Chris Boulton 25 th April 2013 CABoultonexeteracuk cboulton89 cboulton89wordpresscom Tipping Points A bifurcation in the system where a stable equilibrium becomes unstable Consequently the system moves to a new stable state or regime ID: 262813
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Slide1
Early Warning Signals of Environmental Tipping Points
Chris
Boulton
25
th
April 2013
C.A.Boulton@exeter.ac.uk
@cboulton_89
cboulton89.wordpress.com Slide2
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. Slide3
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
should
not work in this case.Slide4
A Simple ExampleSlide5
ExampleSlide6
ExampleSlide7
ExampleSlide8
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).Slide9
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’.Slide10
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 264-271Slide11
Using a Generic IndicatorSlide12
Using AR(1) Coefficient EstimationSlide13
Using AR(1) Coefficient EstimationSlide14
Using AR(1) Coefficient EstimationSlide15
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’.Slide16
Hypothesis Testing
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.
p = 0.006Slide17
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 (
Dakos
et al
, 2008).
Dakos
et al. (
2008)
PNAS
105, pp. 14308-14312Slide18
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...?Slide19
How Much Warning?Slide20
How Much Warning?Slide21
How Much Warning?Slide22
How Much Warning?Slide23
How Much Warning?Slide24
How Much Warning?Slide25
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.Slide26
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.Slide27
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. BSlide28
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.Slide29
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.