Quantifying resilience and the risk of regime shifts under strong correlated noise

Martin Heßler; Oliver Kamps

doi:10.1093/pnasnexus/pgac296

. 2022 Dec 24;2(2):pgac296. doi: 10.1093/pnasnexus/pgac296

Quantifying resilience and the risk of regime shifts under strong correlated noise

Martin Heßler ^1,^2,^✉, Oliver Kamps ³

Editor: Derek Abbott

PMCID: PMC9896148 PMID: 36743473

Abstract

Early warning indicators often suffer from the shortness and coarse-graining of real-world time series. Furthermore, the typically strong and correlated noise contributions in real applications are severe drawbacks for statistical measures. Even under favourable simulation conditions the measures are of limited capacity due to their qualitative nature and sometimes ambiguous trend-to-noise ratio. In order to solve these shortcomings, we analyze the stability of the system via the slope of the deterministic term of a Langevin equation, which is hypothesized to underlie the system dynamics close to the fixed point. The open-source available method is applied to a previously studied seasonal ecological model under noise levels and correlation scenarios commonly observed in real world data. We compare the results to autocorrelation, standard deviation, skewness, and kurtosis as leading indicator candidates by a Bayesian model comparison with a linear and a constant model. We show that the slope of the deterministic term is a promising alternative due to its quantitative nature and high robustness against noise levels and types. The commonly computed indicators apart from the autocorrelation with deseasonalization fail to provide reliable insights into the stability of the system in contrast to a previously performed study in which the standard deviation was found to perform best. In addition, we discuss the significant influence of the seasonal nature of the data to the robust computation of the various indicators, before we determine approximately the minimal amount of data per time window that leads to significant trends for the drift slope estimations.

Keywords: ecology, regime shift, early warning signals, leading indicator, critical transition

Significance Statement.

Commonly proposed statistical early warning measures are scarcely applicable to realistic problems in which limited data availability, coarse-grained sampling and strong correlated noise are typical. Even under favourable simulation conditions they are difficult to interpret due to their qualitative nature, slight trend changes or too-late-increase for policymakers to avoid undesired consequences. Not only in ecology or climate science the development of a robust quantitative resilience measure would be of great importance to guide decision-making processes. Therefore, we propose a novel quantitative measure of resilience, which is system comparable to some extent and outperforms the common indicators under strong correlated noise. It is easy-to-interpret because of credibility intervals and exhibits trends early enough for decision makers to intervene successfully.

Introduction

Even if the idea of universal early warning indicators (1–4) for critical transitions is a fascinating and attractive vision throughout the fields of ecology, climate research, biology, power grids (5–11), its potential for social and economical sciences (12,13), and much more (14–16), the research done over the years in this field has discovered plenty of problems, drawbacks, and limitations of the proposed leading indicators (17–20). The difficulties and limitations result from the sometimes mentioned problematic claim of “universality,” which is hard or impossible to achieve. Just by definition the mentioned universality is already limited to special cases of regime shifts as bifurcation-induced tipping events (21–23), because the leading indicators are a consequence of the commonly observed phenomenon of critical slowing down prior to a bifurcation or flickering in noisy bistable systems (24–26). Critical slowing down is the increased relaxation time of perturbations near a bifurcation, whereas flickering determines jumps of a system between two alternative stable states. Furthermore, a successful detection of a critical transition depends on the eigen-direction in which the transition takes place and the time series at hand (27). Apart from that it remains difficult to get an impression of the leading indicators’ quality applied to real world systems because the tests are often performed with historical test data for which is known that a transition is present (28).

Following this argumentation it is proposed to design specialized indicators in specific fields of research or systems that are known at least in part (29–31). One of those research areas is the field of ecology in which standard leading indicators as autocorrelation at a lag of one (AR1), the standard deviation (std) Inline graphic , the skewness γ or the kurtosis ω are often very limited in their applicability due to high correlated noise contributions and low sampled short time series that are characteristic because of the limits imposed by the experimental and funding resources as stated in (29,32). However, due to the rapid developments in sensor and information processing techniques at least the sampling limitations might be partially overcome in future ecological studies e.g., due to deep-learning image recognition techniques of animal-tracking camera or satellite data (33–35) or acoustic telemetry systems (36). Furthermore, even in simulations in which the afore-mentioned practical limitations do not play a role, the inherent design of the indicators raises problems. As discussed in (37) the standard leading indicator candidates are difficult to interpret because of their qualitative nature: they are designed upon trend changes, which can be too gradual and ambiguous to rely on for decision makers. And in addition, unfortunately these changes are often realized too late for policymakers to adapt management and avoid uprising transitions. Therefore, in the case that a developed early warning measure should be applicable in practice the authors of (37) claim that it

would rely on: (i) defining critical levels of the regime shift indicators, (ii) linking these critical levels to long-term sustainable impact levels, and (iii) finding or developing indicators that have critical levels that are relatively transferable across different ecosystem types.

Based on these demands (37) and the poor performance of standard leading indicator candidates under strong correlated noise found in (29), we want to introduce the alternative drift slope estimation (38–40) to tackle the problem of anticipating an ecological regime shift and compare it to the above mentioned indicators. Similar to (41) the alternative approach considers the data to be generated by a stochastic differential equation of the Langevin form (42)

(1)

where the drift h(x(t), t) captures the deterministic part of the system dynamics under the stochastic influence of a Gaussian and δ-correlated noise process Γ(t) that scales with the diffusion g(x(t), t). The method estimates the parameterized drift and diffusion terms via Markov chain Monte Carlo (MCMC) sampling and calculates the drift slope ζ in the fixed point x* in rolling windows as a resilience measure of the system. The drift slope ζ is negative for stable systems and increases with proceeding destabilization. A zero crossing of the drift slope corresponds to a regime shift (38). In principle, drift and diffusion can also be directly estimated via the Kramers–Moyal expansion (43,44). Nevertheless, this direct drift estimation does not result in stable estimates under the noise conditions of the investigated ecological model [cf. Supplementary Material (45), Figures S5 and S6]. For that reason, we have chosen a fully Bayesian approach, extending the maximum likelihood formulation in (46), since it allows for a significantly more stable estimation of the slope, includes a straight-forward calculation of credibility bands without approximation by Wilks’ theorem and accounts for possibly multipeaked probability densities.

In this study we show that in contrast to the common qualitative leading indicator candidates, the method provides a quantitative and easy-to-interpret resilience measure, which is able to fulfill the requirements stated in (37) for the system discussed there under noise conditions typical for ecological experimental time series data, i.e., correlated strong noise influence (29). In addition, we discuss the important role of the seasonal nature of the data that affects the trend quality of the time series analysis methods. The performance of the early warning signals is tested by comparing the probability that the data might be explained by a linear trend or a constant model with a Bayesian model comparison. The drift slope ζ and — if the seasonality is taken into account — the AR1 provide reliable results in our study. Interestingly, in contrast to previous results regarding the same system (29) the AR1 seems to be preferred to the insignificant standard deviation. In this context it is important to mention that the AR1 would be only considered a reliable indicator if both the AR1 and std Inline graphic would increase at the same time. Apart from this we can reproduce the findings of a generally poor performance of the standard leading indicator candidates (29). In the end, the drift slope seems to be a promising alternative to common leading indicators because of its quantitative nature, easy interpretation and robustness to strong and colored noise contributions. However, its applicability remains limited to situations in which it is possible to generate the necessary amount of data, which is around 50 data points per year for the investigated ecological model.

The ecological system is presented in the section “Ecological model.” The results of the applied leading indicators are discussed in the section “Results,” which is divided into three parts: first, the drift slope results are presented in the subsection “Drift slope analysis.” Second, the performance of the drift slope as leading indicator is compared to established candidates via a Bayesian model comparison in subsection “Comparison of leading indicators’ performance,” before the needed minimum amount of data per window for the model at hand is defined for the drift slope estimation in subsection “Window size limits.” Finally, we summarize our findings in the section “Summary and conclusion.”

Ecological model

In order to investigate the performance of the Bayesian stability analysis tool under typical noise strengths and types in the field of ecology the multispecies model derived in (47), described in detail in (37) and used as a basis of leading indicator performance tests in (29,37) is simulated via the Euler–Maruyama scheme. The ecological system (cf. Fig. 1 and section "Materials and Methods") consists of three parties: juvenile piscivores (J), adult piscivores (A) and planktivores (F). The model contains a continuous “monitoring interval,” which corresponds to observation and measurement intervals of the continuous dynamics and a discrete annual “maturation interval,” which accounts for the discrete cycles of the species’ reproduction. The harvest rate of the adult piscivores is determined via the product of the catchability q and the effort E. Depending on the harvest rate the system settles into a piscivore-dominated or planktivore-dominated state. In the first mentioned scenario the planktivore abundance is kept at a low level because of a large occurrence of adult piscivores, whereas in the second scenario, the large population of planktivores hinders the piscivore population to grow because of the planktivores’ predation of the juvenile group. We focus on regime shifts from the piscivore- into the planktivore-dominated state due to increasing harvest rate or angling pressure qE. In Fig. 2 the key features of the dynamics are illustrated in state space for a more detailed mathematical description.

