{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,8]],"date-time":"2026-01-08T22:23:44Z","timestamp":1767911024892,"version":"3.49.0"},"reference-count":38,"publisher":"Springer Science and Business Media LLC","issue":"10","license":[{"start":{"date-parts":[[2025,7,16]],"date-time":"2025-07-16T00:00:00Z","timestamp":1752624000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,7,16]],"date-time":"2025-07-16T00:00:00Z","timestamp":1752624000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Int. J. Mach. Learn. & Cyber."],"published-print":{"date-parts":[[2025,10]]},"DOI":"10.1007\/s13042-025-02737-3","type":"journal-article","created":{"date-parts":[[2025,7,16]],"date-time":"2025-07-16T20:21:37Z","timestamp":1752697297000},"page":"8523-8550","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Stochastic support vector machine upon unbounded distribution functions for multi-class classification"],"prefix":"10.1007","volume":"16","author":[{"given":"Tara","family":"Mohammadi","sequence":"first","affiliation":[]},{"given":"Hadi","family":"Jabbari","sequence":"additional","affiliation":[]},{"given":"Sohrab","family":"Effati","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,7,16]]},"reference":[{"key":"2737_CR1","doi-asserted-by":"crossref","unstructured":"Boser BE, Guyon IM, Vapnik VN (1992) A training algorithm for optimal margin classifiers. In: Proceedings of the fifth annual workshop on computational learning theory. pp 144\u2013152","DOI":"10.1145\/130385.130401"},{"key":"2737_CR2","doi-asserted-by":"publisher","first-page":"273","DOI":"10.1023\/A:1022627411411","volume":"20","author":"C Cortes","year":"1995","unstructured":"Cortes C, Vapnik V (1995) Support-vector networks. Mach Learn 20:273\u2013297","journal-title":"Mach Learn"},{"key":"2737_CR3","doi-asserted-by":"crossref","unstructured":"Knerr S, Personnaz L, Dreyfus G (1990) Single-layer learning revisited: a stepwise procedure for building and training a neural network. In: Neurocomputing: algorithms, architectures and applications. pp 41\u201350","DOI":"10.1007\/978-3-642-76153-9_5"},{"issue":"5","key":"2737_CR4","doi-asserted-by":"publisher","first-page":"988","DOI":"10.1109\/72.788640","volume":"10","author":"VN Vapnik","year":"1999","unstructured":"Vapnik VN (1999) An overview of statistical learning theory. IEEE Trans Neural Netw 10(5):988\u2013999","journal-title":"IEEE Trans Neural Netw"},{"issue":"2","key":"2737_CR5","first-page":"233","volume":"33","author":"M Gjorgji","year":"2009","unstructured":"Gjorgji M, Dejan G, Ivan C (2009) A multiclass SVM classifier utilizing binary decision tree. Informatica 33(2):233\u2013241","journal-title":"Informatica"},{"key":"2737_CR6","unstructured":"Platt J, Cristianini N, Shawe-Taylor J (1999) Large margin DAGs for multiclass classification. Adv Neural Inf Process Syst 12"},{"key":"2737_CR7","doi-asserted-by":"crossref","unstructured":"Kijsirikul B, Ussivakul N (2002) Multiclass support vector machines using adaptive directed acyclic graph. In: Proceedings of the 2002 international joint conference on neural networks, vol 1. pp 980\u2013985","DOI":"10.1109\/IJCNN.2002.1005608"},{"key":"2737_CR8","first-page":"265","volume":"2","author":"K Crammer","year":"2001","unstructured":"Crammer K, Singer Y (2001) On the algorithmic implementation of multiclass kernel-based vector machines. J Mach Learn Res 2:265\u2013292","journal-title":"J Mach Learn Res"},{"key":"2737_CR9","unstructured":"Weston J, Watkins C (1999) Support vector machines for multi-class pattern recognition. In Esann, vol 99. pp 