate the algorithm's

versatility and robustness across different domains.

ACKNOWLEDGEMENTS

This research work is funded by SFIMediaFutures

Partners and the Research Council of Norway (Grant

number 309339).

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