Fig. 1. — A scheme of the considered foodweb model. In the predation area the adult piscivores A hunt the juvenile piscivores J and the planktivores F, which only hunt juvenile piscivores J. Both, the juvenile piscivores J and the planktivores F can hide themselves in a refuge area in order to retire. External white or colored stochastic influence Z is added to the planktivore population with the noise level σ. We discuss the possibility of regime shifts due to high angling pressure represented by the harvest rate qE, which is given as the product of catchability q and the effort E. Here, the model is restricted to fish, but in general other animals, as e.g., some seabirds, are included in the term “piscivores.”

Fig. 2. — The destabilizing journey from the piscivore regime to the planktivore regime is shown in (A) with time color-coded from red to blue. The harvest rate qE is increased linearly to obtain the trajectory. The system starts in the stable piscivore-dominated regime. One realization is exemplarily shown in (B) for harvest rate qE = 1.38. Only a tiny amount of planktivores 0 < F < 1 is present and therefore the dynamics are reduced to the A-J plane. Juveniles J and adults A of the piscivore species decrease over one year (vector line with time color-coded from green to blue with start and end points given by the violet markers) before the population is updated by the species-intrinsic reproduction rate at the beginning of the new year mirrored by the discrete maturation map (red vector line). The overall surrounding flow is shown as a green vector field with vector lengths reflecting the flow strength. These periodic dynamics are stable until the harvest rate reaches the attractor switch point qE_switch ≈ 1.78. For qE > qE_switch the periodic cycle opens up as observable in (C) for an exemplary harvest rate qE = 2.42. The discrete maturation step cannot compensate the loss of the piscivore population completely anymore as visible by comparison of the start and end positions (violet points) of the one-year dynamics, which leads to always stronger reduction of the piscivore population over the years. At the same time the planktivore population starts to increase slightly as observable in figure (D) for the same qE. The figure is aligned to the planktivore F-axis. The green signed critical transition in (A) is subjectively defined for the planktivores F > 21 for the first time, which coincides with the region after which the planktivore species grows drastically. The point of no return around qE ≈ 2.23 is defined inside of the transition interval between the attractor switch point and the transition point as the point after which even an abrupt reduction of the angling pressure to qE < 0.1 cannot avoid the transition to the planktivore-dominated state anymore. At last the new stable state is illustrated in three dimensions in figure (E). The blue point marks the stable planktivore regime with F ≈ 100, whereas the piscivores are extinct. The flow field strength is color-coded for better resolution.

Results

Drift slope analysis

The drift slope estimation method that is shortly summarized in section “Materials and Methods” and described in detail in (38) is applied to time series simulations of the seasonal ecological model with white, pink, and red noise each of which is realized for three noise levels σ = {0.1, 2.2, 4.5}. The data are evaluated in windows of 750 data points that are shifted by 30 points per step and analysed in two scenarios: first, without preprocessing of the data by deseasonalization and second, with a deseasonalization before computing the drift slope. The analysis results are presented in Fig. 3. The results of the first approach are marked in blue with orange credibility bands defined as the 16% to 84% and 1% to 99% percentile of the drift slope posterior modelled by a kernel density estimate of the sampled parameters (48). The second ansatz is shown in red with the corresponding green credibility bands. Both estimation results can be compared to the analytical values of the smoothed partial derivative of the planktivore F-drift in planktivore F-direction shown as a black dotted line and computed with the data of the model realizations (cf. Supplementary material (45), Figure S4) . The green dotted and orange solid vertical lines are defined equivalently to (29) as the attractor switch point and the “point of no return,” respectively, which is defined as the year in which even a reduction of the harvest rate to qE = 0.1 does not inhibit the destabilization process of the ecological system. The beginning of the gray shaded area is a subjectively defined time at which the previously small planktivore population exceeds 21 individuals and serves as an orientation for the ongoing destabilization process. Each column from left to right belongs to one of the three noise levels σ = {0.1, 2.2, 4.5}. The first (A to C), second (D to F) and third row (G to I) contain the drift slope results of the realizations of the model with additional white, pink, and red noise, respectively. By comparing the results of the analyses with and without deseasonalization over various noise environments of the model we gain valuable insights into the capacities an limits of the methodological concept: In the Fig. 3 (A to C) the deseasonalized cases perform rather similar to the cases without deseasonalization apart from the weak noise case (A) with σ = 0.1. This leads to the conclusion that in the weak noise case (A) the seasonal effects in the data are interpreted by the model probably in terms of noise fluctuations because the parameterization cannot capture the predominant seasonality. With increasing noise the seasonal effects become insignificant as visible in the Fig. 3(B and C) because the noise level covers and hides the seasonal component of the data. The drift slope indicator seems to be suitable to provide information about the resilience of this ecological model with white noise, whereas seasonal aspects should be treated carefully for small noise levels. A comparison with the analytical partial derivative of the planktivore F-drift reveals that the deseasonalized estimates are accurate in the white noise cases.

Fig. 3. — Results of the drift slope analysis for the ecological model with white (A to C), pink (D to F), and red noise (G to I). The columns from left to right correspond to the noise levels σ = {0.1, 2.2, 4.5}. The computations are performed on the time series without deseasonalization (blue lines with orange credibility bands) and with preparation by deseasonalizing the data (red lines with green credibility bands). The green dotted and the orange solid vertical lines indicate the attractor switch point of the deterministic system and the point of no return, respectively, that is defined as the time at which the destabilization cannot be stopped by reducing the harvest rate to qE = 0.1. A comparison of the drift slope estimates to the analytical partial planktivore F-drift slopes (black dotted lines) confirms that the estimates are quantivatively accurate in the case of white noise and qualitatively reasonable for correlated noise. The deseasonalized versions exhibit clear trends and reach the critical zero marked by the red dotted horizontal line around the transition time that is approximately signed by the beginning of the gray shaded area that is defined as the time at which the small planktivore population counts more than 21 individuals. Although the method is not designed to deal with correlated noise and non-Markovian time series, the seasonality of the data has much more influence than the correlated noise. The seasonality reduces clearness of the trends and leads to an earlier zero crossing of the drift slope for weak white and all correlated noise scenarios. In the weak white noise case the numerical method seems to interpret the seasonal effects incorrectly, probably as noise influence. For bigger noise levels the seasonal effects become insignificant for the white noise cases, but not for the correlated noise scenarios. The strong fluctuations of the drift slope estimates in the post transition region of the subfigures (D and G) are probably due to the small correlated noise contributions in the new stable state.

Inline graphic — Results of the drift slope analysis for the ecological model with white (A to C), pink (D to F), and red noise (G to I). The columns from left to right correspond to the noise levels σ = {0.1, 2.2, 4.5}. The computations are performed on the time series without deseasonalization (blue lines with orange credibility bands) and with preparation by deseasonalizing the data (red lines with green credibility bands). The green dotted and the orange solid vertical lines indicate the attractor switch point of the deterministic system and the point of no return, respectively, that is defined as the time at which the destabilization cannot be stopped by reducing the harvest rate to qE = 0.1. A comparison of the drift slope estimates to the analytical partial planktivore F-drift slopes (black dotted lines) confirms that the estimates are quantivatively accurate in the case of white noise and qualitatively reasonable for correlated noise. The deseasonalized versions exhibit clear trends and reach the critical zero marked by the red dotted horizontal line around the transition time that is approximately signed by the beginning of the gray shaded area that is defined as the time at which the small planktivore population counts more than 21 individuals. Although the method is not designed to deal with correlated noise and non-Markovian time series, the seasonality of the data has much more influence than the correlated noise. The seasonality reduces clearness of the trends and leads to an earlier zero crossing of the drift slope for weak white and all correlated noise scenarios. In the weak white noise case the numerical method seems to interpret the seasonal effects incorrectly, probably as noise influence. For bigger noise levels the seasonal effects become insignificant for the white noise cases, but not for the correlated noise scenarios. The strong fluctuations of the drift slope estimates in the post transition region of the subfigures (D and G) are probably due to the small correlated noise contributions in the new stable state.

