Background/Question/Methods The recog­ni­tion that ecosys­tems can undergo sud­den shifts to alter­nate, less desir­able sta­ble states has led to the desire to iden­tify early warn­ing signs of these impend­ing col­lapses. This search has been moti­vated by the math­e­mat­ics of bifur­ca­tions, in which sud­den shifts result not from direct per­tur­ba­tions to the state (i.e. the pop­u­la­tion abun­dance, through mech­a­nisms such as over-harvesting) but to a slowly chang­ing para­me­ter that impacts the sys­tem sta­bil­ity. While these col­lapses can­not be antic­i­pated by observ­ing only the mean dynam­ics (as described by a deter­min­is­tic model), signs of the impend­ing col­lapse are expressed in the ran­dom per­tur­ba­tions, or noise, inher­ent in real sys­tems. The math­e­mat­i­cal the­ory of early warn­ing signs exploits this fact by seek­ing to detect pat­terns such as “crit­i­cal slow­ing down” of these per­tur­ba­tions due to the grad­ual loss of sta­bil­ity which leads to a bifurcation. While much atten­tion has been given to empha­siz­ing the exis­tence both of sud­den col­lapses and of signs of crit­i­cal slow­ing down, lit­tle atten­tion has been paid to its detec­tion. Faced with only finite data, any method risks both false alarms and failed detec­tion events. We believe that weigh­ing these risks must be the bur­den of man­age­ment pol­icy, while research must first pro­vide a reli­able way to quan­tify the rel­a­tive risks of each. We present a method which quan­ti­fies this risk and show how to decrease the uncer­tainty inher­ent in com­mon summary-statistic approaches through the use of a like­li­hood based mod­el­ing approach. Results/Conclusions We demon­strate that com­monly used cor­re­la­tion tests applied to sum­mary sta­tis­tics such as auto­cor­re­la­tion and vari­ance are both inap­pro­pri­ate and insuf­fi­cient tests of early warn­ing signals. Our method esti­mates directly the para­me­ters of a gen­er­al­ized model of the bifur­ca­tion pos­tu­lated by early warn­ing sig­nals the­ory, with and with­out the pres­ence of a grad­ual change lead­ing towards col­lapse. Using Monte Carlo sim­u­la­tion we gen­er­ate the dis­tri­b­u­tion of warning-signal sta­tis­tics expected under each model. From this we can quan­tify the risk of false alarms and missed detec­tion. We then show how apply­ing this approach to the data directly rather than the sum­mary sta­tis­tic increases the power of detec­tion. We illus­trate the approach in both sim­u­lated and empir­i­cal data of sud­den eco­log­i­cal shifts.

1. Limits to the detection of early warning
signals of population collapse
Carl Boettiger & Alan Hastings
UC Davis
cboettig@ucdavis.edu
August 10, 2011
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2. Tipping points: Sudden dramatic changes or regime
shifts. . .
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3. Some catastrophic transitions have already happened
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4. Some catastrophic transitions have already happened
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5. But, what if we could predict such sudden collapse?
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6. But, what if we could predict such sudden collapse?
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7. Can we?
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8. A simple theory built on the mechanism of bifurcations
Scheffer et al. 2009
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9. Early warning indicators
e.g. Variance: Carpenter & Brock 2006;
or Autocorrelation: Dakos et al. 2008; etc.
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10. Let’s give it a try. . .
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11. Carl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 10/77

12. Carl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 11/77

13. Carl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 12/77

14. Carl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 13/77

15. Carl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 14/77

16. Carl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 15/77

17. Carl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 16/77

18. Carl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 17/77

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20. Carl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 19/77

21. Prediction Debrief. . .
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22. Prediction Debrief. . .
So what’s an increase?
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23. Prediction Debrief. . .
So what’s an increase?
Do we have enough data to tell?
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24. Prediction Debrief. . .
So what’s an increase?
Do we have enough data to tell?
Which indicators to trust most?
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25. Empirical examples of early warning
Have relied on comparison to a control system:
Carpenter et al. 2011
Drake & Griffen 2010
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26. We don’t have a control system. . .
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27. All we have is a squiggle
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28. All we have is a squiggle
Making predictions from squiggles is hard
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29. A pattern isn’t enough
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30. We need a framework
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31. A framework for predicting catastrophe
A pattern
Wissel 1984, Carpenter & Brock 2006, Dakos et al. 2008, Guttal et al. 2008, Scheffer et al. 2009, Dakos et
al. 2009, Brock & Carpenter 2010, Drake & Griffen 2010, Carpenter et al. 2011, Carpenter & Brock 2011 . . .
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32. A framework for predicting catastrophe
A pattern
A statistic
Dakos et al. 2008, Dakos et al. 2009,
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33. A framework for predicting catastrophe
A pattern
A statistic
Not approaching
transition
Dakos et al. 2008
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34. A framework for predicting catastrophe
A pattern
A statistic
Not approaching Approaching
transition transition
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35. A framework for predicting catastrophe
A pattern
A statistic
Not approaching Approaching
transition transition
Select a threshold
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36. What’s an increase?
Carl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 32/77

