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

Atish Mitra; Buddha Nath Sharma; Md Nurujjaman; Santanu Nandi; Sushovan Majhi

arxiv: 2604.15262 · v1 · submitted 2026-04-16 · 🧮 math.DS · math.AT

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

Sushovan Majhi , Atish Mitra , Santanu Nandi , Md Nurujjaman , Buddha Nath Sharma This is my paper

Pith reviewed 2026-05-10 09:32 UTC · model grok-4.3

classification 🧮 math.DS math.AT

keywords regime transitionsdynamical systemsEuler characteristicdelay embeddingmonsoon onsetpermutation testtopological data analysistime series

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The pith

The Mixup Euler Characteristic Profile detects regime transitions in dynamical systems by providing a stable topological statistic from delay-embedded trajectories.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops the Mixup Euler Characteristic Profile as a way to spot shifts between different dynamical regimes, including gradual ones hidden by noise. It constructs the profile from the Euler characteristic of ball-union intersections on adjacent segments of a delay-embedded trajectory, then treats changes across filtration scales as a detection signal. A low-side permutation test supplies the statistic with a built-in null model and stability properties. When the profile is paired with complexity variance, Higuchi fractal dimension, and a rolling-mean baseline, the combined detector reaches 9.50 days mean absolute error on Indian monsoon onset dates for Nepal, improving 32 percent over the rolling-mean baseline and 9 percent over CUSUM. The same approach is tested on the Lorenz system, the logistic map, ENSO, Western North Pacific monsoon, and synthetic EEG data.

Core claim

The Mixup ECP is defined as the Euler characteristic of the geometric intersection of ball unions around adjacent delay-embedded trajectory segments, viewed as a function of filtration scale. This profile supplies a detection statistic with a built-in null and guaranteed stability. Regime detection is formalized as a low-side permutation test whose validity and consistency are established, and a multi-delay extension automatically selects the most informative dynamical timescale. Integrated with three other signals, the method records 9.50 days MAE on Nepal monsoon onset, a 32 percent gain over the rolling-mean baseline.

What carries the argument

The Mixup Euler Characteristic Profile (Mixup ECP): the Euler characteristic of the geometric intersection of ball unions around adjacent delay-embedded trajectory segments, tracked as a function of filtration scale.

If this is right

The low-side permutation test on the Mixup ECP is valid and consistent for regime detection.
A multi-delay version automatically selects the most informative dynamical timescale.
The four-signal combination reaches 9.50 days MAE on Indian monsoon onset, improving 32 percent over rolling mean and 9 percent over CUSUM.
The framework performs on the Lorenz attractor, logistic map, ENSO, three monsoon systems, and synthetic EEG data, adding value most when transitions are gradual or noise-obscured.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The stability properties could support real-time early-warning systems for climate or physiological regime shifts.
The method may generalize to other noisy time series with slow transitions, such as financial markets or ecological systems.
Hybrid detectors that replace the rolling-mean baseline with additional topological features could be tested on the same validation sets.

Load-bearing premise

That delay embedding of trajectory segments followed by ball-union intersections produces an Euler characteristic profile whose changes reliably mark regime transitions without embedding parameters or filtration choices creating artifacts.

What would settle it

A controlled test on the Lorenz system with known gradual transitions and added noise in which the Mixup ECP profile shows no detectable shift at the transition time or produces false detections when only the embedding dimension or filtration scale is varied.

Figures

Figures reproduced from arXiv: 2604.15262 by Atish Mitra, Buddha Nath Sharma, Md Nurujjaman, Santanu Nandi, Sushovan Majhi.

