{
 "id": "epytwamcrg",
 "slug": "fuzzy-clustering",
 "title": "Fuzzy clustering",
 "updated": "2026-09-30",
 "topic_path": [
  {
   "id": "physical",
   "label": "Physical world and mathematics",
   "api_url": "https://www.edgepedia.org/api/v1/topics/physical"
  },
  {
   "id": "physical.math",
   "label": "Mathematics and statistics",
   "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math"
  },
  {
   "id": "physical.math.statistics",
   "label": "Statistics and probability",
   "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.statistics"
  },
  {
   "id": "physical.math.statistics.multivariate-analysis",
   "label": "Multivariate association and dimension reduction",
   "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.statistics.multivariate-analysis"
  },
  {
   "id": "physical.math.statistics.multivariate-analysis.alpha-a-to-me",
   "label": "Titles A to Me",
   "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.statistics.multivariate-analysis.alpha-a-to-me"
  }
 ],
 "geo": [
  {
   "id": "geo.nongeo.t1946.physical.math.statistics.multivariate-analysis",
   "label": "Non-geographic · 1946 to 2000: Multivariate association and dimension reduction",
   "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics.multivariate-analysis",
   "path": [
    {
     "id": "geo.nongeo",
     "label": "Non-geographic",
     "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo"
    },
    {
     "id": "geo.nongeo.t1946",
     "label": "Non-geographic · 1946 to 2000",
     "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946"
    },
    {
     "id": "geo.nongeo.t1946.physical",
     "label": "Physical world and mathematics",
     "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical"
    },
    {
     "id": "geo.nongeo.t1946.physical.math",
     "label": "Mathematics and statistics",
     "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math"
    },
    {
     "id": "geo.nongeo.t1946.physical.math.statistics",
     "label": "Statistics and probability",
     "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics"
    },
    {
     "id": "geo.nongeo.t1946.physical.math.statistics.multivariate-analysis",
     "label": "Multivariate association and dimension reduction",
     "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics.multivariate-analysis"
    }
   ]
  }
 ],
 "excerpt": "Fuzzy clustering is a family of clustering methods assigning each point a membership degree between 0 and 1 in every cluster, rather than one hard label.",
 "snippet": "Fuzzy clustering is a family of clustering methods assigning each point a membership degree between 0 and 1 in every cluster, rather than one hard label.",
 "node": "physical.math.statistics.multivariate-analysis.alpha-a-to-me",
 "markdown": "# Fuzzy clustering\n\nFuzzy clustering is a family of clustering methods that assign each data point a membership degree in every cluster, a number between 0 and 1, instead of the single hard label produced by k-means and similar algorithms.<sup>[1](https://wires.onlinelibrary.wiley.com/doi/10.1002/wics.1480)</sup> The best-known member is fuzzy c-means (FCM), which alternates between updating memberships and cluster centers to minimize a weighted least-squares objective.<sup>[2](https://journal.r-project.org/articles/RJ-2019-017/RJ-2019-017.pdf)</sup><sup> • </sup><sup>[3](http://www.scholarpedia.org/article/Fuzzy_C-means_cluster_analysis)</sup> Fuzzy clustering is a form of [soft clustering](https://www.edgepedia.org/soft-clustering), which also includes possibilistic methods, where memberships need not sum to one, and model-based methods such as Gaussian mixtures, where posterior component probabilities play a role analogous to membership degrees.