{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,15]],"date-time":"2025-12-15T19:50:13Z","timestamp":1765828213679,"version":"build-2065373602"},"reference-count":38,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2022,7,13]],"date-time":"2022-07-13T00:00:00Z","timestamp":1657670400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Over the years, research on fuzzy clustering algorithms has attracted the attention of many researchers, and they have been applied to various areas, such as image segmentation and data clustering. Various fuzzy clustering algorithms have been put forward based on the initial Fuzzy C-Means clustering (FCM) with Euclidean distance. However, the existing fuzzy clustering approaches ignore two problems. Firstly, clustering algorithms based on Euclidean distance have a high error rate, and are more sensitive to noise and outliers. Secondly, the parameters of the fuzzy clustering algorithms are hard to determine. In practice, they are often determined by the user\u2019s experience, which results in poor performance of the clustering algorithm. Therefore, considering the above deficiencies, this paper proposes a novel fuzzy clustering algorithm by combining the Gaussian kernel function and Grey Wolf Optimizer (GWO), called Kernel-based Picture Fuzzy C-Means clustering with Grey Wolf Optimizer (KPFCM-GWO). In KPFCM-GWO, the Gaussian kernel function is used as a symmetrical measure of distance between data points and cluster centers, and the GWO is utilized to determine the parameter values of PFCM. To verify the validity of KPFCM-GWO, a comparative study was conducted. The experimental results indicate that KPFCM-GWO outperforms other clustering methods, and the improvement of KPFCM-GWO is mainly attributed to the combination of the Gaussian kernel function and the parameter optimization capability of the GWO. What is more, the paper applies KPFCM-GWO to analyzes the value of an airline\u2019s customers, and five levels of customer categories are defined.<\/jats:p>","DOI":"10.3390\/sym14071442","type":"journal-article","created":{"date-parts":[[2022,7,14]],"date-time":"2022-07-14T00:12:40Z","timestamp":1657757560000},"page":"1442","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["A Novel Adaptive Kernel Picture Fuzzy C-Means Clustering Algorithm Based on Grey Wolf Optimizer Algorithm"],"prefix":"10.3390","volume":"14","author":[{"given":"Can-Ming","family":"Yang","sequence":"first","affiliation":[{"name":"Library, Guilin University of Technology, Guilin 541004, China"}]},{"given":"Ye","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Business, Central South University, Changsha 410083, China"}]},{"given":"Yi-Ting","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Business, Central South University, Changsha 410083, China"}]},{"given":"Yan-Ping","family":"Li","sequence":"additional","affiliation":[{"name":"School of Business, Central South University, Changsha 410083, China"}]},{"given":"Wen-Hui","family":"Hou","sequence":"additional","affiliation":[{"name":"School of Business, Central South University, Changsha 410083, China"}]},{"given":"Sheng","family":"Duan","sequence":"additional","affiliation":[{"name":"College of Computer and Artificial Intelligence, Xiangnan University, Chenzhou 423038, China"}]},{"given":"Jian-Qiang","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Business, Central South University, Changsha 410083, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,7,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"264","DOI":"10.1145\/331499.331504","article-title":"Data clustering: A review","volume":"31","author":"Jain","year":"1999","journal-title":"ACM Comput. Surv."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1016\/0098-3004(84)90020-7","article-title":"FCM: The fuzzy c-means clustering algorithm","volume":"10","author":"Bezdek","year":"1984","journal-title":"Comput. Geosci."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"531","DOI":"10.1016\/j.procs.2017.08.318","article-title":"Enhanced 3D segmentation techniques for reconstructed 3D medical volumes: Robust and accurate intelligent system","volume":"113","author":"Shadi","year":"2017","journal-title":"Procedia Comput. Sci."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"978","DOI":"10.1016\/j.compeleceng.2017.10.011","article-title":"FCM technique for efficient intrusion detection system for wireless networks in cloud environment","volume":"71","author":"Chen","year":"2018","journal-title":"Comput. Electr. Eng."