Predictability of critical transitions

Critical transitions in multistable systems have been discussed as models for a variety of phenomena ranging from the extinctions of species to socioeconomic changes and climate transitions between ice ages and warm ages. From bifurcation theory we can expect certain critical transitions to be preceded by a decreased recovery from external perturbations. The consequences of this critical slowing down have been observed as an increase in variance and autocorrelation prior to the transition. However, especially in the presence of noise, it is not clear whether these changes in observation variables are statistically relevant such that they could be used as indicators for critical transitions. In this contribution we investigate the predictability of critical transitions in conceptual models. We study the quadratic integrate-and-fire model and the van der Pol model under the influence of external noise. We focus especially on the statistical analysis of the success of predictions and the overall predictability of the system. The performance of different indicator variables turns out to be dependent on the specific model under study and the conditions of accessing it. Furthermore, we study the influence of the magnitude of transitions on the predictive performance.

Figure 1

The dynamics of the QIF model illustrated by the phase portrait. The parameters $(\epsilon ,\delta ,{\sigma }_1)=(0.02,0.5,0.2)$. The black curves represent the critical manifold $C_0^{\text{QIF}}=\left\{(x,y):y=x^2\right\}$, a solid line for the attracting part and dashed line for the repelling part. The fold point $(0,0)$ is marked as a red (gray) circle. The gray dashed double arrows indicate the fast subsystem flow, i.e., when the system is not near $C_0^{\text{QIF}}$. A numerical solution is presented as a blue (gray) solid line, and its path of reset as a yellow (light gray) dot-dashed line.

Figure 2

Time series of the fast variable $x$ and the slow variable $y$ of the QIF model, simulated using the Euler-Maruyama method with parameter values $(\epsilon ,\delta ,{\sigma }_1)=(0.02,0.5,0.2)$.

Figure 3

The dynamics of the vdP model. The parameters $(\epsilon ,\delta ,{\sigma }_1)=(0.02,0.5,0.1)$. Analogous to Fig. 1, the critical manifold (black lines, solid for the attracting part and dashed for the repelling part), the fold points at $(-\frac{2}{3}\delta ,1)$ and $(0,0)$ (red circles) and the numerical solution trajectory [blue (gray) solid line] are plotted in state space. The dashed double arrows indicate the orientation of the relaxations in the noisy case and the noise-induced transitions for the fast variable.

Figure 4

Time series of the fast variable $x$ and the slow variable $y$ of the vdP model, simulated using the Euler-Maruyama method with parameter values $(\epsilon ,\delta ,{\sigma }_1)=(0.02,0.5,0.1)$.

Figure 5

Time series $\left\{x_n\right\}$ and tracer time series $\left\{{\chi }_n\right\}$ of the QIF model with $(\epsilon ,\delta ,{\sigma }_1)=(0.02,0.5,0.2)$. In the upper panel, the threshold for reset $x=-\delta$ is shown as a red solid line. Both time series are aligned such that the time points $t_n$ when the CTs occur (dashed lines) and the ones $t_m$ when the predictions are made [orange (light gray) impulses of the tracer time series] can be compared. The forecast lead time between $t_n$ and $t_m$ is set to $k=500$ for demonstration.

Figure 6

Time series $\left\{x_n\right\}$ and tracer time series $\left\{{\chi }_n\right\}$ of the vdP model with $(\epsilon ,\delta ,{\sigma }_1)=(0.02,0.5,0.1)$. In the upper panel, the transition phases are shaded with yellow (very light gray). Both time series are aligned such that the time points $t_n$ when the CTs occur (dashed lines) and the ones $t_n$ when the predictions are made [orange (light gray) impulses of the tracer time series] can be compared. The forecast lead time $k$ is set to $k=1000$ for demonstration.

Figure 7

The detection method for positive and negative increments in the vdP model. The time series $\left\{x_n\right\}$ is shown as blue (gray) lines and the finite-time average prior to CTs as black steps. The finite-time average is used to locate the onset of the transitions by comparing it to the thresholds (dashed lines, ${\theta }_0$ for the lower state and ${\theta }_1$ for the upper state). The narrowed down transition phases (upwards-jumping in the left panel and downwards-jumping in the right panel) are colored yellow (light gray). The values of the parameters $(M,w_{id},w_f,{\theta }_0,{\theta }_1)$ are chosen empirically as $(300,5,20,-0.2,-0.1)$.

Figure 8

Time series of tested indicator variables. From top to bottom, shown in the panels are the time series of $x$ (vdP model with $\delta =0.5)$, of tracer variable $\chi$, and of the four proposed indicators. The standard deviations and the autocorrelations are calculated in a sliding window with length $w=50$ time steps. The window width for averaging standard deviations $w_{\text{avg}}=100$ time steps. The tracer time series in the second panel is shown with a forecast lead time $k=500$ time steps. The time when predictions are made is marked with a dashed line in the four panels of indicators. The indicators within the intervals marked yellow (light gray) are excluded from the statistics for prediction.

Figure 9

An example for decision making via thresholds of CPDFs, as specified in Eq. (27). By gradually lowering the discrimination threshold and correspondingly extending the alarm volume, we can generate an ROC curve from a CPDF. The solid line with hollow squares represents a numerically determined CPDF. The discrimination threshold choosing high-performance indicators is shown as a dashed line and the corresponding (disconnected) alarm volumes (filled rectangles) are indicated by black arrows.

