Geometric Mode-Selection Scores for Delay-Coordinates Dynamic Mode Decomposition

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 frame this as a mode-selection scoring problem: each mode receives a score that ranks it as true or spurious; any hard selection (threshold or clustering) is a downstream choice. We show that mode selection in DC-DMD is fundamentally a problem of subspace geometry. True modes are characterized by concentration within a low-dimensional signal subspace, whereas spurious modes tend to retain non-negligible components outside any moderate overestimate of that subspace. This geometric distinction defines true and spurious modes and motivates fully data-driven robust scoring criteria. The framework yields two complementary scores. The first uses a data-driven proxy of the signal subspace to compute a residual. The second comes 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, with true modes distinguished by the degree of conformity to it. This deviation is governed by the geometric residual. Our analysis further explains the empirical behavior of magnitude- and norm-based heuristics and clarifies when and why they fail under delay coordinates. Numerical experiments, evaluated by precision-recall AUC of true-vs-spurious ranking, show that the proposed scores outperform the tested baselines across most of the small-spatial-dimension regime.