Graph Theory Approach Probability Distribution Based Intuitionistic Fuzzy C-means Clustering Using LINEX Hellinger Distance and Its Applications

Senthil; Nithya; Bhuvaneswari

doi:doi:10.11648/j.ijmfs.20261202.11

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Graph Theory Approach Probability Distribution Based Intuitionistic Fuzzy C-means Clustering Using LINEX Hellinger Distance and Its Applications

Senthil^*

, Nithya

, Bhuvaneswari

Published in International Journal of Management and Fuzzy Systems (Volume 12, Issue 2)

Received: 21 April 2026 Accepted: 8 May 2026 Published: 12 November 2026

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Abstract

Clustering is an essential task to unsupervised learning, its used to partition a dataset into groups of similar elements. In addition to traditional clustering methods, intuitionistic fuzzy c-means (IFCM) clustering algorithm has been studied to capturing uncertain information in the dataset. However, the IFCM algorithm has limitations such as the random initialization of cluster centers and the unrestricted influence of all samples on all cluster centers. Therefore, a robust algorithm named probability distribution of the minimal spanning tree based intuitionistic fuzzy c-means clustering algorithm using LINEX Hellinger distance is proposed for improving shortcomings of the IFCM. Firstly, the probability distribution of the minimal spanning tree algorithm organizes the initial cluster center based on the contribution of local density information from the data. Secondly, MST node probability distribution based IFCM using Hellinger distance, the membership degree boundary is assigned for the data samples on all cluster centers in the clustering process. Finally, the performance of the proposed modified IFCM algorithm is numerically validated using random dataset, partition coefficient and validation function are used to evaluate the validity of clustering also this paper compares the results of proposed method with the results of existing basic intuitionistic fuzzy c-means.

Published in	International Journal of Management and Fuzzy Systems (Volume 12, Issue 2)
DOI	10.11648/j.ijmfs.20261202.11
Page(s)	20-31
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2026. Published by Science Publishing Group

Keywords

Intuitionistic Fuzzy C-means, Probability Distribution, Minimal Spanning Tree, Hellinger Distance, LINEX Function

1. Introduction

Clustering aims to group a set of objects into clusters such that the objects of the same cluster are similar, and objects belonging to diferent clusters are dissimilar. Clustering is an active research topic in statistics, pattern recognition, machine learning, and data mining. Clustering helps in finding natural boundaries in the data whereas fuzzy clustering can be used to handle the problem of vague boundaries of clusters. In fuzzy clustering, the requirement of crisp partition of the data is replaced by a weaker requirement of fuzzy partition, where the associations among data are represented by fuzzy relations.

In Atanassov

[1]

presented “Intuitionistic fuzzy sets” that considers vulnerability in the definition of the membership value. Different approaches have been designed with the Intuitionistic theory for the process of image segmentation.

Chaira

[3]

has given a novel Intuitionistic fuzzy c-means (IFCM) clustering algorithm for the process of partitioning an image into segments. It includes hesitation degree and a new parameter termed as intuitionistic fuzzy entropy (IFE) in the objective function. Although this strategy utilizes the irregular initialization of cluster prototype which produces inaccurate results when chosen randomly. This method does not include the local information and the spatial information, hence is sensitive to disturbances and other image artifacts

[2, 10]

. In order to make clustering more robust, Intuitionistic Fuzzy C-means clustering

[6]

assume that a pixel belongs to multiple clusters with different membership degrees

[12]

. These algorithms utilized only intensity of pixel as the only feature for the segmentation of images and failed to classify noisy pixels accurately. The pixels in an image are exceedingly associated, i.e. the every pixel in the prompt vicinity have about the equivalent feature information unless there is some curve or contour. Subsequently, integrating spatial information along with the membership value results in more homogenous regions as compared to other methods. However, these techniques use the arbitrary initialization of cluster centers which give inaccurate outcomes and more time for optimization.