Technically, the drift slope estimation is not designed to deal with correlated noise and thus, with non-Markovianity. As expected by that fact, the drift slope estimates in (D to I) exhibit a systematic quantitative estimation error compared to the analytical partial planktivore F-derivative. Nevertheless, the qualitative trends remain unchanged. Since the trend resolution toward a potentially zero-crossing is the crucial feature in terms of leading indicator use of the drift slope ζ the findings in (D to I) show that it can be still a helpful early warning tool even in highly correlated and noisy situations. Similar to the results in the weak white noise case (A) the results reach the critical zero line around the attractor switch point and exhibit less clear trends as their deseasonalized counterparts that reach the critical zero around the actual transition that is approximately marked by the beginning of the gray shaded area. In contrast to the white noise cases the seasonality in the correlated noise cases influences the results for all noise levels in a similar way: without deseasonalization the drift slope reaches zero around the attractor switch point, whereas it reaches zero around the point of no return in the absence of seasonality. The critical zero crossing of the drift slope in the deseasoned versions of the correlated noise cases seems to be a bit earlier than the crossings of the white noise counterparts. The high impact of the seasonality in the strong correlated noise cases compared to the strong white noise cases (B and C) is due to the correlation of the noise itself: the noise correlation tends to amplify or weaken the annual amplitudes, whereby the seasonal component of the time series is not hidden by the noise, but more or less preserved. Note also that there is no clear formal reason for the zero crossing of the blue drift slopes at the attractor switch point or for the red drift slopes reaching the critical zero around the point of no return in the correlated noise cases. Besides, the strong fluctuating slope estimates after the transition time in Fig. 3(D and G) without deseasonalization are numerical artefacts probably caused by the small correlated noise contributions in the new stable state.

In conclusion, the drift slope trends are rather robust in the presented model cases and provide reliable information about the resilience and destabilization of the ecological system. The method is relatively complicated to implement in contrast to leading indicator candidates as the AR1 or the std Inline graphic . Anyhow, its performance and robustness could be important advantages in the field of ecology and other data-driven research as outlined in the next subsection “Comparison of leading indicators’ performance” in which the performance of the drift slope in this dynamical rolling window setting is compared to common leading indicator candidates.

Comparison of leading indicators’ performance

In order to compare the performance of the drift slope indicator with established early warning candidates as the autocorrelation at lag-1 (AR1), the standard deviation (std) Inline graphic , the skewness γ or the kurtosis ω we use a Bayesian model comparison in which we compute the Bayes factors BF_ij with i, j ∈ {1, 2} and i ≠ j that are defined as the ratio

(2)

of the evidences Inline graphic that a linear trend model (model ) or a constant model (model ) explain the leading indicator datasets up to the “point of no return.” The BF_ij are calculated for each of the above mentioned noise levels, noise types and the datasets without and with deseasonalization. A Bayes factor is declared to be significant for BF_ij > 100 (49) to account for the fact that most of the Bayes factors lie in the range 10 < BF_ij < 100 or are significantly bigger than 100. The results of the comparison without deseasonalizing the data are summarized in Table 1 where the color code follows (38) with a significant BF₁₂ or BF₂₁ marked by green and orange tiles, respectively, and gray tiles denote cases in which none of the models is favorable. The results of the kurtosis ω are excluded from further discussion, because of the ambiguous, nonmonotone and very noisy trends with jumps, which cannot be reliably interpreted by eye or captured by the linear model Inline graphic of the Bayes model comparison. In some cases the constant model was erroneously preferred or the results were not significant. The corresponding curves of the leading indicators of each case can be found in the Supplementary Material (45) (cf. Figures S1 and S2). Bayes factor pairs with infinite and zero entries correspond to one of the two models with evidence of zero and thus, the model with finite evidence is preferred. Without deseasonalization the common leading indicators AR1, std Inline graphic and skewness γ do not exhibit a significant slope following the Bayesian model comparison in most of the cases, although the time series resolution is relatively high (29) and the time windows are chosen as big as in the last subsection “Drift slope analysis.” Without deseasonalization the AR1 just performs well in the white noise cases with σ = {2.2, 4.5}, whereas the skewness γ does not exhibit any reliable pattern of applicability. Note that these results remain unchanged if the data are only detrended, but not deseasonalized. The corresponding analysis can be found in the Supplementary Material (45) (cf. Figure S3 and Tables S1 and S2). If the results are compared to the deseasonalized counterparts of Table 2 the green tile of the std Inline graphic and the significant white noise cases of the skewness γ turn out to be artefacts caused by the seasonal nature of the time series. Interestingly, the deseasonalization leads to a consistent significance pattern of the skewness γ if only the correlated noise cases are considered. Therefore, the general applicability of the skewness γ as leading indicator is ill-advised since it is rather sensitive to noise types, seasonality and e.g., bistability of the system. Nevertheless, it could be useful under specific conditions as the correlated noise cases considered here or in flickering regimes of bistable systems. Only the recently proposed drift slope and the AR1 with deseasonalization seem to yield reliable results. The performance of the AR1 is significantly improved by deseasonalization that leads to significant trends in all cases as suggested by a comparison of the tables 2 and 1. Under the same conditions the drift slope turns out to be not very sensitive to the seasonal character of the data apart from the early plateaus discussed in subsection “Drift slope analysis.” The drift slope ζ leads to significant positive trends in all considered cases without distinction of non-deseasonalized and deseasonalized data. The results confirm in most instances the results of (29) where a very poor applicability of the standard leading indicator candidates to the ecological test dataset is observed. The most robust leading indicator under strong noise was found to be the variance or std Inline graphic in (29). The Bayes factor analysis proposes AR1 to be the most reliable indicator of the standard measures and rejects the std as a robust indicator.

Table 1.

Summary of the Bayes factors comparing a linear model Inline graphic with positive slope to a constant model for the drift slope ζ, the AR1, the std and the skewness γ for various noise types and levels without deseasonalization of the data.

	White noise (P_white ≈ 7.5)			Pink noise (P_pink ≈ 2 · P_white)			Red noise (P_red ≈ P_pink)
Noise level	σ = 0.1	σ = 2.2	σ = 4.5	σ = 0.1	σ = 2.2	σ = 4.5	σ = 0.1	σ = 2.2	σ = 4.5
Indicator
Indicator
Slope ζ	∞	∞	∞	∞	∞	∞	∞	3.3 · 10⁹⁹	3.0 · 10⁹²
Slope ζ	0	0	0	0	0	0	0	3.0 · 10⁻¹⁰⁰	3.3 · 10⁻⁹³
AR1	12	8.8 · 10⁷	3.8 · 10⁷	9.4	8.7	9.1	9.5	3.1	1.2
AR1	8.3 · 10⁻²	1.1 · 10⁻⁸	2.6 · 10⁻⁸	0.11	0.11	0.11	0.11	0.32	0.85
Std	19	18	57	18	18	39	19	10	8.3 · 10⁴
Std	5.3 · 10⁻²	5.5 · 10⁻²	1.8 · 10⁻²	5.5 · 10⁻²	5.5 · 10⁻²	2.6 · 10⁻²	5.4 · 10⁻²	0.10	1.2 · 10⁻⁵
Skewness γ	6.1 · 10⁶	1.3 · 10⁶	0.91	1.1 · 10⁷	9.5 · 10⁵	8.3 · 10⁸	1.1 · 10⁷	61	3.4 · 10⁴
Skewness γ	1.6 · 10⁻⁷	7.9 · 10⁻⁷	1.1	9.1 · 10⁻⁸	1.1 · 10⁻⁶	1.2 · 10⁻⁹	8.8 · 10⁻⁸	1.6 · 10⁻²	3.0 · 10⁻⁵

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The kurtosis ω is excluded because of nonmonotone behavior. Green tiles mark a BF₁₂ > 100 (49), which is the threshold for a significant leading indicator trend. Gray tiles mark insignificant results. The constant model Inline graphic is never preferred in the analysis. Infinite Bayes factors result from one model with evidence zero, which leads to preferring the finite evidence model. Only the drift slope ζ performs well in the considered cases. The AR1 has a very limited applicability and the skewness γ is not reliable over all cases. The green tile of the std Inline graphic is an artefact of the seasonality, which is confirmed by a comparison with Table 2. For completeness, the same analysis is performed for the skewness with a linear model with negative slope in the Supplementary Material (45) (cf. Table S3) .

Table 2.

Same as Table 1 with deseasonalization of the data. The drift slope ζ applies as before to all test cases.