219\u2013224"},{"issue":"1\u20132","key":"2737_CR10","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1016\/S0925-2312(03)00435-1","volume":"55","author":"C Angulo","year":"2003","unstructured":"Angulo C, Parra X, Catala A (2003) K-SVCR. A support vector machine for multi-class classification. Neurocomputing 55(1\u20132):57\u201377","journal-title":"Neurocomputing"},{"key":"2737_CR11","doi-asserted-by":"publisher","first-page":"580","DOI":"10.1007\/s12559-012-9179-7","volume":"5","author":"Y Xu","year":"2013","unstructured":"Xu Y, Guo R, Wang L (2013) A twin multi-class classification support vector machine. Cogn Comput 5:580\u2013588","journal-title":"Cogn Comput"},{"issue":"5","key":"2737_CR12","doi-asserted-by":"publisher","first-page":"905","DOI":"10.1109\/TPAMI.2007.1068","volume":"29","author":"R Khemchandani","year":"2007","unstructured":"Khemchandani R, Chandra S (2007) Twin support vector machines for pattern classification. IEEE Trans Pattern Anal Mach Intell 29(5):905\u2013910","journal-title":"IEEE Trans Pattern Anal Mach Intell"},{"key":"2737_CR13","doi-asserted-by":"crossref","unstructured":"Bishwas AK, Mani A, Palade V (2017) An all-pair approach for big data multiclass classification with quantum SVM. CoRR arXiv:1704.07664","DOI":"10.1007\/s11128-018-2046-z"},{"issue":"2","key":"2737_CR14","doi-asserted-by":"publisher","first-page":"18","DOI":"10.1109\/MCI.2018.2806997","volume":"13","author":"A Rosales-Perez","year":"2018","unstructured":"Rosales-Perez A, Garc\u00eda S, Terashima-Marin H, Coello C, Herrera F (2018) MC2ESVM: multiclass classification based on cooperative evolution of support vector machines. IEEE Comput Intell Mag 13(2):18\u201329","journal-title":"IEEE Comput Intell Mag"},{"key":"2737_CR15","doi-asserted-by":"publisher","first-page":"132775","DOI":"10.1109\/ACCESS.2019.2941321","volume":"7","author":"W Shu","year":"2019","unstructured":"Shu W, Cai K (2019) An SVM multi-class image classification method based on DE and KNN in smart city management. IEEE Access 7:132775\u2013132785","journal-title":"IEEE Access"},{"key":"2737_CR16","doi-asserted-by":"publisher","first-page":"324","DOI":"10.1016\/j.ins.2018.12.026","volume":"480","author":"L Tang","year":"2019","unstructured":"Tang L, Tian Y, Pardalos PM (2019) A novel perspective on multiclass classification: regular simplex support vector machine. Inf Sci 480:324\u2013338","journal-title":"Inf Sci"},{"key":"2737_CR17","doi-asserted-by":"publisher","DOI":"10.1016\/j.knosys.2021.106801","volume":"216","author":"L Tang","year":"2021","unstructured":"Tang L, Tian Y, Li W, Pardalos PM (2021) Valley-loss regular simplex support vector machine for robust multiclass classification. Knowl Based Syst 216:106801","journal-title":"Knowl Based Syst"},{"issue":"2","key":"2737_CR18","doi-asserted-by":"publisher","first-page":"433","DOI":"10.1007\/s13042-019-00984-9","volume":"11","author":"DR Don","year":"2020","unstructured":"Don DR, Iacob IE (2020) DCSVM: fast multi-class classification using support vector machines. Int J Mach Learn Cybern 11(2):433\u2013447","journal-title":"Int J Mach Learn Cybern"},{"key":"2737_CR19","doi-asserted-by":"publisher","first-page":"430","DOI":"10.1016\/j.neucom.2016.04.024","volume":"205","author":"Y Xu","year":"2016","unstructured":"Xu Y (2016) K-nearest neighbor-based weighted multi-class twin support vector machine. Neurocomputing 205:430\u2013438","journal-title":"Neurocomputing"},{"key":"2737_CR20","doi-asserted-by":"publisher","first-page":"454","DOI":"10.1016\/j.neucom.2020.02.132","volume":"459","author":"M Tanveer","year":"2021","unstructured":"Tanveer M, Sharma