A recent development is to use kernel method to construct the evolutionary kernel intuitionistic fuzzy c-means clustering algorithm, the new probability similarity measurement method and clustering technique are also proposed to design adaptive weights for intuitionistic fuzzy distances

[7]

. In

[13]

the intuitionistic fuzzy set and rough set are combined with the statistical feature extraction technology. The intuitionistic fuzzy set is used to extract the regions of interest, and the gray scale co-occurrence matrix is used for feature extraction. Pop PC

[9]

proposed clustering algorithms based on the minimum spanning tree (MST) are able to detect clusters with arbitrary shapes. In

[4]

the algorithm uses a new distance between local density peaks based on shared neighbors to construct a minimum spanning tree on the local density peaks, which excludes the interference of noise points and reduces the running time of MST-based clustering algorithms. A modified non-membership function to generate intuitionistic fuzzy set, which highlights the effect of uncertainty and makes good use of image information. A method of determining initial clustering centers based on pixel characteristics is also proposed by Jun Kong, JianHou, Min Jiang &Jinhua Sun

[5]

. Intuitionistic fuzzy C-means (IFCM) algorithm, as a successful extension and variant of FCM, has attracted extensive attention and has been widely used in many fields such as image processing and pattern recognition by

[8, 11, 14]

. Although the above methods are claimed to be robust to noise, they are confronted with the problem of selecting the parameters that control the role of the spatial constraints.

In this paper, to overcome the defects of the algorithms as mentioned, we proposed weighted–LINEX HELLINGER distance using IFCM, which requires the determination of the edge density and degree of minimal spanning, named the MST using weighted LINEX_IFCM. The intuitionistic fuzziness is embedded into the calculation process of similarity between the pixel and cluster centers, which achieves more accurate segmentation in the organization boundary. The algorithm is minimal spanning tree initialization method using LINEX HELLINGER_IFCM, which helps to speed up the convergence of the algorithm.

2. Minimal Spanning Tree Algorithm

In this section, we proposed LINEX HELLINGER distance measures for construct the minimal spanning tree.

2.1. HellingerDistance

Let M and N discrete probabilistic distributions with

and

. Then Hellinger distance will be

So Hellinger distance is directly related to the Euclidean norm

[17]

The Hellinger distance posseses the following characteristics:

(i). Its values range between 0 and 1, where 0 signifies complete similarity between two distributions and 1 indicates complete dissimilarity.

(ii). When two distributions are very similar, the Hellinger distance approcahes 0.

(iii). The Hellinger distance is symmetric, meaning

. Compared to other distance metrices, such as Kullback-Leibler (KL) divergence ot total variation distance, Hellinger distance is more robust to outliers and in certain cases more accessible to compute. It finds widespread application in probability distribution comparisons and model fitting.

2.2. LINEX Hellinger Distance Based MST

Given the grayscale point set D, the hierarchical methods starts by constructing a minimal spanning tree (MST) from the points in D. In

and

are two points of a MST and

is an edge between x and y then LINEXHellinger distance function between x and y is denoted by

and calculated using equation (1).

(1)

2.3. Edge Density Weight of MST

is the minimum spanning tree generated from dataset X of size N, where

are vertices corresponding to objects,

are edges corresponding to connection relationship between objects.

Edge density weight

(2)

where

deg(x) = degree of x from MST and

deg(y) = degree of y from MST.

2.4. Algorithm for Optimal Number of Clusters

The Algorithm: LINEX HELLINGER BASED MST

Input: Data points

Output: optimal number of cluster centers

Let e1 be an edge in the LINEX measure MSTconstructed from Data points

Let

be the set of disjoint subtrees of LINEX measure MST.

1) Create a node v, for each data points.

2) Compute the edge weight using equation (1)

3) Construct an LINEX measure MST from 2.

4) Compute the Edge density weight using equation (2)

6) For each

7) To remove longest edge (or)

from MST

is new disjoint subtrees (regions).

10) Compute the center

using average of points.

11)

12)

= Minimum standard deviation of edge density

13)

= Maximum standard deviation of edge density

14)

15) Until

16) Update the clusters points, repeat step 7 to step 14.

17) Finally we obtain the cluster centers.