**Figure 2.** Figure 2: Why signal-level methods miss topological transitions. (a) Two segments of the logistic [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Figure 3: Detection pipeline. A scalar time series is delay-embedded via Takens coordinates, then a [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

**Figure 4.** Figure 4: The Alpha complex and the Euler characteristic curve. (a) The Delaunay triangulation [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗

**Figure 5.** Figure 5: The Mixup ECP as a regime detection statistic. (a) Both windows sample the same attractor: [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗

**Figure 6.** Figure 6: Mixup ECP detection of the Lorenz system bifurcation at [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗

**Figure 7.** Figure 7: Logistic map: (a) bifurcation diagram, (b) Mixup ECP detection statistic [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗

**Figure 8.** Figure 8: Noise robustness on the logistic map (λ : 3.2 → 3.8). (a) Mixup ECP detection statistic S vs. noise level—the topological signal increases with moderate noise and saturates for σ ≳ 0.15. (b) Complexity Variance |∆Var| vs. noise—the geometric signal is stable at moderate noise but increases at σ = 0.50. Error bars: ±1 standard deviation over 20 trials. 5 Applications to Regime Transitions 5.1 Indian Monsoo… view at source ↗

**Figure 9.** Figure 9: ENSO regime detection. (a) Oceanic Nino Index with El Ni ˜ no (red shading) and La Ni ˜ na˜ (blue shading) episodes. (b) Mixup ECP detection statistic S(t). (c) Complexity Variance change ∆Var. Vertical dashed lines mark major transitions. 5.3 EEG Seizure-Like Transitions Seizure onset is a prototypical neural regime transition: the brain abruptly transitions from normal background activity to hypersynchro… view at source ↗

**Figure 10.** Figure 10: EEG seizure detection (synthetic data). (a) EEG signal with seizure epoch (pink shading, [PITH_FULL_IMAGE:figures/full_fig_p021_10.png] view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The Mixup ECP gives a new topological statistic for regime detection that improves monsoon onset forecasts, but the parameter choices and missing robustness checks leave the central claims shaky.

read the letter

The main takeaway is that this paper builds a detection statistic from the Euler characteristic of intersections between ball unions on adjacent delay-embedded segments, then wraps it in a permutation test and an automatic multi-delay selector. They combine it with three other signals and report a 9.5-day MAE on Indian monsoon onset for Nepal, which beats the rolling mean baseline by 32 percent and CUSUM by 9 percent. That empirical number is the clearest concrete result, and the method appears to help most when transitions are gradual or noisy, which matches real climate data problems. They also run it on Lorenz, logistic map, ENSO, and an EEG set, so the scope is broader than one application. The built-in null from the permutation test is a clean feature that lowers the usual circularity risk in change-point work. What the paper does well is turn a topological construction into something usable for forecasting with measurable gains on a high-stakes dataset. The multi-delay extension that picks the timescale automatically is a practical addition that prior topological methods often leave to the user. The soft spots sit mainly in the validation layer. The abstract states formal validity, consistency, and guaranteed stability, yet the text supplies no derivations, no sensitivity sweeps on embedding dimension m, lag tau, or filtration scale, and no cross-validation protocol for the automatic selector. If those choices are tuned even modestly to the monsoon or ENSO examples, the reported improvement could be partly artifact rather than signal from the Euler characteristic itself. The stress-test concern about parameter-induced profile changes holds up here because nothing in the provided material rules it out. Data preprocessing steps for the time series are also not detailed enough to replicate the exact MAE. This work is aimed at people who already use delay embedding or topological tools on dynamical systems, especially those working on climate or physiological time series. A reader who wants a new statistic to try on gradual transitions will get a clear construction and some numbers to benchmark against, even if they have to add their own robustness tests. It deserves a serious referee because the idea is original, the application is practical, and the permutation-test framing is sound in principle. I would send it to review but ask the authors for explicit bounds or sweeps on the embedding parameters, full experimental protocols, and checks that the automatic selection does not overfit the reported cases.

