Tracking the distance to criticality in systems with unknown noise

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.