Critical slowing down associated with critical transition and risk of collapse in crypto-currency

The year 2017 saw the rise and fall of the crypto-currency market, followed by high variability in the price of all crypto-currencies. In this work, we study the abrupt transition in crypto-currency residuals, which is associated with the critical transition (the phenomenon of critical slowing down) or the stochastic transition phenomena. We find that, regardless of the specific crypto-currency or rolling window size, the autocorrelation always fluctuates around a high value, while the standard deviation increases monotonically. Therefore, while the autocorrelation does not display the signals of critical slowing down, the standard deviation can be used to anticipate critical or stochastic transitions. In particular, we have detected two sudden jumps in the standard deviation, in the second quarter of 2017 and at the beginning of 2018, which could have served as the early warning signals of two major price collapses that have happened in the following periods. We finally propose a mean-field phenomenological model for the price of crypto-currency to show how the use of the standard deviation of the residuals is a better leading indicator of the collapse in price than the time-series' autocorrelation. Our findings represent a first step towards a better diagnostic of the risk of critical transition in the price and/or volume of crypto-currencies.


Recommendation?
Major revision is needed (please make suggestions in comments)

Comments to the Author(s)
Dear Authors, I hope that you find the feedback below to be helpful.

METHODS & ARGUMENT
-Please prove that the residuals metrics are leading rather than coincidental --or even lagging -the price. I understand that the Gaussian smoothing used is not causal. Is the smoothing updated with each time step, or is it only done once on the full dataset hence using future information? To visualize, the window around the 2018 crash in Fig 1A and 2A should be zoomed in and compared with the price in a figure. Further, there have been many "mini" and a few not so mini bubbles in the history of Bitcoin. However, volatility of returns (and I guess filter residuals) tends to spike after rather than before. How does this reconcile with the one instance argued in the manuscript? Further, in Fig 3A the AR1 signal seems far too noisy to provide any predictive information. Not to mention that no threshold or hazard function for regime changes is specified.
-In figure 1, as the authors suggest, 2017 was a year of high volatility. This is reflected in the residuals (as well as in the returns --not shown). Mixing these two low and high volatility regimes as done in Fig 2B is probably mostly the cause of the increasing trending of residual standard deviation. This concern is largely confirmed by Fig 3A. In terms of the jumps in the residual standard deviation, it seems that there are plenty of jumps in the short history of Bitcoin and other cryptos which are not predictive of a tipping point. Hence Fig 1B seems not to be predictive. Please resolve.
-These metrics rely heavily on the validity of the detrending by filter, as the Std and AR1 measures largely just measure the size and persistence of deviations from the filter line. How can you justify this filtering as consistently recovering the trend, where it is often thought that bubbles form super-exponential trajectories? E.g., the jumps in autocorrelation and standard deviation observed in Fig 2A can simply mean that there is growth that is too fast for your "30 day bandwidth" filter (unsurprising given the periods of explosive growth). Hence, the approach of [2] is more direct.
More minor points: -Is AR(1) sufficient? How much structure is left in the residuals of the AR(1) fit? -De-trending is easier said than done and can introduce spurious autocorrelations (see: Slutsky-Yule effect). Further, the unit root tests are far from perfect, and not very powerful, meaning that they may miss consequential non-stationarities left behind by the filter. COMPOSITION Intro: -I would leave the background on cryptocurrency to an appendix, or better still, simply point to a more standard reference if available.
-The paper is a bit long on summarizing the ElBaharawy el al. findings, whose prediction of the bitcoin market share has proven to be pretty far off. You may wish to mention the view given in [1], which makes different assumptions.
-Further, in [2] similar authors have looked at cryptocurrency bubbles, with a more specific view on speculative bubbles as a critical phenomena, with tipping point predictability. This should be reflected.

4
Overall, the authors carefully apply an existing methodology to a specific dataset of cryptocurrencies, provide a statistical analysis of the time series of prices, and find a partial indication of a possibly interesting direction of research in the field of cryptocurrencies.
Based on these observations, I would suggest the authors to revise and resubmit the paper to a journal more specifically interested in the quantitative study of the cryptocurrencies market.

