• Bifurcation theory studies how systems change qualitatively as parameters vary
• In the 1970s, René Thom developed catastrophe theory
  • Describes how (low-dimensional) systems can change suddenly
  • Popularized by Christopher Zeeman (1976), who applied it to everything

• There are a lot of core concepts in catastrophe theory:
  • Multiple stable states, critical slowing down, sudden jumps, hysteresis etc.

• However, proponents of catastrophe theory have pushed it too far
  • Catastrophe theory has been described as a great intellectual bubble
  • Zahler & Sussmann (1977) offer a prominent critique (but see Han’s reply)

• One issue was an excessive reliance on qualitative model evaluation
  • There are improvements to this (e.g., Grasman, van der Maas, & Wagenmakers, 2009)
  • Interesting work using catastrophe theory in psychology (e.g., van der Maas et al., 2003; van der Maas et al., 2020)

• 1) Introduction to some core concepts
  • Multiple stable states & bifurcations
  • Resilience & stability
  • Critical slowing down & early warning signals

• 2) Nuances and limitations of critical slowing down
  • Turns out things are (much) more complicated than they seem

• 3) Investigating the performance of early warning signals in online-monitoring settings

• 4) Moving early warning signals forward

• 4) Summary & Conclusion

• We are interested in dynamical systems, that is, systems that change over time
• Differential equations are a prominent way of modeling how systems change

\[ \frac{\mathrm{d}x}{\mathrm{d}t} = f(x) \]

• What we want is \(x(t)\), that is, the state of the system at any time point \(t\)
• The equation above implicitly encodes \(x(t)\), which we can numerically approximate:

\[ x_{n + 1} = x_n + \Delta_t \cdot f(x_n) \enspace . \] - Given an initial condition, \(x(t = 0) = x_0\), we know where the system is at any point \(t\)

• Suppose a population \(x\) grows according to the Malthusian growth model: \[ \frac{\mathrm{d}x}{\mathrm{d}t} = x \]

• Suppose a population \(x\) grows according to the Malthusian growth model: \[ \frac{\mathrm{d}x}{\mathrm{d}t} = x \]

• Suppose a population \(x\) grows according to the Malthusian growth model: \[ \frac{\mathrm{d}x}{\mathrm{d}t} = x \]

• Suppose a population \(x\) grows according to the logistic equation (Verhulst, 1838): \[ \frac{\mathrm{d}x}{\mathrm{d}t} = x \left(1 - \frac{x}{10}\right) \]

• Suppose a population \(x\) grows according to the logistic equation (Verhulst, 1838): \[ \frac{\mathrm{d}x}{\mathrm{d}t} = x \left(1 - \frac{x}{10}\right) \]

• Suppose a population \(x\) grows according to the logistic equation (Verhulst, 1838): \[ \frac{\mathrm{d}x}{\mathrm{d}t} = x \left(1 - \frac{x}{10}\right) \]

• Add a predation term depending on \(p\) to our model (e.g., May, 1977):

\[ \frac{\mathrm{d}x}{\mathrm{d}t} = x \left(1 - \frac{x}{10}\right) - p\frac{x^2}{1 + x^2} \]

• Add a predation term depending on \(p\) to our model (e.g., May, 1977):

\[ \frac{\mathrm{d}x}{\mathrm{d}t} = x \left(1 - \frac{x}{10}\right) - p\frac{x^2}{1 + x^2} \]

• Add a predation term depending on \(p\) to our model (e.g., May, 1977):

\[ \frac{\mathrm{d}x}{\mathrm{d}t} = x \left(1 - \frac{x}{10}\right) - p\frac{x^2}{1 + x^2} \]

• Add a predation term depending on \(p\) to our model (e.g., May, 1977):

\[ \frac{\mathrm{d}x}{\mathrm{d}t} = x \left(1 - \frac{x}{10}\right) - p\frac{x^2}{1 + x^2} \]

• Add a predation term depending on \(p\) to our model (e.g., May, 1977):

\[ \frac{\mathrm{d}x}{\mathrm{d}t} = x \left(1 - \frac{x}{10}\right) - p\frac{x^2}{1 + x^2} \]

• Add a predation term depending on \(p\) to our model (e.g., May, 1977):

\[ \frac{\mathrm{d}x}{\mathrm{d}t} = x \left(1 - \frac{x}{10}\right) - p\frac{x^2}{1 + x^2} \]

• Add a predation term depending on \(p\) to our model (e.g., May, 1977):

\[ \frac{\mathrm{d}x}{\mathrm{d}t} = x \left(1 - \frac{x}{10}\right) - p\frac{x^2}{1 + x^2} \]

