{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,9]],"date-time":"2026-06-09T14:50:15Z","timestamp":1781016615236,"version":"3.54.1"},"reference-count":20,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2024,2,16]],"date-time":"2024-02-16T00:00:00Z","timestamp":1708041600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,2,16]],"date-time":"2024-02-16T00:00:00Z","timestamp":1708041600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100004769","name":"Universit\u00e0 degli Studi di Pavia","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100004769","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Int J Data Sci Anal"],"published-print":{"date-parts":[[2025,8]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>Artificial Intelligence relies on the application of machine learning models which, while reaching high predictive accuracy, lack explainability and robustness. This is a problem in regulated industries, as authorities aimed at monitoring the risks arising from the application of Artificial Intelligence methods may not validate them. No measurement methodologies are yet available to jointly assess accuracy, explainability and robustness of machine learning models. We propose a methodology which fills the gap, extending the Forward Search approach, employed in robust statistical learning, to machine learning models. Doing so, we will be able to evaluate, by means of interpretable statistical tests, whether a specific Artificial Intelligence application is accurate, explainable and robust, through a unified methodology. We apply our proposal to the context of Bitcoin price prediction, comparing a linear regression model against a nonlinear neural network model.<\/jats:p>","DOI":"10.1007\/s41060-024-00512-1","type":"journal-article","created":{"date-parts":[[2024,2,16]],"date-time":"2024-02-16T14:02:37Z","timestamp":1708092157000},"page":"1043-1050","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Robust machine learning models: linear and nonlinear"],"prefix":"10.1007","volume":"20","author":[{"given":"Paolo","family":"Giudici","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Emanuela","family":"Raffinetti","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Marco","family":"Riani","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2024,2,16]]},"reference":[{"key":"512_CR1","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-1160-0","volume-title":"Robust Diagnostic Regression Analysis","author":"AC Atkinson","year":"2000","unstructured":"Atkinson, A.C., Riani, M.: Robust Diagnostic Regression Analysis. Springer-Verlag, New York (2000)"},{"key":"512_CR2","doi-asserted-by":"publisher","first-page":"939","DOI":"10.1093\/biomet\/89.4.939","volume":"89","author":"AC Atkinson","year":"2002","unstructured":"Atkinson, A.C., Riani, M.: Forward search added-variable $$t$$ tests and the effect of masked outliers on model selection. Biometrika 89, 939\u2013946 (2002)","journal-title":"Biometrika"},{"key":"512_CR3","doi-asserted-by":"publisher","first-page":"117","DOI":"10.1016\/j.jkss.2010.02.007","volume":"39","author":"AC Atkinson","year":"2010","unstructured":"Atkinson, A.C., Riani, M., Cerioli, A.: The forward search: theory and data analysis (with discussion). J. Korean Stat. Soc. 39, 117\u2013134 (2010). https:\/\/doi.org\/10.1016\/j.jkss.2010.02.007","journal-title":"J. Korean Stat. Soc."},{"key":"512_CR4","doi-asserted-by":"crossref","unstructured":"Bracke, P., Datta, A., Jung, C., Shayak, S.: Machine learning explainability in finance: an application to default risk analysis. Staff Working Paper No. 816, Bank of England. (2019). Available at https:\/\/www.bankofengland.co.uk\/-\/media\/boe\/files\/working-paper\/2019\/machine-learning-explainability-in-finance-an-application-to-default-risk-analysis.pdf","DOI":"10.2139\/ssrn.3435104"},{"key":"512_CR5","doi-asserted-by":"publisher","first-page":"1","DOI":"10.3389\/frai.2020.00026","volume":"326","author":"N Bussmann","year":"2020","unstructured":"Bussmann, N., Giudici, P., Marinelli, D., Papenbrock, J.: Explainable AI in credit risk management. Front. Artif. Intell. 326, 1\u20135 (2020). https:\/\/doi.org\/10.3389\/frai.2020.00026","journal-title":"Front. Artif. Intell."},{"key":"512_CR6","doi-asserted-by":"publisher","DOI":"10.3389\/frai.2020.00026","volume":"213","author":"F Cabitza","year":"2023","unstructured":"Cabitza, F., Campagner, A., Malgieri, G., Natali, C., Schneeberger, D., Stoeger, K., Holzinger, A.: Quod erat demonstrandum? Towards a typology of the concept of explanation for the design of explainable AI. Expert Syst. Appl. 213, 118888 (2023). https:\/\/doi.org\/10.3389\/frai.2020.00026","journal-title":"Expert Syst. Appl."},{"key":"512_CR7","doi-asserted-by":"publisher","unstructured":"Christodoulou, E., Ma, J., Collins, G.S., Steyerberg, E.W., Verbakel, J.Y., Van Calster, B.: A systematic review shows no performance benefit of machine learning over logistic regression for clinical prediction models. J. Clin. Epidemiol. 