{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,6]],"date-time":"2026-05-06T16:13:51Z","timestamp":1778084031309,"version":"3.51.4"},"reference-count":15,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2026,3,1]],"date-time":"2026-03-01T00:00:00Z","timestamp":1772323200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2026,3,6]],"date-time":"2026-03-06T00:00:00Z","timestamp":1772755200000},"content-version":"vor","delay-in-days":5,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100011104","name":"Universitat Aut\u00f2noma de Barcelona","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100011104","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Mach Learn"],"published-print":{"date-parts":[[2026,3]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    Ordinal classification addresses prediction tasks where class labels have a natural order but are not necessarily equally spaced. While traditional approaches typically assume symmetric misclassification costs, many real-world applications exhibit asymmetric, label-dependent penalties. This paper extends previous work on Bayes-optimal decision rules for ordinal classification under symmetric loss (Delgado, 2025) to this more general, cost-sensitive setting. Within a unified decision-theoretic framework, we formalize the interplay between three fundamental components of classification: the\n                    <jats:italic>loss function<\/jats:italic>\n                    , which encodes misclassification severity; the\n                    <jats:italic>scoring rule<\/jats:italic>\n                    , used to evaluate probabilitic predictions and shown here to satisfy regularity and properness; and the\n                    <jats:italic>decision criterion<\/jats:italic>\n                    that maps predictive distributions to class labels. We prove that the proposed scoring rule coincides with the expected loss up to a change of sign \u2013a result of independent interest\u2013 and we explicitly characterize sufficient structural conditions under which the resulting decision criterion is well defined and Bayes-optimal. Special attention is given to the interval-scale case, where class distances are explicitly incorporated into the loss, the score, and the decision rule. We show that, depending on the structure of the loss function, Bayes optimality may hold either globally or locally in the space of predictive probability distributions. Empirical results on two real-world datasets, covering both interval-scale and fully ordinal cost-sensitive scenarios and different classifier families, illustrate the practical implications of the proposed approach.\n                  <\/jats:p>","DOI":"10.1007\/s10994-026-07023-z","type":"journal-article","created":{"date-parts":[[2026,3,6]],"date-time":"2026-03-06T13:14:08Z","timestamp":1772802848000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Ordinal Classification with Label-Dependent Loss"],"prefix":"10.1007","volume":"115","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1208-9236","authenticated-orcid":false,"given":"Rosario","family":"Delgado","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2026,3,6]]},"reference":[{"key":"7023_CR1","doi-asserted-by":"publisher","first-page":"0957","DOI":"10.1016\/j.eswa.2023.122277","volume":"122277","author":"A Alcacer","year":"2024","unstructured":"Alcacer, A., Martinez-Garcia, M., & Epifanio, I. (2024). Ordinal classification for interval-valued data and interval-valued functional data, Expert Systems with Applications, Volume 238, Part F. ISSN, 122277, 0957\u20134174. https:\/\/doi.org\/10.1016\/j.eswa.2023.122277","journal-title":"ISSN"},{"key":"7023_CR2","doi-asserted-by":"publisher","DOI":"10.1007\/s10994-024-06654-4","author":"G Binotto","year":"2025","unstructured":"Binotto, G., & Delgado, R. (2025). Adapting performance metrics for ordinal classification to interval scale: Length matters. Machine Learning. https:\/\/doi.org\/10.1007\/s10994-024-06654-4","journal-title":"Machine Learning"},{"key":"7023_CR3","doi-asserted-by":"publisher","first-page":"864","DOI":"10.1016\/j.asoc.2015.02.035","volume":"35","author":"JL Garc\u00eda-Lapresta","year":"2015","unstructured":"Garc\u00eda-Lapresta, J. L., & P\u00e9rez-Rom\u00e1n, D. (2015). Ordinal proximity measures in the context of unbalanced qualitative scales and some applications to consensus and clustering. Applied Soft Computing, 35, 864\u2013872. https:\/\/doi.org\/10.1016\/j.asoc.2015.02.035","journal-title":"Applied Soft Computing"},{"key":"7023_CR4","doi-asserted-by":"publisher","unstructured":"Delgado, R. (2025). Ord-MAP criterion: extending MAP for Ordinal Classification. Knowledge-Based Systems, Volume 324, 113837, ISSN 0950\u20137051, https:\/\/doi.org\/10.1016\/j.knosys.2025.113837.","DOI":"10.1016\/j.knosys.2025.113837"},{"key":"7023_CR5","doi-asserted-by":"publisher","first-page":"28453","DOI":"10.1038\/s41598-024-77386-7","volume":"14","author":"R Delgado","year":"2024","unstructured":"Delgado, R., Fern\u00e1ndez-Pel\u00e1ez, F., Pallar\u00e9s, N., et al. (2024). Predictive risk models for COVID-19 patients using the multi-thresholding meta-algorithm. Scientific Reports, 14, 28453. https:\/\/doi.org\/10.1038\/s41598-024-77386-7","journal-title":"Scientific