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Early warning signals of stochastic
switching
John M. Drake
Odum School of Ecology, University of Georgia, Athens, GA, USA
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Comment
Cite this article: Drake JM. 2013 Early
warning signals of stochastic switching. Proc R
Soc B 280: 20130686.
http://dx.doi.org/10.1098/rspb.2013.0686
Received: 18 March 2013
Accepted: 15 April 2013
Author for correspondence:
John M. Drake
e-mail: jdrake@uga.edu
The accompanying reply can be viewed at
http://dx.doi.org/10.1098/rspb.2013.1372.
The development of statistical signals for anticipating changes of state (‘regime
shifts’) is an active area of research in ecology [1,2], earth and atmospheric
science [3], and biology [4,5]. Causes of regime shifts include noise-induced
transitions [6], discontinuities in the underlying dynamics [7], stochastic fluctuations dominated by low frequencies [8], stochastic switching between basins of
attraction [9] and dynamic bifurcations (also known as ‘critical transitions’
[1,10]). Of these, only the last two have been studied in detail. While it is
widely accepted that (under the right conditions) dynamic bifurcations may
be anticipated through the measurement of critical slowing down [1,9], it is generally believed that stochastic switching cannot be anticipated because there is
no change in the shape of the potential function [3,9]. Here, I suggest to the contrary that stochastic switching can be anticipated if data are collected at a
sufficiently high frequency. The proposed cause for this phenomenon is again
slowing down. In this case, however, slowing down is due to the local shape
of the potential function, not the proximity of a critical point. Finally, I argue
that for these or any other statistics to be used as an early warning signal,
one also requires a decision theory to balance the strength of evidence against
the costs and benefits of early warnings and false alarms.
Certainly, there is no disagreement that the causes of regime shift are different in the two cases of dynamic bifurcation and stochastic switching.
Particularly, in stochastic switching there is no critical point, no change in the
shape of the potential function and no change in the eigenvalue of the mean
field model (the fast system in Kuehn’s sense [10]), all of which are different
ways of expressing the same phenomenon. Rather, when stochastic switching
occurs, it is because the stationary system undergoes a low probability excursion from one basin of attraction, which we may think of as constituting one
quasi-stationary distribution of states, to another.
My argument depends on the premise that this excursion requires climbing
and crossing the potential barrier between the different basins of attraction,
which is true for stochastic processes that only move among adjacent states,
including Gaussian processes (which have continuous sample paths) and
birth–death processes (which have lattice sample paths). This is important
because one heuristic for understanding the phenomenon of critical slowing
down is that the potential function becomes flat as the (fast) system approaches
criticality (figure 1). But if a system must climb the potential barrier (as it must
in Gaussian processes and birth–death processes), then at some time it will be
at the top of the potential barrier, and for a time preceding this it will be in the
vicinity of this peak. What portion of the potential function is perceived by the
system when it is in the vicinity of the peak depends on the time scale of observation. For some time scale, the potential function will indeed be perceived to
be flat (figure 1). Of course, the system will not persist in this transient state for
long. Most often, it will drop back into the basin of attraction from which it originated. Nonetheless, if there are stochastic fluctuations on a time scale faster
than the characteristic time to return to the steady state (as assumed by the
models of Boettiger & Hastings [11,12] and Kuehn [10]), and if these are observable, then it is, in principle, possible to detect local changes in the sample
autocorrelation and variance that are very much like the increases in autocorrelation and variance owing to divergence of the variance at a critical point, even
though there is no critical point in the sense of a deterministic bifurcation. Put
& 2013 The Author(s) Published by the Royal Society. All rights reserved.
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(a)
(b)
2
potential function at the critical point
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potential
time 1
180
0
100
200
220
260
300
300
400
500
population size
380
600
700
0
100
200
420
300
400
500
population size
460
500
600
700
Figure 1. A dynamical system perched at a potential barrier appears like a system undergoing a critical transition. Both panels depict the same potential function for
a biological population with an Allee effect (the model given by eqns (2.1) and (2.2) in [11]). In (a), the system is perched on the potential barrier, whereas in (b)
the same bistable system approaches a critical point, but the state is near its mean field solution. Inset plots show that although the potential function is concave in
(a) and convex in (b) in the vicinity of the system’s current state, in a local neighbourhood around the current state the potential function is, nonetheless, nearly flat
in both cases.
differently, many stochastic systems situated at the top of a
potential barrier behave (at some time scale) very much like
systems undergoing a dynamic bifurcation in the mean
field (figure 1). This argument presupposes the potential
function to be continuous and relatively smooth. Thus,
counter-examples such as those developed by Hastings &
Wysham [13] are not expected to exhibit this phenomenon.