	White noise (P_white ≈ 7.5)			Pink noise (P_pink ≈ 2 · P_white)			Red noise (P_red ≈ P_pink)
Noise level	σ = 0.1	σ = 2.2	σ = 4.5	σ = 0.1	σ = 2.2	σ = 4.5	σ = 0.1	σ = 2.2	σ = 4.5
Indicator
Indicator
Slope ζ	∞	∞	∞	∞	∞	∞	∞	∞	∞
Slope ζ	0	0	0	0	0	0	0	0	0
AR1	3.4 · 10⁵	1.6 · 10⁷	6.1 · 10⁶	4.3 · 10⁸	1.5 · 10⁸	1.4 · 10⁹	4.3 · 10⁸	1.4 · 10⁸	1.2 · 10⁹
AR1	3.0 · 10⁻⁶	6.2 · 10⁻⁸	1.6 · 10⁻⁷	2.3 · 10⁻⁹	6.6 · 10⁻⁹	7.3 · 10⁻¹⁰	2.3 · 10⁻⁹	7.2 · 10⁻⁹	8.6 · 10⁻¹⁰
Std	1.0	5.8	40	1.1	1.1	1.1	1.1	1.1	1.3
Std	1.0	0.17	2.5 · 10⁻²	0.89	0.89	0.88	0.89	0.92	0.76
Skewness γ	14	0.58	0.44	1.6 · 10⁹⁰	3.6 · 10⁸⁹	1.1 · 10⁸⁹	1.8 · 10⁹⁰	2.1 · 10⁸⁹	2.3 · 10⁹³
Skewness γ	7.1 · 10⁻²	1.7	2.8	6.3 · 10⁻⁹¹	2.8 · 10⁻⁹⁰	8.8 · 10⁻⁹⁰	5.7 · 10⁻⁹¹	4.7 · 10⁻⁹⁰	4.4 · 10⁻⁹⁴

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However, deseasonalization improves the performance of the AR1 as leading indicator significantly as it also works for all deseasonalized cases, whereas the std Inline graphic does not work. This leads to the conclusion that the fragmentary applicability of the std in Table 1 is an artefact due to a misinterpretation of the seasonal character of the time series. Confronted with these results it is important to keep in mind that the AR1 is only a reliable indicator for a bifurcation-induced critical transition if the std Inline graphic increases at the same time. We could not find one case in which the Bayes factors of both AR1 and std are decisive at the same time. The skewness becomes a reliable indicator for the considered correlated noise cases, whereas its positive trends disappear in the white noise cases due to the deseasonalization procedure. The same analysis for the skewness with a linear model Inline graphic with negative slope can be found in the Supplementary Material (45) (cf. Table S4).

Following the results of this study the drift slope ζ is a possible leading indicator candidate also in very noisy situations, provided that a suitable sampling rate of the time series is guaranteed. In the next subsection “Window size limits” the limitations of the drift slope estimates Inline graphic and their sensitivity to small window sizes are investigated because, as stated in (29), ecological time series are often short and possible window sizes are strongly limited by that fact.

Window size limits

In order to ensure comparability of the results to (29) the drift slope estimates are calculated for comparable window sizes and the corresponding BF_{12, 21} are calculated to get an impression of the minimal necessary amount of data per window that yields significant results. In (29) a low-sampled time series variant with one measurement per year and a high-sampled variant of the time series with 50 data points per year is investigated. Here, we will focus on the high-sampled variants because the discussed indicators including the proposed drift slope are only applicable if the information level in terms of available data is high enough to resolve the considered dynamics. This remains a common limitation of the discussed indicators.

However, focusing on the high-sampled datasets with 50 points per year the BF_{12, 21} are calculated for window sizes {150, 100, 50, 25} in decreasing order until the BF₁₂ is no longer significant (BF₁₂ ≤ 100). The results for the discussed noise levels and types are summarized in Table 3 without deseasonalization and in Table 4 with deseasonalization. The color scheme is defined as in subsection “Drift slope analysis.” The tile is signed to be “inadequate” if both model evidences are numerically zero. A Bayes factor pair of infinite an zero indicates that one model has an evidence of zero and thus, does not fit the data at all. Without deseasonalization significant results are mainly generated for windows bigger than 50 and less or equal to 100 data points except for white noise with σ = 0.1 where windows less or equal to 50 data points are sufficient and pink noise with σ = 4.5 where windows have to be bigger than 100 data points. Thus, most of the significant windows include a time interval of one up to two years, which is mostly comparable to the computations in (29) assuming windows of one year. Furthermore, a suitable deseasonalization is able to decrease the necessary window size for significant drift slope trends even below one year between more than 25 and less or equal to 50 data points for pink and red noise. The performance for small windows tends to become slightly worse for the cases with small and strong white noise σ = {0.1, 4.5}. This is a sign for the difficulties of deseasonalization without removing valuable information for the drift slope estimation at the same time. It has to be mentioned that the drift slope trends for small window sizes as in these limit cases are volatile and thus, less appropriate for an on-line analysis approach.

Table 3.

Summary of the Bayes factors comparing a linear model Inline graphic to a constant model for the small window sizes of {150, 100, 50} for various noise types and levels without deseasonalization of the data.

	White noise (P_white ≈ 7.5)			Pink noise (P_pink ≈ 2 · P_white)			Red noise (P_red ≈ P_pink)
Noise level	σ = 0.1	σ = 2.2	σ = 4.5	σ = 0.1	σ = 2.2	σ = 4.5	σ = 0.1	σ = 2.2	σ = 4.5
Window size
Window size
50	2.6 · 10⁵	Inadequate	Inadequate	1.6 · 10⁻²	1.2 · 10⁻²	(∞)	79	1.2	Inadequate
50	3.9 · 10⁻⁶			63	84	(0)	1.3 · 10⁻²	0.85
100	∞	∞	∞	∞	∞	∞	∞	∞	Inadequate
100	0	0	0	0	0	0	0	0
150	∞	∞	∞	∞	∞	∞	∞	∞	∞
150	0	0	0	0	0	0	0	0	0

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Model Inline graphic is preferred upon the threshhold BF₁₂ > 100 (49) colored in green. Gray tiles are insignificant results. The constant model is never preferred. If both models have an evidence that results in a numerical zero, the tile is marked as “inadequate,” because none of the models is adequate to fit the data. In the case that one evidence is finite and one zero the Bayes factor ratio becomes infinite indicating that the model with an evidence of zero does not fit the data at all and thus, the other one is preferred. The results tend to be significant for more than 50 and less than or equal to 100 data points per window except for the red noise with σ = 4.5 that becomes significant for more than 100 and less than or equal to 150 data points. This corresponds to a period in time between one and two years of high-sampled observation of the ecological system apart from the red noise case with σ = 4.5. The white noise case with σ = 0.1 is already significant for less than or equal to 50 data points, i.e., less than one year of high-sampled observation. The infinite BF₁₂ of the pink noise system with noise level σ = 4.5 is written in brackets and colored in gray, because the trend is very noisy.

Table 4.

Same as Table 3 for window sizes of {150, 100, 50, 25} with deseasonalization of the data.

	White noise (P_white ≈ 7.5)			pink noise (P_pink ≈ 2 · P_white)			red noise (P_red ≈ P_pink)
Noise level	σ = 0.1	σ = 2.2	σ = 4.5	σ = 0.1	σ = 2.2	σ = 4.5	σ = 0.1	σ = 2.2	σ = 4.5
Window size
Window size
25	−	−	−	Inadequate	Inadequate	Inadequate	Inadequate	Inadequate	Inadequate
25	−	−	−
50	Inadequate	Inadequate	Inadequate	∞	∞	∞	∞	∞	∞
50				0	0	0	0	0	0
100	∞	∞	inadequate	∞	∞	∞	∞	∞	∞
100	0	0		0	0	0	0	0	0
150	∞	∞	∞	∞	∞	∞	∞	∞	∞
150	0	0	0	0	0	0	0	0	0

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A deseasonalization deceases the necessary data per window to generate significant results to less than or equal to 50 and more than 25 data points for all pink and red noise cases. This corresponds to time periods of six months up to one year of observation in a high-sampled manner. The slightly worse results for the white noise cases give a hint that the method reacts sensitive to the deseasonalization in that noise case.

Summary and Conclusion

Our investigations are based on the destabilizing ecological model previously considered in (29) with white, correlated and weak up to strong noise geared to real world experimental data. The simulations are almost comparable except for a slightly longer period of data sampling before the “point of no return.”

The main difficulties stated in (29) concerning the applicability of established leading indicator candidates as AR1, std Inline graphic , skewness γ and kurtosis ω are given by the conditions of ecological data acquisition: normally, just short time series with a low sampling rate and strong noise are available. Furthermore, the systems tend to be influenced by correlated pink or red noise and seasonality. The above mentioned early warning signals fail under these circumstances especially due to low data availability for their estimation and high noise levels. Besides, even under favourable simulation conditions the leading indicator candidates are not as reliable as necessary for management decisions (37). In the course of this work we have introduced an alternative leading indicator, the so-called “drift slope,” and evaluated its performance in comparison to the common leading indicators mentioned above. The drift slope is derived from the MCMC-estimated parameters of the drift term of a stochastic differential equation while the drift term is approximated by a third-order Taylor polynomial.

We could show that the drift slope gives reliable trends to estimate the resilience of the system almost regardless of the noise level and type and it fulfills the demands for an early warning signal stated by Biggs et al. (37), which we cite in the section “The drift slope”: the drift slope

exhibits a clear threshold of destabilization at zero and the relative distance to zero measures the level of resilience,
provides trends, which are easy-to-interpret regarding the necessity of management action,
is comparable across systems in similar contexts because of its parametric ansatz and quantitative nature.