3. Formulation of LinexHellinger Distance Based Intuitionistic Fuzzy C Means Clustering

3.1. Intuitionistic Fuzzy C-means (IFCM) Algorithm

Intuitionistic fuzzy set given by Atanassov

[1]

considers both membership

and non-membership

. An intuitionistic fuzzy set A in X, is written as

where

are the membership and non-membership degrees of an element in the set A with the condition

when

for every x in the set A, then the set A becomes a fuzzy set. Also indicated a hesitation degree,

which arises due to lack of knowledge in defining the membership degree of each element x in the set A and is given by

[1]

intuitionistic fuzzy c-means, minimizes the objective function as:

(3)

, where

denotes the intuitionistic fuzzy membership and

(4)

(5)

(6)

(7)

This iteration will stop when

where

is a termination criterion between 0 and 1, where as k is the iteration steps. This procedure converges to a local minimum or a saddle point of

3.2. LINEX Hellinger Distance Based Intuitionistic Fuzzy C-means (IFCM) Algorithm

In this section, we want to use the LINEX HELLINGER distance based Intuitionistic fuzzy c-means (IFCM) algorithm when the over estimating and the under estimating are not of the same importance. The procedures are the same as aIntuitionistic fuzzy c-means algorithm. All the entities are assigned to their nearest centroid from MST, using a LINEX loss function as the dissimilarity distance. The procedure continues until there is no change in clusters. Now consider the following optimization problem,

Objective function:

(8)

where

and

3.3. Updating Membership

To obtain equation for calculating membership we minimizing the objective function

with constraint conditions

Can be solved by using the Lagrangian multiplication as follows:

To find min

it is sufficient to minimize the following inner sum for fixed k:

put

Let

and

is stationary for F only if

that yields to

(9)

(10)

Then

(11)

From (9)

(12)

Substitute (12) in (11). we obtain

(13)

The general equation is used to obtain membership ranks for objects in data for finding meaningful groups.

3.4. Obtaining Cluster Prototype Updating

To find min

it is sufficient to minimize the following inner sum for fixed i:

Taking the partial derivative of objective function with respect to

and setting the result to zero, we have the general form of updating center as

(14)

where

, where

denotes the intuitionistic fuzzy membership and

(15)

(16)

The LINEX HELLINGER distance is suitable for clustering in which it can actually induce the necessary conditions.

This iteration will stop when

The MST based LINEX HELLINGER _IFCM algorithm iteratively optimizes

by continuous updating

and

until the difference in successive

values is very small

, where

is a small positive value between 0 and 1.

4. Efficient LINEX HELLINGER Distance Based IFCM

4.1. Efficient LINEX HELLINGER_IFCM Algorithm

Stage 1: Set the cluster centroids

by using

LINEX measure MST initialization method.

Stage 2: Compute the membership function using (13)

Stage 3: Update the cluster centroids using (14)

Stage 4: Go to stage (3)-(5), repeat until convergence.

Stage 7: Image segmentation after defuzzification and then

a region labeling procedure is proposed.

Stage 8: The termination criterion is as follows

, where m is the iteration count,

is a small number that can be set by the user.

The proposed efficient LINEX HELLINGER distance based MST obtained cluster centers; the LINEX HELLINGER distance based IFCM algorithm continues iteratively updates, membership and centroids with these values. When this improved, Efficient LINEX HELLINGER distance based IFCM algorithm has converged, another defuzzification process takes place in order to convert the fuzzy partition matrix to a crisp partition matrix that is segmented.

4.2. Validation Function Based on Feature Structures

Two representative functions for the fuzzy partition namely; Partition coefficient

and Validation function

are used to evaluate the validity of clustering

[15, 16]

(17)

(18)

The proposed efficient LINEX HELLINGER distance based MST obtained cluster centers; the LINEX HELLINGER distance based IFCM algorithm continues iteratively updates, membership and centroids with these values. When this improved, Efficient MHIFCM algorithm has converged, another defuzzification process takes place in order to convert the fuzzy partition matrix to a crisp partition matrix that is segmented.

5. Results and Discussion

Download: Download full-size image

Figure 1. Scatter diagram for random dataset.

Table 1. Random Data.

Data			Intensity	Data			Intensity
S.No	X	Y	I(v)	S.No	X	Y	I(v)
1	2.50	3.00	0.75	11	15.80	20.50	0.25
2	3.80	3.50	0.50	12	11.80	12.50	0.35
3	7.00	1.80	0.15	13	15.50	14.50	0.80
4	3.10	4.80	0.18	14	5.50	10.50	0.70
5	5.50	7.50	0.45	15	18.50	19.50	0.40
6	8.50	9.50	0.75	16	12.50	13.80	0.25
7	10.80	11.50	0.60	17	21.80	12.50	0.95
8	4.20	3.80	0.25	18	19.80	20.50	0.25
9	2.80	1.80	0.45	19	19.00	20.00	0.60
10	12.50	20.50	0.65	20	11.00	12.00	0.30

Table 2. Dissimilarity matrix.