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Lee, Z.J., Lee, C.Y., Chang, L.Y., and Sano, N. (2021). Clustering and classification based on distributed automatic feature engineering for customer segmentation. Symmetry, 13.","DOI":"10.3390\/sym13091557"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"114121","DOI":"10.1016\/j.eswa.2020.114121","article-title":"A population based hybrid FCM-PSO algorithm for clustering analysis and segmentation of brain image","volume":"167","author":"Hanuman","year":"2021","journal-title":"Expert Syst. Appl."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"517","DOI":"10.1109\/TFUZZ.2004.840099","article-title":"A possibilistic fuzzy c-means clustering algorithm","volume":"13","author":"Pal","year":"2005","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1328","DOI":"10.1109\/TIP.2010.2040763","article-title":"A robust fuzzy local information c-means clustering algorithm","volume":"19","author":"Krinidis","year":"2010","journal-title":"IEEE Trans. Image Process."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"3549","DOI":"10.1007\/s00500-015-1712-7","article-title":"Picture fuzzy clustering: A new computational intelligence method","volume":"20","author":"Thong","year":"2016","journal-title":"Soft Comput."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1109\/TFUZZ.2006.889763","article-title":"Uncertain fuzzy clustering: Interval type-2 fuzzy approach to c-means","volume":"15","author":"Hwang","year":"2007","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"580","DOI":"10.3969\/j.issn.1004-4132.2010.04.009","article-title":"Intuitionistic fuzzy c-means clustering algorithms","volume":"21","author":"Xu","year":"2010","journal-title":"J. Syst. Eng. Electron."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"3681","DOI":"10.3233\/JIFS-191973","article-title":"Hesitant fuzzy c-means algorithm and its application in image segmentation","volume":"39","author":"Zeng","year":"2020","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"859","DOI":"10.1007\/s13042-020-01206-3","article-title":"Intuitionistic fuzzy c-means clustering algorithm based on a novel weighted proximity measure and genetic algorithm","volume":"12","author":"Hou","year":"2021","journal-title":"Int. J. Mach. Learn. Cyb."},{"key":"ref_14","first-page":"409","article-title":"Picture fuzzy sets","volume":"30","author":"Cuong","year":"2014","journal-title":"J. Comput. Sci. Cybern."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1016\/j.eswa.2014.07.026","article-title":"DPFCM","volume":"42","author":"Son","year":"2015","journal-title":"Expert Syst. Appl."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1016\/j.knosys.2016.06.023","article-title":"A novel automatic picture fuzzy clustering method based on particle swarm optimization and picture composite cardinality","volume":"109","author":"Thong","year":"2016","journal-title":"Knowl. Based Syst."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1016\/j.engappai.2016.08.009","article-title":"Picture fuzzy clustering for complex data","volume":"56","author":"Thong","year":"2016","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"105888","DOI":"10.1016\/j.asoc.2019.105888","article-title":"Adaptive entropy weighted picture fuzzy clustering algorithm with spatial information for image segmentation","volume":"86","author":"Wu","year":"2020","journal-title":"Appl. Soft Comput."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"522","DOI":"10.1016\/j.fss.2009.10.021","article-title":"Kernel-based fuzzy clustering and fuzzy clustering: A comparative experimental study","volume":"161","author":"Graves","year":"2010","journal-title":"Fuzzy Sets Syst."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"15507","DOI":"10.1007\/s00500-020-04879-8","article-title":"Kernel intuitionistic fuzzy c-means and state transition algorithm for clustering problem","volume":"24","author":"Zhou","year":"2020","journal-title":"Soft Comput."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"102963","DOI":"10.1016\/j.dsp.2021.102963","article-title":"Noise distance driven fuzzy clustering based on adaptive weighted local information and entropy-like divergence kernel for robust image segmentation","volume":"111","author":"Wu","year":"2021","journal-title":"Digit. Signal Process"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1074","DOI":"10.1109\/TFUZZ.2013.2280141","article-title":"A novel evolutionary kernel intuitionistic fuzzy c-means clustering algorithm","volume":"22","author":"Lin","year":"2014","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"298","DOI":"10.1016\/j.asoc.2017.03.014","article-title":"Hybrid genetic algorithm and fuzzy clustering for bankruptcy prediction","volume":"56","author":"Chou","year":"2017","journal-title":"Appl. Soft Comput."