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Limits to Detection for Early Warning Signals of Population Collapse
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Limits to Detection for Early Warning Signals of Population Collapse

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Background/Question/Methods …

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.

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  • 1. Limits to the detection of early warning signals of population collapse Carl Boettiger & Alan Hastings UC Davis cboettig@ucdavis.edu August 10, 2011Carl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 1/77
  • 2. Tipping points: Sudden dramatic changes or regimeshifts. . .Carl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 2/77
  • 3. Some catastrophic transitions have already happenedCarl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 3/77
  • 4. Some catastrophic transitions have already happenedCarl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 4/77
  • 5. But, what if we could predict such sudden collapse?Carl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 5/77
  • 6. But, what if we could predict such sudden collapse?Carl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 5/77
  • 7. Can we?Carl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 6/77
  • 8. A simple theory built on the mechanism of bifurcations Scheffer et al. 2009Carl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 7/77
  • 9. Early warning indicators e.g. Variance: Carpenter & Brock 2006; or Autocorrelation: Dakos et al. 2008; etc.Carl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 8/77
  • 10. Let’s give it a try. . .Carl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 9/77
  • 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
  • 19. Carl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 18/77
  • 20. Carl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 19/77
  • 21. Prediction Debrief. . .Carl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 20/77
  • 22. Prediction Debrief. . . So what’s an increase?Carl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 20/77
  • 23. Prediction Debrief. . . So what’s an increase? Do we have enough data to tell?Carl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 20/77
  • 24. Prediction Debrief. . . So what’s an increase? Do we have enough data to tell? Which indicators to trust most?Carl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 20/77
  • 25. Empirical examples of early warning Have relied on comparison to a control system: Carpenter et al. 2011Drake & Griffen 2010 Carl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 21/77
  • 26. We don’t have a control system. . .Carl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 22/77
  • 27. All we have is a squiggleCarl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 23/77
  • 28. All we have is a squiggle Making predictions from squiggles is hardCarl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 24/77
  • 29. A pattern isn’t enoughCarl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 25/77
  • 30. We need a frameworkCarl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 26/77
  • 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 . . .Carl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 27/77
  • 32. A framework for predicting catastrophe A pattern A statistic Dakos et al. 2008, Dakos et al. 2009,Carl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 28/77
  • 33. A framework for predicting catastrophe A pattern A statistic Not approaching transition Dakos et al. 2008Carl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 29/77
  • 34. A framework for predicting catastrophe A pattern A statistic Not approaching Approaching transition transitionCarl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 30/77
  • 35. A framework for predicting catastrophe A pattern A statistic Not approaching Approaching transition transition Select a thresholdCarl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 31/77
  • 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.Carl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 32/77
  • 38. Unfortunately. . . Both patterns come from a stable process!Carl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 33/77
  • 39. Typical? False alarm!Carl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 34/77
  • 40. Typical? False alarm! How often do we see false alarms?Carl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 34/77
  • 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)Carl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 35/77
  • 42. Another way to be wrong Warning Signal? Failed Detection?Carl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 36/77
  • 43. Another way to be wrong Warning Signal? Failed Detection?Carl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 36/77
  • 44. τ can take any value in a collapsing system (Using a novel, general stochastic model to estimate)Carl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 37/77
  • 45. Carl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 38/77
  • 46. How much data is necessary?Carl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 39/77
  • 47. Beyond the SquigglesCarl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 40/77
  • 48. Beyond the Squiggles general models by likelihood: stable and criticalCarl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 40/77
  • 49. Beyond the Squiggles general models by likelihood: stable and critical simulated replicates for null and test casesCarl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 40/77
  • 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)Carl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 40/77
  • 51. So how are we doing?Carl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 41/77
  • 52. False Alarm?Carl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 42/77
  • 53. Failed Detection?Carl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 43/77
  • 54. Do we have enough data to tell?Carl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 44/77
  • 55. How about Type I/II error?Carl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 45/77
  • 56. Formally, identical.Carl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 46/77
  • 57. Linguistically, a disaster.Carl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 47/77
  • 58. Instead: focus on trade-offCarl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 48/77
  • 59. Receiver-operator characteristics (ROCs): Visualize the trade-off between false alarms and failed detectionCarl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 49/77
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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 TimeCarl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 72/77
  • 83. Carl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 73/77
  • 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 PositiveCarl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 74/77
  • 85. Carl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 75/77
  • 86. Conclusions Estimate false alarms & failed detections Identify which indicators are best Explore the influence of more data on these rates.Carl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 76/77
  • 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 outCarl Boettiger & Alan Hastings, UC Davis cboettig@ucdavis.edu Early Warning Signs 77/77

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