Boettiger & Hastings [11] have presented a set of simulations that demonstrates this phenomenon. The goal of
their study was to understand the early warning problem
in the context of an evidentiary decision theory. They
asked: when is an early warning signal evidence of an
approaching state shift? They pointed to the ‘prosecutor’s fallacy’: mistakenly equating the probability of an event given
some evidence with the probability of evidence given the
event. They concluded, correctly, that determining the probability of a state shift from an early warning signal requires
knowing the prior probability of the switch occurring, an
issue elsewhere described as the base rate problem [14].
That is, the conditional probability that a critical transition
is occurring given some early warning signal cannot be calculated without knowing the unconditional probability of
transition. This may be illustrated by an example.
From Bayes’s theorem, we have
PðtransitionjsignalÞ ¼
PðsignaljtransitionÞPðtransitionÞ
:
PðsignalÞ
ð1Þ
For our example, we assume the signal is perfectly sensitive
ðPðsignaljtransitionÞ 1Þ, that critical transitions occur
at rate p, and that the false alarm rate is q (signal specificity
is 1– q). What is the probability that the cause of an observed
signal is the approach of a critical transition? We rewrite
P(signal) as P(signaljtransition)P(transition) þ P(signaljno
transition)P(no transition) ¼ p þ q(12p). Substituting into
equation (1), we have P(transitionjsignal) ¼ p/( p þ q(12p)).
Clearly, the conditional probability of transition depends on
both p and q. As an example, let p ¼ 0.0001 and q ¼ 0.02
(false alarms happen in 2% of non-transitions). Substituting,
we have PðtransitionjsignalÞ 0:005. In addition, now suppose
that we are mistaken about the false alarm rate: although
it really is q ¼ 0.01, we think it is q* ¼ 0.0001. Now, we have
Pq ðtransitionjsignalÞ 0:5. The point of Boettiger & Hastings
[11] is that historical studies such as that of Dakos et al. [15]
select time series for analysis in such a way that q is inflated
(we think the false error rate is q* when in fact it is q). As a
result, we mistakenly conclude that the evidence for a critical
transition is high ðPq ðtransitionjsignalÞ 0:5Þ when in fact it
is low ðPðtransitionjsignalÞ 0:0099Þ.
But, I argue, the results of Boettiger & Hastings show
something more: they show that systems undergoing stochastic switching can exhibit slowing down. The evidence for this
comes not from the argument about q and q* (which is correct), but from the simulations that were performed to
show that historical time series selected in this way do in
fact have inflated false alarm rates, as may be seen in their
figs 2 and 3 [11]. These figures show the result of simulations
for two different scenarios in which stochastic switching
occurs in bistable ecological models. The first case is an
Allee effect, a nonlinear birth –death process in which the
birth rate is density-dependent, such as might occur if there
is mate limitation at small population sizes. The second case
is logistic-like population growth of a prey species subject to
predation by a predator exhibiting a saturating functional
response, represented as a stochastic system evolving in discrete
time. The predator-induced bistability in the second case has
sometimes been considered to be a class of Allee effects called
predator-induced Allee effects [16].
Boettiger & Hastings sought to ‘test whether selecting systems that have experienced spontaneous transitions
could bias the analysis towards false-positive detection of
early warning signals’ ( p. 4737 of [11]). Accordingly, they
selected simulated sample paths conditional on subsequent
stochastic switching. They then selected a window ending
just before the switch occurred, calculated variance and autocorrelation coefficients in a moving window of half the length
Proc R Soc B 280: 20130686
potential
potential
time 0
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(a) 1.0
true-positive rate
0.6
3
variance
AUC = 0.72
Wilcoxon test: p ª 0
autocorrelation
AUC = 0.69
Wilcoxon test: pª 0
autocorrelation
AUC = 0.71
Wilcoxon test: pª 0
0.4
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0.8
(b)
variance
AUC = 0.8
Wilcoxon test: pª 0
0
0.2
0.4
0.6
false-positive rate
0.8
1.0
0
0.2
0.4
0.6
false-positive rate
0.8
1.0
Figure 2. Receiver – operator characteristics for two early warning signals of a stochastic switch. (a) Allee effect, (b) predator – prey. Results in (a) are for a
population with an Allee effect and correspond to fig. 2 in [11]. Results in (b) are for a population with a predator-induced Allee effect and correspond to
fig. 3 in [11]. The AUC values greater than 0.5 indicate that these signals can discriminate systems at risk of stochastic switching from those that are not at
risk. Statistically significant Wilcoxon rank sum tests show that the correlation between the computed statistic and time elapsed is significantly greater in
those systems on the verge of a stochastic switch than in those that are not. (Online version in colour.)