The standard measures skewness γ and kurtosis ω turn out to usually fail to predict the destabilization process, which coincides with the observations in (29). The kurtosis ω exhibits nonmonotone or ambiguous behavior and is not suited to be applied as leading indicator in this study. Without deseasonalization the skewness γ shows only fragmentary significant results and thus, is not reliable over the range of the considered cases. With deseasonalization the skewness γ yields at least significant results under correlated noise conditions. In contrast to the results of (29) the std Inline graphic also fails to generate significant results, whereas the AR1 seems to be the most robust of the standard measures. Nevertheless, the AR1 is very sensitive to the seasonality of the time series that seems to play an important role in the calculations of the leading indicators in general. Deseasonalization has to be taken into account to achieve optimal results, if the noise intensity does not hide the seasonal component. Accordingly, the applicability of the AR1 is enlarged to situations with correlated noise and the clearness of the drift slope trends could be improved. Furthermore, the necessary minimum amount of data per window for the drift slope estimation could be diminished due to a deseasonalization of the time series. The minima of necessary data for the pink and red noise cases are decreased from between 50 and 100 to 25 to 50 data points except for the red noise case with σ = 4.5 and thus lie in the observation range of one year or less. The white noise cases do not benefit in that way from a deseasonalization.

We considered the destabilization due to a bifurcation, but in principle the Langevin estimation lends itself to monitor changes of the noise level at the same time, which can be crucial for systems with the threat of noise-induced transitions [cf. fig. 5 (38)].

In the end, the drift slope could be an interesting alternative in order to deal with very noisy correlated data under typical circumstances in ecology and other fields, but it is limited due to the available amount of data. The low-sampled scenarios with one point per year are impossible to handle neither with the drift slope estimation nor with the standard measures. However, in some cases the opportunities of tracking resilience with the drift slope measure might be an attractive reason to improve sampling-rates and data collection — e.g., by using deep-learning image recognition techniques (33–36) for experimental design and management purposes — wherever possible.

Materials and Methods

Ecological model

The model contains a continuous “monitoring interval”

(3)

(4)

(5)

and a discrete annual “maturation interval” realized as the map equations

(6)

(7)

(8)

where the index y; t = 1 denotes the abundance of each party at the end of the monitoring interval (i.e., t = 1) of the corresponding year y. In the map s determines the survivorship between maturation intervals and f the fecundity rate of the adult piscivores A. The harvest rate of the adult piscivores is given by the product of the catchability q and the effort E. The planktivores exchange between a protected area, the so-called refuge reservoir F_R and the foraging arena D_F. The parameters c_{i, j} with i, j = {A, F, J} model the consumption or control rates of i by j. Besides, the piscivores become vulnerable to planktivores with the rate ν and enter their refuge with h. Environmental stochasticity of the lower level of the food web is incorporated via Z. White noise Z_white corresponds simply to a Wiener process Z_white = dW with spectral power P_white ≈ 7.5. Pink noise Z_pink is obtained by the following procedure:

Fourier transform of a white noise signal ξ(t) with f denoting the frequencies,
adjusting the obtained power spectrum by a power law with β_pink = 0.8 [cf. (29) for comparability],
and finally an inverse Fourier transform of the adjusted power spectrum . The pink noise data are thus .

To keep close to the former studies of Perretti et al. (29) the red noise signal Z_red is computed via the Ornstein–Uhlenbeck process

(9)

with ϕ = 0.53, which results in a spectral exponent β_red ≈ 1.6 [cf. (29) for comparability]. The total powers P of the correlated signals are adjusted to be approximately equal P_pink ≈ P_red ≈ 15.4. For each of the three noise types the model is evaluated for three different noise intensities, explicitly

(10)

(11)

(12)

with a time step of dt = 1/50. The realizations of the model are computed with the parameters explicitly listed in Table 5 and chosen analogously to (29) apart from the initial harvest rate qE_init that is chosen to be qE_init = 1 instead of qE_init = 1.5 in order to widen the temporal resolution of the stable regime. The parameter choice in (29) follows approximately experimentally observed values in ecological systems of that kind, especially for the noise strength (50), the noise power law exponents (51,52), and the increase in angling pressure (53). Note that as stated in Table 5, the rate of linear destabilization Δ(qE) as well as all the other parameters are chosen analogously to (29) and thus, the choice qE_init = 1 does not affect the comparability.

Table 5.

The parameter values of the ecological model with short definitions.

Parameter	Value	Short definition
qE _init	1	Initial harvest rate
Δ(qE)	0.013	Change of harvest rate per year
F_R	100	Refuge reservoir for planktivores
D_F	0.1	Foraging arena
c_FA	0.3	Rate at which adult piscivores consume planktivores
c_JA	0.001	Control of juvenile piscivores by adult piscivores
c_JF	0.5	Rate at which planktivores consume juvenile piscivores
ν	1	Rate at which juvenile piscivores become vulnerable against planktivores
h	8	Rate at which juvenile planktivores enter the refuge
f	2	Fecundity rate of adult piscivores
s	0.5	Survival rate of adult and juvenile piscivores over the winter period

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Deseasonalization

For each considered year of the simulated data with additive white noise [Fig. 3 (A to C)] a deseasonalization is performed by subtracting the corresponding values of the previous year. In the correlated noise cases [Fig. 3 (D to I)] the seasonality component is removed by estimating the seasonality by a Gaussian kernel smoothing with a width of 2.5 and subtracting it from the data. Therefore, the Python function scipy.ndimage.filters.gaussian_filter (54) is used. Note that this procedure is a form of deseasonalization and detrending at the same time. However, the trend component of the data is weak and does not affect the conclusions [cf. Supplementary material (45), Figure S3, Tables S1 and S2].

Numerical method

Starting with the Langevin equation Eq. 1 we parameterize the drift and diffusion as h (x(t), t) ≡ h (x(t)) and g (x(t), t) ≡ const. ≕ σ. Since we assume to be in a fixed point and close to a bifurcation we develop h (x, t) into a Taylor series up to order three, which is sufficient to describe the normal forms of simple bifurcation scenarios (55). This results in

(13)

so that the information on the linear stability is incorporated in α₁. For practical reasons Eq. 13 is used in the form

(14)

in the numerical approach, where an arbitrary fixed point x* is incorporated in the coefficients Inline graphic by algebraic transformation and comparison of coefficients. A change of the negative sign of the slope

(15)

of the nonlinear drift at the fixed point x*, which is estimated to be the data mean, corresponds to a loss of stability via the formalism of linear stability analysis.

The task is now to estimate the parameters Inline graphic . Their posterior distribution is given by applying Bayes’ theorem

(16)

The likelihood Inline graphic is given as the transition probability of the process defined by Eq. (1) [see (38)] and the prior knowledge is incorporated in . The evidence normalizes the posterior probability density function (pdf) . One advantage of this procedure is the consistent definition of credibility bands of the estimated parameters based on the posterior pdf. The posterior distribution of the parameters can be estimated via MCMC sampling with the flat Jeffreys’ priors

(17)

and

(18)

for the scale variable σ (56). Gaussian priors

(19)

centred around the mean μ = 0 with standard deviations Inline graphic in an adequate range are used for the rest of the parameters. We use the MCMC sampling algorithm implemented in the python package emcee (57). The method is applied in rolling windows in order to resolve the time evolution of the drift slope. A detailed description of the presented algorithm and its implementation steps can be found in (38).

Prior choice

The flat Jeffreys’ priors are chosen broadly as (−50, 50) for θ_{0, 1} and (0, 50) for θ₄ except for the analysis of the deseasonalized versions of the correlated models. In these cases (red lines in D to I) the prior range is chosen even broader as (−70, 70) for θ_{0, 1} and (0, 70) for θ₄ to make sure that the available data determine the posterior distribution. The Gaussian priors for θ_{2, 3} are implemented with σ = {4, 8}, respectively.

Bayes factors and model comparison

The Bayes factors

(20)

are computed in the same way for the different leading indicator candidates. The prior parameter ranges of the linear model Inline graphic and the constant model are adapted to the specific leading indicator time series data based on the following procedure:

The initial value of each leading indicator time series is used as mean of a Gaussian distribution . The distribution is used to draw the parameter of the constant model , which is also used as the intercept b of the linear model .
The slope a of the linear model is drawn from a uniform distribution in the range . The calculations of BFs in the case of deceasing skewness is performed by drawing uniformly from the interval .
The logarithmic noise is drawn uniformly from the interval [log (0.5), log (5)].

The convergence of the results is guaranteed by drawing 10⁷ realizations of each model in each BF calculation.

Statistical leading indicators

The biased autocorrelation at lag-1 is computed via statsmodels.tsa.stattools.acf (58) and the biased standard deviation Inline graphic via numpy.std (59). The skewness γ and kurtosis ω calculations are performed with the biased uncorrected estimators of the python package scipy.stats (54). The biased versions are used because of the large sample sizes, which provide sufficient accuracy. The skewness definition follows the not-adjusted Fisher–Pearson estimator and the kurtosis is defined via the Pearson estimator corresponding to a kurtosis ω = 3 for a Gaussian distribution.

