Table 1.
Early warning signals for critical transitions.
Figure 1.
Two simulated paths towards a critical transition to overexploitation that resulted in the critical slowing down and flickering datasets used in the study.
(A) Bifurcation diagram of an ecological model of a logistically growing resource under harvesting. As grazing rate c increases (x axis), resource biomass gradually declines up to a critical grazing threshold that the resource undergoes a critical transition through a fold bifurcation (F1). At this bifurcation the resource collapses to the alternative overexploited state. If grazing rate c is restored, resource biomass returns to the previous underexploited state at another threshold (F2). [solid lines represent equilibria, dashed line marks the boundary between the two basins of attraction between the underexploited (cyan) and overexploited (yellow) states] (B) Critical slowing down simulated dataset of resource biomass (blue line) for gradually increasing grazing rate (green line). (C) Flickering simulated dataset of resource biomass (blue line) for gradually increasing grazing rate (green line).
Figure 2.
Metric-based rolling window indicators estimated on the critical slowing down and flickering datasets.
(A, B) Time series of the state variable. (C) Residual time series after applying Gaussian filtering. (D) Standardized time series after log-transforming the flickering dataset. (E–I) Autocorrelation at-lag-1 (AR1), standard deviation, and skewness estimated within rolling windows of half the size of either the original, filtered or transformed time series. The Kendall τ indicate the strength of the trend in the indicators along the time series. [red line is the Gaussian filtering; black lines correspond to the metrics estimated on the original data, blue lines correspond to the metrics estimated on the residual or transformed data].
Figure 3.
Detrended fluctuation analysis exponents (DFA) estimated on the critical slowing down and flickering datasets.
(A, C) Time series of the state variable. (B, D). DFA estimated within rolling windows of half the size of the original time series applied after linear detrending. (E, F) Distributions of Kendall τ rank correlations indicate a positive trend in the indicators along the time series for different sizes of rolling windows.
Figure 4.
Conditional heteroskedasticity estimated on the critical slowing down and flickering datasets.
(A, B) Time series of the state variable. (C, D) CH estimated within rolling windows of 10% the size of the original time series. CH was applied to the residuals of the best fit AR(p) on the original datasets. Values of CH above the dashed red line are significant (P = 0.1).
Table 2.
BDS statistic estimated on the critical slowing down and flickering datasets with measurement error.
Figure 5.
Nonparametric drift-diffusion-jump metrics in the critical slowing down dataset.
(A) Time series of the state variable (resource biomass). (B, F) Conditional variance versus time and resource biomass respectively. (C, G) Total variance versus time and resource biomass respectively. (D, H) Diffusion versus time and resource biomass respectively. (G, I) Jump intensity versus time and resource biomass respectively.
Figure 6.
Nonparametric drift-diffusion-jump metrics in the flickering dataset.
(A) Time series of the state variable (resource biomass). (B, F) Conditional variance versus time and biomass respectively. (C, G) Total variance versus time and resource biomass respectively. (D, H) Diffusion versus time and resource biomass respectively. (G, I) Jump intensity versus time and resource biomass respectively.
Figure 7.
Fitting time-varying AR(n) models to the critical slowing down and flickering datasets.
(A) Time-varying AR(1) model fit to the critical slowing down dataset. Differences between the fitted trajectory (blue line) and the simulated data (black dots) are attributed to measurement error. The green line gives the time-varying estimate of b0(t) from the AR(1). Parameter estimates are: b0 = 1.263, b1 = 0.278, σε = 0.154, σα = 0.113, and σ1 = 0.015, and the log likelihood is 150.838. (B) Time-varying AR(3) model fit to the critical slowing down dataset. Parameter estimates are: b0 = 1.284, b1 = 0.342, b2 = 0.02, b3 = 0.139, σε = 0.116, σα = 0.141, σ1 = 0.019, σ2 = 0.015, and σ3<0.001, and the log likelihood is 154.102. (C, D) The inverse of the characteristic root for the AR(1) and AR(3) time-varying models respectively.
Figure 8.
Fitting a threshold AR(3) model to the flickering dataset.
Differences between the fitted trajectory (blue line) and the simulated data (black dots) are attributed to measurement error. The green line gives the estimates of b0(t) and b0'(t), and the yellow line gives the threshold c which separates the two AR(3) processes. Parameter estimates are: b0 = −0.941, b0' = 0.797, b1 = 1.192, b1' = 1.22, b2 = 0.069, b2' = −0.231, b3 = −0.326, b3' = −0.135, c = 0.1, σε = 0.125, and σε = 0.054, and the log likelihood = 238.954.
Figure 9.
Potential analysis for the critical slowing down and flickering datasets.
The potential contour plot represents the number of detected wells (states) of the system potential (x-axis corresponds to the time scale of the series, and y-axis is the size of the rolling window for detection). A change in the color of the potential plot along all time scales (vertically) denotes a critical transition in the time series.
Figure 10.
Sensitivity analysis for rolling window metrics (autocorrelation (AR1), standard deviation, and skewness) for the critical slowing down dataset.
Contour plots show the effect of the width of the rolling window and Gaussian filtering on the observed trend in the metrics as measured by the Kendall’s τ (A, C, E). Upside triangles indicate the parameter choice used in the analyses presented in the text. The histograms give the frequency distribution of the trend statistic (B, D, F).
Figure 11.
Significance testing for rolling window metrics (autocorrelation at-lag-1 (AR1), standard deviation, and skewness) for the critical slowing down dataset.
(A, C, E) Contour plots of P values estimated from distributions of Kendall trend statistics derived from surrogate datasets for different rolling window lengths and sizes of Gaussian filtering. The surrogate datasets were produced from the best-fit ARMA model on the residual records of the critical slowing down dataset. P values were derived from probability distributions of the estimated trend statistic for a set of 1,000 surrogate datasets for a combination of a rolling window size and Gaussian filtering. For example, panels B, D, F show the distribution of Kendall trends estimated on 1,000 surrogates of the original residual dataset for rolling window size and Gaussian filtering as the one presented in the text. Black vertical lines indicate the P = 0.1 significance level and the upside open triangle is the actual Kendall trend estimated on the original residual dataset for rolling window size and Gaussian filtering as the one presented in the text (upside solid triangle in A, C, E).
Figure 12.
Flowchart for detecting early warning signals for critical transitions in time series.
Solid arrows represent the procedure presented in the text. Dotted arrows represent interactions that affect different steps in the detection of early warning and that need to be taken into account in the interpretation of the signals.
Table 3.
Rules of thumb for avoiding bottlenecks in detecting early warning signals in time series.