{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:07:21Z","timestamp":1760058441567,"version":"build-2065373602"},"reference-count":33,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2025,4,10]],"date-time":"2025-04-10T00:00:00Z","timestamp":1744243200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this study, the cumulative effect of the empirical probability distribution of a random variable is identified as a factor that amplifies the occurrence of extreme events in datasets. To quantify this observation, a corresponding information measure is introduced, drawing upon Shannon entropy for joint probabilities. The proposed approach is validated using selected market data as case studies, encompassing various instances of extreme events. In particular, the results indicate that the introduced cumulative measure exhibits distinctive signatures of such events, even when the data are relatively noisy. These findings highlight the potential of the discussed concept for developing a new class of related indicators or classifiers.<\/jats:p>","DOI":"10.3390\/e27040410","type":"journal-article","created":{"date-parts":[[2025,4,10]],"date-time":"2025-04-10T10:47:41Z","timestamp":1744282061000},"page":"410","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Signatures of Extreme Events in Cumulative Entropic Spectrum"],"prefix":"10.3390","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8689-5658","authenticated-orcid":false,"given":"Ewa A.","family":"Drzazga-Szcz\u0229\u015bniak","sequence":"first","affiliation":[{"name":"Department of Physics, Faculty of Production Engineering and Materials Technology, Czestochowa University of Technology, 19 Armii Krajowej Ave., 42200 Czestochowa, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0956-8623","authenticated-orcid":false,"given":"Adam Z.","family":"Kaczmarek","sequence":"additional","affiliation":[{"name":"Institute of Physics, Faculty of Science and Technology, Jan Dlugosz University in Czestochowa, 13\/15 Armii Krajowej Ave., 42200 Czestochowa, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Marta","family":"Kielak","sequence":"additional","affiliation":[{"name":"Institute of Physics, Faculty of Science and Technology, Jan Dlugosz University in Czestochowa, 13\/15 Armii Krajowej Ave., 42200 Czestochowa, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shivam","family":"Gupta","sequence":"additional","affiliation":[{"name":"EntropyX Labs Pvt. Ltd., Ghaziabad 201010, Uttar Pradesh, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0007-4355-1514","authenticated-orcid":false,"given":"Jakub T.","family":"Gnyp","sequence":"additional","affiliation":[{"name":"Condensed Matter Spectroscopy Division, Faculty of Mathematics, Physics and Informatics, University of Gdansk, 57 Wita Stwosza Str., 80308 Gdansk, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Katarzyna","family":"Pluta","sequence":"additional","affiliation":[{"name":"Institute of Physics, Faculty of Science and Technology, Jan Dlugosz University in Czestochowa, 13\/15 Armii Krajowej Ave., 42200 Czestochowa, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zygmunt","family":"Ba\u0327k","sequence":"additional","affiliation":[{"name":"Institute of Physics, Faculty of Science and Technology, Jan Dlugosz University in Czestochowa, 13\/15 Armii Krajowej Ave., 42200 Czestochowa, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0009-0245-8414","authenticated-orcid":false,"given":"Piotr","family":"Szczepanik","sequence":"additional","affiliation":[{"name":"Institute of Pricing and Market Analysis, Analitico, 49\/8 Krolewska Str., 47400 Raciborz, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1880-1255","authenticated-orcid":false,"given":"Dominik","family":"Szcz\u0229\u015bniak","sequence":"additional","affiliation":[{"name":"Institute of Physics, Faculty of Science and Technology, Jan Dlugosz University in Czestochowa, 13\/15 Armii Krajowej Ave., 42200 Czestochowa, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,4,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Ding, D., Zhang, M., Pan, X., Yang, M., and He, X. (2019, January 4\u20138). Modeling Extreme Events in Time Series Prediction. Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining. Association for Computing Machinery, KDD \u201919, Anchorage, AK, USA.","DOI":"10.1145\/3292500.3330896"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"3159","DOI":"10.1109\/TNSE.2020.3017495","article-title":"Distance dependent competitive interactions in a frustrated network of mobile agents","volume":"7","author":"Chowdhury","year":"2020","journal-title":"IEEE Trans. Netw. Sci. Eng."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"268","DOI":"10.1016\/0016-3287(80)90076-2","article-title":"Sudden events","volume":"12","author":"Ramage","year":"1980","journal-title":"Futures"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"121732","DOI":"10.1016\/j.techfore.2022.121732","article-title":"Sudden shock and stock market network structure characteristics: A comparison of past crisis events","volume":"180","author":"He","year":"2022","journal-title":"Technol. Forecast. Soc. Change"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"183","DOI":"10.3847\/1538-4357\/837\/2\/183","article-title":"Equilibrium and sudden events in chemical evolution","volume":"837","author":"Weinberg","year":"2017","journal-title":"Astrophys. J."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2022.04.001","article-title":"Extreme events in dynamical systems and random walkers: A review","volume":"966","author":"Chowdhury","year":"2022","journal-title":"Phys. Rep."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"023028","DOI":"10.1103\/PhysRevResearch.4.023028","article-title":"Predicting extreme events from data using deep machine learning: When and where","volume":"4","author":"Jiang","year":"2022","journal-title":"Phys. Rev. Res."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"043082","DOI":"10.1103\/PhysRevResearch.6.043082","article-title":"Prediction of chaotic dynamics and extreme events: A recurrence-free quantum reservoir computing approach","volume":"6","author":"Ahmed","year":"2024","journal-title":"Phys. Rev. Res."