Figure 10

An example for an ROC curve (red dots connected by solid lines) and its summary indices. The AUC (area under the curve, shaded gray), and the KSD (Kolmogorov Smirnov distance, blue dotted line representing the distance between the upper left corner $(0,1)$ and the ROC curve) summarize the overall behavior of the ROC curves. The two blue (gray) dashed lines connect the closest and the second closest points on the curve to the $(0,1)$ point, which are needed in the calculation of KSD. The corresponding likelihood ratio $\mathrm{\Lambda }\left({\psi }_n^{*}\right)$, i.e., the slope of the ROC curve in the vicinity of $(0,0)$, is shown as a yellow dot-dashed line. The black dashed diagonal represents an ROC curve generated by random predictions.

Figure 11

ROC curves generated by the indicators introduced in Sec. 4a used to predict CTs in the QIF model. In general, we observe an increase in prediction quality with decreasing lead time $k$ and increasing window width $w$.

Figure 12

ROC curves generated by the indicators introduced in Sec. 4a used to predict CTs in the VdP model. In general, we observe an increase in prediction quality with decreasing lead time $k$ and increasing window width $w$.

Figure 13

Heat maps of the ROC indices of the indicator $a_n\left(1\right)$ and $s_n$ in the $k\text{−}w$ space for the QIF model. AUCs (areas under curve) are shown in the upper row and KSDs (Kolmogorov-Smirnov distances) in the lower row.

Figure 14

Heat maps of the ROC indices of the indicator $a_n\left(1\right)$ and ${\langle s\rangle }_n$ in the $k\text{−}w$ space for the vdP model. AUCs are shown in the upper row and KSDs in the lower row.

Figure 15

Magnitude dependence of the time interval between CTs. Dashed lines represent the average interval and solid lines the minimal interval between CTs. Green (gray) lines are for the vdP model and dark blue (black) ones for the QIF model.

Figure 16

Magnitude dependence of the ROC curves (a)–(d) of $s_n$ in the QIF model for different combinations of lead times $k$ and window widths $w$. For small lead times we observe an increase in prediction quality with an increasing CT magnitude $\delta$.

Figure 17

Magnitude dependence of the predictability indices of the QIF model. ROC indices, AUC and KSD, are plotted in (a) and (b), and the best indicator $s_n^{*}$ and the likelihood ratio evaluated at the best indicator in (c) and (d). The indices for $k=1,w=10$ are shown with solid lines $\left(--\right)$, for $k=1,w=500$ with dot-dashed lines $\left(-·-\right)$, for $k=500,w=10$ with dashed lines $\left(--\right)$, and for $k=500,w=500$ with dotted lines $\left(\cdots \right)$. For small lead times we observe an increase in prediction quality with an increasing CT magnitude $\delta$.

Figure 18

Magnitude dependence of the ROC curves (a)–(d) of $s_n$ in the vdP model for different combinations of lead times $k$ and window widths $w$. For large lead times and large window widths (d) and small lead times and small window widths (a) we observe a mild decrease in prediction quality with an increasing CT magnitude $\delta$.

Figure 19

Magnitude dependence of the predictability indices of the vdP model. ROC indices, AUC and KSD, are shown in (a) and (b), and the best indicators $s_n^{*}$ and the likelihood ratio evaluated at the best indicator in (c) and (d). The indices for $k=1,w=10$ are shown with solid lines $\left(--\right)$, for $k=1,w=500$ with dot-dashed lines $\left(-·-\right)$, for $k=500,w=10$ with dashed lines $\left(--\right)$, and for $k=500,w=500$ with dotted lines $\left(\cdots \right)$. For large lead times and large window widths and small lead times and small window widths we observe a mild decrease in prediction quality with an increasing CT magnitude $\delta$ in (a) and (b).

Figure 20

Magnitude dependence of the QIF model according to the condition $c({\psi }_n,\delta )$ [see Eq. (30)] for four representative combinations of $k$ and $w$. The values of $c({\psi }_n,\delta )$ are represented as grayscale-coded dots with a black border as long as they can be evaluated. The location of the optimal indicator are marked as a “$\times$” if the value of $c({\psi }_n^{*}\left(\delta \right),\delta )$ is available, otherwise and it is marked as a black “$+$”. Note that $c({\psi }_n,{\delta }_i)$ can only be evaluated if PDFs, CPDFs, and their derivatives could be estimated with sufficient accuracy for all values of ${\psi }_n,{\delta }_i$, and ${\delta }_{i+1}$, where the index $i$ enumerates discrete steps in $\delta$.

Figure 21

Magnitude dependence of the vdP model according to the condition $c({\psi }_n,\delta )$ [see Eq. (30)] for four representative combinations of $k$ and $w$. Color coding as in Fig. 20.

Figure 22

Magnitude dependence of the QIF model using combined alarm volumes $c^{*}:=c_V[V^{*},\chi \left(\delta \right)]$ [Eq. (34)] for four representative values of $k$ and $w$.

Figure 23

Magnitude dependence of the vdP model using combined alarm volumes $c^{*}:=c_V[V^{*},\chi \left(\delta \right)]$ [Eq. (34)] for four representative values of $k$ and $w$.