S.No	Co-ordinate		intensity
S.No	x	y	I(v)	vertex	1	2	3	4	5	6	7	8	9	10
1	2.50	3.00	0.75	1	0.0000	0.0408	0.2892	0.0942	0.3091	0.6035	0.8693	0.0867	0.0449	1.5020
2	3.80	3.50	0.50	2		0.0000	0.1761	0.0490	0.1841	0.4338	0.6711	0.0140	0.0780	1.3012
3	7.00	1.80	0.15	3			0.0000	0.2876	0.3871	0.5657	0.7566	0.1521	0.2048	1.4524
4	3.10	4.80	0.18	4				0.0000	0.1455	0.4181	0.6427	0.0337	0.1618	1.2158
5	5.50	7.50	0.45	5					0.0000	0.1029	0.2566	0.1525	0.4303	0.7407
6	8.50	9.50	0.75	6						0.0000	0.0539	0.3941	0.7374	0.4416
7	10.80	11.50	0.60	7							0.0000	0.6146	1.0036	0.2621
8	4.20	3.80	0.25	8								0.0000	0.1144	1.2360
9	2.80	1.80	0.45	9									0.0000	1.6847
10	12.50	20.50	0.65	10										0.0000

Table 3. LINEX measure based minimal spanning tree edges.

S.No	Edges	LINEX HELLINGER distance	S.No	Edges	LINEX HELLINGER distance
1	(1,2)	0.0408	11	(20,12)	0.0051
2	(2,8)	0.0140	12	(12,16)	0.0120
3	(8,4)	0.0337	13	(16,13)	0.0714
4	(8,9)	0.1144	14	(13,15)	0.1204
5	(8,3)	0.1521	15	(15,19)	0.0064
6	(8,5)	0.1525	16	(15,18)	0.0124
7	(5,14)	0.0619	17	(15,11)	0.0313
8	(14,6)	0.0767	18	(11,10)	0.0635
9	(6,7)	0.0539	19	(11,17)	0.3183
10	(7,20)	0.0136

Table 4. Degree of minimal spanning tree edges.

S.No	Degree of MST	S.No	Degree of MST
1	1	11	3
2	2	12	2
3	1	13	2
4	1	14	2
5	2	15	4
6	2	16	2
7	2	17	1
8	5	18	1
9	1	19	1
10	1	20	2

Download: Download full-size image

Figure 2. LINEX HELLINGER distance based Minimal spanning tree connected through points.

Table 5. Edge density minimal spanning tree edges.

S.No	Edges	Edge density	S.No	Edges	Edge density
1	(1,2)	0.0577	11	(20,12)	0.0102
2	(2,8)	0.0443	12	(12,16)	0.0240
3	(8,4)	0.0754	13	(16,13)	0.1428
4	(8,9)	0.2558	14	(13,15)	0.3405
5	(8,3)	0.3401	15	(15,19)	0.0128
6	(8,5)	0.4822	16	(15,18)	0.0248
7	(5,14)	0.1238	17	(15,11)	0.1084
8	(14,6)	0.1534	18	(11,10)	0.1100
9	(6,7)	0.1078	19	(11,17)	0.5513
10	(7,20)	0.0272

Our weighted LINEX Hellinger distance based minimal spanning tree algorithm constructs MST from the dissimilarity matrix is shown Figure 2. Figure 2 shows a typical example of weighted LINEX Hellinger measure MST1 constructed from point set (from Dissimilarity matrix), in which inconsistent edges are removed to create subtree (clusters/regions). our algorithm finds the center of each clusters, which will be useful in many applications. Generally in most of the clustering algorithm data points can be represented as dissimilarity matrix representation. It contains the distance values between the data points represented as lower or upper triangular matrix. First to identify the longest edge or

in the MST to generate subtree (clusters). Table 4, the longest edge weight 0.5513 connecting the data points 11 and 17 is find to be inconsistent one. By removing the inconsistent edge from the weighted LINEX HELLINGER_MST, data points partitioned into two subtrees or clusters

and

namely.