Referee Report

3 major / 2 minor

Summary. The manuscript introduces the Mixup Euler Characteristic Profile (Mixup ECP) as the Euler characteristic of the geometric intersection of ball unions around adjacent delay-embedded trajectory segments, viewed as a function of filtration scale. It positions this as a detection statistic with a built-in null for regime transitions, formalized via a low-side permutation test whose validity and consistency are asserted, along with a multi-delay extension that automatically selects the most informative timescale. The topological signal is combined with Complexity Variance, Higuchi fractal dimension, and a rolling mean baseline, yielding a four-signal method that reports 9.50-day MAE on Indian monsoon onset (Nepal target), with claimed 32% improvement over rolling mean and 9% over CUSUM. Validation is presented on the Lorenz system, logistic map, multiple monsoon systems, ENSO, and synthetic EEG data, with emphasis on utility for gradual or noisy transitions.

Significance. If the stability guarantees, permutation-test consistency, and robustness to embedding parameters hold, the framework would supply a new topological tool for regime detection that complements existing statistical methods precisely where transitions are gradual or noise-obscured. The built-in null and automatic multi-delay selection are potentially valuable strengths for dynamical-systems applications in climate and neuroscience, provided the empirical gains on monsoon onset generalize beyond the chosen datasets and preprocessing.

major comments (3)

[Abstract] Abstract: the assertion of 'formal validity and consistency' for the low-side permutation test together with 'guaranteed stability' of the Mixup ECP is load-bearing for the central mathematical claim, yet the abstract supplies no derivation, error bounds, or proof sketch; without these details the soundness of the detection statistic cannot be assessed.
[Multi-delay extension] Multi-delay extension and experimental sections: the automatic selection of delays, weights, and filtration scales is data-driven and applied to the specific monsoon/ENSO examples; absent explicit cross-validation protocols, sensitivity sweeps over (m, τ), or ablation on held-out data, the reported 9.50-day MAE improvement risks being an artifact of favorable parameter choices rather than a property of the topological construction.
[Experimental validation] Validation on monsoon onset and gradual transitions: the core assumption that delay-embedded ball-union intersections produce profile changes that reliably signal true regime shifts (especially gradual ones) is undermined by the lack of demonstrated invariance to embedding dimension, lag, and scale range; the skeptic's concern that parameter artifacts could drive the statistic is therefore unresolved and directly affects the empirical claim.

minor comments (2)

[Abstract] Abstract: the MAE figure is reported to two decimal places; clarify whether this is mean absolute error, the exact number of test years, and the precise definition of the rolling-mean and CUSUM baselines for reproducibility.
[Methods] Notation: the term 'Mixup ECP' is introduced without an explicit equation linking the Euler characteristic of the intersection to the filtration parameter; a compact definition early in the methods would improve clarity.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thoughtful and constructive report. We address each major comment in turn, clarifying the theoretical foundations presented in the full manuscript and strengthening the experimental validation sections. Revisions have been made to incorporate additional details, sensitivity analyses, and invariance checks as detailed below.

read point-by-point responses

Referee: [Abstract] Abstract: the assertion of 'formal validity and consistency' for the low-side permutation test together with 'guaranteed stability' of the Mixup ECP is load-bearing for the central mathematical claim, yet the abstract supplies no derivation, error bounds, or proof sketch; without these details the soundness of the detection statistic cannot be assessed.

Authors: The abstract is intentionally concise, but the full manuscript (Section 3) provides the formal definitions, stability proof via continuity of the Euler characteristic under Hausdorff perturbations of the filtered complexes, and the permutation test consistency argument based on exchangeability under the null of stationary dynamics. We have added a one-sentence proof sketch to the revised abstract referencing these results and the supporting lemmas. revision: yes
Referee: [Multi-delay extension] Multi-delay extension and experimental sections: the automatic selection of delays, weights, and filtration scales is data-driven and applied to the specific monsoon/ENSO examples; absent explicit cross-validation protocols, sensitivity sweeps over (m, τ), or ablation on held-out data, the reported 9.50-day MAE improvement risks being an artifact of favorable parameter choices rather than a property of the topological construction.

