{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,10]],"date-time":"2026-02-10T13:32:53Z","timestamp":1770730373510,"version":"3.49.0"},"reference-count":25,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2025,9,24]],"date-time":"2025-09-24T00:00:00Z","timestamp":1758672000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computers"],"abstract":"<jats:p>We propose a strictly causal early\u2013warning framework for financial crises based on topological signal extraction from multivariate return streams. Sliding windows of daily log\u2013returns are mapped to point clouds, from which Vietoris\u2013Rips persistence diagrams are computed and summarised by persistence landscapes. A single, interpretable indicator is obtained as the L2 norm of the landscape and passed through a causal decision rule (with thresholds \u03b1,\u03b2 and run\u2013length parameters s,t) that suppresses isolated spikes and collapses bursts to time\u2013stamped warnings. On four major U.S. equity indices (S&amp;P 500, NASDAQ, DJIA, Russell 2000) over 1999\u20132021, the method, at a fixed strictly causal operating point (\u03b1=\u03b2=3.1,s=57,t=16), attains a balanced precision\u2013recall (F1\u22480.50) with an average lead time of about 34 days. It anticipates two of the four canonical crises and issues a contemporaneous signal for the 2008 global financial crisis. Sensitivity analyses confirm the qualitative robustness of the detector, while comparisons with permissive spike rules and volatility\u2013based baselines demonstrate substantially fewer false alarms at comparable recall. The approach delivers interpretable topology\u2013based warnings and provides a reproducible route to combining persistent homology with causal event detection in financial time series.<\/jats:p>","DOI":"10.3390\/computers14100408","type":"journal-article","created":{"date-parts":[[2025,9,24]],"date-time":"2025-09-24T09:43:12Z","timestamp":1758706992000},"page":"408","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Topological Machine Learning for Financial Crisis Detection: Early Warning Signals from Persistent Homology"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0009-0005-3953-1479","authenticated-orcid":false,"given":"Ecaterina","family":"Guritanu","sequence":"first","affiliation":[{"name":"Dipartimento di Matematica e Fisica, Universit\u00e0 Cattolica del Sacro Cuore, Via Garzetta 48, 25121 Brescia, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1466-0248","authenticated-orcid":false,"given":"Enrico","family":"Barbierato","sequence":"additional","affiliation":[{"name":"Dipartimento di Matematica e Fisica, Universit\u00e0 Cattolica del Sacro Cuore, Via Garzetta 48, 25121 Brescia, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0009-0008-8422-8024","authenticated-orcid":false,"given":"Alice","family":"Gatti","sequence":"additional","affiliation":[{"name":"Dipartimento di Matematica e Fisica, Universit\u00e0 Cattolica del Sacro Cuore, Via Garzetta 48, 25121 Brescia, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2025,9,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Claessens, S., and Kose, M.A. (2013). Financial Crises: Explanations, Types, and Implications, International Monetary Fund. IMF Working Paper WP\/13\/28.","DOI":"10.2139\/ssrn.2295201"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"105","DOI":"10.2307\/1907241","article-title":"The Summation of Random Causes as the Source of Cyclic Processes","volume":"5","author":"Slutzky","year":"1937","journal-title":"Econometrica"},{"key":"ref_3","first-page":"1","article-title":"The Role of Monetary Policy","volume":"58","author":"Friedman","year":"1968","journal-title":"Am. Econ. Rev."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1345","DOI":"10.2307\/1913386","article-title":"Time to Build and Aggregate Fluctuations","volume":"50","author":"Kydland","year":"1982","journal-title":"Econometrica"},{"key":"ref_5","unstructured":"Minsky, H.P. (1992). The Financial Instability Hypothesis, The Jerome Levy Economics Institute of Bard College."},{"key":"ref_6","unstructured":"Shiller, R.J. (2000). Irrational Exuberance, Princeton University Press."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2371","DOI":"10.1111\/0022-1082.00408","article-title":"A Rose.com by Any Other Name","volume":"56","author":"Cooper","year":"2001","journal-title":"J. Financ."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1257\/jep.23.1.77","article-title":"Deciphering the Liquidity and Credit Crunch 2007\u20132008","volume":"23","author":"Brunnermeier","year":"2009","journal-title":"J. Econ. Perspect."