Reports"},{"key":"7023_CR6","doi-asserted-by":"publisher","unstructured":"Elkan, C. (2001). The Foundations of Cost-Sensitive Learning. In Proceedings of the 17th International Joint Conference of Artificial Intelligence (IJCAI\u201901) 2, 973\u2013978. Seattle, Washington: Morgan Kaufmann https:\/\/doi.org\/10.5555\/1642194.1642224","DOI":"10.5555\/1642194.1642224"},{"key":"7023_CR7","doi-asserted-by":"publisher","first-page":"149","DOI":"10.1016\/j.ins.2018.03.034","volume":"448\u2013449","author":"JL Garc\u00eda-Lapresta","year":"2018","unstructured":"Garc\u00eda-Lapresta, J. L., Gonz\u00e1lez del Pozo, R., & P\u00e9rez-Rom\u00e1n, D. (2018). Metrizable ordinal proximity measures and their aggregation. Information Sciences, 448\u2013449, 149\u2013163. https:\/\/doi.org\/10.1016\/j.ins.2018.03.034","journal-title":"Information Sciences"},{"key":"7023_CR8","doi-asserted-by":"publisher","first-page":"652","DOI":"10.1016\/j.asoc.2017.05.064","volume":"67","author":"JL Garc\u00eda-Lapresta","year":"2018","unstructured":"Garc\u00eda-Lapresta, J. L., & P\u00e9rez-Rom\u00e1n, D. (2018). Aggregating opinions in non-uniform ordered qualitative scales. Applied Soft Computing, 67, 652\u2013657. https:\/\/doi.org\/10.1016\/j.asoc.2017.05.064","journal-title":"Applied Soft Computing"},{"key":"7023_CR9","doi-asserted-by":"publisher","DOI":"10.1198\/016214506000001437","author":"T Gneiting","year":"2007","unstructured":"Gneiting, T., & Raftery, A. E. (2007). Strictly proper scoring rules, prediction and estimation. Journal of the American Statistical Association. https:\/\/doi.org\/10.1198\/016214506000001437","journal-title":"Journal of the American Statistical Association"},{"issue":"140","key":"7023_CR10","first-page":"5","volume":"22","author":"R Likert","year":"1932","unstructured":"Likert, R. (1932). A Technique for the Measurement of Attitudes. Archives of Psychology, 22(140), 5\u201355.","journal-title":"Archives of Psychology"},{"issue":"1","key":"7023_CR11","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1561\/1500000011","volume":"2","author":"B Pang","year":"2008","unstructured":"Pang, B., & Lee, L. (2008). Opinion mining and sentiment analysis. Foundations and Trends in Information Retrieval, 2(1), 1\u2013135. https:\/\/doi.org\/10.1561\/1500000011","journal-title":"Foundations and Trends in Information Retrieval"},{"key":"7023_CR12","doi-asserted-by":"publisher","first-page":"204","DOI":"10.1186\/s12879-024-08986-x","volume":"24","author":"A Robert","year":"2024","unstructured":"Robert, A., Chapman, L. A. C., Grah, R., Niehus, R., Sandmann, F., Prasse, B., Funk, S., & Kucharski, A. J. (2024). Predicting subnational incidence of COVID-19 cases and deaths in EU countries. BMC Infectious Diseases, 24, 204. https:\/\/doi.org\/10.1186\/s12879-024-08986-x","journal-title":"BMC Infectious Diseases"},{"issue":"336","key":"7023_CR13","doi-asserted-by":"publisher","first-page":"783","DOI":"10.1080\/01621459.1971.10482346","volume":"66","author":"LJ Savage","year":"1971","unstructured":"Savage, L. J. (1971). Elicitation of Personal Probabilities and Expectations. Journal of the American Statistical Association, 66(336), 783\u2013801. https:\/\/doi.org\/10.1080\/01621459.1971.10482346","journal-title":"Journal of the American Statistical Association"},{"key":"7023_CR14","doi-asserted-by":"publisher","first-page":"2729","DOI":"10.1007\/s10994-021-06010-w","volume":"110","author":"JL Su\u00e1rez","year":"2021","unstructured":"Su\u00e1rez, J. L., Garc\u00eda, S., & Herrera, F. (2021). Ordinal regression with explainable distance metric learning based on ordered sequences. Machine Learning, 110, 2729\u20132762. https:\/\/doi.org\/10.1007\/s10994-021-06010-w","journal-title":"Machine Learning"},{"key":"7023_CR15","doi-asserted-by":"publisher","DOI":"10.1029\/2023WR036710","author":"JA Vrugt","year":"2024","unstructured":"Vrugt, J. A. (2024). Distribution-based model evaluation and diagnostics: Elicitability, propriety, and scoring rules for hydrograph functionals. Water Resource Research. https:\/\/doi.org\/10.1029\/2023WR036710","journal-title":"Water Resource Research"}],"container-title":["Machine Learning"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10994-026-07023-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10994-026-07023-z","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10994-026-07023-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,5,6]],"date-time":"2026-05-06T15:30:53Z","timestamp":1778081453000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10994-026-07023-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,3]]},"references-count":15,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2026,3]]}},"alternative-id":["7023"],"URL":"https:\/\/doi.org\/10.1007\/s10994-026-07023-z","relation":{},"ISSN":["0885-6125","1573-0565"],"issn-type":[{"value":"0885-6125","type":"print"},{"value":"1573-0565","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,3]]},"assertion":[{"value":"3 November 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 January 2026","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 February 2026","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 March 2026","order":4,"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 Conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"60"}}