<sup>[1](https://wires.onlinelibrary.wiley.com/doi/10.1002/wics.1480)</sup>\n\n| Fact | Detail |\n|---|---|\n| Output | A \\( c \\times n \\) membership matrix \\( U \\) with entries in [0,1] whose column sums equal 1, plus c cluster centers V <sup>[3](http://www.scholarpedia.org/article/Fuzzy_C-means_cluster_analysis)</sup> |\n| Objective | \\( J_{m} = \\sum_{i} \\sum_{j} u_{ij}^{m} \\cdot d_{ij}^{2} \\), minimized by alternating optimization <sup>[4](https://community.intel.com/cipcp26785/attachments/cipcp26785/fortran-compiler/158561/1/1-s2.0-0098300484900207-main.pdf)</sup> |\n| Fuzzifier m | \\( m = 1 \\) gives hard partitions; higher \\( m \\) gives fuzzier memberships; \\( m = 2 \\) is the usual default <sup>[5](http://fuzzy.cs.ovgu.de/wiki/pmwiki.php/Lehre/FS1112?action=downloadman&upname=fs2011_ch09_clustering.pdf)</sup> |\n| Typical convergence | 10–25 iterations with a tolerance on the change in U <sup>[4](https://community.intel.com/cipcp26785/attachments/cipcp26785/fortran-compiler/158561/1/1-s2.0-0098300484900207-main.pdf)</sup> |\n| First formulation | Ruspini, 1969, as minimization of an objective function over fuzzy partitions <sup>[6](https://dl.acm.org/doi/10.1109/MCI.2018.2881643)</sup> |\n| FCM algorithm | Dunn (m = 2 case) and Bezdek (general m > 1), 1973; consolidated in Bezdek's 1981 monograph <sup>[3](http://www.scholarpedia.org/article/Fuzzy_C-means_cluster_analysis)</sup> |\n| Main failure mode | Unit-sum constraints force outliers into clusters; noise points equidistant from two centers get equal memberships <sup>[7](https://comp.ita.br/%7Eforster/CC-222/material/fuzzyclust/fuzzy01492404.pdf)</sup> |\n\n## How it works\n\n[Fuzzy c-means](https://www.edgepedia.org/fuzzy-c-means) minimizes a weighted least-squared-error functional. With data points \\( x_{j} \\), cluster centers \\( v_{i} \\), memberships \\( u_{ij} \\), and squared distances \\( d_{ij}^{2} = \\lVert x_{j} - v_{i} \\rVert^{2} \\), the objective is\n\n\\[ J_m = \\sum_{i=1}^{c} \\sum_{j=1}^{n} u_{ij}^{m} \\, d_{ij}^{2} \\]\n\nsubject to \\( \\sum_{i} u_{ij} = 1 \\) for each point, with \\( m > 1 \\) the fuzzifier.<sup>[8](http://fuzzy.cs.ovgu.de/wiki/pmwiki.php/Lehre/FS0809?action=downloadman&upname=fs08.pdf)</sup> The weight attached to each squared error is \\( u_{ij}^{m} \\), the \\( m \\)-th power of the point's membership in cluster i, so the centers act as centers of mass of the partitioning subsets.<sup>[4](https://community.intel.com/cipcp26785/attachments/cipcp26785/fortran-compiler/158561/1/1-s2.0-0098300484900207-main.pdf)</sup> A norm matrix \\( A \\) in the distance \\( \\lVert x_{j} - v_{i} \\rVert_{A}^{2} \\) generalizes the metric; a covariance norm yields ellipsoidal clusters with [Mahalanobis distance](https://www.edgepedia.org/mahalanobis-distance).<sup>[9](https://www.sciencedirect.com/science/article/abs/pii/S156849461630686X)</sup>\n\nThe fuzzifier controls how soft the partition is. For \\( m = 1 \\) the objective reduces to the hard c-means criterion and minimization always yields a hard partition, so fuzziness plays no role.<sup>[10](https://www.jehps.net/Decembre2008/Bock.pdf)</sup> Larger m pushes memberships away from 0 and 1, and \\( m = 2 \\) is the value used in most applications.<sup>[4](https://community.intel.com/cipcp26785/attachments/cipcp26785/fortran-compiler/158561/1/1-s2.0-0098300484900207-main.pdf)</sup><sup> • </sup><sup>[11](https://real.mtak.hu/29902/1/196_1015_1_PB_u.pdf)</sup> Useful values range roughly over [1, 30], and for most data \\( 1.5 \\leq m \\leq 3.0 \\) gives good results.