Review form: Reviewer 3
Is the manuscript scientifically sound in its present form? No

Comments to the Author(s)
The authors aim at anticipating the cryptocurrency market crisis by analyzing the prices series of few selected currencies. Specifically, they look for evidence for the "critical slowing down" phenomenon, which occur in any continuous model approaching a fold bifurcation. Their analysis consists in measuring the autocorrelation and the standard deviation of the residuals time series over a sliding window. The problem of anticipating the cryptocurrency market crisis is extremely interesting within the literature and relevant for a variety of stakeholders. The approach adopted is potentially interesting but the design of the article and the analysis is extremely limited in the current state.
First, the authors provide no theoretical framework justifying the choice to adopt the "critical slowing down" phenomenon approach. Critical slowing down is observed as a system approaches a fold bifurcation point, but, from the current article, it is unclear why and when exactly the cryptocurrency market would undergo such a transition. Literature in finance has suggested that critical slowing down indicators are weak to predict to global financial crisis, so it is not evident that critical transitions in financial markets are preceded by critical slowing down. The authors should definitely provide a section reviewing literature in finance and a more solid theoretical background.
Then, I find the scope of the analysis unclear and the tools used very limited. The authors should provide a clear description of what they define as a "critical transitions" in the cryptocurrency market, and how long before a transition their method allows to anticipate it. The current method is qualitative, since the detection of periods where the standard deviation and autocorrelation are supposedly "increasing" is not automatized. The analysis focuses only on few selected currencies, and the role played by the temporal sampling of data is not discussed.
Finally, there is little care for the figures design. Figure  Manuscript ID RSOS-181097 entitled "Critical slowing down associated with critical transition and risk of collapse in cryptocurrency" which you submitted to Royal Society Open Science, has been reviewed. The comments from reviewers are included at the bottom of this letter.
In view of the criticisms of the reviewers, the manuscript has been rejected in its current form. However, a new manuscript may be submitted which takes into consideration these comments.
Please note that resubmitting your manuscript does not guarantee eventual acceptance, and that your resubmission will be subject to peer review before a decision is made.
You will be unable to make your revisions on the originally submitted version of your manuscript. Instead, revise your manuscript and upload the files via your author centre.
Once you have revised your manuscript, go to https://mc.manuscriptcentral.com/rsos and login to your Author Center. Click on "Manuscripts with Decisions," and then click on "Create a Resubmission" located next to the manuscript number. Then, follow the steps for resubmitting your manuscript.
Your resubmitted manuscript should be submitted by 04-Jul-2019. If you are unable to submit by this date please contact the Editorial Office.
Please note that Royal Society Open Science will introduce article processing charges for all new submissions received from 1 January 2018. Charges will also apply to papers transferred to Royal Society Open Science from other Royal Society Publishing journals, as well as papers submitted as part of our collaboration with the Royal Society of Chemistry (http://rsos.royalsocietypublishing.org/chemistry). If your manuscript is submitted and accepted for publication after 1 Jan 2018, you will be asked to pay the article processing charge, unless you request a waiver and this is approved by Royal Society Publishing. You can find out more about the charges at http://rsos.royalsocietypublishing.org/page/charges. Should you have any queries, please contact openscience@royalsociety.org.
We look forward to receiving your resubmission. Comments to the Author: Three reviewers have assessed your paper. The broad view is that it is not yet ready for publication. However, as each reviewer provides extensive feedback for improvement, the journal is willing to consider a substantially revised manuscript that fully addresses the concerns raised by the referees. If, for whatever reason, you cannot incorporate their requested changes, you should provide a fully reasoned scientific rebuttal for excluding the change. If you resubmit, the manuscript will be returned to at least these reviewers for their advice. If they remain unsatisfied by your changes, we will not be able to consider any further revisions unless their are exceptional reasons for doing so. Good luck and we look forward to receiving the resubmission.
Reviewers' Comments to Author: Reviewer: 1 Comments to the Author(s) Dear Authors, I hope that you find the feedback below to be helpful.