• Add a predation term depending on \(p\) to our model (e.g., May, 1977):

\[ \frac{\mathrm{d}x}{\mathrm{d}t} = x \left(1 - \frac{x}{10}\right) - p\frac{x^2}{1 + x^2} \]

• Add a predation term depending on \(p\) to our model (e.g., May, 1977):

\[ \frac{\mathrm{d}x}{\mathrm{d}t} = x \left(1 - \frac{x}{10}\right) - p\frac{x^2}{1 + x^2} \]

• Wissel (1984) found a “universal law” of slowing down near saddle-node bifurcations
• Dai et al. (2012) study yeast population collapse
  • Map out the bifurcation diagram experimentally
  • Dillution amounts to increasing death rate
  • Find early warning signals before the saddle-node bifurcation

1. Some systems can exhibit multiple stable equilibria
2. Equilibria may collide and vanish \(\rightarrow\) saddle-node bifurcation (tipping point; van Nes et al., 2016)
   • Critical transitions, hysteresis
3. Resilience of a system as the extent of the perturbation from which it still can recover
4. Stability of a system as the rate with which it recovers to the stable equilibrium
5. As the system approaches a saddle-node bifurcation, stability decreases
   • The system returns more slowly to the stable equilibrium \(\rightarrow\) critical slowing down (Wissel, 1984)
6. Critical slowing down gives rise to a zoo of early warning signals
   • \(\uparrow\) Autocorrelation, variance, kurtosis, skewness (Scheffer et al., 2009; Guttal & Jayaprakash., 2008)
   • \(\uparrow\) Cross-correlation, spatial variance, etc. (Kefi et al., 2014; Dakos et al., 2010)

• Critical transitions are notoriously hard to predict and reverse
• Early warning signals have the potential to signal critical transitions
• What types of errors could we make?

• First, we could signal a critical transition even though no critical transition occurs
  • Chance pattern: early warning signals just happened to increase
  • Systematic pattern: early warning signal also for non-critical transitions

• Second, we could fail to signal a critical transition even though it does occur
  • This can be due to many reasons — I focus on a single one here

• Not all system variables express critical slowing down equally strongly or at all
• Boerlijst et al. (2013) study a staged predator-prey system
  • Predator preys on adult prey but not on juvenile prey
  • \(\mu_p \approx 0.553\) is the bifurcation point for which the predators collapse

1. The theory of critical slowing down is both general and nuanced
2. CSD is not specific to critical transitions
   • CSD occurs prior to all zero-eigenvalue bifurcations (saddle-node, transcritical, etc.)
     • CSD can thus occur during smooth transitions
     • This may not be what we desire; little predictive advantage over e.g. the mean
   • CSD not specific to bifurcations, can occur due to non-linear forcing (Kefi et al., 2013)
   • CSD indicate instability in a very general sense
3. CSD can fail to occur for a number of reasons
   • Early warning signals are not expressed equally or at all in all variables (Boerlijst et al., 2013)
   • For more, see https://psyarxiv.com/5wc28

• Early warning signals are being discussed as a potential online-monitoring tool in clinical practice
  • Monitor patient \(\rightarrow\) signal potential bad transition \(\rightarrow\) intervene to prevent transition
  • Monitor patient \(\rightarrow\) signal potential good transition \(\rightarrow\) intervene to bring about transition

• Difficult statistical challenge!
  • Have to monitor early warning signals in real-time — when do we signal a transition?
  • What factors influence the performance of such an approach in practice?

• How well do early warning indicators anticipate transitions in this system?
  • Type of early warning signal
  • Noise level: \(\sigma_{\epsilon} \in [4, 6, 8, 10]\)
  • Extent of baseline data: 25, 50, or 100 days
  • Sampling frequency: 10, 5, or 1 observation per day
  • Time to tipping point: 10, 25, 50 days
  • Decision threshold: \(\sigma \in [0.25, 0.50, \ldots, 6]\)

• Generalized Lotka-Volterra model not a realistic model of any psychological system
  • Never see such data in practice!
  • Time scale arbitrary

• Two ways one can interpret our simulation study
• First, as an illustration of how to study early warning signals in simulation
  • Given a (crude) model of a system, shows how one can study the performance of indicators
  • Relevant factors to keep in mind in practice
    • Type of indicator, noise intensity, sampling frequency
    • Extent of baseline, time to tipping point, decision threshold