<sup>[4](https://community.intel.com/cipcp26785/attachments/cipcp26785/fortran-compiler/158561/1/1-s2.0-0098300484900207-main.pdf)</sup>\n\nMinimizing \\( J_{m} \\) over \\( U \\) with fixed centers gives the membership update\n\n\\[ u_{ij} = \\frac{d_{ij}^{\\,2/(1-m)}}{\\sum_{k=1}^{c} d_{kj}^{\\,2/(1-m)}} \\]\n\nand minimizing over centers with fixed memberships gives\n\n\\[ v_i = \\frac{\\sum_{j=1}^{n} u_{ij}^{m} \\, x_j}{\\sum_{j=1}^{n} u_{ij}^{m}} \\]\n\nBoth updates follow from the first-order necessary conditions for the stated objective with the squared [Euclidean distance](https://www.edgepedia.org/euclidean-distance); for other distance measures different update formulas are required.<sup>[8](http://fuzzy.cs.ovgu.de/wiki/pmwiki.php/Lehre/FS0809?action=downloadman&upname=fs08.pdf)</sup>\n\n## How it is done\n\nThe practitioner fixes c, m, a distance (norm matrix), and an initial membership matrix U(0), then alternates the centroid and membership updates until convergence.<sup>[4](https://community.intel.com/cipcp26785/attachments/cipcp26785/fortran-compiler/158561/1/1-s2.0-0098300484900207-main.pdf)</sup> Four phases are usually distinguished: initialization, centroid calculation, classification (membership update), and convergence checking. Four convergence criteria appear in the literature: the largest difference of membership values between two consecutive iterations, the largest centroid movement, the difference of the objective function between successive iterations, and a limit on the number of iterations.<sup>[12](https://www.mdpi.com/2075-1680/13/9/592)</sup> Numerical convergence is usually achieved in 10 to 25 iterations.<sup>[4](https://community.intel.com/cipcp26785/attachments/cipcp26785/fortran-compiler/158561/1/1-s2.0-0098300484900207-main.pdf)</sup>\n\nInitialization matters. MATLAB's fcm starts from a random guess of centers and random membership grades.<sup>[13](https://www.mathworks.com/help/fuzzy/fuzzy-c-means-clustering.html)</sup> Comparative work found that MaxMin Linear initialization achieved the best average ranking for eight of nine quality indices across 22 datasets.<sup>[14](https://ar5iv.labs.arxiv.org/html/1808.00197)</sup> [Classification](https://www.edgepedia.org/classification) of points, when a hard label is needed, assigns each point to the cluster for which it has the highest membership.<sup>[13](https://www.mathworks.com/help/fuzzy/fuzzy-c-means-clustering.html)</sup>\n\nBecause FCM requires c in advance, the number of clusters is usually chosen by running the algorithm for several values and scoring each partition with a validity index.<sup>[8](http://fuzzy.cs.ovgu.de/wiki/pmwiki.php/Lehre/FS0809?action=downloadman&upname=fs08.pdf)</sup> The partition coefficient \\( PC = (1/n) \\sum_{i} \\sum_{j} u_{ij}^{2} \\) favors memberships near 0 or 1 and ranges from 1/c to 1 (maximized);<sup>[8](http://fuzzy.cs.ovgu.de/wiki/pmwiki.php/Lehre/FS0809?action=downloadman&upname=fs08.pdf)</sup><sup> • </sup><sup>[14](https://ar5iv.labs.arxiv.org/html/1808.00197)</sup> the partition entropy is its companion;<sup>[4](https://community.intel.com/cipcp26785/attachments/cipcp26785/fortran-compiler/158561/1/1-s2.0-0098300484900207-main.pdf)</sup> the Xie–Beni index and the Fukuyama–Sugeno index are minimized;<sup>[14](https://ar5iv.labs.arxiv.org/html/1808.00197)</sup> and the fuzzy silhouette adapts the hard silhouette to soft partitions.<sup>[2](https://journal.r-project.org/articles/RJ-2019-017/RJ-2019-017.pdf)</sup> Bezdek, Ehrlich, and Full described the partition coefficient and entropy as the most reliable indicants of cluster validity for the FCM algorithms,<sup>[4](https://community.intel.com/cipcp26785/attachments/cipcp26785/fortran-compiler/158561/1/1-s2.0-0098300484900207-main.pdf)</sup> but in a 2024 benchmark of eight indices on five standard datasets with \\( m = 2 \\), PC, PE, and Xie–Beni failed to identify the correct c on at least one dataset, and the Fukuyama–Sugeno index was described as much more unreliable than the others.