Supplementary Material

pgac296_Supplemental_File

Click here for additional data file.^{(21.9MB, pdf)}

ACKNOWLEDGEMENTS

We acknowledge support from the Open Access Publication Fund of the University of Muenster. We thank colleagues and friends for proofreading the manuscript. M.H. thanks the Studienstiftung des deutschen Volkes for a scholarship including financial support.

Notes

Competing Interest: The authors declare no competing interest.

Contributor Information

Martin Heßler, Institute for Theoretical Physics, Westphalian Wilhelms–University Münster, Wilhelm-Klemm-Straße 9 48149, North Rhine-Westphalia, Germany; Center for Nonlinear Science, Westphalian Wilhelms-University Münster, Corrensstraße 2 48149, North Rhine-Westphalia, Germany.

Oliver Kamps, Center for Nonlinear Science, Westphalian Wilhelms-University Münster, Corrensstraße 2 48149, North Rhine-Westphalia, Germany.

Funding

The authors declare no funding.

Authors' Contributions

M.H. and O.K. designed research. M.H. performed research, analyzed data, designed graphics, and wrote the paper in consultation with O.K.

Preprints

A preprint of this article is published at https://doi.org/10.48550/arXiv.2204.03403.

Data Availability

The simulated data and Python codes are available on github via https://github.com/MartinHessler/Quantifying_resilience_under_realistic_noise (DOI: 10.5281/zenodo.7529049) under a GNU General Public License v3.0. The open source python-implementation is named antiCPy and can be found at https://github.com/MartinHessler/antiCPy (DOI: 10.5281/zenodo.7401015) under a GNU General Public License v3.0.

References

1. Dakos V, et al. 2012. Methods for detecting early warnings of critical transitions in time series illustrated using simulated ecological data. PLoS One. 7(7):e41010. [DOI] [PMC free article] [PubMed] [Google Scholar]
2. Dakos V, van Nes EH, Donangelo R, Fort H, Scheffer M., 2009. Spatial correlation as leading indicator of catastrophic shifts. Theor Ecol. 3(3):163–174. [Google Scholar]
3. Scheffer M, Carpenter SR, Dakos V, van Nes EH., 2015. Generic indicators of ecological resilience: inferring the chance of a critical transition. Ann Rev Ecol Evol Syst. 46(1):145–167. [Google Scholar]
4. Liang J, Hu Y, Chen G, Zhou T. 2017. A universal indicator of critical state transitions in noisy complex networked systems. Sci Rep. 7(1): 1–9. [DOI] [PMC free article] [PubMed] [Google Scholar]
5. Veraart AJ, et al. , 2011. Recovery rates reflect distance to a tipping point in a living system. Nature. 481(7381):357–359. [DOI] [PubMed] [Google Scholar]
6. Drake JM, Griffen BD., 2010. Early warning signals of extinction in deteriorating environments. Nature. 467(7314):456–459. [DOI] [PubMed] [Google Scholar]
7. Dakos V, Glaser SM, hao Hsieh C, Sugihara G., 2017. Elevated nonlinearity as an indicator of shifts in the dynamics of populations under stress. J Roy Soc Int. 14(128):20160845. [DOI] [PMC free article] [PubMed] [Google Scholar]
8. Livina VN, Kwasniok F, Lenton TM., 2010. Potential analysis reveals changing number of climate states during the last 60 kyr. Climate Past. 6(1):77–82. [Google Scholar]
9. Livina VN, Martins TMV, Forbes AB., 2015. Tipping point analysis of atmospheric oxygen concentration. Chaos Int J Nonlinear Sci. 25(3):036403. [DOI] [PubMed] [Google Scholar]
10. Lenton TM., 2012. Arctic climate tipping points. AMBIO. 41(1):10–22. [DOI] [PMC free article] [PubMed] [Google Scholar]
11. Cotilla-Sanchez E, Hines PDH, Danforth CM., 2012. Predicting critical transitions from time series synchrophasor data. IEEE Trans Smart Grid. 3(4):1832–1840. [Google Scholar]
12. Jusup M, et al. , 2022. Social physics. Phys Rep. 948:1–148. [Google Scholar]
13. Helbing D, et al. , 2014. Saving human lives: what complexity science and information systems can contribute. J Stat Phys. 158(3):735–781. [DOI] [PMC free article] [PubMed] [Google Scholar]
14. Izrailtyan I, et al. , 2000. Early detection of acute allograft rejection by linear and nonlinear analysis of heart rate variability. J Thorac Cardiovas Surg. 120(4):737–745. [DOI] [PubMed] [Google Scholar]
15. Chadefaux T., 2014. Early warning signals for war in the news. J Peace Res. 51(1):5–18. [Google Scholar]
16. van de Leemput IA, et al. , 2013. Critical slowing down as early warning for the onset and termination of depression. Proc Natl Acad Sci. 111(1):87–92. [DOI] [PMC free article] [PubMed] [Google Scholar]
17. Clements CF, Drake JM, Griffiths JI, Ozgul A., 2015. Factors influencing the detectability of early warning signals of population collapse. Am Naturalist. 186(1):50–58. [DOI] [PubMed] [Google Scholar]
18. Hastings A, Wysham DB., 2010. Regime shifts in ecological systems can occur with no warning. Ecol Lett. 13(4):464–472. [DOI] [PubMed] [Google Scholar]
19. Ditlevsen PD, Johnsen SJ. 2010. Tipping points: early warning and wishful thinking. Geophys Res Lett. 37(L19703): 1–4. [Google Scholar]
20. Wilkat T, Rings T, Lehnertz K., 2019. No evidence for critical slowing down prior to human epileptic seizures. Chaos: Int J Nonlinear Sci. 29(9):091104. [DOI] [PubMed] [Google Scholar]
21. Scheffer M, et al. , 2009. Early-warning signals for critical transitions. Nature. 461(7260):53–59. [DOI] [PubMed] [Google Scholar]
22. Ritchie P, Sieber J. 2017. Probability of noise- and rate-induced tipping. Phys Rev E. 95(5): 052209. [DOI] [PubMed] [Google Scholar]
23. Ashwin P, Wieczorek S, Vitolo R, Cox P., 2012. Tipping points in open systems: bifurcation, noise-induced and rate-dependent examples in the climate system. Phil Trans Roy Soc A: Math Phys Eng Sci. 370(1962):1166–1184. [DOI] [PubMed] [Google Scholar]
24. Scheffer M, et al. , 2012. Anticipating critical transitions. Science. 338(6105):344–348. [DOI] [PubMed] [Google Scholar]
25. Wissel C., 1984. A universal law of the characteristic return time near thresholds. Oecologia. 65(1):101–107. [DOI] [PubMed] [Google Scholar]
26. Schröder A, Persson L, de Roos AM, Lundbery P., 2005. Direct experimental evidence for alternative stable states: a review. Oikos. 110(1):3–19. [Google Scholar]
27. Boerlijst MC, Oudman T, de Roos AM., 2013. Catastrophic collapse can occur without early warning: examples of silent catastrophes in structured ecological models. PLoS One. 8(4):e62033. [DOI] [PMC free article] [PubMed] [Google Scholar]
28. Boettiger C, Hastings A., 2012. Early warning signals and the prosecutor’s fallacy. Proc Roy Soc B Biol Sci. 279(1748):4734–4739. [DOI] [PMC free article] [PubMed] [Google Scholar]
29. Perretti CT, Munch SB., 2012. Regime shift indicators fail under noise levels commonly observed in ecological systems. Ecol Appl. 22(6):1772–1779. [DOI] [PubMed] [Google Scholar]
30. Gsell AS, et al. , 2016. Evaluating early-warning indicators of critical transitions in natural aquatic ecosystems. Proc Natl Acad Sci USA. 113(50):E8089–E8095. [DOI] [PMC free article] [PubMed] [Google Scholar]
31. Dablander F, Pichler A, Cika A, Bacilieri A. 2020. Anticipating critical transitions in psychological systems using early warning signals: theoretical and practical considerations. Psychol Methods. 1–34. [DOI] [PubMed] [Google Scholar]
32. Bissonette JA., 1999. Small sample size problems in wildlife ecology: a contingent analytical approach. Wildlife Biol. 5(2):65–71. [Google Scholar]
33. Francisco FA, Nührenberg P, Jordan A. 2020. High-resolution, non-invasive animal tracking and reconstruction of local environment in aquatic ecosystems. Movement Ecol. 8(1):1–12. [DOI] [PMC free article] [PubMed] [Google Scholar]
34. Duporge I, Isupova O, Reece S, Macdonald DW, Wang T., 2020. Using very-high-resolution satellite imagery and deep learning to detect and count African elephants in heterogeneous landscapes. Remote Sensing Ecol Conserv. 7(3):369–381. [Google Scholar]
35. Zhao P, et al. , 2020. Estimating animal population size with very high-resolution satellite imagery. Conserv Biol. 35(1):316–324. [DOI] [PubMed] [Google Scholar]
36. Aspillaga E, et al. 2021. Performance of a novel system for high-resolution tracking of marine fish societies. Animal Biotel. 9(1):1–14. [Google Scholar]
37. Biggs R, Carpenter SR, Brock WA., 2009. Turning back from the brink: detecting an impending regime shift in time to avert it. Proc Natl Acad Sci USA. 106(3):826–831. [DOI] [PMC free article] [PubMed] [Google Scholar]
38. Heßler M, Kamps O. 2021a. Bayesian on-line anticipation of critical transitions. New J Phys. 24:063021. [Google Scholar]
39. Heßler M. 2021b. antiCPy. ( 10.5281/zenodo.7401015). [accessed 2023 Jan 11]. https://github.com/MartinHessler/antiCPy. [DOI]
40. Heßler M, 2021c. antiCPy’s documentation. [accessed 2023 Jan 11]. https://anticpy.readthedocs.io.
41. Carpenter SR, Brock WA., 2011. Early warnings of unknown nonlinear shifts: a nonparametric approach. Ecology. 92(12):2196–2201. [DOI] [PubMed] [Google Scholar]
42. Kloeden PE, Platen E. 1992. Numerical solution of stochastic differential equations. Berlin Heidelberg: Springer. [Google Scholar]
43. Friedrich R, Peinke J., 1997. Description of a turbulent cascade by a Fokker–Planck equation. Phys Rev Lett. 78:863–866. [Google Scholar]
44. Friedrich R, et al. , 2000. Extracting model equations from experimental data. Phys Lett A. 271(3):217–222. [Google Scholar]
45. Heßler M, 2022. Supplementary Information. PNAS Nexus. 1(5):1–8.. ( 10.1093/pnasnexus/pgac296). [DOI] [Google Scholar]
46. Kleinhans D. 2011. Estimation of drift and diffusion functions from time series data: A maximum likelihood framework. Phys Rev E Stat Nonlinear Soft Matter Phys. 10: 85. [DOI] [PubMed] [Google Scholar]
47. Carpenter SR, Brock WA., 2004. Spatial complexity, resilience and policy diversity: fishing on lake-rich landscapes. Ecol Soc. 9(1):8. [Google Scholar]
48. Pedregosa F, et al. , 2011. Scikit-learn: machine learning in Python. J Mach Learn Res. 12(85):2825–2830. [Google Scholar]
49. Jeffreys H. 1998. Theory of probability. Oxford: OUP. [Google Scholar]
50. Reed DH, Hobbs GR., 2004. The relationship between population size and temporal variability in population size. Animal Conserv. 7(1):1–8. [Google Scholar]
51. Steele JH., 1985. A comparison of terrestrial and marine ecological systems. Nature. 313(6001):355–358. [Google Scholar]
52. Vasseur DA, Yodzis P., 2004. The color of environmental noise. Ecology. 85(4):1146–1152. [Google Scholar]
53. Pope J., 1996. An evaluation of the stock structure of North Sea cod, haddock, and whiting since 1920, together with a consideration of the impacts of fisheries and predation effects on their biomass and recruitment. ICES J Marine Sci. 53(6):1157–1169. [Google Scholar]
54. Virtanen P, et al. , 2020. SciPy 1.0: fundamental algorithms for scientific computing in Python. Nat Methods. 17:261–272. [DOI] [PMC free article] [PubMed] [Google Scholar]
55. Strogatz SH. 2018. Nonlinear dynamics and chaos. With applications to physics, biology, chemistry and engineering. 2nd edn. Boulder: Westview Press. [Google Scholar]
56. von der Linden W, Dose V, von Tussaint U. Bayesian probability theory. Applications in the physical sciences. Cambridge: Cambridge University Press. [Google Scholar]
57. Foreman-Mackey D, Hogg DW, Lang D, Goodman J., 2013. emcee: The MCMC hammer. Publ Astron Soc Pacific. 125(925):306–312. [Google Scholar]
58. Seabold S, Perktold J. 2010. Statsmodels: Econometric and statistical modeling with python. 9th Python Sci Conf. 57: 92–96. [Google Scholar]
59. Harris CR, et al. , 2020. Array programming with NumPy. Nature. 585(7825):357–362. [DOI] [PMC free article] [PubMed] [Google Scholar]

