Sampling pseudospectrum for data-driven matrices

TL;DR

This paper introduces a probabilistic sampling pseudospectrum and an estimator to assess the reliability of eigenvalues derived from finite data in complex systems, distinguishing true signals from noise.

Contribution

It presents a novel sampling pseudospectrum and an efficient estimator for statistically evaluating eigenvalues from data-driven matrices.

Findings

Provides a probabilistic method to identify genuine eigenvalues.

Offers a computationally efficient estimator for finite-data eigenvalue analysis.

Enables rigorous assessment of spectral features as signal or noise.

Abstract

Many complex systems can be reduced to their key components through spectrally decomposing matrices that capture their dynamics. These matrices can in turn be constructed from data, often by least-squares fitting: examples of algorithms to do this include Dynamical Mode Decomposition and variants, subspace identification and eigenvalue realisation algorithms. Typical outputs of these algorithms include a range of isolated, peripheral eigenvalues capturing persistent emergent patterns in the system. However, there is no objective way to assess which of these discrete eigenvalues are artefacts of finite data error, and which are reflections of a fully sampled operator. n this paper, we present a sampling pseudospectrum $P(\lambda)$, that provides probabilistic information on the behaviour of finite-data eigenvalues in the complex plane, and an estimator $\hat P(\lambda)$, which can be obtained by reprocessing our finite data sample. The estimator, which is computationally efficient to implement, allows us to test statistically for the location of the true eigenvalues. This gives us a rigorous and very general way to assess whether the patterns we extract from finite data are likely to be signal or noise.