<sup>[15](https://arxiv.org/pdf/2407.06774)</sup>\n\n## Origin\n\nThe first formal treatment of fuzzy clustering as an optimization problem was Enrique H. Ruspini's 1969 paper \"A new approach to clustering\" in [Information](https://www.edgepedia.org/information) and Control,<sup>[16](https://doi.org/10.1016/s0019-9958%2869%2990591-9)</sup> which established the structure for fuzzy partitioning and described the first algorithm for accomplishing it;<sup>[6](https://dl.acm.org/doi/10.1109/MCI.2018.2881643)</sup> Numerical techniques were developed for optimizing functionals over fuzzy classifications.<sup>[17](https://www.sciencedirect.com/science/article/abs/pii/S0020025570800561)</sup> Ruspini argued that assigning each point a degree of belongingness to each cluster characterizes bridges, strays, and undetermined points, which is especially useful for scattered data.<sup>[17](https://www.sciencedirect.com/science/article/abs/pii/S0020025570800561)</sup> The fuzzy-set foundation came from L.A. Zadeh's 1965 paper \"Fuzzy sets\".<sup>[18](https://doi.org/10.1016/s0019-9958%2865%2990241-x)</sup>\n\nFuzzy c-means itself was first reported by J. C. Dunn in \"A fuzzy relative of the ISODATA process and its use in detecting compact well-separated clusters\" (Journal of [Cybernetics](https://www.edgepedia.org/cybernetics), 1973).<sup>[19](https://doi.org/10.1080/01969727308546046)</sup> The general case for any \\( m > 1 \\) was developed by James C. Bezdek in his 1973 PhD thesis at Cornell University.<sup>[3](http://www.scholarpedia.org/article/Fuzzy_C-means_cluster_analysis)</sup><sup> • </sup><sup>[20](https://doi.org/10.1080/01969727308546047)</sup> Bezdek's 1981 monograph Pattern Recognition with Fuzzy Objective Function Algorithms consolidated the theory and became the most referenced early source,<sup>[21](https://doi.org/10.1007/978-1-4757-0450-1)</sup><sup> • </sup><sup>[3](http://www.scholarpedia.org/article/Fuzzy_C-means_cluster_analysis)</sup> and his 1984 paper with Robert Ehrlich and William Full supplied a FORTRAN implementation.<sup>[22](https://doi.org/10.1016/0098-3004%2884%2990020-7)</sup>\n\n## Variants\n\n**Distance extensions.** FCM with Euclidean distance uses spherical distance geometry and tends to find similarly sized clusters.<sup>[8](http://fuzzy.cs.ovgu.de/wiki/pmwiki.php/Lehre/FS0809?action=downloadman&upname=fs08.pdf)</sup> The Gustafson–Kessel algorithm replaces the Euclidean distance with a cluster-specific Mahalanobis distance based on fuzzy covariance matrices, finding hyper-ellipsoidal clusters of the same size; implementations constrain the determinant or the condition number of each covariance matrix to avoid singularity.<sup>[2](https://journal.r-project.org/articles/RJ-2019-017/RJ-2019-017.pdf)</sup><sup> • </sup><sup>[23](https://www.mathworks.com/help/fuzzy/fuzzy-clustering.html)</sup> The Gath–Geva algorithm uses a fuzzy maximum likelihood exponential distance and detects hyper-ellipsoidal clusters of varying shapes, sizes, and densities.<sup>[24](https://cran.r-universe.dev/ppclust/doc/manual.html)</sup>\n\n**Possibilistic methods.** Possibilistic c-means (PCM), due to R. Krishnapuram and J. M. Keller,<sup>[25](https://doi.org/10.1109/91.531779)</sup> drops the column-sum constraint so memberships become degrees of typicality depending only on the distance to the belonging cluster's prototype; noisy data and outliers can then belong to no cluster.<sup>[7](https://comp.ita.br/%7Eforster/CC-222/material/fuzzyclust/fuzzy01492404.pdf)</sup> PCM's price is high sensitivity to initialization and a tendency to generate coincident clusters, since its objective is truly minimized when all centers coincide.