Bayesian analysis of early warning signals using a time-dependent model

Myrvoll-Nilsen, Eirik; Hallali, Luc; Rypdal, Martin

doi:https://doi.org/10.5194/egusphere-2024-436

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https://doi.org/10.5194/egusphere-2024-436

Preprints

19 Feb 2024

| 19 Feb 2024

Bayesian analysis of early warning signals using a time-dependent model

Eirik Myrvoll-Nilsen, Luc Hallali, and Martin Rypdal

Abstract. A tipping point is defined by the IPCC as a critical threshold beyond which a system reorganizes, often abruptly and/or irreversibly. Tipping points can be crossed solely by internal variation in the system or by approaching a bifurcation point where the current state loses stability and forces the system to move to another stable state. It is currently debated whether or not Dansgaard-Oeschger (DO) events, abrupt warmings occurring during the last glacial period, are noise-induced or caused by the system reaching a bifurcation point. It can be shown that before a bifurcation point is reached there are observable changes in the statistical properties of the state variable. These are known as early warning signals and include increased fluctuation and correlation time. To express this behaviour we propose a new model based on the well-known first order autoregressive process (AR), with modifications to the correlation parameter such that it depends linearly on time. In order to estimate the time evolution of the correlation parameter we adopt a hierarchical Bayesian modeling framework, from which Bayesian analysis can be performed using the methodology of integrated nested Laplace approximations. We then apply the model to segments of the oxygen isotope ratios from the Northern Greenland Ice Core Project record corresponding to 17 DO events. Early warning signals were detected and found statistically significant for a number of DO events, suggesting that such events could indeed be caused by approaching a bifurcation point. The methodology developed to perform the given early warning analyses can be applied more generally, and is publicly available as the R-package INLA.ews.

Received: 14 Feb 2024 – Discussion started: 19 Feb 2024

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Journal article(s) based on this preprint

24 Sep 2025

Bayesian analysis of early warning signals using a time-dependent model

Eirik Myrvoll-Nilsen, Luc Hallali, and Martin Rypdal

Earth Syst. Dynam., 16, 1539–1556, https://doi.org/10.5194/esd-16-1539-2025,https://doi.org/10.5194/esd-16-1539-2025, 2025

Short summary

Eirik Myrvoll-Nilsen, Luc Hallali, and Martin Rypdal

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2024-436', Anonymous Referee #1, 14 Apr 2024
Referee Report on EGUsphere-2024-436
‘Bayesian analysis of early warning signals using a time-dependent
model’

General comments

The ms ‘Bayesian analysis of early warning signals using a time-dependent
model’ interprets geoscience time series containing DO events through the lens of an AR1 process and assumes a low-order Taylor expansion time-dependent propagator. The propagator’s parameters are determined through Bayesian learning from the time series’ segments that are quasi-stationary. From the analysis, the authors identify some of the DO events as bifurcations, while others are seen as merely noise-induced.

To the best of my knowledge, utilizing a time-dependent AR1 process to diagnose a system approaching a bifurcation is novel. By definition, data are exploited best by utilizing a statistic that explicitly contains the time-dependence, as against utilizing a moving window in combination with a static AR1 process. Here the ms provides a great service to the geoscience community in demonstrating such an approach can be implemented. I very much would like to see this ms being published with a ‘peer-reviewed’ status in an EGU journal.

However, the ms should be modified in two main respects. Firstly, it should become clearer what is the domain of applicability of the utilized method. When analyzing time series through the lens of an AR1 process, one lives in a quadratic approximation of the potential as shown in Fig1. This in turn can only be justified in a small noise expansion. However, the ms provides no hint why the small noise expansion might be justified. Quite the contrary, the Conclusions section suggests that a subset of DO events is rather noise- than bifurcation-induced. So, the noise level is seen as large enough to trigger jumping to another equilibrium. This raises doubts whether a small noise expansion is compatible with the time series at hand.

Secondly, I would expect that most readers of EGUsphere are no trained statisticians. Most natural scientists might have heard of Bayes’ formula. However, it might be useful to recover the Bayes’ principle in a short Appendix. I personally found some of the wording on page 8 inaccessible, such as ‘latent field’. I find it necessary that anything transcending elementary Bayesian learning is clearly defined somewhere – either in the main text (preferred) or an Appendix. So here I am asking for a didactical upgrade of the statistical method used in view of a natural science audience.

Overall, a timely and exciting to read article which is apparently on a very high technical level.

