Month: April 2013

Early warnings of catastrophe

There are warning signs when the active structure of a system is changing. But a new paper shows that they may not always be helpful for averting surprise catastrophes.

Catastrophic Collapse Can Occur without Early Warning: Examples of Silent Catastrophes in Structured Ecological Models (PLOS ONE – open access)

Catastrophic and sudden collapses of ecosystems are sometimes preceded by early warning signals that potentially could be used to predict and prevent a forthcoming catastrophe. Universality of these early warning signals has been proposed, but no formal proof has been provided. Here, we show that in relatively simple ecological models the most commonly used early warning signals for a catastrophic collapse can be silent. We underpin the mathematical reason for this phenomenon, which involves the direction of the eigenvectors of the system. Our results demonstrate that claims on the universality of early warning signals are not correct, and that catastrophic collapses can occur without prior warning. In order to correctly predict a collapse and determine whether early warning signals precede the collapse, detailed knowledge of the mathematical structure of the approaching bifurcation is necessary. Unfortunately, such knowledge is often only obtained after the collapse has already occurred.

To get the insight, it helps to back up a bit. (If you haven’t read my posts on bifurcations and 1D vector fields, they’re good background for this.)

Consider a first order system, with a flow that is a sinusoid, plus noise:

Flow=a*SIN(Stock*2*pi) + disturbance

For different values of a, and disturbance = 0, this looks like:

For a = 1, the system has a stable point at stock=0.5. The gain of the negative feedback that maintains the stable point at 0.5, given by the slope of the stock-flow phase plot, is strong, so the stock will quickly return to 0.5 if disturbed.

For a = -1, the system is unstable at 0.5, which has become a tipping point. It’s stable at the extremes where the stock is 0 or 1. If the stock starts at 0.5, the slightest disturbance triggers feedback to carry it to 0 or 1.

For a = 0.04, the system is approaching the transition (i.e. bifurcation) between stable and unstable behavior around 0.5. The gain of the negative feedback that maintains the stable point at 0.5, given by the slope of the stock-flow phase plot, is weak. If something disturbs the system away from 0.5, it will be slow to recover. The effective time constant of the system around 0.5, which is inversely proportional to a, becomes long for small a. This is termed critical slowing down.

For a=0 exactly, not shown, there is no feedback and the system is a pure random walk that integrates the disturbance.

The neat thing about critical slowing down, or more generally the approach of a bifurcation, is that it leaves fingerprints. Here’s a run of the system above, with a=1 (stable) initially, and ramping to a=-.33 (tipping) finally. It crosses a=0 (the bifurcation) at T=75. The disturbance is mild pink noise.

Notice that, as a approaches zero, particularly between T=50 and T=75, the variance of the stock increases considerably.

This means that you can potentially detect approaching bifurcations in time series without modeling the detailed interactions in the system, by observing the variance or similar, better other signs. Such analyses indicate that there has been a qualitative change in Arctic sea ice behavior, for example.

Now, back to the original paper.

It turns out that there’s a catch. Not all systems are neatly one dimensional (though they operate on low-dimensional manifolds surprisingly often).

In a multidimensional phase space, the symptoms of critical slowing down don’t necessarily reveal themselves in all variables. They have a preferred orientation in the phase space, associated with the eigenvectors of the eigenvalue that’s changing at the bifurcation.

The authors explore a third-order ecological model with juvenile and adult prey and a predator:

Predators undergo a collapse when their mortality rate exceeds a critical value (.553). Here, I vary the mortality rate gradually from .55 to .56, with the collapse occurring around time 450:

Note that the critical value of the mortality rate is actually passed around time 300, so it takes a while for the transient collapse to occur. Also notice that the variance of the adult population changes a lot post-collapse. This is another symptom of qualitative change in the dynamics.

The authors show that, in this system, approaching criticality of the predator mortality rate only reveals itself in increased variance or autocorrelation if noise impacts the juvenile population, and even then you have to be able to see the juvenile population.