<sup>[7](https://comp.ita.br/%7Eforster/CC-222/material/fuzzyclust/fuzzy01492404.pdf)</sup><sup> • </sup><sup>[5](http://fuzzy.cs.ovgu.de/wiki/pmwiki.php/Lehre/FS1112?action=downloadman&upname=fs2011_ch09_clustering.pdf)</sup> Remedies include initializing PCM with FCM and re-estimating the scale parameters \\( \\eta_{i} \\) between runs,<sup>[5](http://fuzzy.cs.ovgu.de/wiki/pmwiki.php/Lehre/FS1112?action=downloadman&upname=fs2011_ch09_clustering.pdf)</sup> adding a repulsion term on the distances between centers (Timm, Borgelt, Döring, and Kruse, 2003),<sup>[26](https://doi.org/10.1016/j.fss.2003.11.009)</sup> and hybrid models: FPCM produces memberships and typicalities, while PFCM produces memberships and possibilities simultaneously and solves FCM's noise sensitivity and PCM's coincident-cluster problems.<sup>[7](https://comp.ita.br/%7Eforster/CC-222/material/fuzzyclust/fuzzy01492404.pdf)</sup>\n\n**Noise and robustness.** Noise clustering, introduced by Rajesh N. Dave in 1991,<sup>[27](https://doi.org/10.1016/0167-8655%2891%2990002-4)</sup> instead assigns outliers to an extra noise cluster at a distance δ chosen in advance, lowering their memberships in the proper clusters without affecting the partition.<sup>[8](http://fuzzy.cs.ovgu.de/wiki/pmwiki.php/Lehre/FS0809?action=downloadman&upname=fs08.pdf)</sup><sup> • </sup><sup>[2](https://journal.r-project.org/articles/RJ-2019-017/RJ-2019-017.pdf)</sup> Robust-learning fuzzy c-means (Yang and Nataliani, 2017) estimates the number of clusters automatically.<sup>[28](https://doi.org/10.1016/j.patcog.2017.05.017)</sup>\n\n**Kernels and relational data.** Kernel fuzzy c-means maps data implicitly into a higher-dimensional feature space and can outperform FCM on nonspherical data, though it remains noise-sensitive; kernel possibilistic c-means (Rhee, Choi, and Choi, 2009) applies the kernel approach to PCM.<sup>[29](https://onlinelibrary.wiley.com/doi/10.1002/int.20336)</sup> For dissimilarity data, the NEFRC algorithm uses a general fuzzifier and suits all kinds of dissimilarities.<sup>[2](https://journal.r-project.org/articles/RJ-2019-017/RJ-2019-017.pdf)</sup>\n\n## Applications\n\nFuzzy c-means is applied to gene expression data because genes can be co-regulated in different ways under different conditions, so non-exclusive clusters with multiple memberships fit the biology.<sup>[30](https://pmc.ncbi.nlm.nih.gov/articles/PMC2673503/)</sup> GO Fuzzy c-means initializes memberships from [Gene Ontology](https://www.edgepedia.org/gene-ontology) annotations with \\( m = 2 \\) and a membership cutoff of 0.05, guaranteeing repeatable results and removing the need to pre-specify c; it performed significantly better than regular fuzzy c-means, FuzzySOM, and FLAME.<sup>[30](https://pmc.ncbi.nlm.nih.gov/articles/PMC2673503/)</sup> The Gustafson–Kessel algorithm has been widely applied in image segmentation, geospatial analysis, and medical imaging, and kernel-based fuzzy clustering in gene expression analysis, web mining, social network analysis, and multimedia retrieval.<sup>[31](https://www.rsisinternational.org)</sup> A recent deep-learning use embeds a differentiable FCM pooling layer in convolutional networks, computing soft memberships within each receptive field.<sup>[32](https://www.techscience.com/cmc/v87n2/66577)</sup>\n\n## Limitations and alternatives\n\nFCM-type algorithms are not robust to outliers: the unit-sum constraints force every point, including outliers, to be assigned to clusters, and centroids as weighted means are dragged by anomalous points.<sup>[33](https://iris.uniroma1.it/retrieve/e383532d-bd8d-15e8-e053-a505fe0a3de9/Ferraro_fuzzy%20k-means_2021.pdf)</sup> A related degeneracy is that a noise point equidistant from two cluster centers receives equal membership in both; in one documented example an added far-away point got 0.50 in each cluster while other points' memberships changed by at most 0.05.