Detecting Regime Transitions in Dynamical Systems via the Mixup Euler Characteristic Profile

TL;DR

This paper introduces the Mixup Euler Characteristic Profile (Mixup ECP), a topological method for detecting regime shifts in dynamical systems, validated on climate and synthetic data with improved accuracy.

Contribution

The paper presents a novel topological detection statistic, Mixup ECP, with a formal null model, stability guarantees, and a multi-delay extension for identifying regime transitions.

Findings

Achieves 9.50 days MAE on Indian monsoon onset prediction

Outperforms baseline methods by 32% and 9% in MAE

Validated on multiple dynamical systems and climate datasets

Abstract

We develop a framework for detecting regime transitions in dynamical systems using the Mixup Euler Characteristic Profile (Mixup ECP) -- the Euler characteristic of the geometric intersection of ball unions around adjacent delay-embedded trajectory segments, viewed as a function of filtration scale. The Mixup ECP provides a detection statistic with a built-in null and guaranteed stability. We formalize regime detection as a low-side-permutation test, establish its validity and consistency, and introduce a multi-delay extension that automatically selects the most informative dynamical timescale. Complementing the topological signal with Complexity Variance, Higuchi fractal dimension, and a rolling mean baseline, the four-signal combined method achieves $9.50$ days MAE on Indian monsoon onset (Nepal target) -- a $32\%$ improvement over the rolling mean baseline and $9\%$ over CUSUM. Validated on the Lorenz system, logistic map, and three monsoon systems spanning both hemispheres (Indian/Nepal, Indian/Kerala, Western North Pacific), plus ENSO and a synthetic EEG dataset, the framework adds value precisely when the transition is gradual or obscured by noise.