Technical corrections

P4: On the history of early warning systems, the following additions might be in order. (1) First mentioning of a noise-induced precursor of a bifurcation: Wiesenfeld (1985), (2) first extension to a complex system, justifying 1D AR1: Held and Kleinen (2004), (3) first utilization on real data (in fact, ice core data): Dakos et al. (2008).

L90-100: How does this § relate the other parts of the ms? Later on, an AR1 process is utilized, while this § seems to suggest that it should not be utilized. Furthermore, the Green’s function as of Eq9 might be interpreted as a superposition of Green’s functions as of Eq7 which could easily occur in multi-dimensional systems. Should one then simply expand the presented formalism to larger dimensions than one? So, I am confused about the logical positioning of that § in the overall ms.

I have issues following the rationale of Section 3.1. In the context of Bayesian learning, I would expect that an observation variable is defined (is it x or is x observed through an additive layer of noise – please clarify), and the conditional probability of an observation (in our case a time-correlated time series) on uncertain parameters is presented. In this context, I do not understand the definitions of eta, beta, z, y, for what reason I need to postpone my review of that part to another iteration.

Eq29: Why do we need non-constant time-steps? Is it due to missing data?

Caption of Fig5: What are the intervals showing? Are they 2.5-97.5% quantiles of the posteriors of those variables? Or are they confidence intervals in the frequentist’s sense? Would the latter logically consistent with a Bayesian setup?

L110: Why are you utilizing a Bayesian approach at all? You emphasize the benefit of having a PDF as output, which has tremendous advantages when later being utilized in economic decision theory. However, you do not mention the drawback of the Bayesian approach: That you need to justify the choice of a prior. What is your justification and what prior was chosen? Is it a ‘vague’ prior (L185, whatever that means). In the whole ms, I could not find a single reason why a Bayesian approach as against a frequentist’s approach was necessary. The utilized likelihoods could likewise have been utilized for a frequentist’s approach. Hence, some more motivation of the choice would be helpful.

Specific comments

L168: stochastic -> probabilistic? (to distinguish a situation from aleatoric uncertainty?)

Define ‘hierarchical Bayesian model’.

Define kappa before it is utilized in Eq23.

Activate ‘day month year’ (eg L330, to be found more than once in the ms).

Literature

Dakos, V.; M. Scheffer; E.H. Van Nes; V. Brovkin; V. Petoukhov; and H. Held. 2008. Slowing down as an early warning signal for abrupt climate change. Proceedings of the National Academy of Sciences 105:14308-14312.
Held, H. and T. Kleinen. 2004. Detection of climate system bifurcations by degenerate fingerprinting. Geophysical Research Letters 31:L23207.
Wiesenfeld, K. 1985. Virtual Hopf phenomenon: A new precursor of period-doubling bifurcations. Physical Review A 32:1744.
Citation: https://doi.org/10.5194/egusphere-2024-436-RC1
- AC1: 'Reply on RC1', Eirik Myrvoll-Nilsen, 15 May 2024
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2024/egusphere-2024-436/egusphere-2024-436-AC1-supplement.pdf
  
  Citation: https://doi.org/10.5194/egusphere-2024-436-AC1
RC2:
'Comment on egusphere-2024-436', Anonymous Referee #2, 15 Apr 2024

The review is provided as a pdf and a word document in the attached zip folder. The design of the study and the structure of the manuscript are generally sound. However, the overall assessment 'reconsider after major revisions' is due to the large number of technical inaccuracies.

Citation: https://doi.org/10.5194/egusphere-2024-436-RC2
- AC2: 'Reply on RC2', Eirik Myrvoll-Nilsen, 15 May 2024
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2024/egusphere-2024-436/egusphere-2024-436-AC2-supplement.pdf
  
  Citation: https://doi.org/10.5194/egusphere-2024-436-AC2

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2024-436', Anonymous Referee #1, 14 Apr 2024
Referee Report on EGUsphere-2024-436
‘Bayesian analysis of early warning signals using a time-dependent
model’

General comments

The ms ‘Bayesian analysis of early warning signals using a time-dependent
model’ interprets geoscience time series containing DO events through the lens of an AR1 process and assumes a low-order Taylor expansion time-dependent propagator. The propagator’s parameters are determined through Bayesian learning from the time series’ segments that are quasi-stationary. From the analysis, the authors identify some of the DO events as bifurcations, while others are seen as merely noise-induced.

To the best of my knowledge, utilizing a time-dependent AR1 process to diagnose a system approaching a bifurcation is novel. By definition, data are exploited best by utilizing a statistic that explicitly contains the time-dependence, as against utilizing a moving window in combination with a static AR1 process. Here the ms provides a great service to the geoscience community in demonstrating such an approach can be implemented. I very much would like to see this ms being published with a ‘peer-reviewed’ status in an EGU journal.

