The current issue of Theoretical Ecology is dedicated on Early Warnings and Tipping Points in Ecology. This special issue was co-edited by Vasilis Dakos and Alan Hastings and contains 11 original research papers from key contributors of the topic. You can find a complete list of content here. We hope it will have a strong impact in the further development of anticipating regime shifts in complex systems. It will be highlighted in the upcoming ESA conference in Minneapolis.
Our paper that launched the idea of the Early Warning Signals Toolbox has appeared in PloS One. It is a pure methodological paper that summarizes in a protocol most of the methods being presented for estimating early-warnings in timeseries. In a short time it will be matched with a similar paper on methods using spatial data. Most of the content of both papers will be found in this webpage in a simpler format. Stay in tune!
Just like threshold AR(p) models, potential analysis in essence identifies flickering and serves as warning of the existence of alternative states. Potential analysis is a technique for deriving the shape of the underlying potential of a system. It assumes that a time series may be approximated by a stochastic potential equation
dZ = -[dU/dz]dt + σdW
where dU/dz is a polynomial potential of even order (2nd for one-well potential, 4th for double-well potential, etc.), dW is white noise of unit variance and intensity σ. The order of the best-fit polynomial in essence reflects the number of potential system states identified along the time series.
Contrary to the threshold AR(p) model fitting, potential analysis is performed within rolling windows of different size (ranging from 10 to half the size of the dataset). We applied it on untransformed data for a flickering time series as we did for the threshold AR(p) models (panel a). We identified one state for most of the time series, except from the last 2,000 points onwards when multiple states where identified (panel b). Such high number of detected states meant that, in principle, the data were on the edge of having no clear potential.
Threshold AR(p) models are not, strictly speaking, early warnings for critical transitions. They constitute, however, a method that is suitable for detecting flickering just like potential analysis does. Flickering occurs when a time series repeatedly crosses the domains of attraction of two alternative states. Flickering detection methods can robustly indicate the presence of alternative regimes during the period that the system has not permanently shifted to the alternative attractor, and, thus, serve as an early warning for a permanent shift to an alternative state.
The difficulty in identifying flickering lies in robustly estimating that a time series is jumping among two (or more) distinct states. Threshold AR(p) models are designed to identify these occasional transitions. These models assume there are two underlying processes governing the dynamics in a time series, with the possibility that the state variable switches between them when it crosses a threshold. The two processes are described by two AR(p) models. As with the time-varying AR(p) models, we can also incorporate measurement error, and use the Kalman filter to compute likelihoods that can be used for parameter estimation and model selection.
We fitted the threshold AR(p) model to a time series that does approach a critical transition, but nonetheless under a strong stochastic regime that caused the the time series to jump between multiple states (panel a). The threshold AR(p) model was applied on log-transformed and standardized data. Assuming that the time series was produced by two AR(p) processes of the same order, we fitted AR(p) models of order 1, 2, and 3 and found that the best-fitting model was an AR(3). The fitted threshold AR(p) models showed that the dataset was characterized by two distinct states, which suggested that the system would eventually stabilize in the alternative state.
Time-varying AR(p) [autoregressive] models provide a model-based approach for estimating time-dependent return rates in time series, which can act as an early warning of a critical transition. In time-invariant AR(p) models, the inverse of the characteristic root, λ, of a fitted AR(p) model is similar in magnitude to the dominant eigenvalue of the Jacobian matrix computed at a stationary point of a deterministic discrete-time model. Values of λ near zero imply that the state variable returns rapidly towards the mean; this central tendency diminishes as values approach one.
Time-varying AR(p) models assume that the coefficients of the AR(p) model can change through time, thereby allowing estimation of the time-dependent characteristic root as it varies along a time series up to a transition. To the general form of time-varying AR(p) models we can incorporate measurement error to construct a state-space model that can be fit using a Kalman filter. Fitting with a Kalman filter gives maximum likelihood parameter estimates, and likelihood ratio tests (LRT) can be used for statistical inference about the parameter estimates, such as Akaike’s Information Criterion (AIC) for model selection.
We fitted time-varying AR(p) models with p = 1, 2, and 3 to the time series after log-transforming and standardizing the data. For all cases, we computed time-varying AR(p) models for which only the mean was allowed to vary through time and compared them to AR(p) models for which both the mean and the autoregressive coefficients were allowed to vary with time. The log-likelihood ratio test (LRT) indicated that the models with varying autoregressive coefficients were significantly better than the mean-varying-only models. Comparing across models, the best fit was derived with the time-varying AR(1) model (panel a). We computed the inverse of the characteristic root λ of time-varying AR(1) models at each point in the time series from the estimates of their autoregressive coefficients (panel ). Values of λ approaching 1 imply critical slowing down, while values of λ > 1 imply loss of stationarity. We found a clear increasing trend in λ (τ = 0.736) in the time-varying AR(1) model (panel ), as the time series approached the transition.