37. What’s an increase?
τ ∈ [−1, 1] quantifies the trend.
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38. Unfortunately. . .
Both patterns come from a stable process!
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39. Typical? False alarm!
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40. Typical? False alarm!
How often do we see false alarms?
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41. Often. τ can take any value in a stable system
(We introduce a method to estimate this distribution on given
data, ∼ Dakos et al. 2008)
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42. Another way to be wrong
Warning Signal? Failed Detection?
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43. Another way to be wrong
Warning Signal? Failed Detection?
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44. τ can take any value in a collapsing system
(Using a novel, general stochastic model to estimate)
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46. How much data is necessary?
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47. Beyond the Squiggles
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48. Beyond the Squiggles
general models by likelihood: stable and critical
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49. Beyond the Squiggles
general models by likelihood: stable and critical
simulated replicates for null and test cases
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50. Beyond the Squiggles
general models by likelihood: stable and critical
simulated replicates for null and test cases
Use model likelihood as an indicator (Cox 1962)
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51. So how are we doing?
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52. False Alarm?
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53. Failed Detection?
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54. Do we have enough data to tell?
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55. How about Type I/II error?
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56. Formally, identical.
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57. Linguistically, a disaster.
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58. Instead: focus on trade-off
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59. Receiver-operator characteristics (ROCs):
Visualize the trade-off between
false alarms and failed detection
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82. (a) Stable (b) Deteriorating (c) Daphnia (d) Glaciation III
30
750
Data
4
600
20
0
650
450
7010
-4
τ = 0.72
τ = -0.7 (p = 0.0059) τ = 0.93
τ = 0.22
1400
(p = <2e-16)
6
(p = 1e-05) (p = 0.18)
2500
Var
50
4
30
0.65 1500
0.00 800
0.70 2
τ = -0.15
τ = 0.7 τ=0
Autocor
(p = 1.6e-06) (p = 0.35) (p = 1) τ = 0.64
0.3
(p = 3.6e-13)
1.6 0.60
-0.20
0.8 0.0
0.50
0.4
τ = 0.72 τ = -0.15 τ = 0.61 τ = -0.54
(p = 5.6e-06) (p = 0.35) (p = 0.025) (p = 9.2e-10)
0.2
Skew
-0.2
1.2
0.4
-0.2
0.8
6-0.8
4.00.0
τ = -0.67 τ = 0.31 τ = 0.72 τ = 0.11
(p = 2.3e-05) (p = 0.049) (p = 0.0059) (p = 0.21)
1000
5
1.8
CV
2.5
4
-500
1.2
3
1.0
0 400 800 0 400 800 160 200 240 0 10000 25000
Time
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84. (a) Simulation (b) Daphnia (c) Glaciation III
1.0
1.0
1.0
0.8
0.8
0.8
True Positive
True Positive
True Positive
0.6
0.6
0.6
0.4
0.4
0.4
Likelihood, 0.85 Likelihood, 0.87 Likelihood, 1
Variance, 0.8 Variance, 0.59 Variance, 0.46
0.2
0.2
0.2
Autocorr, 0.51 Autocorr, 0.56 Autocorr, 0.4
Skew, 0.5 Skew, 0.56 Skew, 0.48
CV, 0.81 CV, 0.65 CV, 0.49
0.0
0.0
0.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
False Positive False Positive False Positive
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86. Conclusions
Estimate false alarms & failed detections
Identify which indicators are best
Explore the influence of more data on these rates.
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87. Acknowledgements
Visit code development site
Vasilis Dakos
Sebastian Schreiber
Marissa Baskett
Marcel Holyoak
Center for Population Biology
DoE Computational Science
Graduate Fellowship
& try it out
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