A, Suganthan PN (2021) Least squares KNN-based weighted multiclass twin SVM. Neurocomputing 459:454\u2013464","journal-title":"Neurocomputing"},{"issue":"7","key":"2737_CR21","doi-asserted-by":"publisher","first-page":"873","DOI":"10.1007\/s10472-021-09747-1","volume":"90","author":"H Moosaei","year":"2022","unstructured":"Moosaei H, Hlad\u00edk M (2022) Least squares approach to K-SVCR multi-class classification with its applications. Ann Math Artif Intell 90(7):873\u2013892","journal-title":"Ann Math Artif Intell"},{"key":"2737_CR22","doi-asserted-by":"publisher","DOI":"10.1016\/j.eswa.2021.114885","volume":"176","author":"S Dong","year":"2021","unstructured":"Dong S (2021) Multi class SVM algorithm with active learning for network traffic classification. Expert Syst Appl 176:114885","journal-title":"Expert Syst Appl"},{"key":"2737_CR23","doi-asserted-by":"publisher","DOI":"10.1016\/j.knosys.2021.107123","volume":"226","author":"Z Gao","year":"2021","unstructured":"Gao Z, Fang SC, Gao X, Luo J, Medhin N (2021) A novel kernel-free least squares twin support vector machine for fast and accurate multi-class classification. Knowl Based Syst 226:107123","journal-title":"Knowl Based Syst"},{"key":"2737_CR24","doi-asserted-by":"publisher","first-page":"448","DOI":"10.1016\/j.neucom.2021.04.115","volume":"454","author":"Y Duan","year":"2021","unstructured":"Duan Y, Zou B, Xu J, Chen F, Wei J, Tang YY (2021) OAA-SVM-MS: a fast and efficient multi-class classification algorithm. Neurocomputing 454:448\u2013460","journal-title":"Neurocomputing"},{"issue":"2","key":"2737_CR25","first-page":"467","volume":"194","author":"HS Yazdi","year":"2007","unstructured":"Yazdi HS, Effati S, Saberi Z (2007) The probabilistic constraints in the support vector machine. Appl Math Comput 194(2):467\u2013479","journal-title":"Appl Math Comput"},{"key":"2737_CR26","doi-asserted-by":"publisher","first-page":"145","DOI":"10.1007\/s10107-010-0415-1","volume":"127","author":"A Ben-Tal","year":"2011","unstructured":"Ben-Tal A, Bhadra S, Bhattacharyya C, Saketha Nath J (2011) Chance constrained uncertain classification via robust optimization. Math Program 127:145\u2013173","journal-title":"Math Program"},{"key":"2737_CR27","doi-asserted-by":"publisher","first-page":"243","DOI":"10.1007\/s10489-017-0964-6","volume":"48","author":"M Abaszade","year":"2018","unstructured":"Abaszade M, Effati S (2018) Stochastic support vector regression with probabilistic constraints. Appl Intell 48:243\u2013256","journal-title":"Appl Intell"},{"key":"2737_CR28","doi-asserted-by":"publisher","first-page":"152","DOI":"10.1007\/s00357-018-9282-x","volume":"36","author":"M Abaszade","year":"2019","unstructured":"Abaszade M, Effati S (2019) A new method for classifying random variables based on support vector machine. J Classif 36:152\u2013174","journal-title":"J Classif"},{"issue":"3","key":"2737_CR29","doi-asserted-by":"publisher","first-page":"586","DOI":"10.1080\/10618600.2019.1585260","volume":"28","author":"X Wang","year":"2019","unstructured":"Wang X, Zhang HH, Wu Y (2019) Multiclass probability estimation with support vector machines. J Comput Graph Stat 28(3):586\u2013595","journal-title":"J Comput Graph Stat"},{"issue":"12","key":"2737_CR30","doi-asserted-by":"publisher","first-page":"3415","DOI":"10.1007\/s13042-021-01368-8","volume":"12","author":"L Yang","year":"2021","unstructured":"Yang L, Li G, Wu Z, Wu C (2021) Robust truncated $$L_2$$-norm twin support vector machine. Int J Mach Learn Cybern 12(12):3415\u20133436","journal-title":"Int J Mach Learn