of the time series, and computed Kendall’s t statistic for the
correlation between variance or autocorrelation and time
elapsed during the pre-switch interval [12]. Their figs 2 and
3 in [11] show that the distribution of Kendall’s t for intervals
of the sample path obtained in this conditional way (i.e.
biased to be close to the top of the potential barrier) were
different from those computed in the same way but selected
unconditionally. These results can be further analysed by performing a Wilcoxon rank sum test for a difference of means (a
test of the hypothesis that Kendall’s t for time series conditionally selected to go over the potential barrier are
greater than for unconditionally selected time series) and
by computing the receiver –operator characteristic (ROC), a
common device for evaluating the performance of a binary
classifier. In our case, the ROC curve shows the trade-off
between false alarms (false-positive rate) and sensitivity
(true-positive rate) of the early warning statistic as the conservatism of the early warning signal is varied. The integrated
area under curve (AUC) provides a measure of the ability
of Kendall’s t to discriminate those trajectories that did
switch from those that did not. An AUC of 1.0 corresponds
to a signal exhibiting perfect sensitivity and perfect specificity, whereas an AUC of 0.5 is equivalent to random
guessing with a fair coin. Neither of these tests concerns the
estimation of the probability that the system will in fact
undergo a stochastic switch, but both indicate, unequivocally,
that in both models the stochastic switching is preceded by a
period of slowing down (figure 2).
I suggest that these simulation results are also evidence
that early warning signals may be exhibited before a stochastic switch. Indeed, these results show that the ability to
anticipate stochastic switching applies to an even larger
class of systems than I argued for initially. In the first case,
the model for the Allee effect, the model does not exhibit
any stochastic fluctuations on a time scale faster than the
time scale of the dynamics. (The stochastic fluctuations in
this case are caused by demographic stochasticity, which
occur on the same time scale as the change of state. The
state of the system is unchanging in the time between demographic events.) In the second case, the model for the
predator-induced Allee effect, there is no smooth potential.
This interpretation of these results is consistent with
the point of Boettiger & Hastings [11] that a statistical indication must avoid the prosecutor’s fallacy if the indicator is to
be interpreted as evidence that a regime shift is probable
(say, more probably than not), but expands the implications of their study, particularly by showing that such
stochastic switching can indeed be anticipated, contrary to
Ditlevsen & Johnsen [9]. Importantly, Boettiger & Hastings
( p. 4737 of [11]) anticipate this interpretation:
It seems tempting to argue that this bias towards positive detection in historical examples is not problematic—each of these
systems did indeed collapse; so the increased probability of exhibiting warning signals could be taken as a successful detection.
Unfortunately, this is not the case. At the moment the forecast
is made, these systems are not likely to transition, because they
experience a strong pull towards the original stable state.
But this confuses the phenomenon (slowing down near the
potential barrier), the evidentiary problem (how much evidence is there of an approaching state shift?) and the
decision-making problem (how should I act, given the evidence that I have?). In my view, to go from the existence of
the phenomenon to a decision to act requires (i) that we
know the positive predictive value (which requires knowing
the base rate, which depends on the mean field and the noise,
and, so far as I know, cannot be calculated in any generic or
non-parametric way), and (ii) a decision theory saying what
are the costs of false alarms and failure to issue alarms,
and how these should be balanced. Particularly when these
costs are not equal, such as in catastrophic shifts in ecosystems or the Earth’s climate system, it may be desirable
to issue an early warning even if the odds of the switch
occurring are considerably less than one (cf. [17]).
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