and

. Now minimal spanning tree produced outlier is {17}. Next to find minimum and maximum standard deviation of

then Cluster Separation value less than 0.5625 is valid. Secondly to remove next longest edge, weighted LINEX HELLINGER distance based MST partitioned into Three subtrees or clusters

and

namely

and

. Next to find minimum and maximum edge and to calculate standard deviation of

then Cluster separation value less than 0.5625 is valid. Here standard deviation of edge density

, Standard deviation of edge density

and Standard deviation of edge density

then

then we stop the removing inconsistent edges.. Finally minimal spanning tree produced three cluster center and 1 outliers. Then the center of the cluster and its convergence of standard FCM and LINEX HELLINGER distance based IFCM are determined under successive interations of experiments using data points. With the new efficient objective function based LINEX Hellinger distance induced weighted measure the termination value is achieved, with very less iteration and with much better performance in getting membership (Table 6) than standard FCM. Table 7 gives the number of iteration to achieve the results of cluster on the data points by standard FCM and weighted LINEX_IFCM. It is clear from the final cluster, membership (Table 6), scatter diagram (Figure 1), minimal spanning tree (Figure 2) that our proposed LINEX Hellinger _IFCM induced weighted degree of minimal spanning tree is much faster than the standard FCM and the method is converged fast to terminate condition with less run time. To test the effectiveness of weighted LINEX_IFCM, the edge density minimal spanning tree based IFCM is used as center. This is done to find out the fuzzy membership and appropriate number of clusters. Thus, we have concluded the final optimal clusters formed as 3 (Figure 3). This algorithm has also reduced the number of iterations. Best result is achieved by this measure fuzzy partition coefficient

maximum and validation function

minimum (Table 8). The weighted LINEX_IFCM clustering algorithm has the following membership value intimacy (Table 6),

Table 6. Final membership of three clusters of LINEX HELLINGER distance based Intuitionistic FCM method and object allocation.

Co-ordinate (x,y)			intensity				appropriate cluster
S.No	x	y	I(v)	Mem-1	Mem-2	Mem-3	appropriate cluster
1	2.50	3.00	0.75	1	0.9213	0.0523	0.0265
2	3.80	3.50	0.50	2	0.9895	0.0072	0.0032
3	7.00	1.80	0.15	1	0.7820	0.1444	0.0736
4	3.10	4.80	0.18	1	0.8808	0.0830	0.0362
5	5.50	7.50	0.45	2	0.4630	0.4292	0.1077
6	8.50	9.50	0.75	2	0.0794	0.8548	0.0658
7	10.80	11.50	0.60	2	0.0046	0.9858	0.0096
8	4.20	3.80	0.25	5	0.9710	0.0202	0.0087
9	2.80	1.80	0.45	1	0.9149	0.0551	0.0299
10	12.50	20.50	0.65	1	0.0559	0.2632	0.6809
11	15.80	20.50	0.25	3	0.0199	0.0788	0.9013
12	11.80	12.50	0.35	2	0.0269	0.8865	0.0866
13	15.50	14.50	0.80	2	0.0480	0.3319	0.6201
14	5.50	10.50	0.70	2	0.2367	0.6315	0.1318
15	18.50	19.50	0.40	4	0.0026	0.0093	0.9882
16	12.50	13.80	0.25	2	0.0490	0.7068	0.2442
17	21.80	12.50	0.95	1	0.0852	0.2753	0.6395
18	19.80	20.50	0.25	1	0.0161	0.0513	0.9326
19	19.00	20.00	0.60	1	0.0050	0.0171	0.9779
20	11.00	12.00	0.30	2	0.0171	0.9420	0.0410

Table 7. Comparison of iteration count.

	No.of iterations	No.of clusters
FCM	11	3
KFCM	8	3
Edge density MST initialization method basedLINEX Hellingerdistance based Intuitionistic FCM	3	3

Table 8. Cluster validity function.

	V_pc	V_p
FCM	0.8325	0.1825
KFCM	0.8238	0.1720
MST initialization method based LINEXHellinger Intuitionistic FCM	0.8889	0.0661

Download: Download full-size image

Figure 3. MST initialization method based LINEX Hellinger distance using Intuitionistic FCM, final cluster three.