However, the ms should be modified in two main respects. Firstly, it should become clearer what is the domain of applicability of the utilized method. When analyzing time series through the lens of an AR1 process, one lives in a quadratic approximation of the potential as shown in Fig1. This in turn can only be justified in a small noise expansion. However, the ms provides no hint why the small noise expansion might be justified. Quite the contrary, the Conclusions section suggests that a subset of DO events is rather noise- than bifurcation-induced. So, the noise level is seen as large enough to trigger jumping to another equilibrium. This raises doubts whether a small noise expansion is compatible with the time series at hand.

Secondly, I would expect that most readers of EGUsphere are no trained statisticians. Most natural scientists might have heard of Bayes’ formula. However, it might be useful to recover the Bayes’ principle in a short Appendix. I personally found some of the wording on page 8 inaccessible, such as ‘latent field’. I find it necessary that anything transcending elementary Bayesian learning is clearly defined somewhere – either in the main text (preferred) or an Appendix. So here I am asking for a didactical upgrade of the statistical method used in view of a natural science audience.

Overall, a timely and exciting to read article which is apparently on a very high technical level.

Technical corrections

P4: On the history of early warning systems, the following additions might be in order. (1) First mentioning of a noise-induced precursor of a bifurcation: Wiesenfeld (1985), (2) first extension to a complex system, justifying 1D AR1: Held and Kleinen (2004), (3) first utilization on real data (in fact, ice core data): Dakos et al. (2008).

L90-100: How does this § relate the other parts of the ms? Later on, an AR1 process is utilized, while this § seems to suggest that it should not be utilized. Furthermore, the Green’s function as of Eq9 might be interpreted as a superposition of Green’s functions as of Eq7 which could easily occur in multi-dimensional systems. Should one then simply expand the presented formalism to larger dimensions than one? So, I am confused about the logical positioning of that § in the overall ms.

I have issues following the rationale of Section 3.1. In the context of Bayesian learning, I would expect that an observation variable is defined (is it x or is x observed through an additive layer of noise – please clarify), and the conditional probability of an observation (in our case a time-correlated time series) on uncertain parameters is presented. In this context, I do not understand the definitions of eta, beta, z, y, for what reason I need to postpone my review of that part to another iteration.

Eq29: Why do we need non-constant time-steps? Is it due to missing data?

Caption of Fig5: What are the intervals showing? Are they 2.5-97.5% quantiles of the posteriors of those variables? Or are they confidence intervals in the frequentist’s sense? Would the latter logically consistent with a Bayesian setup?

L110: Why are you utilizing a Bayesian approach at all? You emphasize the benefit of having a PDF as output, which has tremendous advantages when later being utilized in economic decision theory. However, you do not mention the drawback of the Bayesian approach: That you need to justify the choice of a prior. What is your justification and what prior was chosen? Is it a ‘vague’ prior (L185, whatever that means). In the whole ms, I could not find a single reason why a Bayesian approach as against a frequentist’s approach was necessary. The utilized likelihoods could likewise have been utilized for a frequentist’s approach. Hence, some more motivation of the choice would be helpful.

Specific comments

L168: stochastic -> probabilistic? (to distinguish a situation from aleatoric uncertainty?)

Define ‘hierarchical Bayesian model’.

Define kappa before it is utilized in Eq23.

Activate ‘day month year’ (eg L330, to be found more than once in the ms).

Literature

Dakos, V.; M. Scheffer; E.H. Van Nes; V. Brovkin; V. Petoukhov; and H. Held. 2008. Slowing down as an early warning signal for abrupt climate change. Proceedings of the National Academy of Sciences 105:14308-14312.
Held, H. and T. Kleinen. 2004. Detection of climate system bifurcations by degenerate fingerprinting. Geophysical Research Letters 31:L23207.
Wiesenfeld, K. 1985. Virtual Hopf phenomenon: A new precursor of period-doubling bifurcations. Physical Review A 32:1744.
Citation: https://doi.org/10.5194/egusphere-2024-436-RC1
- AC1: 'Reply on RC1', Eirik Myrvoll-Nilsen, 15 May 2024
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2024/egusphere-2024-436/egusphere-2024-436-AC1-supplement.pdf
  
  Citation: https://doi.org/10.5194/egusphere-2024-436-AC1
RC2:
'Comment on egusphere-2024-436', Anonymous Referee #2, 15 Apr 2024

The review is provided as a pdf and a word document in the attached zip folder. The design of the study and the structure of the manuscript are generally sound. However, the overall assessment 'reconsider after major revisions' is due to the large number of technical inaccuracies.