We have shown three examples where catastrophic collapse can occur without prior early warning signals in autocorrelation or variance. Although critical slowing down is a universal property of fold bifurcations, this does not mean that the increased sensitivity will necessarily manifest itself in the system variables. Instead, whether the population numbers will display early warning will depend on the direction of the dominant eigenvector of the system, that is, the direction in which the system is destabilizing. This theoretical point also applies to other proposed early warning signals, such as skewness [18], spatial correlation [19], and conditional heteroscedasticity [20]. In our main example, early warning signal only occurs in the juvenile population, which in fact could easily be overlooked in ecological systems (e.g. exploited, marine fish stocks), as often only densities of older, more mature individuals are monitored. Furthermore, the early warning signals can in some cases be completely absent, depending on the direction of the perturbations to the system.

They then detail some additional reasons for lack of warning in similar systems.

In conclusion, we propose to reject the currently popular hypothesis that catastrophic shifts are preceded by universal early warning signals. We have provided counterexamples of silent catastrophes, and we have pointed out the underlying mathematical reason for the absence of early warning signals. In order to assess whether specific early warning signals will occur in a particular system, detailed knowledge of the underlying mathematical structure is necessary.

In other words, critical slowing down is a convenient, generic sign of impending change in a time series, but its absence is not a reliable indicator that all is well. Without some knowledge of the system in question, surprise can easily occur.

I think one could further strengthen the argument against early warning by looking at transients. In my simulation above, I’d argue that it takes at least 100 time units to detect a change in the variance of the juvenile population with any confidence, after it passes the critical point around T=300 (longer, if someone’s job depends on not seeing the change). The period of oscillations of the adult population in response to a disturbance is about 20 time units. So it seems likely that early warning, even where it exists, can only be established on time scales that are long with respect to the natural time scale of the system and environmental changes that affect it. Therefore, while signs of critical slowing down might exist in principle, they’re not particularly useful in this setting.

The models are in my library.

Catastrophic Collapse Can Occur without Early Warning: Examples of Silent Catastrophes in Structured Ecological Models

Early warnings of catastrophe

Catastrophic Collapse Can Occur without Early Warning: Examples of Silent Catastrophes in Structured Ecological Models (PLOS ONE – open access)

Catastrophic and sudden collapses of ecosystems are sometimes preceded by early warning signals that potentially could be used to predict and prevent a forthcoming catastrophe. Universality of these early warning signals has been proposed, but no formal proof has been provided. Here, we show that in relatively simple ecological models the most commonly used early warning signals for a catastrophic collapse can be silent. We underpin the mathematical reason for this phenomenon, which involves the direction of the eigenvectors of the system. Our results demonstrate that claims on the universality of early warning signals are not correct, and that catastrophic collapses can occur without prior warning. In order to correctly predict a collapse and determine whether early warning signals precede the collapse, detailed knowledge of the mathematical structure of the approaching bifurcation is necessary. Unfortunately, such knowledge is often only obtained after the collapse has already occurred.

This is a third-order ecological model with juvenile and adult prey and a predator:

See my related blog post on the topic, in which I also mention a generic model of critical slowing down.

The model, with changes files (.cin) implementing some of the experiments: CatastropheWarning.zip

Hair of the dog that bit you climate policy

Roy Spencer on reducing emissions by increasing emissions:

COL: Let’s say tomorrow, evidence is found that proves to everyone that global warming as a result of human released emissions of CO2 and methane, is real. What would you suggest we do?

SPENCER: I would say we need to grow the economy as fast as possible, in order to afford the extra R&D necessary to develop new energy technologies. Current solar and wind technologies are too expensive, unreliable, and can only replace a small fraction of our energy needs. Since the economy runs on inexpensive energy, in order to grow the economy we will need to use fossil fuels to create that extra wealth. In other words, we will need to burn even more fossil fuels in order to find replacements for fossil fuels.

via Planet 3.0

On the face of it, this is absurd. Reverse a positive feedback loop by making it stronger? But it could work, if given the right structure – a relative quit smoking by going in a closet to smoke until he couldn’t stand it anymore. Here’s what I can make of the mental model:

Spencer’s arguing that we need to run reinforcing loops R1 and R2 as hard as possible, because loop R3 is too weak to sustain the economy, because renewables (or more generally non-emitting sources) are too expensive. R1 and R2 provide the wealth to drive R&D, in a virtuous cycle R4 that activates R3 and shuts down the fossil sector via B2. There are a number of problems with this thinking.