METHODS & ARGUMENT
-Please prove that the residuals metrics are leading rather than coincidental --or even lagging -the price. I understand that the Gaussian smoothing used is not causal. Is the smoothing updated with each time step, or is it only done once on the full dataset hence using future information? To visualize, the window around the 2018 crash in Fig 1A and 2A should be zoomed in and compared with the price in a figure. Further, there have been many "mini" and a few not so mini bubbles in the history of Bitcoin. However, volatility of returns (and I guess filter residuals) tends to spike after rather than before. How does this reconcile with the one instance argued in the manuscript? Further, in Fig 3A the AR1 signal seems far too noisy to provide any predictive information. Not to mention that no threshold or hazard function for regime changes is specified.
-In figure 1, as the authors suggest, 2017 was a year of high volatility. This is reflected in the residuals (as well as in the returns --not shown). Mixing these two low and high volatility regimes as done in Fig 2B is probably mostly the cause of the increasing trending of residual standard deviation. This concern is largely confirmed by Fig 3A. In terms of the jumps in the residual standard deviation, it seems that there are plenty of jumps in the short history of Bitcoin and other cryptos which are not predictive of a tipping point. Hence Fig 1B seems not to be predictive. Please resolve.
-These metrics rely heavily on the validity of the detrending by filter, as the Std and AR1 measures largely just measure the size and persistence of deviations from the filter line. How can you justify this filtering as consistently recovering the trend, where it is often thought that bubbles form super-exponential trajectories? E.g., the jumps in autocorrelation and standard deviation observed in Fig 2A can simply mean that there is growth that is too fast for your "30 day bandwidth" filter (unsurprising given the periods of explosive growth). Hence, the approach of [2] is more direct.
More minor points: -Is AR(1) sufficient? How much structure is left in the residuals of the AR(1) fit? -De-trending is easier said than done and can introduce spurious autocorrelations (see: Slutsky-Yule effect). Further, the unit root tests are far from perfect, and not very powerful, meaning that they may miss consequential non-stationarities left behind by the filter. COMPOSITION Intro: -I would leave the background on cryptocurrency to an appendix, or better still, simply point to a more standard reference if available.
-The paper is a bit long on summarizing the ElBaharawy el al. findings, whose prediction of the bitcoin market share has proven to be pretty far off. You may wish to mention the view given in [1], which makes different assumptions.
-Further, in [2] similar authors have looked at cryptocurrency bubbles, with a more specific view on speculative bubbles as a critical phenomena, with tipping point predictability. This should be reflected.
Reviewer: 2 Comments to the Author(s) Review of "Critical slowing down associated with critical transition and risk of collapse in cryptocurrency" The paper investigates how simple metrics, coming from a general theory of critical transitions, can serve as a early-warning signal for the risk of large changes in cryptocurrencies. The authors apply a statistical analysis to a set of time series of prices of cryptocurrencies, aimed at identifying warning signals of catastrophic changes, as inspired by ecological literature, Three main comments arise from reading this paper on cryptocurrencies: -the authors don't make any theoretical contribution to the theory of critical slowing down for critical transitions detection, -the authors don't make specific efforts in understanding why the small set of cryptocurrencies studied, which represent a very specific context for testing such general theory, should be particularly relevant or informative for the general theory, and how their results quantitatively relate to results in other fields, e.g. ecology, -the authors appear to identify in total two individual cases, which might be considered as "true positive" predictions of the theory in this context, although they don't specify a quantitative binary criterion for predicting large changes in a near future, so it is not clear at this stage whether these two cases are actually predicted by their theory and whether the theory would predict also other large changes that did not occur.
Overall, the authors carefully apply an existing methodology to a specific dataset of cryptocurrencies, provide a statistical analysis of the time series of prices, and find a partial indication of a possibly interesting direction of research in the field of cryptocurrencies.
Based on these observations, I would suggest the authors to revise and resubmit the paper to a journal more specifically interested in the quantitative study of the cryptocurrencies market.
Reviewer: 3 Comments to the Author(s) The authors aim at anticipating the cryptocurrency market crisis by analyzing the prices series of few selected currencies. Specifically, they look for evidence for the "critical slowing down" phenomenon, which occur in any continuous model approaching a fold bifurcation. Their analysis consists in measuring the autocorrelation and the standard deviation of the residuals time series over a sliding window.

8
The problem of anticipating the cryptocurrency market crisis is extremely interesting within the literature and relevant for a variety of stakeholders. The approach adopted is potentially interesting but the design of the article and the analysis is extremely limited in the current state.
First, the authors provide no theoretical framework justifying the choice to adopt the "critical slowing down" phenomenon approach. Critical slowing down is observed as a system approaches a fold bifurcation point, but, from the current article, it is unclear why and when exactly the cryptocurrency market would undergo such a transition. Literature in finance has suggested that critical slowing down indicators are weak to predict to global financial crisis, so it is not evident that critical transitions in financial markets are preceded by critical slowing down. The authors should definitely provide a section reviewing literature in finance and a more solid theoretical background.
Then, I find the scope of the analysis unclear and the tools used very limited. The authors should provide a clear description of what they define as a "critical transitions" in the cryptocurrency market, and how long before a transition their method allows to anticipate it. The current method is qualitative, since the detection of periods where the standard deviation and autocorrelation are supposedly "increasing" is not automatized. The analysis focuses only on few selected currencies, and the role played by the temporal sampling of data is not discussed.
Finally, there is little care for the figures design. Figure

Are the interpretations and conclusions justified by the results? No
Is the language acceptable? Yes