Cybern"},{"issue":"1","key":"2737_CR31","doi-asserted-by":"publisher","first-page":"268","DOI":"10.1016\/j.ejor.2023.09.027","volume":"314","author":"S Ben\u00edtez-Pe\u00f1a","year":"2024","unstructured":"Ben\u00edtez-Pe\u00f1a S, Blanquero R, Carrizosa E, Ram\u00edrez-Cobo P (2024) Cost-sensitive probabilistic predictions for support vector machines. Eur J Oper Res 314(1):268\u2013279","journal-title":"Eur J Oper Res"},{"key":"2737_CR32","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2024.117434","volume":"432","author":"X Yang","year":"2024","unstructured":"Yang X, Zhang Y, Zhao J, Jiang W (2024) A novel active learning method based on matrix-operation RBF model for high-dimensional reliability analysis. Comput Methods Appl Mech Eng 432:117434","journal-title":"Comput Methods Appl Mech Eng"},{"key":"2737_CR33","doi-asserted-by":"crossref","unstructured":"Mohammadi T, Jabbari H, Effati S (2024) Stochastic multi-class support vector machine based on regular simplex. Communications in Statistics: Case Studies, Data Analysis and Applications","DOI":"10.1080\/23737484.2025.2550445"},{"key":"2737_CR34","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611971255","volume-title":"Nonlinear programming","author":"OL Mangasarian","year":"1994","unstructured":"Mangasarian OL (1994) Nonlinear programming. Society for Industrial and Applied Mathematics, Providence"},{"key":"2737_CR35","doi-asserted-by":"publisher","first-page":"569","DOI":"10.1007\/978-1-4612-4380-9_41","volume-title":"Breakthroughs in statistics: methodology and Distribution","author":"B Efron","year":"1992","unstructured":"Efron B (1992) Bootstrap methods: another look at the jackknife. In: New York NY (ed) Breakthroughs in statistics: methodology and Distribution. Springer, New York, pp 569\u2013593"},{"key":"2737_CR36","doi-asserted-by":"publisher","DOI":"10.4135\/9781412985116","volume-title":"Monte Carlo simulation","author":"CZ Mooney","year":"1997","unstructured":"Mooney CZ (1997) Monte Carlo simulation, vol 116. Sage, New York"},{"issue":"11","key":"2737_CR37","doi-asserted-by":"publisher","first-page":"4212","DOI":"10.1007\/s10489-018-1204-4","volume":"48","author":"D Gupta","year":"2018","unstructured":"Gupta D, Richhariya B (2018) Entropy based fuzzy least squares twin support vector machine for class imbalance learning. Appl Intell 48(11):4212\u20134231","journal-title":"Appl Intell"},{"key":"2737_CR38","unstructured":"Asuncion A, Newman D (2007) UCI Machine learning repository. https:\/\/archive.ics.uci.edu\/"}],"container-title":["International Journal of Machine Learning and Cybernetics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s13042-025-02737-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s13042-025-02737-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s13042-025-02737-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,15]],"date-time":"2025-10-15T17:00:05Z","timestamp":1760547605000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s13042-025-02737-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,16]]},"references-count":38,"journal-issue":{"issue":"10","published-print":{"date-parts":[[2025,10]]}},"alternative-id":["2737"],"URL":"https:\/\/doi.org\/10.1007\/s13042-025-02737-3","relation":{},"ISSN":["1868-8071","1868-808X"],"issn-type":[{"value":"1868-8071","type":"print"},{"value":"1868-808X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,7,16]]},"assertion":[{"value":"24 July 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 July 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 July 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}]}}