6. Conclusion

In this paper, we have proposed a modified intuitionistic fuzzy c-means algorithm (IFCM) induced weighted LINEX Hellinger distance measure using degree of MST and solved analytically the objective function of the weighted LINEX Hellinger_IFCM method using Lagrange method of undetermined multiplier. Besides, the intuitionistic fuzzy set and non-membership degree are generated by an improved method, which highlight the role of uncertainty effectively. The algorithm overcomes problems involved with membership values of objects to each cluster by generalizing degrees of membership of objects to each cluster. For the initial partition matrix, if the membership degrees and non-membership degrees of the classified objects to the categories are obviously different, the iteration times will be reduced accordingly. Compared with FCM clustering algorithm, Experimental results demonstrate that the performance of random image data points, the pixel clustering effect and the robustness to noise of the proposed algorithm are all significantly better than other algorithms method.

Abbreviations

MST	Minimal Spanning Tree
IFCM	Intuitionistic Fuzzy C-means
LINEX	Linear Exponential

Acknowledgments

We would like to thank the reviewers for their constructive comments. We thank to Tmt. K. Jayagandhi, Additional Director (AGC), Department of Economics and Statistics, Government of Tamilnadu, Chennai, for his encouragement and support given. I also thank Dr. A. Sathya, Assistant Professor, Department of Mathematics, Government College of Daman, Daman & Diu and Dr. Annamalai, Statistical Inspector (Agri), Department of Economics and Statistics, Chennaifor her timely help throughout this work.

Author Contributions

Senthil: Data curation, Methodology

Nithya: Conceptualization, Resources

Bhuvaneswari: Conceptualization, Resources

Conflicts of Interest

The authors declare no conflicts of interest.

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Senthil, Nithya, Bhuvaneswari. (2026). Graph Theory Approach Probability Distribution Based Intuitionistic Fuzzy C-means Clustering Using LINEX Hellinger Distance and Its Applications. International Journal of Management and Fuzzy Systems, 12(2), 20-31. https://doi.org/10.11648/j.ijmfs.20261202.11

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Senthil; Nithya; Bhuvaneswari. Graph Theory Approach Probability Distribution Based Intuitionistic Fuzzy C-means Clustering Using LINEX Hellinger Distance and Its Applications. Int. J. Manag. Fuzzy Syst. 2026, 12(2), 20-31. doi: 10.11648/j.ijmfs.20261202.11

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Senthil, Nithya, Bhuvaneswari. Graph Theory Approach Probability Distribution Based Intuitionistic Fuzzy C-means Clustering Using LINEX Hellinger Distance and Its Applications. Int J Manag Fuzzy Syst. 2026;12(2):20-31. doi: 10.11648/j.ijmfs.20261202.11

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@article{10.11648/j.ijmfs.20261202.11,
  author = {Senthil and Nithya and Bhuvaneswari},
  title = {Graph Theory Approach Probability Distribution Based Intuitionistic Fuzzy C-means Clustering Using LINEX Hellinger Distance and Its Applications},
  journal = {International Journal of Management and Fuzzy Systems},
  volume = {12},
  number = {2},
  pages = {20-31},
  doi = {10.11648/j.ijmfs.20261202.11},
  url = {https://doi.org/10.11648/j.ijmfs.20261202.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijmfs.20261202.11},
  abstract = {Clustering is an essential task to unsupervised learning, its used to partition a dataset into groups of similar elements. In addition to traditional clustering methods, intuitionistic fuzzy c-means (IFCM) clustering algorithm has been studied to capturing uncertain information in the dataset. However, the IFCM algorithm has limitations such as the random initialization of cluster centers and the unrestricted influence of all samples on all cluster centers. Therefore, a robust algorithm named probability distribution of the minimal spanning tree based intuitionistic fuzzy c-means clustering algorithm using LINEX Hellinger distance is proposed for improving shortcomings of the IFCM. Firstly, the probability distribution of the minimal spanning tree algorithm organizes the initial cluster center based on the contribution of local density information from the data. Secondly, MST node probability distribution based IFCM using Hellinger distance, the membership degree boundary is assigned for the data samples on all cluster centers in the clustering process. Finally, the performance of the proposed modified IFCM algorithm is numerically validated using random dataset, partition coefficient and validation function are used to evaluate the validity of clustering also this paper compares the results of proposed method with the results of existing basic intuitionistic fuzzy c-means.},
 year = {2026}
}

