Standard early warning signals for regime shifts — increasing variance, critical slowing down, autocorrelation growth — work by detecting changes in the time series. As a system approaches a tipping point, its statistical properties change in predictable ways. Measure the variance over a window. Watch it grow. Sound the alarm.
This fails in noisy environments. Arctic under-ice phytoplankton blooms transition between two states: low-biomass and high-biomass. The transitions are real and consequential. But the environmental variability is so high that the statistical indicators drown in noise. Variance is already large. Autocorrelation fluctuates wildly. The time series contains the transition but not a clean signal preceding it.
Shi et al. built a geometric indicator instead. In phase space, the two stable states are basins separated by a stochastic separatrix — the isocommittor surface where the system is equally likely to fall into either basin. The width of the transition layer around this separatrix and the distance of the current state from the separatrix provide warnings that don't depend on statistical properties of the time series at all.
The geometric indicators scale linearly with noise strength. This is the opposite of the statistical problem — noise makes statistics harder but makes the geometry more visible, because the stochastic separatrix is defined by the noise structure. Higher noise broadens the transition layer in a predictable, measurable way. The indicator works better in the regime where statistics work worst.
The through-claim: time series and phase-space geometry contain the same information about the system, but extract it differently. Statistical indicators compress the trajectory into moments (variance, autocorrelation) that become unreliable under noise. Geometric indicators use the trajectory's position relative to phase-space structures that noise actually helps delineate. The same data, read as statistics, fails. Read as geometry, it succeeds. The problem was never the data. It was the representation.