Do you have any ethical concerns with this paper? No
Have you any concerns about statistical analyses in this paper? Yes

Recommendation?
Major revision is needed (please make suggestions in comments)

Comments to the Author(s)
The authors present a methodology to anticipate collapses in cryptocurrency prices. The method is based on the theory of "Critical Slowing Down" phenomenon, which suggests that tipping points are preceded by an increase in autocorrelation and standard deviation of price fluctuations. The authors find no evidence for increase in autocorrelation, but they find some examples in which collpases of cryptocurrency prices were preceeded by an increase of standard deviation.
Anticipating cryptocurrency prices and their market behaviour is of paramount importance for stakeholder and investors in the cryptocurrency market. The article is clearly written and suitable for the broad readership of Royal Society Open Science. However, I have some concerns regarding the methodology that should be addressed for the article to be ready for publication. Namely, I find the author should implement a clear and well-described procedure to identify "price collapse" events. Then, they should evaluate the ability of their presented methodology in identifying such events. The way the article is written now (results of these procedure are presented based on few successful examples), there is no way to evaluate if the methodology is meaningful or not. Also, there is an overlap between the period considered for the "early detection" and the effective "price collapse event" (see table 2). This should be resolved to make sure the method is effective.
Below, I present some more detailed comments: Page 3, line 7. "These results illustrate that the random drift and the creation at random times of new crypto-currencies (speciation) underlie the emergence of neutral conditions." Please add "may underlie".  Table 2. Please define more clearly how you identified "events". There is an overlap between the periods corresponding to the warning signal and the event. Are you sure that the early warning signal detected effectively occurs before the drop in price? Page 9, line 27. Compilation error.

Review form: Reviewer 4
Is the manuscript scientifically sound in its present form? Yes

Are the interpretations and conclusions justified by the results? Yes
Is the language acceptable? Yes Do you have any ethical concerns with this paper? No

Recommendation? Accept as is
Comments to the Author(s) The paper presents an interesting analysis of cryptocurrency volatility, drawing parallels with ecological systems. The claim is that by utilizing rigorous statistical analysis, one extracts the amount of volatility by computing the standard deviation, autocorrelation coefficients, and other statistics in the cyrptocurrency pricing data, and uses it to predict downturns in the broader market. The authors use daily pricing data on several major cryptocurrencies, including Bitcoin, in their analysis. The pricing data is run through preprocessing before analysis which seems consistent with typical data analysis. The conclusion is that the resulting autocorrelation coefficients exhibit a period of increased volatility before a downturn in prices.