• Second, as dampening enthusiasm
  • The model is in some sense “ideal”
  • Poor performance under noise also found by others (e.g., Peretti & Munch, 2012)
  • If poor performance here, why would we have good performance in practice?
  • But set of all possible models extremely large — generalization difficult

1. Dynamical systems theory describes how systems change over time
   • Some systems can exhibit multiple stable equilibria and critical transitions
   • Critical slowing down \(\rightarrow\) slower recovery after a perturbation when close to the tipping point
2. Critical slowing down is pretty nuanced
   • Can occur during smooth transitions or fail to occur prior to critical transitions
   • Things always turn out to be more complicated than they initially seem
3. Somewhat ironically, need to build up a sufficient understanding of the system
   • Multiple stable equilibria & critical transitions, linear or nonlinear driver, what time scales?
   • Build formal models — that’s hard! Maybe use the cusp as a first approximation — test it!
4. Early warning indicators performed rather poorly in simulation mimicking online-monitoring settings
5. Theoretical understanding will have practical implications
   • Moving from “generic” indicators to system-specific indicators will increase performance

• Data pre-processing
  • Detrending and filtering
  • Conduct sensitivity analyses (e.g., Lenton 2011; Dakos et al. 2012)

• Statistical modeling
  • Conduct sensitivity analyses (e.g., Lenton 2011; Dakos et al. 2012)
  • Combine multiple indicators into a standardized indicator (Drake et al., 2010)
    • Can help mitigate false positives (e.g., Ditlevsen & Johnson, 2010)
  • If a (crude) model is available, conduct simulation studies and ROC analyses
  • Move away from generic indicators to system-specific indicators (e.g., Boettiger & Hastings, 2013)
  • Could also try to machine learn the heck out of it! (e.g., Jacobsen & Chung, 2020)

• For more details, see: https://psyarxiv.com/5wc28

• Does the system exhibit multiple stable equilibria?
• Are transitions in the system smooth, abrupt, or even hysteretic?
• Importantly, what drives these transitions?

• Litzow & Hunsicker (2016) review 94 early warning signals studies

• Does the system exhibit multiple stable equilibria?
• Are transitions in the system smooth, abrupt, or even hysteretic?
• Importantly, what drives these transitions?

• Does the system exhibit multiple stable equilibria?
• Are transitions in the system smooth, abrupt, or even hysteretic?
• Importantly, what drives these transitions?

• Not enough to look at the state variables, need to think about drivers
• One heuristic way to do this is using the cusp model
  • Allows for bimodality, sudden transitions, hysteresis, critical slowing down, etc.
  • van der Maas et al. (2003) demonstrate this on attitude research

• Not enough to look at the state variables, need to think about drivers
• One heuristic way to do this is using the cusp model
  • Allows for bimodality, sudden transitions, hysteresis, critical slowing down, etc.
  • van der Maas et al. (2003) demonstrate this on attitude research

• O’Regan & Drake (2013)
  • Critical slowing down occurs in the basic SIS and SIR compartmental models

• O’Regan et al. (2016)
  • Elimination of vector-borne diseases (e.g., malaria) through gradually deployed control measures
  • They show that in theory, critical slowing down is expected to occur, but nuances exist:
    • \(\uparrow\) Autocorrelation & \(\uparrow\) variance when eliminating malaria by reducing biting rate of the mosquitoes or their population size
    • Autocorrelation does not anticipate elimination by reducing human infectious period or the per-capita mosquito mortality rate, but \(\downarrow\) variance in both cases

• O’Dea et al. (2018)
  • In a SIR model, the autocorrelation of the number of infected provides a better estimate of the distance to the epidemic threshold than the autocorrelation of the number of susceptibles

• Brett et al. (2018)
  • Case reports are lagging behind
  • Deaths lag behind substantially
  • Estimating \(R_0\) is extremely difficult
  • Show that several early warning indicators are robust to reporting errors and aggregation in anticipating epidemic transitions

• Brett et al. (2020)
  • Critical slowing down occurs in high-dimensional models

• Critical slowing down is a result of a loss in stability
• We usually take this to indicate a loss in resilience
• However, the positive correlation between stability and resilience need not hold

• Dai et al. (2015) show this
  • Let a yeast population collapse by (a) increasing the dillution or (b) reducing nutrients

• Several blog posts on statistics and other topics at https://fabiandablander.com/
  • An introduction to Causal inference
  • A gentle introduction to dynamical systems theory
  • Estimating the risks of partying during a pandemic
  • Bayesian modeling using Stan: A case study, etc.

• Colloquium on “Simulation-based Science”
  • Starts Friday 15th of January, always 4-5pm; planning to have a panel discussion on tipping points