Topological Machine Learning for Financial Crisis Detection: Early Warning Signals from Persistent Homology

We propose a strictly causal early–warning framework for financial crises based on topological signal extraction from multivariate return streams. Sliding windows of daily log–returns are mapped to point clouds, from which Vietoris–Rips 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 α,β and run–length parameters s,t) that suppresses isolated spikes and collapses bursts to time–stamped warnings. On four major U.S. equity indices (S&P 500, NASDAQ, DJIA, Russell 2000) over 1999–2021, the method, at a fixed strictly causal operating point (α=β=3.1,s=57,t=16), attains a balanced precision–recall (F1≈0.50) 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–based baselines demonstrate substantially fewer false alarms at comparable recall. The approach delivers interpretable topology–based warnings and provides a reproducible route to combining persistent homology with causal event detection in financial time series.