12-Dec-2019
Dear Dr Tu, The Subject Editor assigned to your paper ("Critical slowing down associated with critical transition and risk of collapse in cryptocurrency") has now received comments from reviewers. We would like you to revise your paper in accordance with the referee and Associate Editor suggestions which can be found below (not including confidential reports to the Editor). Please note this decision does not guarantee eventual acceptance.
Please submit a copy of your revised paper before 04-Jan-2020. Please note that the revision deadline will expire at 00.00am on this date. If we do not hear from you within this time then it will be assumed that the paper has been withdrawn. In exceptional circumstances, extensions may be possible if agreed with the Editorial Office in advance. We do not allow multiple rounds of revision so we urge you to make every effort to fully address all of the comments at this stage. If deemed necessary by the Editors, your manuscript will be sent back to one or more of the original reviewers for assessment. If the original reviewers are not available we may invite new reviewers.
To revise your manuscript, log into http://mc.manuscriptcentral.com/rsos and enter your Author Centre, where you will find your manuscript title listed under "Manuscripts with Decisions." Under "Actions," click on "Create a Revision." Your manuscript number has been appended to denote a revision. Revise your manuscript and upload a new version through your Author Centre.
When submitting your revised manuscript, you must respond to the comments made by the referees and upload a file "Response to Referees" in "Section 6 -File Upload". Please use this to document how you have responded to each of the comments, and the adjustments you have made. In order to expedite the processing of the revised manuscript, please be as specific as possible in your response.
In addition to addressing all of the reviewers' and editor's comments please also ensure that your revised manuscript contains the following sections before the reference list: • Ethics statement If your study uses humans or animals please include details of the ethical approval received, including the name of the committee that granted approval. For human studies please also detail whether informed consent was obtained. For field studies on animals please include details of all permissions, licences and/or approvals granted to carry out the fieldwork.
• Data accessibility It is a condition of publication that all supporting data are made available either as supplementary information or preferably in a suitable permanent repository. The data accessibility section should state where the article's supporting data can be accessed. This section should also include details, where possible of where to access other relevant research materials such as statistical tools, protocols, software etc can be accessed. If the data has been deposited in an external repository this section should list the database, accession number and link to the DOI for all data from the article that has been made publicly available. Data sets that have been deposited in an external repository and have a DOI should also be appropriately cited in the manuscript and included in the reference list.
If you wish to submit your supporting data or code to Dryad (http://datadryad.org/), or modify your current submission to dryad, please use the following link: http://datadryad.org/submit?journalID=RSOS&manu=RSOS-191450 • Competing interests Please declare any financial or non-financial competing interests, or state that you have no competing interests.
• Authors' contributions All submissions, other than those with a single author, must include an Authors' Contributions section which individually lists the specific contribution of each author. The list of Authors should meet all of the following criteria; 1) substantial contributions to conception and design, or acquisition of data, or analysis and interpretation of data; 2) drafting the article or revising it critically for important intellectual content; and 3) final approval of the version to be published.
All contributors who do not meet all of these criteria should be included in the acknowledgements.
We suggest the following format: AB carried out the molecular lab work, participated in data analysis, carried out sequence alignments, participated in the design of the study and drafted the manuscript; CD carried out the statistical analyses; EF collected field data; GH conceived of the study, designed the study, coordinated the study and helped draft the manuscript. All authors gave final approval for publication.
• Acknowledgements Please acknowledge anyone who contributed to the study but did not meet the authorship criteria.
• Funding statement Please list the source of funding for each author.
Once again, thank you for submitting your manuscript to Royal Society Open Science and I look forward to receiving your revision. If you have any questions at all, please do not hesitate to get in touch. Comments to the Author(s) The authors present a methodology to anticipate collapses in cryptocurrency prices. The method is based on the theory of "Critical Slowing Down" phenomenon, which suggests that tipping points are preceded by an increase in autocorrelation and standard deviation of price fluctuations. The authors find no evidence for increase in autocorrelation, but they find some examples in which collpases of cryptocurrency prices were preceeded by an increase of standard deviation.
Anticipating cryptocurrency prices and their market behaviour is of paramount importance for stakeholder and investors in the cryptocurrency market. The article is clearly written and suitable for the broad readership of Royal Society Open Science. However, I have some concerns regarding the methodology that should be addressed for the article to be ready for publication. Namely, I find the author should implement a clear and well-described procedure to identify "price collapse" events. Then, they should evaluate the ability of their presented methodology in identifying such events. The way the article is written now (results of these procedure are presented based on few successful examples), there is no way to evaluate if the methodology is meaningful or not. Also, there is an overlap between the period considered for the "early detection" and the effective "price collapse event" (see table 2). This should be resolved to make sure the method is effective.
Below, I present some more detailed comments: Page 3, line 7. "These results illustrate that the random drift and the creation at random times of new crypto-currencies (speciation) underlie the emergence of neutral conditions." Please add "may underlie".  Table 2. Please define more clearly how you identified "events". There is an overlap between the periods corresponding to the warning signal and the event. Are you sure that the early warning signal detected effectively occurs before the drop in price? Page 9, line 27. Compilation error.

Reviewer: 4
Comments to the Author(s) The paper presents an interesting analysis of cryptocurrency volatility, drawing parallels with ecological systems. The claim is that by utilizing rigorous statistical analysis, one extracts the amount of volatility by computing the standard deviation, autocorrelation coefficients, and other statistics in the cyrptocurrency pricing data, and uses it to predict downturns in the broader market. The authors use daily pricing data on several major cryptocurrencies, including Bitcoin, in their analysis. The pricing data is run through preprocessing before analysis which seems consistent with typical data analysis. The conclusion is that the resulting autocorrelation coefficients exhibit a period of increased volatility before a downturn in prices.

Comments to the Author(s) My comments have been addressed.
Decision letter (RSOS-191450.R1)

27-Feb-2020
Dear Dr Tu, It is a pleasure to accept your manuscript entitled "Critical slowing down associated with critical transition and risk of collapse in cryptocurrency" in its current form for publication in Royal Society Open Science. The comments of the reviewer(s) who reviewed your manuscript are included at the foot of this letter.
Please ensure that you send to the editorial office an editable version of your accepted manuscript, and individual files for each figure and table included in your manuscript. You can send these in a zip folder if more convenient. Failure to provide these files may delay the processing of your proof. You may disregard this request if you have already provided these files to the editorial office.
You can expect to receive a proof of your article in the near future. Please contact the editorial office (openscience_proofs@royalsociety.org) and the production office (openscience@royalsociety.org) to let us know if you are likely to be away from e-mail contact --if you are going to be away, please nominate a co-author (if available) to manage the proofing process, and ensure they are copied into your email to the journal.
Due to rapid publication and an extremely tight schedule, if comments are not received, your paper may experience a delay in publication.
Please see the Royal Society Publishing guidance on how you may share your accepted author manuscript at https://royalsociety.org/journals/ethics-policies/media-embargo/.