Associated Data

This section collects any data citations, data availability statements, or supplementary materials included in this article.

Supplementary Materials

pgac296_Supplemental_File

Click here for additional data file.^{(21.9MB, pdf)}

Data Availability Statement

[bib1] 1. Dakos V, et al. 2012. Methods for detecting early warnings of critical transitions in time series illustrated using simulated ecological data. PLoS One. 7(7):e41010. [DOI] [PMC free article] [PubMed] [Google Scholar]

[bib2] 2. Dakos V, van Nes EH, Donangelo R, Fort H, Scheffer M., 2009. Spatial correlation as leading indicator of catastrophic shifts. Theor Ecol. 3(3):163–174. [Google Scholar]

[bib3] 3. Scheffer M, Carpenter SR, Dakos V, van Nes EH., 2015. Generic indicators of ecological resilience: inferring the chance of a critical transition. Ann Rev Ecol Evol Syst. 46(1):145–167. [Google Scholar]

[bib4] 4. Liang J, Hu Y, Chen G, Zhou T. 2017. A universal indicator of critical state transitions in noisy complex networked systems. Sci Rep. 7(1): 1–9. [DOI] [PMC free article] [PubMed] [Google Scholar]

[bib5] 5. Veraart AJ, et al. , 2011. Recovery rates reflect distance to a tipping point in a living system. Nature. 481(7381):357–359. [DOI] [PubMed] [Google Scholar]

[bib6] 6. Drake JM, Griffen BD., 2010. Early warning signals of extinction in deteriorating environments. Nature. 467(7314):456–459. [DOI] [PubMed] [Google Scholar]

[bib7] 7. Dakos V, Glaser SM, hao Hsieh C, Sugihara G., 2017. Elevated nonlinearity as an indicator of shifts in the dynamics of populations under stress. J Roy Soc Int. 14(128):20160845. [DOI] [PMC free article] [PubMed] [Google Scholar]

[bib8] 8. Livina VN, Kwasniok F, Lenton TM., 2010. Potential analysis reveals changing number of climate states during the last 60 kyr. Climate Past. 6(1):77–82. [Google Scholar]

[bib9] 9. Livina VN, Martins TMV, Forbes AB., 2015. Tipping point analysis of atmospheric oxygen concentration. Chaos Int J Nonlinear Sci. 25(3):036403. [DOI] [PubMed] [Google Scholar]

[bib10] 10. Lenton TM., 2012. Arctic climate tipping points. AMBIO. 41(1):10–22. [DOI] [PMC free article] [PubMed] [Google Scholar]

[bib11] 11. Cotilla-Sanchez E, Hines PDH, Danforth CM., 2012. Predicting critical transitions from time series synchrophasor data. IEEE Trans Smart Grid. 3(4):1832–1840. [Google Scholar]

[bib12] 12. Jusup M, et al. , 2022. Social physics. Phys Rep. 948:1–148. [Google Scholar]

[bib13] 13. Helbing D, et al. , 2014. Saving human lives: what complexity science and information systems can contribute. J Stat Phys. 158(3):735–781. [DOI] [PMC free article] [PubMed] [Google Scholar]

[bib14] 14. Izrailtyan I, et al. , 2000. Early detection of acute allograft rejection by linear and nonlinear analysis of heart rate variability. J Thorac Cardiovas Surg. 120(4):737–745. [DOI] [PubMed] [Google Scholar]

[bib15] 15. Chadefaux T., 2014. Early warning signals for war in the news. J Peace Res. 51(1):5–18. [Google Scholar]

[bib16] 16. van de Leemput IA, et al. , 2013. Critical slowing down as early warning for the onset and termination of depression. Proc Natl Acad Sci. 111(1):87–92. [DOI] [PMC free article] [PubMed] [Google Scholar]

[bib17] 17. Clements CF, Drake JM, Griffiths JI, Ozgul A., 2015. Factors influencing the detectability of early warning signals of population collapse. Am Naturalist. 186(1):50–58. [DOI] [PubMed] [Google Scholar]

[bib18] 18. Hastings A, Wysham DB., 2010. Regime shifts in ecological systems can occur with no warning. Ecol Lett. 13(4):464–472. [DOI] [PubMed] [Google Scholar]

[bib19] 19. Ditlevsen PD, Johnsen SJ. 2010. Tipping points: early warning and wishful thinking. Geophys Res Lett. 37(L19703): 1–4. [Google Scholar]

[bib20] 20. Wilkat T, Rings T, Lehnertz K., 2019. No evidence for critical slowing down prior to human epileptic seizures. Chaos: Int J Nonlinear Sci. 29(9):091104. [DOI] [PubMed] [Google Scholar]