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"012033","DOI":"10.1088\/1742-6596\/394\/1\/012033","article-title":"Entropy: A new measure of stock market volatility?","volume":"394","author":"Bentes","year":"2012","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"056112","DOI":"10.1103\/PhysRevE.86.056112","article-title":"Alternate entropy measure for assessing volatility in financial markets","volume":"86","author":"Bose","year":"2012","journal-title":"Phys. Rev. E\u2014Stat. Nonlinear Soft Matter Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"655","DOI":"10.1016\/S2212-5671(15)00279-8","article-title":"Entropy measures for assessing volatile markets","volume":"22","author":"Sheraz","year":"2015","journal-title":"Procedia Econ. Financ."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Sheraz, M., and Nasir, I. (2021). Information-theoretic measures and modeling stock market volatility: A comparative approach. Risks, 9.","DOI":"10.3390\/risks9050089"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Drzazga-Szcz\u0229\u015bniak, E.A., Szczepanik, P., Kaczmarek, A.Z., and Szcz\u0229\u015bniak, D. (2023). Entropy of financial time series due to the shock of war. Entropy, 25.","DOI":"10.3390\/e25050823"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"623","DOI":"10.1002\/j.1538-7305.1948.tb00917.x","article-title":"A Mathematical Theory of Communication","volume":"27","author":"Shannon","year":"1948","journal-title":"Bell Syst. Tech. J."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1140\/epjb\/e2006-00113-2","article-title":"An econophysics approach to analyse uncertainty in financial markets: An application to the Portuguese stock market","volume":"50","author":"Dionisio","year":"2006","journal-title":"Eur. Phys. J. B"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Delgado-Bonal, A. (2019). Quantifying the randomness of the stock markets. Sci. Rep., 9.","DOI":"10.1038\/s41598-019-49320-9"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"125770","DOI":"10.1016\/j.physa.2021.125770","article-title":"Quantifying the randomness of the forex market","volume":"569","year":"2021","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Rosser, J.B. (2021). Econophysics and the entropic foundations of economics. Entropy, 23.","DOI":"10.3390\/e23101286"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"112403","DOI":"10.1016\/j.chaos.2022.112403","article-title":"Measuring market efficiency: The Shannon entropy of high-frequency financial time series","volume":"162","author":"Shternshis","year":"2022","journal-title":"Chaos Solitons Fractals"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Ormos, M., and Zibriczky, D. (2014). Entropy-based financial asset pricing. PLoS ONE, 9.","DOI":"10.1371\/journal.pone.0115742"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Gupta, R., Drzazga-Szcz\u0229\u015bniak, E.A., Kais, S., and Szcz\u0229\u015bniak, D. (2024). Entropy corrected geometric Brownian motion. Sci. Rep., 14.","DOI":"10.1038\/s41598-024-79714-3"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1029\/2018EA000464","article-title":"Global seismic nowcasting with Shannon information entropy","volume":"6","author":"Rundle","year":"2019","journal-title":"Earth Space Sci."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Guel-Cortez, A.J., and Kim, E.J. (2021). Information geometric theory in the prediction of abrupt changes in system dynamics. Entropy, 23.","DOI":"10.3390\/e23060694"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"eadn1721","DOI":"10.1126\/sciadv.adn1721","article-title":"Causes of extreme events revealed by R\u00e9nyi information transfer","volume":"10","author":"Manshour","year":"2024","journal-title":"Sci. Adv."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"121104","DOI":"10.1063\/5.0232645","article-title":"Complexity measure of extreme events","volume":"34","author":"Das","year":"2024","journal-title":"Chaos"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Cover, T.M., and Thomas, J.A. (2006). Elements of Information Theory, Wiley.","DOI":"10.1002\/047174882X"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"828","DOI":"10.1214\/aoms\/1177706212","article-title":"A generalization of the Glivenko-Cantelli theorem","volume":"30","author":"Tucker","year":"1959","journal-title":"Ann. Math. Stat."},{"key":"ref_28","first-page":"421","article-title":"Sulla determinazione empirica delle leggi di probabilita","volume":"4","author":"Cantelli","year":"1933","journal-title":"G. Dell\u2019Istituto Ital. Degli Attuari"},{"key":"ref_29","first-page":"92","article-title":"Sulla determinazione empirica delle leggi di probabilita","volume":"4","author":"Glivenko","year":"1933","journal-title":"G. Dell\u2019Istituto Ital. Degli Attuari"},{"key":"ref_30","first-page":"77","article-title":"Determination of the number of bins\/classes used in histograms and frequency tables: A short bibliography","volume":"7","year":"2010","journal-title":"Istat. Ara\u015ft\u0131rma Dergisi"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"136","DOI":"10.1080\/13504851.2017.1302057","article-title":"A first look at Brexit and global equity markets","volume":"25","author":"Burdekin","year":"2018","journal-title":"Appl. Econ. Lett."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"101690","DOI":"10.1016\/j.frl.2020.101690","article-title":"COVID-19 and the march 2020 stock market crash. Evidence from S&P1500","volume":"38","author":"Mazur","year":"2021","journal-title":"Financ. Res. Lett."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"035021","DOI":"10.1088\/2632-072X\/ac221f","article-title":"Extreme events in globally coupled chaotic maps","volume":"2","author":"Chowdhury","year":"2021","journal-title":"J. Phys. Complex."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/4\/410\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:12:22Z","timestamp":1760029942000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/4\/410"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,10]]},"references-count":33,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2025,4]]}},"alternative-id":["e27040410"],"URL":"https:\/\/doi.org\/10.3390\/e27040410","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2025,4,10]]}}}