Reply to the Reviewers' Comments:
Reviewer: 1 First of all, we want to thank the reviewer 1 for her/his careful reading of our work and very relevant and helpful comments. We have considered all the reviewer suggestions, making the related changes in the manuscript. When we have not incorporated the requested changes, we have provided reasoned and referenced explanations, and we have anyway rewritten the related parts in the manuscript to make them clearer. We feel that thank to reviewer 1, now our manuscript is greatly improved. We report reviewer's comments in italic.
-Please prove that the residuals metrics are leading rather than coincidental --or even lagging --the price. Research from complex systems suggest that residuals metrics can be used to compute generic early-warning signals that may indicate for a wide class of systems if a critical threshold is approaching 1 . In particular, it has been observed that fluctuations of population density increased in size and duration near the tipping point, in agreement with the theory 2 . Our work starts from the analogy of crypto-currency market as complex ecological systems and as also suggested in studies of early warning on financial market rising variability (in our case measured by residuals) could signal systemic risk 3 . We have added a paragraph to clarify this point.

I understand that the Gaussian smoothing used is not causal. Is the smoothing updated with each time step, or is it only done once on the full dataset hence using future information?
The Gaussian smoothing/filter is updated at each time step. In fact, we adopted Gaussian filter with radius 30 meaning that it uses the backward and forward local windows of 15 time steps. This is a standard filtering procedure also in other studies of cryptocurrency market 4-6 .
To visualize, the window around the 2018 crash in Fig 1A and 2A should be zoomed in and compared with the price in a figure. Considering the Reviewer's suggestion, we have re-plotted Fig 2. The new panels C and D zoom in the crash windows and compare with the price.

Volatility of returns (and I guess filter residuals) tends to spike after rather than before. How does this reconcile with the one instance argued in the manuscript?
A key difference between variance as we have computed and the measure of volatility is as follows. We computed variance of the residuals, which is obtained by removing longer time scale trends of the stock market time series. Volatility, on the other hand, is a measure of variance in the rate of return calculated from price of stocks without removing any time trends in the data. They both are measures of variability but as the reviewer suggest volatility of returns tends to spike after rather than before, while several studies show that variance of residuals may be used as early warning of critical transition 3,5-9 .
In Fig 3A the AR1 signal seems far too noisy to provide any predictive information.
As the reviewer pointed out, when filtering on short time windows the signal is noisy. However, in this case we can detect local trends that cannot be detected with larger rolling windows. The latter analysis in fact provides useful information only on general trends. Nevertheless, in both cases we can detect two very strong fluctuations that can serve as an early warning signal.
To quantify possible early warning signs, we introduced the threshold  , so that conditions in which ||   Std , where  denotes the difference between consecutive time intervals of 20-day-length are considered as early warning signals of a transition. We focus on the Std of the residuals as possible early warning signal because the previous analysis shows that, unlike AR1, Std exhibits signals of CSD. These thresholding criteria are introduced to quantitatively define and consistently recognize early warning signals of tipping points. Of course this method is highly sensitive to the selected threshold (i.e., if  is "too low", we will detect many early warnings, while if it is "too high" we will detect only few events. However, setting  in a reasonable range with respect to data means selecting  between one or three standard deviation of the residuals time series. In our case we have set the threshold to catch fluctuations greater than one standard deviation,   . Each crypto-currency thus has a different threshold  , depending on the standard deviation of the residual times series. We list all these warning event in the following table.
Events detected using the criteria ||   Std . The first column is the number of events, the second column is the crypto-currency of event, the third column is the duration of early warning signal and the fourth column is the collapse event following the corresponding signal. […]No threshold or hazard function for regime changes is specified.
In this paper, we investigate the Critical Slowing Down (CSD) phenomenon as early warning signals of critical transitions. If the system's dynamics approach a bifurcation point (or 'tipping point'), its lag-1 autocorrelation (AR1) and standard deviation (Std) will increase. In many studies 3,5,6,9 , this approachdifferently from typical use of the threshold or hazard functions to detect regime changesdoes not use a quantitative threshold to define the early warning event. Nevertheless, as explained above, we think that the suggestion of the reviewers is very pertinent, and thus we decided to add this new analysis where we indeed introduce a threshold for the definition of the early warning signals.
There are plenty of jumps in the short history of Bitcoin and other cryptos which are not predictive of a tipping point.
Filtering should help to remove these irrelevant small jumps, and by introducing the threshold criteria we now have an "objective" criterion to define early warning events for tipping points. Of course, the proposed technique may give false positive or true negative errors. In Table 2 and in the revised text we comment these limitations.