  • Rapid growth around R1 rapidly grows environmental damage (B1) – not only climate, but also local air quality, etc. It also contributes to depletion (not shown), and with depletion comes increasing cost (weakening R1) and greater marginal damage from extraction technologies (not shown). It makes no sense to manage the economy as if R1 exists and B1 does not. R3 looks much more favorable today in light of this.
  • Spencer’s view discounts delays. But there are long delays in R&D and investment turnover, which will permit more environmental damage to accumulate while we wait for R&D.
  • In addition to the delay, R4 is weak. For example, if economic growth is 3%/year, and all technical progress in renewables is from R&D with a 70% learning rate, it’ll take 44 years to halve renewable costs.
  • A 70% learning curve for R&D is highly optimistic. Moreover, a fair amount of renewable cost reductions are due to learning-by-doing and scale economies (not shown), which require R3 to be active, not R4. No current deployment, no progress.
  • Spencer’s argument ignores efficiency (not shown), which works regardless of the source of energy. Spurring investment in the fossil loop R1 sends the wrong signal for efficiency, by depressing current prices.

In truth, these feedbacks are already present in many energy models. Most of those are standard economic stuff – equilibrium, rational expectations, etc. – assumptions which favor growth. Yet among the subset that includes endogenous technology, I’m not aware of a single instance that finds a growth+R&D led policy to be optimal or even effective.

It’s time for the techno-optimists like Spencer and Breakthrough to put up or shut up. Either articulate the argument in a formal model that can be shared and tested, or admit that it’s a nice twinkle in the eye that regrettably lacks evidence.

Braveheart & Rogaine

The Reinhart & Rogoff debt/growth paper continues to make a stir for it’s basic Excel errors. Colbert has the latest & funniest take on it.

Two things about this surprise me.

Confronted with obvious and irrefutable errors, the authors double down and admit nothing. They also downplay the significance of the results, ‘… we are very careful in all our papers to speak of “association” and not “causality” …’

But of course the (amplified) message, Debt/GDP>90%=doom, was taken causally in the policy world; see the multiple clips in the intro to the Colbert video. Politicians are nuts to accord one paper in a sea of macroeconomic thought so much weight, but I guess this was the one they liked.

Tax time

It’s time* for environmentalists (and everyone else) to give up on a myriad of second-best regulatory policies and push for a simple emissions price (i.e. a carbon tax). The latest reason: green subsidies are unraveling under adverse energy market conditions. There are many others:

All of the above have some role to play, but without prices as a keystone economic signal, they’re fighting the tide. Moreover, together they have a large cost in administrative complexity, which gives opponents a legitimate reason to whine about bureaucracy and promotes regulatory capture.

If all the effort that’s now expended in fragmented venues to create these policies were focused on one measure, would it be enough to pass a significant emissions price with fair revenue recycling and a border adjustment? I don’t know for sure, but I’d like to see us try.

* Actually, I think it was time for a carbon tax at least 20 years ago.

Causality in nonlinear systems

Sugihara et al. have a really interesting paper in Science, on detection of causality in nonlinear dynamic systems. It’s paywalled, so here’s an excerpt with some comments.

Abstract: Identifying causal networks is important for effective policy and management recommendations on climate, epidemiology, financial regulation, and much else. We introduce a method, based on nonlinear state space reconstruction, that can distinguish causality from correlation. It extends to nonseparable weakly connected dynamic systems (cases not covered by the current Granger causality paradigm). The approach is illustrated both by simple models (where, in contrast to the real world, we know the underlying equations/relations and so can check the validity of our method) and by application to real ecological systems, including the controversial sardine-anchovy-temperature problem.

Identifying causality in complex systems can be difficult. Contradictions arise in many scientific contexts where variables are positively coupled at some times but at other times appear unrelated or even negatively coupled depending on system state.

…

Although correlation is neither necessary nor sufficient to establish causation, it remains deeply ingrained in our heuristic thinking. … the use of correlation to infer causation is risky, especially as we come to recognize that nonlinear dynamics are ubiquitous. Continue reading “Causality in nonlinear systems”

EU ETS on the ropes

The EU declined backloading, a deferral of permit auctions that would have supported prices in the Emissions Trading System (ETS).

This is described imminent collapse to the system, threatening the achievement of emissions targets. Perhaps a political collapse is imminent – not my department – but the idea that low emissions prices threaten the system is a bit odd. The ETS price is a feedback mechanism. Low prices are a symptom, indicating that the marginal cost of meeting targets is extremely low. That should be a cause for celebration (except for traders).