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TY  - JOUR
T1  - Graph Theory Approach Probability Distribution Based Intuitionistic Fuzzy C-means Clustering Using LINEX Hellinger Distance and Its Applications
AU  - Senthil
AU  - Nithya
AU  - Bhuvaneswari
Y1  - 2026/11/12
PY  - 2026
N1  - https://doi.org/10.11648/j.ijmfs.20261202.11
DO  - 10.11648/j.ijmfs.20261202.11
T2  - International Journal of Management and Fuzzy Systems
JF  - International Journal of Management and Fuzzy Systems
JO  - International Journal of Management and Fuzzy Systems
SP  - 20
EP  - 31
PB  - Science Publishing Group
SN  - 2575-4947
UR  - https://doi.org/10.11648/j.ijmfs.20261202.11
AB  - Clustering is an essential task to unsupervised learning, its used to partition a dataset into groups of similar elements. In addition to traditional clustering methods, intuitionistic fuzzy c-means (IFCM) clustering algorithm has been studied to capturing uncertain information in the dataset. However, the IFCM algorithm has limitations such as the random initialization of cluster centers and the unrestricted influence of all samples on all cluster centers. Therefore, a robust algorithm named probability distribution of the minimal spanning tree based intuitionistic fuzzy c-means clustering algorithm using LINEX Hellinger distance is proposed for improving shortcomings of the IFCM. Firstly, the probability distribution of the minimal spanning tree algorithm organizes the initial cluster center based on the contribution of local density information from the data. Secondly, MST node probability distribution based IFCM using Hellinger distance, the membership degree boundary is assigned for the data samples on all cluster centers in the clustering process. Finally, the performance of the proposed modified IFCM algorithm is numerically validated using random dataset, partition coefficient and validation function are used to evaluate the validity of clustering also this paper compares the results of proposed method with the results of existing basic intuitionistic fuzzy c-means.
VL  - 12
IS  - 2
ER  -

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Author Information

Senthil

Department of Economics and Statistics, DMS Campus, Teynampet, Chennai, India

Biography: Senthil was born in 1983. He is a Statistical Inspector (RMIS) in the Department of Economics and Statistics, Government of Tamilnadu, Chennai. He completed her PhD in Mathematics at Anna University, Chennai in 2016. He teaches Engineering Mathematics, Graph Theory and Applications of Data Mining, Statistical Techniques in Data Analysis. She has over 10 years of teaching and research experience. Her research interests are in Data Mining, Data Analytics and Digital Transformation. She is a member of Board of Studies UG and PG Mathematics Mother Teresa Women’s University, Kodaikanal. He had published some of her work in both local and international journals.

Contact Email

http://orcid.org/0000-0002-6608-4580
Nithya

Adi Dravida Welfare Primary School, Dindigul, India

Biography: Nithya was born in 1992. She is a Teacher, Adi Dravida Welfare Primary School, V, Perumalkovil Patty, dindigul and Research scholar at the Department of Mathematics, Mother Teresa Women’s University, Kodaikanal. She completed her Master of Philosophy (M.Phil) in Mathematics at Mother Teresa Women’s University, Kodaikanal in the year 2019 and Master of Science (M.Sc) in Mathematics from the same institution in 2017. His current research interests include digital topology, data mining and image processing.

http://orcid.org/0000-0002-8222-466X
Bhuvaneswari

Department of Mathematics, Mother Teresa Women’s University, Kodaikanal, India

Biography: Bhuvaneswari was born in 1968. She is a Professor of Mathematics in the Department of Mathematics, Mother Teresa Women’s University, Kodaikanal. She completed her PhD in Mathematics at Bharathiar University, Coimbatore in 2005. She teaches Digital Topology and Applications of Data Mining, Techniques in Data Analysis and Principles of Compilers. She has over 28 years of teaching and research experience. Her research interests are in Data Mining, Data Analytics and Digital Transformation. She is a member of Board of Studies – UG and PG Mathematics Tamil Nadu State Council for Higher Education (TANSCHE), Chennai - 11th Dec. 2018. She had published some of her work in both local and international journals and presented in several conferences.

http://orcid.org/0000-0003-4288-9250

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APA Style

Senthil, Nithya, Bhuvaneswari. (2026). Graph Theory Approach Probability Distribution Based Intuitionistic Fuzzy C-means Clustering Using LINEX Hellinger Distance and Its Applications. International Journal of Management and Fuzzy Systems, 12(2), 20-31. https://doi.org/10.11648/j.ijmfs.20261202.11

