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Early Warning Signals of Environmental Tipping Points. Chris Boulton 25 th April 2013 C.A.Boulton@exeter.ac.uk @cboulton_89 cboulton89.wordpress.com . Tipping Points. A bifurcation in the system where a stable equilibrium becomes unstable.
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Early Warning Signals of Environmental Tipping Points Chris Boulton 25th April 2013 C.A.Boulton@exeter.ac.uk @cboulton_89 cboulton89.wordpress.com
Tipping Points • A bifurcation in the system where a stable equilibrium becomes unstable. • Consequently the system moves to a new stable state or regime. • Difficult to predict from observing time series without analysis. • There is usually a sense of irreversibility or hysteresis meaning more work would be needed to return to the previous stable state.
Causes • Forcing the system (climate) causes the approach to the bifurcation. • The system is also perturbed by noise (weather) which pushes the system out of the basin of attraction of the state. • In some cases, noise alone is enough to drive to system to a tipping point without approaching a bifurcation. EWS shouldnot work in this case.
Generic Early Warning Signals • We look for a change in indicators over the time series. • A window length is specified to calculate the indicator in, which moves up one time point, creating a time series for the indicator. • Time series is usually detrended, especially if it drifts. • Kendall tau rank correlation coefficient is used to measure tendency of indicator (1 if always increases and -1 if always decreasing).
AR(1) Coefficient Estimation • The dominant eigenvalue in the system tends to zero approaching a tipping point, causing critical slowing down. • This can be seen as an increase in the AR(1) coefficient ‘a’, where x(t+1) = a*x(t)+e (e - noise). • This can be seen as ‘today is becoming more like yesterday’.
Variance and Skewness • Variance ‘generally’ increases on the approach to the tipping point due to the destabilisation of the state. • In certain rare cases it can decrease (Dakos et al. 2012). • Skewness can increase or decrease depending on the direction the tipping point is going to move the system. Dakos et al. (2012) Ecology 93(2) pp 264-271
Robustness • In the animation, a window length (WL) of 400 (1/2 time series length) and a bandwidth (BW) of 30 was used during detrending. • Tau can be computed for combinations of WL and BW to check for robustness. • Decrease in variance at low bandwidths suggests ‘reddening’.
Hypothesis Testing p = 0.006 • In the case where we have enough knowledge of the dynamics, we can create null models. • We can run an ensemble where the bifurcation is not approached and measure the taus on this. • Important to use same WL and BW.
Hypothesis Testing • Most likely don’t know dynamics of the system and only have time series. • Can also test other null models derived from time series, such as bootstrapping, sampling from the same probability distribution and recreating the original time series using the statistics of the detrended time series (Dakoset al, 2008). Dakos et al. (2008) PNAS 105, pp. 14308-14312
How Much Warning? • Especially in climate systems, it’s very important to know how much time there is for prevention. Maybe only adaptation is possible. • AR(1) coefficient estimation should reach 1 at tipping point but detrending and window length choices can change the value so we can only really look for increases. • Observing trends in hindsight and by confirming these with Kendall’s tau is simple. • What about in real time...?
False and Missed Alarms • As with nearly all forecasting, there is the chance of missed and false alarms. • In reality we don’t have unlimited time series to test early warning signals on. • To combat this, we try to test as many time series of a system as possible, test for robustness and test more than one indicator.
Prosecutor’s Fallacy • In law, a fallacy resulting from confusion that a suspect is guilty due to fitting evidence which is significant, i.e. The probability evidence is satisfied give the defendant is innocent, P(E|I) is small. • However, this doesn’t imply that the defendant is guilty given they match the evidence, P(I|E) is also small. • This can be proven by Bayes’ theorem. • A similar phenomenon occurs when looking for early warnings of tipping points.
Prosecutor’s Fallacy • Boettiger & Hastings (2012) test a model which exhibits a bifurcation and creates an ensemble where this is not approached (no forcing). • Some pass the tipping point by chance (noise induced), the majority do not. Tau is calculated on all members. • Taus from tipping members are in grey histogram. Boettiger & Hastings (2012) Proc. R. Soc. B
Prosecutor’s Fallacy • We might be happy that our early warning works when we are approaching a tipping point. • However these are still false alarms that are caused by correlated noise pushing the system towards the tipping point. These systems tip quickly. • It could be argued that we are more likely to find an early warning signal in a time series which has tipped, because we have singled it out to test.
Conclusions • Early warnings of approaching bifurcations in systems can be determined due to critical slowing down observed in the time series of the system. • We can hypothesis test these the signals we find as well as test their robustness. • There is still the chance of false or missed alarms which we try to combat with the use of more than just one indicator. • We need to be wary of hindsight predictions and remember that observing early warning in real time could be difficult.
