Financial Engineering Seminar Series: Topological Data Analysis for Detecting Critical Transitions in Financial Time Series

Abstract

Critical transitions in financial markets are sudden, non-linear shifts from one stable state to another, analogous to phase transitions in thermodynamics and statistical mechanics. Such transitions can be triggered by various factors, including financial bubbles and crashes, policy-induced regime shifts, or instability driven by investor herding. Based on the theory of phase transitions, the Log-Periodic Power Law Singularity (LPPLS) model describes financial bubbles and crashes. It frames crashes as predictable, finite-time singularities that emerge from positive feedback mechanisms.

Our research introduces a novel framework for identifying these critical transitions using Topological Data Analysis (TDA). This approach analyzes the underlying 'shape' of complex, multi-dimensional, and noisy financial datasets, providing structural information that complements traditional statistical methods. We demonstrate the effectiveness of our TDA method by applying it to the LPPLS model as well as to the real financial market

Biography

Marian Gidea is a Professor of Mathematics and Director of the Graduate Program at Yeshiva University in New York City. He held previous appointments at the Mathematical Sciences Research Institute in Berkeley, the Institute for Advanced Study in Princeton, Centre de Recerca Matematica in Barcelona, Northeastern Illinois University, Northwestern University, and Loyola University in Chicago. He also served at the National Science Foundation as a program director in the Mathematical Sciences Division. His research interests include Dynamical Systems and its applications (such as to Celestial Mechanics, Mathematical Physics, Mathematical Biology, and Financial Mathematics) and Topological Data Analysis.