This paper introduces a simple way to estimate how resilient a changing system is when the system itself has trends or strong seasonal cycles. The resilience is measured by a recovery rate called λ (lambda). The authors show how to recover λ from data that are non-stationary, irregularly sampled, or have gaps — situations that commonly break older methods.

The key idea is to write the system’s short-term change as a linear restoring force plus noise, using a version of the Langevin equation. The authors discretize that equation so that the discrete change ÷ time between observations becomes a linear regression problem. In that regression the slope is directly related to −λ, while terms that describe the moving attractor (for example seasonal cycles) are included as additional regressors. When seasonality is important they add a small Fourier series (sine and cosine terms) to capture the attractor’s shape. They use robust regression and can weight samples by known uncertainties.

This approach avoids heavy pre-processing such as deseasoning or interpolating missing points. It works with irregular time steps and with time-varying measurement uncertainty. The method also gives uncertainty bounds on the estimated stability and can be extended to spatial data. The authors demonstrate it on both synthetic examples and real-world records that previously required extensive pre-processing, including vegetation time series, paleoclimate proxy records, and glacier surge data.

Why this matters: many common early-warning tools for critical transitions — for example lag-1 autocorrelation (AC1) and variance — assume the data fluctuate around a fixed mean and are evenly sampled. Those assumptions fail for seasonal or uneven records and can create misleading signals. The regression-based Langevin approach removes the need to force stationarity first, so it can give more direct and interpretable estimates of the recovery rate λ.