Citation: https://doi.org/10.5194/egusphere-2024-436-RC2
- AC2: 'Reply on RC2', Eirik Myrvoll-Nilsen, 15 May 2024
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2024/egusphere-2024-436/egusphere-2024-436-AC2-supplement.pdf
  
  Citation: https://doi.org/10.5194/egusphere-2024-436-AC2

Peer review completion

AR: Author's response | RR: Referee report | ED: Editor decision | EF: Editorial file upload

ED: Reconsider after major revisions (12 Jul 2024) by Jonathan Donges

AR by Eirik Myrvoll-Nilsen on behalf of the Authors (23 Aug 2024) Author's response Author's tracked changes Manuscript

ED: Referee Nomination & Report Request started (11 Sep 2024) by Jonathan Donges

RR by Anonymous Referee #1 (26 Sep 2024)

RR by Chris Boulton (28 Mar 2025)

Suggestions for revision or reasons for rejection

I found the manuscript ‘Bayesian analysis of early warning signals using a time-dependent model’ by Myrvoll-Nilsen et al. to be an interesting read, especially as someone who uses these types of generic early warning signals often, and has to make decisions over the length of a window used to calculate them on. Furthermore, any method that can provide a sense of certainty through Bayesian statistics is building upon current techniques.

I realise I am seeing this manuscript for the first time after a round of revisions and after looking through the authors’ responses I can see that the manuscript has been improved greatly. I only really have very minor suggestions which I have detailed below:

Generally: It is for sure a personal choice, but I would find ‘autocorrelation parameter’ rather than ‘correlation parameter’ throughout the manuscript to be easier to follow, particularly in the abstract.

Generally: Don’t forget to update the access dates, on the R package particularly.

Figure 1: It might be worth making clear in the figure caption that for panel b, the red and black points are on top of each other in the part where you say they eventually collapse into each other. Also, here the 3rd from bottom line at the end should say ‘The red line represents’, the s is missing.

Figure 2: The start of the caption says ‘n=500’ but it is actually for n=500 and n=1000 for the figure overall so perhaps refer to these only for the specific panels later in the caption. It is worth noting that the outliers are shown as points too.

Section 4.2: Most likely as part of Figure 5, it would be good to know how many data points are used in the analysis for each event.

Figure 5: Again personal choice but I would swap the top and bottom rows so the figure reads from earliest in the top left to latest in the bottom right, this may also mean reordering Figure 6.

Line ~265: Is it possible to delve into how much agreement there is between the Rypdal and Boers results in Table 2? Clearly there is agreement with Event 2, and with Rypdal in Event 5 etc. There is a similar level of agreement with both papers, compared to the agreement the Rypdal and Boers papers have themselves. Without fully counting the instances myself, comparing these agreements across all 3 looks like it could show this method finds things that the other papers themselves disagree with on a number of occasions.

Line 375: Need an ‘s’ on represents.

Line 380: Need an ‘s’ on depends.

Hide

ED: Publish subject to minor revisions (review by editor) (01 Apr 2025) by Jonathan Donges

AR by Eirik Myrvoll-Nilsen on behalf of the Authors (11 Apr 2025) Author's response Author's tracked changes Manuscript

ED: Publish as is (12 Apr 2025) by Jonathan Donges

AR by Eirik Myrvoll-Nilsen on behalf of the Authors (19 Apr 2025)

Post-review adjustments

AA: Author's adjustment | EA: Editor approval

AA by Eirik Myrvoll-Nilsen on behalf of the Authors (08 Sep 2025) Author's adjustment Manuscript

EA: Adjustments approved (22 Sep 2025) by Jonathan Donges

Journal article(s) based on this preprint

24 Sep 2025

Bayesian analysis of early warning signals using a time-dependent model

Eirik Myrvoll-Nilsen, Luc Hallali, and Martin Rypdal

Earth Syst. Dynam., 16, 1539–1556, https://doi.org/10.5194/esd-16-1539-2025,https://doi.org/10.5194/esd-16-1539-2025, 2025

Short summary

Eirik Myrvoll-Nilsen, Luc Hallali, and Martin Rypdal

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Before a climate component reaches a tipping point, there may be observable changes in its statistical properties. These are known as early warning signals and include increased fluctuation and correlation times. We present a Bayesian approach to detect these signals, using a model where the correlation parameter depends linearly on time for which the slope can be estimated directly from the data. The model is then applied to Dansgaard-Oeschger events using Greenland Ice core data.


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