Many real-world systems undergo abrupt changes in dynamics as they move across critical points, often with dramatic and irreversible consequences. Much of the existing theory on identifying the time-series signatures of nearby critical points -- such as increased signal variance and slower timescales -- is derived from analytically tractable systems, typically considering the case of fixed, low-amplitude noise. However, real-world systems are often corrupted by unknown levels of noise which can obscure these temporal signatures. Here we aimed to develop noise-robust indicators of the distance to criticality (DTC) for systems affected by dynamical noise in two cases: when the noise amplitude is either fixed, or is unknown and variable across recordings. We present a highly comparative approach to tackling this problem that compares the ability of over 7000 candidate time-series features to track the DTC in the vicinity of a supercritical Hopf bifurcation. Our method recapitulates existing theory in the fixed-noise case, highlighting conventional time-series features that accurately track the DTC. But in the variable-noise setting, where these conventional indicators perform poorly, we highlight new types of high-performing time-series features and show that their success is underpinned by an ability to capture the shape of the invariant density (which depends on both the DTC and the noise amplitude) relative to the spread of fast fluctuations (which depends on the noise amplitude). We introduce a new high-performing time-series statistic, termed the Rescaled Auto-Density (RAD), that distils these two algorithmic components. Our results demonstrate that large-scale algorithmic comparison can yield theoretical insights and motivate new algorithms for solving important practical problems.

翻译：许多真实世界的系统在穿越临界点时会经历动态的突变，常常导致剧烈且不可逆的后果。现有关于识别临近临界点时间序列特征（如信号方差增大和时间尺度变慢）的理论，大多源于可解析处理的系统，通常考虑固定低振幅噪声的情况。然而，真实世界系统常被未知强度的噪声污染，这类噪声可能掩盖这些时间特征。本研究旨在为受动态噪声影响的系统开发噪声鲁棒的临界距离指标，涵盖两种情形：噪声振幅固定，以及噪声振幅未知且随记录变化。我们提出一种高度比较性的方法来解决该问题，该方法比较了超过7000个候选时间序列特征在超临界Hopf分岔附近追踪临界距离的能力。在固定噪声情况下，我们的方法复现了现有理论，突出显示了能准确追踪临界距离的常规时间序列特征。但在可变噪声环境下，这些常规指标表现不佳，我们转而强调了新型高性能时间序列特征，并揭示其成功源于能够捕捉不变密度形状（取决于临界距离和噪声振幅）相对于快速涨落扩散范围（取决于噪声振幅）的能力。我们引入一个新型高性能时间序列统计量——重标自密度（Rescaled Auto-Density, RAD），它融合了这两个算法组件。我们的结果表明，大规模算法比较能够产生理论洞见，并推动解决重要实际问题的新算法开发。