<sup>[7](https://comp.ita.br/%7Eforster/CC-222/material/fuzzyclust/fuzzy01492404.pdf)</sup> Robust alternatives include the noise cluster, possibilistic methods, fuzzy k-medoids, an exponential-distance variant, and trimmed fuzzy clustering.<sup>[33](https://iris.uniroma1.it/retrieve/e383532d-bd8d-15e8-e053-a505fe0a3de9/Ferraro_fuzzy%20k-means_2021.pdf)</sup>\n\nMinimizing the hard c-means objective is NP-hard and the scheme can get stuck in local minima, requiring several restarts.<sup>[5](http://fuzzy.cs.ovgu.de/wiki/pmwiki.php/Lehre/FS1112?action=downloadman&upname=fs2011_ch09_clustering.pdf)</sup> FCM can converge to either a local minimum or a saddle point of its objective, and results may depend on the initialization, so multiple starts are often used.<sup>[38](https://doi.org/10.1109/fuzz.2002.1004967)</sup><sup> • </sup><sup>[5](http://fuzzy.cs.ovgu.de/wiki/pmwiki.php/Lehre/FS1112?action=downloadman&upname=fs2011_ch09_clustering.pdf)</sup> FCM generalizes k-means: as \\( m \\rightarrow 1 \\) it reduces to k-means.<sup>[33](https://iris.uniroma1.it/retrieve/e383532d-bd8d-15e8-e053-a505fe0a3de9/Ferraro_fuzzy%20k-means_2021.pdf)</sup> In comparative simulations, k-means was always extremely faster than FCM because FCM involves more iterative calculations, and k-means with multiple starts is recommended for well-separated large datasets, while FCM gives better results on noisy clustered datasets.<sup>[11](https://real.mtak.hu/29902/1/196_1015_1_PB_u.pdf)</sup> In a simulation of 2530 datasets with overlapping clusters and outliers, FCM (with fuzziness 2) was very stable, maintaining an average recovery rate over or close to 90% at 40% overlap, where all other tested algorithms degraded substantially.<sup>[34](https://www.comp.nus.edu.sg/~rudys/arnie/som-vs-fuzzy.pdf)</sup> Neither k-means nor FCM handles nested irregular or concave cluster structures above an acceptable failure level; spectral, hierarchical, DBSCAN, or Birch methods are recommended there.<sup>[11](https://real.mtak.hu/29902/1/196_1015_1_PB_u.pdf)</sup> Gaussian mixture models fitted by EM also produce soft partitions, with posterior component probabilities playing the role of membership degrees, but rest on probabilistic assumptions.<sup>[1](https://wires.onlinelibrary.wiley.com/doi/10.1002/wics.1480)</sup>\n\nRecent work targets FCM's standing problems of initialization, cluster-number selection, and noise. HPFCM (2024) achieved on the SPAM dataset an average reduction of 97.65% in iterations and an 82.42% improvement in solution quality relative to standard FCM.<sup>[12](https://www.mdpi.com/2075-1680/13/9/592)</sup> RL-MFCM ([IEEE Transactions on Fuzzy Systems](https://www.edgepedia.org/ieee-transactions-on-fuzzy-systems), 2023) determines initial centers from belief peaks and estimates the number of clusters without initialization.<sup>[35](https://dl.acm.org/doi/10.1109/TFUZZ.2023.3286910)</sup> FKM-L0 (2025) adds an L0 regularization term producing sparse membership matrices, with clear assignments at exactly 1 and 0 and soft degrees retained for unclear ones.<sup>[36](https://link.springer.com/article/10.1007/s10479-025-06502-1)</sup> FedFCD (2026) combines a contrastive autoencoder and an FCM network per client with server-side Bayesian ensemble aggregation, remaining stable under non-IID data.