Geometric Framework for Robust Order Detection in Delay-Coordinates Dynamic Mode Decomposition

TL;DR

This paper introduces a geometric framework for robustly detecting the true order and meaningful modes in delay-coordinates dynamic mode decomposition, addressing noise and overestimation issues with data-driven criteria.

Contribution

It develops a novel geometric and operator-theoretic approach for distinguishing true modes from spurious ones in DC-DMD, with proven robustness and accuracy.

Findings

The proposed methods outperform existing baselines across various conditions.

The geometric residual effectively identifies true modes.

The framework explains the limitations of traditional heuristics.

Abstract

Delay-coordinates dynamic mode decomposition (DC-DMD) is widely used to extract coherent spatiotemporal modes from high-dimensional time series. A central challenge is distinguishing dynamically meaningful modes from spurious modes induced by noise and order overestimation. We show that model order detection and mode selection in DC-DMD are fundamentally problems of subspace geometry. Specifically, true modes are characterized by concentration within a low-dimensional signal subspace, whereas spurious modes necessarily retain non-negligible components outside any moderate overestimate of that subspace. This geometric distinction yields a perturbation-robust definition of true and spurious modes and yields fully data-driven selection criteria. This geometric framework leads to two complementary data-driven selection criteria. The first is derived directly from the geometric distinction and uses a data-driven proxy of the signal-subspace to compute a residual score. The second arises from a new operator-theoretic analysis of delay embedding. Using a block-companion formulation, we show that all modes exhibit a Kronecker-Vandermonde (KV) structure induced by the delay-coordinates, and true modes are distinguished by the degree to which they conform to it. Importantly, we also show that this deviation is governed precisely by the geometric residual. In addition, our analysis provides a principled explanation for the empirical behavior of magnitude- and norm-based heuristics, clarifying when and why they fail under delay-coordinates. Extensive numerical experiments confirm the theoretical predictions and demonstrate that the proposed geometric and structure-based methods achieve robust and accurate order detection and mode selection, consistently better than existing baselines across noise levels, spectral separations, damping regimes, and embedding lengths.