[bib21] 21. Scheffer M, et al. , 2009. Early-warning signals for critical transitions. Nature. 461(7260):53–59. [DOI] [PubMed] [Google Scholar]

[bib22] 22. Ritchie P, Sieber J. 2017. Probability of noise- and rate-induced tipping. Phys Rev E. 95(5): 052209. [DOI] [PubMed] [Google Scholar]

[bib23] 23. Ashwin P, Wieczorek S, Vitolo R, Cox P., 2012. Tipping points in open systems: bifurcation, noise-induced and rate-dependent examples in the climate system. Phil Trans Roy Soc A: Math Phys Eng Sci. 370(1962):1166–1184. [DOI] [PubMed] [Google Scholar]

[bib24] 24. Scheffer M, et al. , 2012. Anticipating critical transitions. Science. 338(6105):344–348. [DOI] [PubMed] [Google Scholar]

[bib25] 25. Wissel C., 1984. A universal law of the characteristic return time near thresholds. Oecologia. 65(1):101–107. [DOI] [PubMed] [Google Scholar]

[bib26] 26. Schröder A, Persson L, de Roos AM, Lundbery P., 2005. Direct experimental evidence for alternative stable states: a review. Oikos. 110(1):3–19. [Google Scholar]

[bib27] 27. Boerlijst MC, Oudman T, de Roos AM., 2013. Catastrophic collapse can occur without early warning: examples of silent catastrophes in structured ecological models. PLoS One. 8(4):e62033. [DOI] [PMC free article] [PubMed] [Google Scholar]

[bib28] 28. Boettiger C, Hastings A., 2012. Early warning signals and the prosecutor’s fallacy. Proc Roy Soc B Biol Sci. 279(1748):4734–4739. [DOI] [PMC free article] [PubMed] [Google Scholar]

[bib29] 29. Perretti CT, Munch SB., 2012. Regime shift indicators fail under noise levels commonly observed in ecological systems. Ecol Appl. 22(6):1772–1779. [DOI] [PubMed] [Google Scholar]

[bib30] 30. Gsell AS, et al. , 2016. Evaluating early-warning indicators of critical transitions in natural aquatic ecosystems. Proc Natl Acad Sci USA. 113(50):E8089–E8095. [DOI] [PMC free article] [PubMed] [Google Scholar]

[bib31] 31. Dablander F, Pichler A, Cika A, Bacilieri A. 2020. Anticipating critical transitions in psychological systems using early warning signals: theoretical and practical considerations. Psychol Methods. 1–34. [DOI] [PubMed] [Google Scholar]

[bib32] 32. Bissonette JA., 1999. Small sample size problems in wildlife ecology: a contingent analytical approach. Wildlife Biol. 5(2):65–71. [Google Scholar]

[bib33] 33. Francisco FA, Nührenberg P, Jordan A. 2020. High-resolution, non-invasive animal tracking and reconstruction of local environment in aquatic ecosystems. Movement Ecol. 8(1):1–12. [DOI] [PMC free article] [PubMed] [Google Scholar]

[bib34] 34. Duporge I, Isupova O, Reece S, Macdonald DW, Wang T., 2020. Using very-high-resolution satellite imagery and deep learning to detect and count African elephants in heterogeneous landscapes. Remote Sensing Ecol Conserv. 7(3):369–381. [Google Scholar]

[bib35] 35. Zhao P, et al. , 2020. Estimating animal population size with very high-resolution satellite imagery. Conserv Biol. 35(1):316–324. [DOI] [PubMed] [Google Scholar]

[bib36] 36. Aspillaga E, et al. 2021. Performance of a novel system for high-resolution tracking of marine fish societies. Animal Biotel. 9(1):1–14. [Google Scholar]

[bib37] 37. Biggs R, Carpenter SR, Brock WA., 2009. Turning back from the brink: detecting an impending regime shift in time to avert it. Proc Natl Acad Sci USA. 106(3):826–831. [DOI] [PMC free article] [PubMed] [Google Scholar]

[bib38] 38. Heßler M, Kamps O. 2021a. Bayesian on-line anticipation of critical transitions. New J Phys. 24:063021. [Google Scholar]

[bib39] 39. Heßler M. 2021b. antiCPy. ( 10.5281/zenodo.7401015). [accessed 2023 Jan 11]. https://github.com/MartinHessler/antiCPy. [DOI]

[bib40] 40. Heßler M, 2021c. antiCPy’s documentation. [accessed 2023 Jan 11]. https://anticpy.readthedocs.io.

[bib41] 41. Carpenter SR, Brock WA., 2011. Early warnings of unknown nonlinear shifts: a nonparametric approach. Ecology. 92(12):2196–2201. [DOI] [PubMed] [Google Scholar]

[bib42] 42. Kloeden PE, Platen E. 1992. Numerical solution of stochastic differential equations. Berlin Heidelberg: Springer. [Google Scholar]

[bib43] 43. Friedrich R, Peinke J., 1997. Description of a turbulent cascade by a Fokker–Planck equation. Phys Rev Lett. 78:863–866. [Google Scholar]

[bib44] 44. Friedrich R, et al. , 2000. Extracting model equations from experimental data. Phys Lett A. 271(3):217–222. [Google Scholar]

[bib45] 45. Heßler M, 2022. Supplementary Information. PNAS Nexus. 1(5):1–8.. ( 10.1093/pnasnexus/pgac296). [DOI] [Google Scholar]

[bib46] 46. Kleinhans D. 2011. Estimation of drift and diffusion functions from time series data: A maximum likelihood framework. Phys Rev E Stat Nonlinear Soft Matter Phys. 10: 85. [DOI] [PubMed] [Google Scholar]

[bib47] 47. Carpenter SR, Brock WA., 2004. Spatial complexity, resilience and policy diversity: fishing on lake-rich landscapes. Ecol Soc. 9(1):8. [Google Scholar]

[bib48] 48. Pedregosa F, et al. , 2011. Scikit-learn: machine learning in Python. J Mach Learn Res. 12(85):2825–2830. [Google Scholar]

[bib49] 49. Jeffreys H. 1998. Theory of probability. Oxford: OUP. [Google Scholar]

[bib50] 50. Reed DH, Hobbs GR., 2004. The relationship between population size and temporal variability in population size. Animal Conserv. 7(1):1–8. [Google Scholar]

[bib51] 51. Steele JH., 1985. A comparison of terrestrial and marine ecological systems. Nature. 313(6001):355–358. [Google Scholar]

[bib52] 52. Vasseur DA, Yodzis P., 2004. The color of environmental noise. Ecology. 85(4):1146–1152. [Google Scholar]

[bib53] 53. Pope J., 1996. An evaluation of the stock structure of North Sea cod, haddock, and whiting since 1920, together with a consideration of the impacts of fisheries and predation effects on their biomass and recruitment. ICES J Marine Sci. 53(6):1157–1169. [Google Scholar]

[bib54] 54. Virtanen P, et al. , 2020. SciPy 1.0: fundamental algorithms for scientific computing in Python. Nat Methods. 17:261–272. [DOI] [PMC free article] [PubMed] [Google Scholar]

[bib55] 55. Strogatz SH. 2018. Nonlinear dynamics and chaos. With applications to physics, biology, chemistry and engineering. 2nd edn. Boulder: Westview Press. [Google Scholar]

[bib56] 56. von der Linden W, Dose V, von Tussaint U. Bayesian probability theory. Applications in the physical sciences. Cambridge: Cambridge University Press. [Google Scholar]

[bib57] 57. Foreman-Mackey D, Hogg DW, Lang D, Goodman J., 2013. emcee: The MCMC hammer. Publ Astron Soc Pacific. 125(925):306–312. [Google Scholar]

[bib58] 58. Seabold S, Perktold J. 2010. Statsmodels: Econometric and statistical modeling with python. 9th Python Sci Conf. 57: 92–96. [Google Scholar]

[bib59] 59. Harris CR, et al. , 2020. Array programming with NumPy. Nature. 585(7825):357–362. [DOI] [PMC free article] [PubMed] [Google Scholar]

PERMALINK

Quantifying resilience and the risk of regime shifts under strong correlated noise

Martin Heßler

Oliver Kamps

Roles

Abstract

Significance Statement.

Introduction

Ecological model

Fig. 1.

Fig. 2.

Results

Drift slope analysis

Fig. 3.

Comparison of leading indicators’ performance

Table 1.

Table 2.

Window size limits

Table 3.

Table 4.

Summary and Conclusion

Materials and Methods

Ecological model

Table 5.

Deseasonalization

Numerical method

Prior choice

Bayes factors and model comparison

Statistical leading indicators

Supplementary Material

ACKNOWLEDGEMENTS

Notes

Contributor Information

Funding

Authors' Contributions

Preprints

Data Availability

References

Associated Data

Supplementary Materials

Data Availability Statement

ACTIONS

PERMALINK

RESOURCES

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