<sup>[37](https://www.ieee-jas.net/en/article/doi/10.1109/JAS.2025.125561)</sup>\n\n## References\n\n1. [Soft clustering (Wiley Interdisciplinary Reviews: Computational Statistics)](https://wires.onlinelibrary.wiley.com/doi/10.1002/wics.1480)\n2. [fclust: An R Package for Fuzzy Clustering (Ferraro, Giordani, Scepi, The R Journal)](https://journal.r-project.org/articles/RJ-2019-017/RJ-2019-017.pdf)\n3. [Fuzzy C-means cluster analysis (Scholarpedia, curated by James C. Bezdek, 2011)](http://www.scholarpedia.org/article/Fuzzy_C-means_cluster_analysis)\n4. [FCM: The fuzzy c-means clustering algorithm (Bezdek, Ehrlich, Full, Computers & Geosciences, 1984; DOI 10.1016/0098-3004(84)90020-7)](https://community.intel.com/cipcp26785/attachments/cipcp26785/fortran-compiler/158561/1/1-s2.0-0098300484900207-main.pdf)\n5. [Fuzzy Systems - Fuzzy Clustering (Kruse & Moewes, chapter 9 lecture notes, 2011)](http://fuzzy.cs.ovgu.de/wiki/pmwiki.php/Lehre/FS1112?action=downloadman&upname=fs2011_ch09_clustering.pdf)\n6. [Fuzzy Clustering: A Historical Perspective (IEEE Computational Intelligence Magazine, Vol 14, No 1)](https://dl.acm.org/doi/10.1109/MCI.2018.2881643)\n7. [A Possibilistic Fuzzy c-Means Clustering Algorithm (Pal, Pal, Keller, Bezdek, IEEE Transactions on Fuzzy Systems, 2005; retrieved copy)](https://comp.ita.br/%7Eforster/CC-222/material/fuzzyclust/fuzzy01492404.pdf)\n8. [Fuzzy Systems - Fuzzy Cluster Analysis (Kruse & Moewes, Otto-von-Guericke University Magdeburg lecture notes)](http://fuzzy.cs.ovgu.de/wiki/pmwiki.php/Lehre/FS0809?action=downloadman&upname=fs08.pdf)\n9. [Generalized Possibilistic Fuzzy C-Means with novel cluster validity indices for clustering noisy data](https://www.sciencedirect.com/science/article/abs/pii/S156849461630686X)\n10. [Origins and extensions of the k-means algorithm in cluster analysis (H.-H. Bock)](https://www.jehps.net/Decembre2008/Bock.pdf)\n11. [Comparison of K-means and Fuzzy C-means Algorithms on Different Cluster Structures (retrieved copy)](https://real.mtak.hu/29902/1/196_1015_1_PB_u.pdf)\n12. [Hybrid Fuzzy C-Means Clustering Algorithm, Improving Solution Quality and Reducing Computational Complexity (HPFCM, Mathematics, 2024)](https://www.mdpi.com/2075-1680/13/9/592)\n13. [Fuzzy C-Means Clustering example (MATLAB Fuzzy Logic Toolbox documentation)](https://www.mathworks.com/help/fuzzy/fuzzy-c-means-clustering.html)\n14. [MaxMin Linear Initialization for Fuzzy C-Means (arXiv:1808.00197)](https://ar5iv.labs.arxiv.org/html/1808.00197)\n15. [A New Validity Measure for Fuzzy C-Means Clustering (arXiv, 2024)](https://arxiv.org/pdf/2407.06774)\n16. [A new approach to clustering (Information and Control, 1969)](https://doi.org/10.1016/s0019-9958%2869%2990591-9)\n17. [Numerical methods for fuzzy clustering (E. H. Ruspini, Information Sciences, 1970)](https://www.sciencedirect.com/science/article/abs/pii/S0020025570800561)\n18. [Fuzzy sets (Information and Control, 1965)](https://doi.org/10.1016/s0019-9958%2865%2990241-x)\n19. [J. C. Dunn (1973). A Fuzzy Relative of the ISODATA Process and Its Use in Detecting Compact Well-Separated Clusters. Journal of Cybernetics.](https://doi.org/10.1080/01969727308546046)\n20. [James C. Bezdek† (1973). Cluster Validity with Fuzzy Sets. Journal of Cybernetics.](https://doi.org/10.1080/01969727308546047)\n21. [James C. Bezdek (1981). Pattern Recognition with Fuzzy Objective Function Algorithms. .](https://doi.org/10.1007/978-1-4757-0450-1)\n22. [FCM: The fuzzy c-means clustering algorithm (Computers & Geosciences, 1984)](https://doi.org/10.1016/0098-3004%2884%2990020-7)\n23. [Fuzzy Clustering - MATLAB & Simulink (MathWorks documentation)](https://www.mathworks.com/help/fuzzy/fuzzy-clustering.html)\n24. [Package 'ppclust' reference manual](https://cran.r-universe.dev/ppclust/doc/manual.html)\n25. [R. Krishnapuram, J.M. Keller (1996). The possibilistic C-means algorithm: insights and recommendations. IEEE Transactions on Fuzzy Systems.](https://doi.org/10.1109/91.531779)\n26. [Heiko Timm and colleagues (2003). An extension to possibilistic fuzzy cluster analysis. Fuzzy Sets and Systems.](https://doi.org/10.1016/j.fss.2003.11.009)\n27. [Characterization