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1386839","DOI":"10.1155\/2020\/1386839","article-title":"Hybrid fuzzy clustering method based on FCM and enhanced logarithmical PSO (ELPSO)","volume":"2020","author":"Zhang","year":"2020","journal-title":"Comput. Intel. Neurosc."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1016\/j.asoc.2009.07.001","article-title":"An efficient hybrid approach based on PSO, ACO and k-means for cluster analysis","volume":"10","author":"Niknam","year":"2010","journal-title":"Appl. Soft Comput."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"687","DOI":"10.1016\/j.asoc.2007.05.007","article-title":"On the performance of artificial bee colony (ABC) algorithm","volume":"8","author":"Karaboga","year":"2008","journal-title":"Appl. Soft Comput."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"46","DOI":"10.1016\/j.advengsoft.2013.12.007","article-title":"Grey wolf optimizer","volume":"69","author":"Mirjalili","year":"2014","journal-title":"Adv. Eng. Softw."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"236","DOI":"10.1016\/S0019-9958(80)90156-4","article-title":"On the measure of fuzziness and negation. II. Lattices","volume":"44","author":"Yager","year":"1980","journal-title":"Inf. Control."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"358","DOI":"10.1007\/s10115-004-0154-9","article-title":"Exact indexing of dynamic time warping","volume":"7","author":"Keogh","year":"2005","journal-title":"Knowl. Inf. Syst."},{"key":"ref_30","unstructured":"Qingshan, L., Rui, H., Hanqing, L., and Songde, M. (2002, January 21). Face recognition using kernel-based fisher discriminant Analysis. Proceedings of the Fifth IEEE International Conference on Automatic Face Gesture Recognition, Washington, DC, USA."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1351","DOI":"10.1109\/TGRS.2005.846154","article-title":"Kernel-based methods for hyperspectral image classification","volume":"43","author":"Bruzzone","year":"2005","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"602","DOI":"10.1016\/j.apenergy.2014.07.064","article-title":"Short-term load forecasting using a kernel-based support vector regression combination model","volume":"132","author":"Che","year":"2014","journal-title":"Appl. Energy"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"584","DOI":"10.1016\/j.jsv.2015.11.008","article-title":"Machine learning algorithms for damage detection: Kernel-based approaches","volume":"363","author":"Santos","year":"2016","journal-title":"J. Sound Vib."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1711","DOI":"10.1016\/j.asoc.2010.05.005","article-title":"A novel intuitionistic fuzzy c means clustering algorithm and its application to medical images","volume":"11","author":"Chaira","year":"2011","journal-title":"Appl. Soft Comput."},{"key":"ref_35","unstructured":"Dua, D.a.G. (2022, January 12). Casey: UCI Machine Learning Repository. Available online: http:\/\/archive.ics.uci.edu\/ml."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1016\/j.patrec.2019.02.017","article-title":"A modified intuitionistic fuzzy c-means algorithm incorporating hesitation degree","volume":"122","author":"Verma","year":"2019","journal-title":"Pattern Recognit. Lett."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Zeng, J., Jing, W., Song, X., and Lu, Z. (2020). Analysis method for customer value of aviation big data based on LRFMC model. Data Science, Proceedings of the 6th International Conference of Pioneering Computer Scientists, Engineers and Educators, ICPCSEE 2020, Taiyuan, China, 18\u201321 September 2020, Springer.","DOI":"10.1007\/978-981-15-7981-3"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1016\/S1525-1578(10)60455-2","article-title":"Analysis of microarray data using Z score transformation","volume":"5","author":"Cheadle","year":"2003","journal-title":"J. Mol. Diagn."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/7\/1442\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:49:43Z","timestamp":1760140183000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/7\/1442"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,7,13]]},"references-count":38,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2022,7]]}},"alternative-id":["sym14071442"],"URL":"https:\/\/doi.org\/10.3390\/sym14071442","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,7,13]]}}}