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ACS Style

Senthil; Nithya; Bhuvaneswari. Graph Theory Approach Probability Distribution Based Intuitionistic Fuzzy C-means Clustering Using LINEX Hellinger Distance and Its Applications. Int. J. Manag. Fuzzy Syst. 2026, 12(2), 20-31. doi: 10.11648/j.ijmfs.20261202.11

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AMA Style

Senthil, Nithya, Bhuvaneswari. Graph Theory Approach Probability Distribution Based Intuitionistic Fuzzy C-means Clustering Using LINEX Hellinger Distance and Its Applications. Int J Manag Fuzzy Syst. 2026;12(2):20-31. doi: 10.11648/j.ijmfs.20261202.11

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@article{10.11648/j.ijmfs.20261202.11,
  author = {Senthil and Nithya and Bhuvaneswari},
  title = {Graph Theory Approach Probability Distribution Based Intuitionistic Fuzzy C-means Clustering Using LINEX Hellinger Distance and Its Applications},
  journal = {International Journal of Management and Fuzzy Systems},
  volume = {12},
  number = {2},
  pages = {20-31},
  doi = {10.11648/j.ijmfs.20261202.11},
  url = {https://doi.org/10.11648/j.ijmfs.20261202.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijmfs.20261202.11},
  abstract = {Clustering is an essential task to unsupervised learning, its used to partition a dataset into groups of similar elements. In addition to traditional clustering methods, intuitionistic fuzzy c-means (IFCM) clustering algorithm has been studied to capturing uncertain information in the dataset. However, the IFCM algorithm has limitations such as the random initialization of cluster centers and the unrestricted influence of all samples on all cluster centers. Therefore, a robust algorithm named probability distribution of the minimal spanning tree based intuitionistic fuzzy c-means clustering algorithm using LINEX Hellinger distance is proposed for improving shortcomings of the IFCM. Firstly, the probability distribution of the minimal spanning tree algorithm organizes the initial cluster center based on the contribution of local density information from the data. Secondly, MST node probability distribution based IFCM using Hellinger distance, the membership degree boundary is assigned for the data samples on all cluster centers in the clustering process. Finally, the performance of the proposed modified IFCM algorithm is numerically validated using random dataset, partition coefficient and validation function are used to evaluate the validity of clustering also this paper compares the results of proposed method with the results of existing basic intuitionistic fuzzy c-means.},
 year = {2026}
}

Copy | Download

TY  - JOUR
T1  - Graph Theory Approach Probability Distribution Based Intuitionistic Fuzzy C-means Clustering Using LINEX Hellinger Distance and Its Applications
AU  - Senthil
AU  - Nithya
AU  - Bhuvaneswari
Y1  - 2026/11/12
PY  - 2026
N1  - https://doi.org/10.11648/j.ijmfs.20261202.11
DO  - 10.11648/j.ijmfs.20261202.11
T2  - International Journal of Management and Fuzzy Systems
JF  - International Journal of Management and Fuzzy Systems
JO  - International Journal of Management and Fuzzy Systems
SP  - 20
EP  - 31
PB  - Science Publishing Group
SN  - 2575-4947
UR  - https://doi.org/10.11648/j.ijmfs.20261202.11
AB  - Clustering is an essential task to unsupervised learning, its used to partition a dataset into groups of similar elements. In addition to traditional clustering methods, intuitionistic fuzzy c-means (IFCM) clustering algorithm has been studied to capturing uncertain information in the dataset. However, the IFCM algorithm has limitations such as the random initialization of cluster centers and the unrestricted influence of all samples on all cluster centers. Therefore, a robust algorithm named probability distribution of the minimal spanning tree based intuitionistic fuzzy c-means clustering algorithm using LINEX Hellinger distance is proposed for improving shortcomings of the IFCM. Firstly, the probability distribution of the minimal spanning tree algorithm organizes the initial cluster center based on the contribution of local density information from the data. Secondly, MST node probability distribution based IFCM using Hellinger distance, the membership degree boundary is assigned for the data samples on all cluster centers in the clustering process. Finally, the performance of the proposed modified IFCM algorithm is numerically validated using random dataset, partition coefficient and validation function are used to evaluate the validity of clustering also this paper compares the results of proposed method with the results of existing basic intuitionistic fuzzy c-means.
VL  - 12
IS  - 2
ER  -

Copy | Download