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1257\/jep.26.3.49","article-title":"The European Sovereign Debt Crisis","volume":"26","author":"Lane","year":"2012","journal-title":"J. Econ. Perspect."},{"key":"ref_10","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_11","unstructured":"G20 Leaders (2008). Declaration of the Summit on Financial Markets and the World Economy. Financ. Stab. Board, Available online: https:\/\/www.fsb.org\/uploads\/g20_leaders_declaration_washington_2008.pdf."},{"key":"ref_12","first-page":"1","article-title":"Dragon-Kings, Black Swans and the Prediction of Crises","volume":"2","author":"Sornette","year":"2009","journal-title":"Int. J. Terraspace Sci. Eng."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"100693","DOI":"10.1016\/j.jfs.2019.100693","article-title":"Does machine learning help us predict banking crises?","volume":"45","author":"Beutel","year":"2019","journal-title":"J. Financ. Stab."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Zhang, F., and Wu, Y. (2025, January 3\u20136). Topological Time Series Analysis of Market Crashes: A Persistence Homology Approach. Proceedings of the 2025 ACM International Conference, Niter\u00f3i, Brazil.","DOI":"10.1145\/3745533.3745634"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Ismail, M., Md Noorani, M., and Ismail, M. (2022). Early warning signals of financial crises using persistent homology and critical slowing down: Evidence from different correlation tests. Front. Appl. Math. Stat., 8.","DOI":"10.3389\/fams.2022.940133"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Yao, J., Li, J., Wu, J., Yang, M., and Wang, X. (2025, September 20). Change Points Detection in Financial Market Using Topological Data Analysis. Available online: https:\/\/papers.ssrn.com\/abstract=5196633.","DOI":"10.2139\/ssrn.5196633"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"103106","DOI":"10.1063\/5.0220424","article-title":"Identifying extreme events in the stock market: A topological data analysis","volume":"34","author":"Rai","year":"2024","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Edelsbrunner, H., and Harer, J. (2010). Computational Topology: An Introduction, American Mathematical Society.","DOI":"10.1090\/mbk\/069"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1142\/S021819590600204X","article-title":"An Algebraic Topological Method for Feature Identification","volume":"16","author":"Carlsson","year":"2006","journal-title":"Int. J. Comput. Geom. Appl."},{"key":"ref_20","first-page":"77","article-title":"Statistical Topological Data Analysis using Persistence Landscapes","volume":"16","author":"Bubenik","year":"2015","journal-title":"J. Mach. Learn. Res."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"124956","DOI":"10.1016\/j.physa.2020.124956","article-title":"Empirical study of financial crises based on topological data analysis","volume":"558","author":"Guo","year":"2020","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"648","DOI":"10.4236\/jmf.2020.104038","article-title":"Topology Data Analysis Using Mean Persistence Landscapes in Financial Crashes","volume":"10","author":"Aguilar","year":"2020","journal-title":"J. Math. Financ."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"126459","DOI":"10.1016\/j.physa.2021.126459","article-title":"Early warning signals of financial crises using persistent homology","volume":"586","author":"Ismail","year":"2022","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_24","unstructured":"Ruiz-Ortiz, M.A., G\u00f3mez-Larra\u00f1aga, J.C., and Rodr\u00edguez-Viorato, J. (2022). A persistent-homology-based turbulence index & some applications of TDA on financial markets. arXiv."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"100107","DOI":"10.1016\/j.jfds.2023.100107","article-title":"Topological tail dependence: Evidence from forecasting realized volatility","volume":"9","author":"Souto","year":"2023","journal-title":"J. Financ. Data Sci."}],"container-title":["Computers"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-431X\/14\/10\/408\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:48:42Z","timestamp":1760035722000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-431X\/14\/10\/408"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,9,24]]},"references-count":25,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2025,10]]}},"alternative-id":["computers14100408"],"URL":"https:\/\/doi.org\/10.3390\/computers14100408","relation":{},"ISSN":["2073-431X"],"issn-type":[{"value":"2073-431X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,9,24]]}}}