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 $P(λ)$, that provides probabilistic information on the behaviour of finite-data eigenvalues in the complex plane, and an estimator $\hat P(λ)$, 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.

翻译：许多复杂系统可通过谱分解表征其动力学的矩阵，从而简化为核心组件。这类矩阵通常基于数据构建，常见方法包括最小二乘拟合：典型算法有动力学模态分解及其变体、子空间辨识和特征值实现算法。这些算法的输出通常包含一系列孤立的外围特征值，用于表征系统中持续的涌现模式。然而，目前尚无客观方法评估这些离散特征值中哪些是有限数据误差的伪影，哪些反映了完全采样算子。本文提出一种采样伪谱 $P(λ)$，可提供复平面上有限数据特征值行为的概率信息，并构建了可通过再处理有限数据样本获得的估计量 $\hat P(λ)$。该估计量计算高效，使我们能够对真实特征值的位置进行统计检验。这为评估从有限数据中提取的模式究竟是信号还是噪声，提供了一套严谨且普适的判定方法。