How can you justify this filtering as consistently recovering the trend, where it is often thought that bubbles form super-exponential trajectories?
We thank the reviewer to justify this point. As we hope now is clearer, early warning based on CSD are based on the use of AR1 and variance of residuals as leading indicators of critical transitions. Detrending is motivated by the fact that time series with strong trends or periodicities tends to exhibit a strong correlation structure, which would affect these quantities. Likewise, high-frequency fluctuations can cause spurious indications of impending transitions. Gaussian filter is a common method to remove such trends and filter out high frequencies.
Is AR(1) sufficient? How much structure is left in the residuals of the AR(1) fit?
If the system's dynamics approach a bifurcation point (or 'tipping point'), the dominant eigenvalue characterizing the rates of return to equilibrium after a 'small' displacement tends to zero. Therefore, its AR1 increases. This is true independently of how much structure is left in the residuals of the AR1. In fact, when the underlying non-linear dynamics of complex systems are not known, this theory can be used to detect the proximity of a system to a critical point 7,10,11 .
De-trending is easier said than done and can introduce spurious autocorrelations. the unit root tests are far from perfect In this paper, we would not like to discuss the drawback of de-trending. Gaussian filter is a common method to remove trends and filter out the high frequencies. Additionally, applying other type of detrending or filtering.
e.g. first-differences, removing running means, loess smoothing, will obtain similar results. We tested for stationarity by augmented Dickey-Fuller (ADF) test and the Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test. The ADF test performs a hypothesis testing on the time series with the null hypothesis that it is unit root and the alternative hypothesis is that the time series is stationary. Smaller p-value are associated with higher probability that the tested time series is stationary. By contrast, the KPSS test is actually a stationarity test, meaning that the null hypothesis is that the time series is stationary. A time series is considered to be stationary, if its ADF test is rejected and KPSS test is accepted.
I would leave the background on cryptocurrency to an appendix, or better still, simply point to a more standard reference if available.
We have re-written this part according to the Reviewer's suggestion.
The paper is a bit long on summarizing the ElBaharawy el al. findings, whose prediction of the bitcoin market share has proven to be pretty far off. You may wish to mention the view given in [1], which makes different assumptions.
Considering the Reviewer's suggestion, we have re-written the summary of ElBaharawy el al. In addition, we also cite this and related important references in Introduction.
Further, in [2] similar authors have looked at cryptocurrency bubbles, with a more specific view on speculative bubbles as a critical phenomena, with tipping point predictability. This should be reflected. Thank you for your suggestion. We have cited this relevant reference.

Reviewer: 2
We want to thank the reviewer 2 for her/his comments that have helped us to better explain the scope and goals of our work, clarify and rewrite some points and add further analysis. We hope that reviewer 2 will find the revised version of our manuscript improved.