and detection of noise in clustering (Pattern Recognition Letters, 1991)](https://doi.org/10.1016/0167-8655%2891%2990002-4)\n28. [Miin-Shen Yang, Yessica Nataliani (2017). Robust-learning fuzzy c-means clustering algorithm with unknown number of clusters. Pattern Recognition.](https://doi.org/10.1016/j.patcog.2017.05.017)\n29. [Kernel approach to possibilistic C-means clustering (Rhee, 2009, International Journal of Intelligent Systems)](https://onlinelibrary.wiley.com/doi/10.1002/int.20336)\n30. [Fuzzy c-means clustering with prior biological knowledge (GO Fuzzy c-means)](https://pmc.ncbi.nlm.nih.gov/articles/PMC2673503/)\n31. [Heuristic-Based Approaches in Fuzzy Clustering: A Comprehensive Review](https://www.rsisinternational.org)\n32. [Fuzzy C-Means Clustering-Driven Pooling for Robust and Generalizable Convolutional Neural Networks (Computers, Materials & Continua, 2026)](https://www.techscience.com/cmc/v87n2/66577)\n33. [Fuzzy k-Means: history and applications (Ferraro, preprint submitted to Econometrics and Statistics, Nov 21, 2021)](https://iris.uniroma1.it/retrieve/e383532d-bd8d-15e8-e053-a505fe0a3de9/Ferraro_fuzzy%20k-means_2021.pdf)\n34. [Comparison among nonhierarchical and hierarchical clustering algorithms including SOM and Fuzzy c-means (Mingoti & Lima, European Journal of Operational Research 174 (2006) 1742–1759; retrieved copy)](https://www.comp.nus.edu.sg/~rudys/arnie/som-vs-fuzzy.pdf)\n35. [A Robust Learning Membership Scaling Fuzzy C-Means Algorithm Based on New Belief Peak (RL-MFCM, IEEE Transactions on Fuzzy Systems, Dec 2023)](https://dl.acm.org/doi/10.1109/TFUZZ.2023.3286910)\n36. [Fuzzy clustering with L0 regularization (FKM-L0), Annals of Operations Research](https://link.springer.com/article/10.1007/s10479-025-06502-1)\n37. [Deep Fuzzy C-Means Clustering in a Federated Heterogeneous Scenario (FedFCD, IEEE/CAA Journal of Automatica Sinica, 2025)](https://www.ieee-jas.net/en/article/doi/10.1109/JAS.2025.125561)\n38. [31wzx2wsmfs (exa.ai)](https://doi.org/10.1109/fuzz.2002.1004967)\n\n---\n*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Statistics and probability › Multivariate association and dimension reduction › Titles A to Me*\n\n*Initially written Sep 29, 2026 · Reviewed: Sep 30, 2026 · Edited: Sep 30, 2026 · Last review: Sep 30, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n",
 "same_as": [],
 "url": "https://www.edgepedia.org/fuzzy-clustering",
 "markdown_url": "https://www.edgepedia.org/fuzzy-clustering.md",
 "license": {
  "name": "Edgepedia Community License 1.0",
  "url": "https://www.edgepedia.org/edgepedia/license",
  "summary": "Free with credit, commercial use included. AI training is open to everyone. For other uses, organizations over USD 100M in revenue or 100M monthly users license separately.",
  "spdx": "LicenseRef-Edgepedia-Community-1.0"
 },
 "credit": "\"Fuzzy clustering\", Edgepedia (EdgeChat), https://www.edgepedia.org/fuzzy-clustering. Edgepedia Community License 1.0.",
 "credit_md": "\"[Fuzzy clustering](https://www.edgepedia.org/fuzzy-clustering)\", Edgepedia (EdgeChat), [https://www.edgepedia.org/fuzzy-clustering](https://www.edgepedia.org/fuzzy-clustering). [Edgepedia Community License 1.0](https://www.edgepedia.org/edgepedia/license).",
 "credit_html": "\"<a href=\"https://www.edgepedia.org/fuzzy-clustering\">Fuzzy clustering</a>\", Edgepedia (EdgeChat), <a href=\"https://www.edgepedia.org/fuzzy-clustering\">https://www.edgepedia.org/fuzzy-clustering</a>. <a href=\"https://www.edgepedia.org/edgepedia/license\">Edgepedia Community License 1.0</a>.",
 "speakable": "Fuzzy clustering is a family of clustering methods assigning each point a membership degree between 0 and 1 in every cluster, rather than one hard label."
}