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.
The BDS test (after the initials of W. A. Brock, W. Dechert and J. Scheinkman) detects nonlinear serial dependence in time series. The BDS test was not developed as a leading indicator, but it can help to avoid false detections of critical transitions due to model misspecification. After detrending (or first-differencing) to remove linear structure from the time series by fitting any linear model (e.g. ARMA(p,q), ARCH(q) or GARCH(p,q) models), the BDS tests the null hypothesis that the remaining residuals are independent and identically distributed (i.i.d.). Rejection of the i.i.d. hypothesis implies that there is remaining structure in the time series, which could include a hidden nonlinearity, hidden nonstationarity or other type of structure missed by detrending or model fitting. As critical transitions are considered to be triggered by strong nonlinear responses, the BDS test is expected to reject the i.i.d. hypothesis in the residual time series from a system that is approaching a critical transition. The BDS test can be helpful as an ad-hoc diagnostic test to detect nonlinearities in time series prior to transitions: if the BDS test rejects the i.i.d. hypothesis and there is another strong leading indicator, then the detected early warning is less likely to be a false positive.
We present the BDS method applied to a simulated time series in which a critical transition is approaching (see panel below). We removed the underlying linear structure by: a. first-differencing, b. fitting an AR(1), or c. fitting a GARCH(0,1)) to the entire time series after log-transforming. The remaining residuals were used to estimate the BDS statistic for embedding dimensions 2 and 3, and ε values 0.5, 0.75, and 1 times the observed standard deviation of the time series (table). For each case, the significance of the BDS statistics was calculated using 1,000 bootstrap iterations. In general, the BDS statistic provided strong evidence for nonlinearity in the time series.
In principle, we can apply the BDS statistic within rolling windows to flag a potentially increasing nonlinearity in a time series that is approaching a transition. However, the fact that the BDS test requires a large number of observations for a reliable estimate and that it is sensitive to data preprocessing and filtering choices currently limits its use as a rolling window metric.
In some cases disturbances push the state of the system towards values that are close to the boundary between the two alternative states. Because the dynamics at the boundary become slow, we may observe a rise in the skewness of a time series- the distribution of the values in the time series will become asymmetric. Just like variance, skewness can also increase because of flickering. Skewness is the standardized third moment around the mean of a distribution. Note that skewness may increase, or decrease, depending on whether the transition is towards an alternative state that is larger or smaller than the present state.
Flickering or strong perturbations also make it more likely that the state of a system may reach more extreme values close to a transition. Such effects can lead to a rise in the kurtosis of a time series prior to the transition; the distribution may become ‘leptokurtic’: the tails of the time series distribution become fatter due to the increased presence of rare values in the time series. Kurtosis is the standardized fourth moment around the mean of a distribution.