The authors don't make any theoretical contribution to the theory of critical slowing down for critical transitions detection
The focus of this work was to investigate possible early warning signs of critical transitions in the cryptocurrency market based on Critical Slowing Down (CSD) phenomenon. To our knowledge there are not previous attempt to apply CSD to this interesting case. Several empirical studies have demonstrated how this method can be successfully applied to a variety of dynamical systems (but without adding any novel theoretical methodological contributions to CSD 2,10,12-15 ). Nevertheless, we appreciate the suggestion of the reviewer about a possible more theoretical contribution to the work, and we thus decided to add an entire section so to illustrate possible existence of early warnings also for a mean-field model of neutral dynamics displaying abrupt transitions.
El-Baharawy et al. 16 showed that from an ecological perspective, despite its simplicity and the assumption of no selective advantage among crypto-currencies, the so-called neutral model of evolution is able to reproduce a number of key empirical observations (see Fig. 4 in El-Baharawy et al. 16 ). Here, we propose a simple model for the crypto-currency price (instead of its market share). In particular, we develop a mean-field phenomenological model for neutral evolution 17 of prices, introducing density dependent fitness through Allee effects 18,19 , a key ecological process observed in many systems, that disfavors rare species with respect to abundant one. In this context, the Allee effect would disfavour crypto-currencies with low market cap, as less appealing by the buyers.
The mean field equations of the proposed neutral model read as: where u is the price of a given cryptocurrency, the parameter m represents the migration rate (i.e., endogenous causes affecting the market price), r the growth rate (intrinsic dynamics of the market price). The deterministic dynamics of this model has two stable points. The bifurcation diagram is obtained by finding the equilibria (u* at which f(u*) = 0; equilibria are stable (unstable) if * / | 0   uu df du (> 0). The state variable u can be in one of the two stable equilibria, which correspond to higher and lower price.
We include stochasticity through a multiplicative noise term in Eq. Error! Reference source not found. and the resulting equation is given by where D is the diffusion constant, and W is the standard uncorrelated Wiener process with zero mean value. By varying the migration rate parameter or the noise intensity, a transition between the two states may abruptly occur, and in this case we have a critical transition from high values (bullish phase) to the lower values (bearish phase).
To derive early warning signals of critical transitions, the idea is to investigate the dynamics of small deviations from the stable equilibrium u* by linearizing 3 ()      f u m r u u around u*. Analytically we cannot derive early warning signs for the residuals times series of Eq. (2), but we numerically study them and we show that thanks to the model we are able to explain why Std is more effective than AR1 to anticipate critical transitions.
We finally note that CSD is a general method that is not specific to the data we are analysing. The gradual change of some underlying conditions may lead to a system closer to a critical point causing a loss of resilience, with smaller perturbations being able to induce a shift to the alternative state. CSD indicates that the system is approaching this critical point, and thus its return time to equilibrium upon a small system perturbation strongly increases. Even when the underlying dynamics of the complex system are unknown or limited time series are available, the early warning signals still exist, and the leading indicators can be used to detect critical points before they shift the state. A large number of studies have now demonstrated the potential application of these leading indicators as warning signs of increased risk for upcoming state transitions.
For all these reasons, and because our mean-field model provides proof of concept showing possible critical transition detected by AR1 and Std, we have decided to use CSD as early warning of critical transition in the crypto-market. We have added all these points in the revised manuscript.
The authors don't make specific efforts in understanding why the small set of cryptocurrencies studied, which represent a very specific context for testing such general theory, should be particularly relevant or informative for the general theory, and how their results quantitatively relate to results in other fields. We thank the reviewer to give the opportunity to clarify this point. We have used the set of cryptocurrencies that between 2016 and 2018 was covering most of the total cryptocurrency market cap. Therefore, we feel that we are considering representative data to describe general behaviour of the crypto world. Moreover, CSD is a general method that is not specific to the data we are analysing. The gradual change of some underlying conditions may lead to a system closer to a critical point causing a loss of resilience, with smaller perturbations being able to induce a shift to the alternative state. CSD indicates that the system is approaching this critical point, and thus its return time to equilibrium upon a small system perturbation strongly increases. Even when the underlying dynamics of the complex system are unknown or limited time series are available, the early warning signatures still exist, and the leading indicators can be used to detect critical points before they shift the state. A large number of studies have now demonstrated the potential application of these leading indicators as warning signals of increased risk for upcoming state transitions.
In particular, the analogy with the neutral ecological models are considering here (and proposed in 17,[20][21][22], allows us to say that our results are also informative and can be related to ecological systems. In particular CSD for the mean field model for neutral of evolution with density dependent Allee effects is a novel result that can be applied also in the context of neutral theory.
The authors appear to identify in total two individual cases, which might be considered as "true positive" predictions of the theory in this context, although they don't specify a quantitative binary criterion for predicting large changes in a near future, so it is not clear at this stage whether these two cases are actually predicted by their theory and whether the theory would predict also other large changes that did not occur.
To quantify possible early warning signs, we introduced the threshold  , so that conditions in which ||   Std , where  denotes the difference between consecutive time intervals of 20-day-length are considered as early warning signals of a transition. We focus on the Std of the residuals as possible early warning signal because the previous analysis shows that, unlike AR1, Std exhibits signals of CSD.
These thresholding criteria are introduced to quantitatively define and consistently recognize early warning signals of tipping points. Of course this method is highly sensitive to the selected threshold (i.e., if  is "too low", we will detect many early warnings, while if it is "too high" we will detect only few events. However, setting  in a reasonable range with respect to data means selecting  between one or three standard deviation of the residuals time series. In our case we have set the threshold to catch fluctuations greater than one standard deviation,   . Each crypto-currency thus has a different threshold  , depending on the standard deviation of the residual times series.
Where possible, we try to identify and comments all the events detected by CSD and comment related limitations. We list all these warning events in the new Table 2.