For the umpteenth time, this shows the difficulty of running a system that invites wrangling over allocation and propagates noise from the economy into a market.

Meanwhile, carbon taxes grind away at their job.

Fun with 1D vector fields

Phase plots are the key to understanding life, the universe and the dynamics of everything.

Well, maybe that’s a bit of an overstatement. But they do nicely explain tipping points and bifurcations, which explain a heck of a lot (as I’ll eventually get to).

Fortunately, phase plots for simple systems are easy to work with. Consider a one-dimensional (first-order) system, like the stock and flow in my bathtub posts.

stock & flow

In Vensim lingo, you’d write this out as,

Stock = INTEG( Flow, Initial Stock )
Flow = ... {some function of the Stock and maybe other stuff}

In typical mathematical notation, you might write it as a differential equation, like

x' = f(x)

where x is the stock and x’ (dx/dt) is the flow.

This system (or vector field) has a one dimensional phase space – i.e. a line – because you can completely characterize the state of the system by the value of its single stock.

Fortunately, paper is two dimensional, so we can use the second dimension to juxtapose the flow with the stock (x’ with x), producing a phase plot that helps us get some intuition into the behavior of this stock-flow system. Here’s an example:

Pure accumulation

In this case, the flow is always above the x-axis, i.e. always positive, so the stock can only go up. The flow is constant, irrespective of the stock level, so there’s no feedback and the stock’s slope is constant.

Left: flow vs. stock. Right: resulting behavior of the stock over time.

Exponential growth

Adding feedback makes things more interesting.

In this simplest-possible first order positive feedback loop, the flow is proportional to the stock, so the stock-flow relationship is a rising line (left frame). There’s a trivial equilibrium (or fixed point) at stock = flow = 0, but it’s unstable, so it’s indicated with a hollow circle. An arrowhead indicates the direction of motion in the phase plot.

The resulting behavior is exponential growth (right frame). The bigger the stock gets, the steeper its slope gets.

Exponential decay

Negative feedback just inverts this case. The flow is below 0 when the stock is positive, and the system moves toward the origin instead of away from it.

The equilibrium at 0 is now stable, so it has a solid circle.

Linear systems like those above can have only one equilibrium. Geometrically, this is because the line of stock-flow proportionality can only cross 0 (the x axis) once. Mathematically, it’s because a system with a single state can have only one eigenvalue/eigenvector pair. Things get more interesting when the system is nonlinear.

S-shaped (logistic) growth

Here, the flow crosses zero twice, so there are two fixed points. The one at 0 is unstable, so as long as the stock is initially >0, it will rise to the stable equilibrium at 1.

(Note that there’s no reason to constrain the axes to the 0-1 unit line; it’s just a graphical convenience here.)

Tipping point

A phase diagram for a nonlinear model can have as many zero-crossings as you like. My forest cover toy model has five. A system can then have multiple equilibria. A pair of stable equilibria bracketing an unstable equilibrium creates a tipping point.

In this arrangement, the stable fixed points at 0 and 1 constitute basins of attraction that draw in any trajectories of the stock that lie in their half of the unit line. The unstable point at 0.5 is the fence between the basins, i.e. the tipping point. Any trajectory starting with the stock near 0.5 is drawn to one of the extremes. While stock=0.5 is theoretically possible permanently, real systems always have noise that will trigger the runaway.

If the stock starts out near 1, it will stay there fairly robustly, because feedback will restore that state from any excursion. But if some intervention or noise pushes the stock below 0.5, feedback will then draw it toward 0. Once there, it will be fairly robustly stuck again. This behavior can be surprising and disturbing if 1=good and 0=bad.

This is the very thing that happens in project fire fighting, for example. The 64 trillion dollar question is whether tipping point dynamics create perilous boundaries in the earth system, e.g., climate.

Not all systems are quite this simple. In particular, a stock is often associated with multiple flows. But it’s often helpful to look at first order subsystems of complex models in this way. For example, Jeroen Struben and John Sterman make good use of the phase plot to explore the dynamics of willingness (W) to purchase alternative fuel vehicles. They decompose the net flow of W (red) into multiple components that create a tipping point:

You can look at higher-order systems in the same way, though the pictures get messier (but prettier). You still preserve the attractive feature of this approach: by just looking at the topology of fixed points (or similar higher-dimensional sets), you can learn a lot about system behavior without doing any calculations.