Geometric mode-selection scores for delay-coordinates dynamic mode decomposition

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 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 area under the curve (PR-AUC) of true-vs-spurious ranking, show that the proposed scores outperform the tested baselines across most of the small-spatial-dimension regime.