Authors: We agree that explicit validation of the data-driven selection is essential. The revised manuscript now includes a dedicated sensitivity sweep over embedding dimensions m = 2..6 and lags τ = 1..10, together with an ablation study that removes the automatic selection and reports performance on held-out segments of each monsoon and ENSO series. Cross-validation folds are described in the new supplementary section; the 9.50-day MAE gain remains stable across these checks, indicating it is not an artifact of a single parameter choice. revision: partial
Referee: [Experimental validation] Validation on monsoon onset and gradual transitions: the core assumption that delay-embedded ball-union intersections produce profile changes that reliably signal true regime shifts (especially gradual ones) is undermined by the lack of demonstrated invariance to embedding dimension, lag, and scale range; the skeptic's concern that parameter artifacts could drive the statistic is therefore unresolved and directly affects the empirical claim.

Authors: The manuscript already contains controlled experiments on the Lorenz system and logistic map that vary embedding parameters while keeping the known transition fixed; detection remains consistent. For the real-world datasets we have added supplementary figures showing that the Mixup ECP profiles and detected onset dates are qualitatively unchanged across m ∈ {3,4,5}, τ ∈ {3,5,7}, and filtration ranges spanning two orders of magnitude. Additional synthetic gradual-transition experiments (linear ramp between regimes) confirm that the statistic responds to the underlying change rather than to specific embedding choices. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper defines the Mixup ECP directly from delay-embedded segments and ball-union intersections as a new detection statistic, then formalizes a permutation test and multi-delay selection procedure whose validity is established via stated assumptions on the dynamical systems. Empirical MAE results (9.50 days on monsoon data) are reported as performance measurements against external baselines (rolling mean, CUSUM) on both synthetic (Lorenz, logistic map) and real datasets (monsoon variants, ENSO, EEG), not as quantities derived by construction from the same inputs. No self-definitional loops, fitted parameters renamed as predictions, or load-bearing self-citations appear in the provided claims; the framework's central statistic and test are independent of the target regime labels.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 1 invented entities

Ledger inferred from abstract description only; full paper would list explicit parameter choices and embedding assumptions.

free parameters (3)

delay embedding parameters
Delay and embedding dimension required to form trajectory segments but not numerically specified in abstract.
filtration scale range
Profile defined as function of filtration scale; range and sampling must be chosen.
multi-delay selection hyperparameters
Automatic selection process still requires underlying criteria or thresholds.

axioms (2)

domain assumption Takens delay embedding theorem preserves dynamical information sufficient for regime detection
Method relies on delay-embedded trajectory segments from time series.
standard math Euler characteristic is well-defined and stable for the geometric intersections of ball unions
Central computation of the Mixup ECP.

invented entities (1)

Mixup Euler Characteristic Profile no independent evidence
purpose: Stable detection statistic with built-in null for regime transitions
Newly defined construction combining delay embedding and topological Euler characteristic.

pith-pipeline@v0.9.0 · 5508 in / 1649 out tokens · 67877 ms · 2026-05-10T09:32:38.321065+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages · 2 internal anchors

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Enrique Alvarado, Daniela Beckelhymer, Joshua Dorrington, Tung Lam, Sushovan Majhi, Jasmine Noory , Mar´ıa S´anchez Muniz, and Kristian Strommen. Detecting the indian monsoon using topolog- ical data analysis.arXiv preprint arXiv:2504.01022,

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Topological Characterization of Churn Flow and Unsupervised Correction to the Wu Flow-Regime Map in Small-Diameter Vertical Pipes

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Salam Rabindrajit Luwang, Sushovan Majhi, Vishal Mandal, Atish J Mitra, Md Nurujjaman, and Bud- dha Nath Sharma. Interpretable classification of time series using euler characteristic surfaces.arXiv preprint arXiv:2603.15079,

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Mixup barcodes: quan- tifying geometric-topological interactions between point clouds.arXiv preprint arXiv:2402.15058,

Hubert Wagner, Nickolas Arustamyan, Matthew Wheeler, and Peter Bubenik. Mixup barcodes: quan- tifying geometric-topological interactions between point clouds.arXiv preprint arXiv:2402.15058,

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