110, 12\u201322 (2019). https:\/\/doi.org\/10.1016\/j.jclinepi.2019.02.004","DOI":"10.1016\/j.jclinepi.2019.02.004"},{"key":"512_CR8","unstructured":"European Commission: On Artificial Intelligence - A European approach to excellence and trust. White Paper, European Commission, Brussels, 19-02-2020. https:\/\/commission.europa.eu\/system\/files\/2020-02\/commission-white-paper-artificial-intelligence-feb2020_en.pdf (2020)"},{"key":"512_CR9","doi-asserted-by":"publisher","first-page":"309","DOI":"10.1016\/j.frl.2018.05.013","volume":"28","author":"P Giudici","year":"2019","unstructured":"Giudici, P., Abu-Hashish, I.: What determines bitcoin exchange prices? A network VAR approach. Financ. Res. Lett. 28, 309\u2013318 (2019). https:\/\/doi.org\/10.1016\/j.frl.2018.05.013","journal-title":"Financ. Res. Lett."},{"key":"512_CR10","doi-asserted-by":"publisher","first-page":"754","DOI":"10.1007\/s00357-019-09358-w","volume":"37","author":"P Giudici","year":"2020","unstructured":"Giudici, P., Raffinetti, E.: Lorenz model selection. J. Classif. 37, 754\u2013768 (2020). https:\/\/doi.org\/10.1007\/s00357-019-09358-w","journal-title":"J. Classif."},{"issue":"114104","key":"512_CR11","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.eswa.2020.114104","volume":"167","author":"P Giudici","year":"2021","unstructured":"Giudici, P., Raffinetti, E.: Shapley\u2013Lorenz eXplainable artificial intelligence. Expert Syst. Appl. 167(114104), 1\u20137 (2021). https:\/\/doi.org\/10.1016\/j.eswa.2020.114104","journal-title":"Expert Syst. Appl."},{"key":"512_CR12","doi-asserted-by":"publisher","DOI":"10.1016\/j.frl.2023.104088","volume":"56","author":"P Giudici","year":"2023","unstructured":"Giudici, P., Raffinetti, E.: SAFE artificial intelligence in finance. Financ. Res. Lett. 56, 104088 (2023). https:\/\/doi.org\/10.1016\/j.frl.2023.104088","journal-title":"Financ. Res. Lett."},{"key":"512_CR13","doi-asserted-by":"publisher","unstructured":"Holzinger, A.: The Next Frontier: AI We Can Really Trust. In: Kamp, M. (ed.) Proceedings of the ECML PKDD 2021, CCIS 1524, pp. 427\u2013440. Springer-Nature, Cham (2021). https:\/\/doi.org\/10.1007\/978-3-030-93736-2_33","DOI":"10.1007\/978-3-030-93736-2_33"},{"key":"512_CR14","doi-asserted-by":"publisher","unstructured":"Kieseberg, P., Weippl, E., Tjoa, A. M., Cabitza, F., Campagner, A. Holzinger, A.: Controllable AI\u2014an alternative to trustworthiness in complex AI systems? Lecture Notes in Computer Science (LNCS) Volume 14065. Springer. 1\u201312 (2023). https:\/\/doi.org\/10.1007\/978-3-031-40837-3_1","DOI":"10.1007\/978-3-031-40837-3_1"},{"key":"512_CR15","doi-asserted-by":"publisher","first-page":"873","DOI":"10.2307\/2291682","volume":"91","author":"G Koshevoy","year":"1996","unstructured":"Koshevoy, G., Mosler, K.: The Lorenz zonoid of a multivariate distribution. J. Am. Stat. Assoc. 91, 873\u2013882 (1996). https:\/\/doi.org\/10.2307\/2291682","journal-title":"J. Am. Stat. Assoc."},{"key":"512_CR16","doi-asserted-by":"publisher","first-page":"986","DOI":"10.1137\/16M1097717","volume":"5","author":"AB Owen","year":"2017","unstructured":"Owen, A.B., Prieur, C.: On Shapley value for measuring importance of dependent inputs. SIAM\/ASA J. Uncertain. Quantif. 5, 986\u20131002 (2017). https:\/\/doi.org\/10.1137\/16M1097717","journal-title":"SIAM\/ASA J. Uncertain. Quantif."},{"key":"512_CR17","doi-asserted-by":"publisher","first-page":"871","DOI":"10.2307\/2288718","volume":"79","author":"PJ Rousseeuw","year":"1984","unstructured":"Rousseeuw, P.J.: Least median of squares regression. J. Am. Stat. Assoc. 79, 871\u2013880 (1984). https:\/\/doi.org\/10.2307\/2288718","journal-title":"J. Am. Stat. Assoc."},{"key":"512_CR18","doi-asserted-by":"crossref","unstructured":"Shapley, L.S.: A value for $$n$$-person games. Contributions to the Theory of Games, 307\u2013317 (1953)","DOI":"10.1515\/9781400881970-018"},{"key":"512_CR19","first-page":"799","volume":"33","author":"S Tonekaboni","year":"2020","unstructured":"Tonekaboni, S., Joshi, S., Campbell, K., Duvenaud, D.K., Goldenberg, A.: What went wrong and when? Instance-wise feature importance for time-series black-box models. Adv. Neural. Inf. Process. Syst. 33, 799\u2013809 (2020)","journal-title":"Adv. Neural. Inf. Process. Syst."},{"key":"512_CR20","unstructured":"Ye, J., Borovykh, A., Hayou, S., Shokri, R.: Leave-one-out Distinguishability in Machine Learning. arXiv preprint. arXiv:org\/abs\/2309.17310 (2023)"}],"container-title":["International Journal of Data Science and Analytics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s41060-024-00512-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s41060-024-00512-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s41060-024-00512-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,5]],"date-time":"2025-09-05T19:54:06Z","timestamp":1757102046000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s41060-024-00512-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,2,16]]},"references-count":20,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2025,8]]}},"alternative-id":["512"],"URL":"https:\/\/doi.org\/10.1007\/s41060-024-00512-1","relation":{"has-preprint":[{"id-type":"doi","id":"10.21203\/rs.3.rs-3306884\/v1","asserted-by":"object"}]},"ISSN":["2364-415X","2364-4168"],"issn-type":[{"value":"2364-415X","type":"print"},{"value":"2364-4168","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,2,16]]},"assertion":[{"value":"29 August 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 January 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 February 2024","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":"Conflict of interest"}}]}}