Alessio Borgi

Alessio Borgi

PhD in Graph Neural Networks and Generative AI 👨🏻‍💻🤖

TDA for Time Series: Topology of Dynamical Systems

3 minute read

Published:

TL;DR: Given a scalar time series x(t), the sliding window embedding SW_{d,τ}(x) = (x(t), x(t+τ), ..., x(t+(d-1)τ)) creates a point cloud in ℝᵈ. For a periodic signal, this cloud traces a loop (S¹); for a quasiperiodic signal, a torus (T²). Persistent H₁ detects these loops: a single long-lived H₁ bar = periodic; two long-lived bars = quasiperiodic. This converts the signal analysis problem into a topology problem.

Sliding Window Embeddings

For a time series \(x: \{0, 1, \ldots, T\} \to \mathbb{R}\) and parameters \(d\) (dimension) and \(\tau\) (lag):

$$\mathrm{SW}_{d,\tau}(x)(t) = (x(t), x(t+\tau), x(t+2\tau), \ldots, x(t+(d-1)\tau)) \in \mathbb{R}^d$$

The sliding window point cloud \(P = \{\mathrm{SW}_{d,\tau}(x)(t) : t = 0, \ldots, T-(d-1)\tau\}\).

Takens’ Theorem: For a generic smooth dynamical system on an \(m\)-dimensional attractor, the sliding window embedding with \(d \geq 2m+1\) gives a diffeomorphism between the attractor and \(P\). Thus, \(P\) has the same topology as the attractor.

Topology of Different Signal Types

Periodic signal \(x(t) = \sin(2\pi t / T_0)\):

  • Sliding window traces an ellipse ≅ \(S^1\).
  • \(H_0\): one component; \(H_1\): one generator (the loop).
  • Persistence diagram: single long-lived \(H_1\) bar.

Quasiperiodic signal \(x(t) = \sin(2\pi t/T_1) + \sin(2\pi t/T_2)\) with \(T_1/T_2 \notin \mathbb{Q}\):

  • Attractor is a 2-torus \(T^2 = S^1 \times S^1\).
  • \(H_1\) has rank 2 (two generators).
  • Persistence diagram: two long-lived \(H_1\) bars.

Chaotic signal (Lorenz attractor):

  • Complex attractor with interesting \(H_1\) (loops around the “wings”).
  • Persistence diagram: a few moderate-lived \(H_1\) bars.

White noise: random point cloud, no persistent features; all \(H_1\) bars short-lived.

The Periodicity Score

Perea & Harer (2015) define the periodicity score using the maximum persistence of \(H_1\):

\[\mathrm{PS}(x, d, \tau) = \max_{(b,d) \in H_1 \text{ diagram}} (d - b) / \mathrm{diam}(P)\]

A high score (close to 1) indicates clear periodicity; a low score indicates noise or aperiodic behaviour.

The optimal parameters \(d, \tau\) can be estimated from the autocorrelation of \(x\).

Applications

EEG analysis: Persistent \(H_1\) of sliding window embeddings detects epileptic seizure onset — the irregular burst activity has a different topological signature than normal brain rhythms.

ECG analysis: Heartbeat irregularities (atrial fibrillation) disrupt the regular \(S^1\) loop of normal heartbeat embeddings, producing topological changes detectable by persistence.

Climate data: Ocean temperature oscillations (ENSO cycles) have periodic/quasiperiodic components identifiable via topological methods that are robust to missing data and measurement noise.

Financial time series: Market cycles and regime changes show up as changes in the topological complexity of the sliding window embedding.

Key Insight: The key advantage of TDA for time series over spectral methods (Fourier, wavelets) is robustness: a signal with a strong frequency component but phase noise, missing samples, or amplitude modulation still produces a clear loop in the sliding window embedding. Fourier analysis would show broad spectral peaks; TDA shows a sharp persistence bar. This makes TDA particularly powerful for biological and physical signals where stationarity and regularity are only approximate.

References

  • J. Perea, J. Harer, “Sliding Windows and Persistence: An Application of Topological Methods to Signal Analysis,” Foundations of Computational Mathematics, 2015. arXiv:1307.6188.
  • F. Takens, “Detecting Strange Attractors in Turbulence,” Lecture Notes in Mathematics, Springer, 1981.
  • J. Perea, “Topological Time Series Analysis,” Notices of the AMS, 2019.

You May Also Enjoy

GAPE: Remember to Forget — Gated Adaptive Positional Encoding

4 minute read

Published:

GAPE is a drop-in RoPE augmentation that adds content-aware attention logit biases: a query-gate suppresses irrelevant distant context while a key-gate preserves salient distant tokens. Provably sharper attention and improved long-context robustness — no architecture changes needed.

PolyNSD: Polynomial Neural Sheaf Diffusion

4 minute read

Published:

PolyNSD replaces the NSD propagation operator with a degree-K Chebyshev polynomial in the normalised sheaf Laplacian, achieving SOTA on homo- and heterophilic benchmarks with only diagonal restriction maps and dramatically lower memory usage.

TDA in Materials Science: Topology of Structure and Phase

3 minute read

Published:

Persistent homology characterises the multi-scale structure of materials — from pore geometry in catalysts to glass transition in amorphous materials. H₀ captures connectivity, H₁ captures channels and rings, H₂ captures enclosed voids. These topological descriptors predict mechanical properties, adsorption capacity, and phase transitions more accurately than geometric averages.

TDA in Drug Discovery: Molecular Topology

3 minute read

Published:

Persistent homology provides multi-scale topological fingerprints of molecular structures — capturing rings, cavities, and protein pockets that traditional cheminformatics descriptors miss. Applications include binding site detection, ADMET property prediction, and protein-ligand interaction modelling.