Limits to Detection for Early Warning Signals of Population Collapse

Limits to the detection of early warningsignals of population collapse

Carl Boettiger & Alan Hastings

UC [email protected]

August 10, 2011

Carl Boettiger & Alan Hastings, UC Davis [email protected] Early Warning Signs 1/77

Tipping points: Sudden dramatic changes or regimeshifts. . .


Some catastrophic transitions have already happened


Some catastrophic transitions have already happened


But, what if we could predict such sudden collapse?


But, what if we could predict such sudden collapse?


Can we?


A simple theory built on the mechanism of bifurcations

Scheffer et al. 2009


Early warning indicators

e.g. Variance: Carpenter & Brock 2006;or Autocorrelation: Dakos et al. 2008; etc.


Let’s give it a try. . .












Prediction Debrief. . .

So what’s an increase?Do we have enough data to tell?Which indicators to trust most?



So what’s an increase?

Do we have enough data to tell?Which indicators to trust most?



So what’s an increase?Do we have enough data to tell?

Which indicators to trust most?



So what’s an increase?Do we have enough data to tell?Which indicators to trust most?


Empirical examples of early warning

Have relied on comparison to a control system:

Drake & Griffen 2010

Carpenter et al. 2011


We don’t have a control system. . .


All we have is a squiggle


All we have is a squiggle

Making predictions from squiggles is hard


A pattern isn’t enough


We need a framework


A framework for predicting catastropheA pattern

Wissel 1984, Carpenter & Brock 2006, Dakos et al. 2008, Guttal et al. 2008, Scheffer et al. 2009, Dakos etal. 2009, Brock & Carpenter 2010, Drake & Griffen 2010, Carpenter et al. 2011, Carpenter & Brock 2011 . . .Carl Boettiger & Alan Hastings, UC Davis [email protected] Early Warning Signs 27/77


A statistic

Dakos et al. 2008, Dakos et al. 2009,



A statistic

Not approaching transition

Dakos et al. 2008



A statistic


Approaching transition



A statistic


Approaching transition

Select a threshold


What’s an increase?

τ ∈ [−1,1] quantifies the trend.


What’s an increase?

τ ∈ [−1,1] quantifies the trend.


Unfortunately. . .

Both patterns come from a stable process!


Typical? False alarm!

How often do we see false alarms?


Typical? False alarm!

How often do we see false alarms?


Often. τ can take any value in a stable system

(We introduce a method to estimate this distribution on givendata, ∼ Dakos et al. 2008)


Another way to be wrong

Warning Signal? Failed Detection?


Another way to be wrong

Warning Signal? Failed Detection?


τ can take any value in a collapsing system

(Using a novel, general stochastic model to estimate)



How much data is necessary?


Beyond the Squiggles

general models by likelihood: stable and criticalsimulated replicates for null and test casesUse model likelihood as an indicator (Cox 1962)



general models by likelihood: stable and critical

simulated replicates for null and test casesUse model likelihood as an indicator (Cox 1962)



general models by likelihood: stable and criticalsimulated replicates for null and test cases

Use model likelihood as an indicator (Cox 1962)



general models by likelihood: stable and criticalsimulated replicates for null and test casesUse model likelihood as an indicator (Cox 1962)


So how are we doing?


False Alarm?


Failed Detection?


Do we have enough data to tell?


How about Type I/II error?


Formally, identical.


Linguistically, a disaster.


Instead: focus on trade-off


Receiver-operator characteristics (ROCs):

Visualize the trade-off betweenfalse alarms and failed detection
























650

750

(a) Stable

Da

ta

450

600

(b) Deteriorating

10

20

30

(c) Daphnia

-40

4

(d) Glaciation III

800

1400 τ = -0.7

(p = 1e-05)

Va

r

150

02500 τ = 0.22

(p = 0.18)

30

50

70

τ = 0.72(p = 0.0059)

24

6

τ = 0.93(p = <2e-16)

-0.2

00.0

0 τ = 0.7(p = 1.6e-06)

Au

toco

r

0.5

00.6

5

τ = -0.15(p = 0.35)

0.0

0.3

τ = 0(p = 1)

0.6

00.7

0

τ = 0.64(p = 3.6e-13)

-0.2

0.2

τ = 0.72(p = 5.6e-06)

Ske

w

-0.8

-0.2

0.4

τ = -0.15(p = 0.35)

0.0

0.4

0.8

τ = 0.61(p = 0.025)

0.8

1.2

1.6 τ = -0.54

(p = 9.2e-10)

0 400 800

1.2

1.8

τ = -0.67(p = 2.3e-05)

CV

0 400 800

34

56 τ = 0.31

(p = 0.049)

160 200 240

1.0

2.5

4.0

τ = 0.72(p = 0.0059)

0 10000 25000

-50

010

00

τ = 0.11(p = 0.21)

Time



0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

(a) Simulation

False Positive

Tru

e P

osi

tive

Likelihood, 0.85Variance, 0.8Autocorr, 0.51Skew, 0.5CV, 0.81

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

(b) Daphnia

False Positive

Tru

e P

osi

tive

Likelihood, 0.87Variance, 0.59Autocorr, 0.56Skew, 0.56CV, 0.65

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

(c) Glaciation III

False Positive

Tru

e P

osi

tive

Likelihood, 1Variance, 0.46Autocorr, 0.4Skew, 0.48CV, 0.49



Conclusions

Estimate false alarms & failed detectionsIdentify which indicators are bestExplore the influence of more data on these rates.


Acknowledgements

Vasilis DakosSebastian SchreiberMarissa BaskettMarcel HolyoakCenter for Population BiologyDoE Computational ScienceGraduate Fellowship

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Limits to Detection for Early Warning Signals of Population Collapse

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