A Brief Introduction to Nonlinear Time Series Analysis and Recurrence Plots

Round 1

Reviewer 1 Report

This is an excellent review of nonlinear time series analysis and the application of recurrence plots in particular. While it is not unique in this regard, it is timely. Nonlinear time series techniques have now matured to a state that clear descriptions of the methods and applications (such as this) will be of great benefit to practitioners and not just specialists. I have only minor suggestions/comments:

- My first comment is that I find the editorial process of this journal a little odd. One must agree to review a paper before being able to see it and then be told that the review is expected within a stipulated timeframe. I don't think that such a time frame is suitable or realistic for a paper of this depth and substance. I feel that the journal is perhaps taking their reviewers for granted and put the onus on the unpaid reviewers to maintain and review schedule they can boast about.

- On to the paper itself: I feel that it is more about nonlinear time series analysis in general than just recurrence plot in particular - I would suggest (although this is not compulsory) that the author modifies the title and abstract to reflect this broader scope.

- There are minor grammatical issues (line 40 should have "as" before "it is"; two "Related" in the same sentence on line 75-6; an apostrophe is required following Python on one 584 - and an appropriate reference to the module in question) throughout which should be more carefully checked.

- Fig. 4: there appears to be quite a bit of phase slip in Fig. 4.A. which is not evident in 4.C.

- Fig. 5: I am a bit conflicted by the message here. Since the surrogate algorithm is destined to exactly preserve the power spectrum for any input signal claiming that power spectra in sufficient for nonlinear dynamics is (on the basis of this evidence) a straw man. The real message is in Fig. 5.C.

- Sec. 5.1. I believe (I could be mistaken) that Eckmann originally used k-nearest neighbours in his recurrence plots (akin to what some authors now do for phase space networks)?

- The author should cite Professor Michael Small's work more often in the references.

Author Response

Response to Reviewer 1 Comments

Point 1: This is an excellent review of nonlinear time series analysis and the application of recurrence plots in particular. While it is not unique in this regard, it is timely. Nonlinear time series techniques have now matured to a state that clear descriptions of the methods and applications (such as this) will be of great benefit to practitioners and not just specialists. I have only minor suggestions/comments: 

Response 1: I thank the reviewer for the extremely positive review of the manuscript. I respond to the comments point-by-point below.

Point 2: My first comment is that I find the editorial process of this journal a little odd. One must agree to review a paper before being able to see it and then be told that the review is expected within a stipulated timeframe. I don’t think that such a time frame is suitable or realistic for a paper of this depth and substance. I feel that the journal is perhaps taking their reviewers for granted and put the onus on the unpaid reviewers to maintain and review schedule they can boast about.

Response 2: I think this comment is addressed to the editorial team. Therefore, I have no response to this point.

Point 3: On to the paper itself: I feel that it is more about nonlinear time series analysis in general than just recurrence plot in particular - I would suggest (although this is not compulsory) that the author modifies the title and abstract to reflect this broader scope. 

Response 3: The title is now changed to: “A Brief Introduction to Nonlinear Time Series Analysis and Recurrence Plots.”

Point 4: There are minor grammatical issues (line 40 should have “as” before “it is”; two “Related” in the same sentence on line 75-6; an apostrophe is required following Python on one 584 - and an appropriate reference to the module in question) throughout which should be more carefully checked. 

Response 4: The above mentioned mistakes were corrected and the entire manuscript was thoroughly checked fr grammatical errors and typos.

Point 5: Fig. 4: there appears to be quite a bit of phase slip in Fig. 4.A. which is not evident in 4.C. 

Response 5: Yes, I agree that there is a phase slip visible in Fig. 4A. However, the phase difference for the coupled systems is ultimately bounded and this bound (in the order of a few radians) is rather small compared to the overall phase increment (axes in Fig $C go up to 500 rad). Hence, the plot in Fig 4C looks like a straight diagonal line.

Point 6: Fig. 5: I am a bit conflicted by the message here. Since the surrogate algorithm is destined to exactly preserve the power spectrum for any input signal claiming that power spectra in sufficient for nonlinear dynamics is (on the basis of this evidence) a straw man. The real message is in Fig. 5.C.

Response 6: I agree with the reviewer that the real message is in Fig. 5C, but I fail to see the confusion here. Even though It might seem like a straw man argument to use an iAAFT surrogate to show that power spectrum is insufficient to pin down nonlinear dynamics, in my understanding, it is typical in signal processing to use power spectra as a preferred mode of classifying and comparing signals without attempting to reconstruct the attractor. The intention of this figure was to point out to people unfamiliar with the paradigm of nonlinear time series analysis as to how relying solely on power spectra can lead to potential pitfalls, and the example was designed accordingly.

Point 7: Sec. 5.1. I believe (I could be mistaken) that Eckmann originally used k-nearest neighbours in his recurrence plots (akin to what some authors now do for phase space networks)? 

Response 7: Yes, the reviewer is correct. Eckmann, Kamphorst and Ruelle used k-nearest neighbour norm. This fact is now mentioned in Sec. 7.1 and phase space networks study by Xu, Zhang and Small (2008) is cited in Sec 7.2.

Point 8: The author should cite Professor Michael Small’s work more often in the references. 

Response 8: The following new references are cited in the revised version of the manuscript:

Moore, J. M., Corrêa, D. C., & Small, M. (2018). Is Bach’s brain a Markov chain? Recurrence quantification to assess Markov order for short, symbolic, musical compositions. Chaos, 28, 085715. Zaitouny, A., Walker, D. M., & Small, M. (2019). Quadrant scan for multi-scale transition detection. Chaos, 29, 103117. Xu, X., Zhang, J., & Small, M. (2008). Superfamily phenomena and motifs of networks induced from time series. Proc. Nat. Acad. Sci., 105, 19601–19605.

Reviewer 2 Report

In this review manuscript, the author summarizes his own story of recurrence plots. If we make the purpose of this review manuscript narrow, it is well written and easy to understand. However, if we look at the manuscript as an introduction to the current state-of-the art of recurrence plots, the manuscript is too biased to the author’s work and his colleagues’. Therefore, if the author intends this review manuscript for his own story of recurrence plots, he should state this fact more explicitly and honestly in the abstract and introduction, and can keep the other contents as they are. In this case, the values for the manuscript would be limited. If the author would like to gain the values for the manuscript more, he should cite the others’ work, and discuss or mention, for example, but not limited to, how much information for the original time series a recurrence plot retains, the fact that we can infer dynamical invariants such as correlation dimensions and correlation entropy from recurrence plots, cross-recurrence plots, order recurrence plots, relationship of recurrence plots with unstable periodic orbits, how we can interpret the nature of deterministic chaos from a recurrence plot, how we can infer driving forces using recurrence plots, and applications of recurrence plots for more exotic data such as point processes, irregularly sampled data, and network data. This is the author’s choice whether he keeps most parts of the current manuscript as they are by stating that this review is his personal story of recurrence plots, or he adds the contents of the others’ work to increase the values of the manuscript. In either case, I will support the publication of the manuscript after an appropriate revision is made.

Author Response

Response to Reviewer 2 Comments

Point 1: In this review manuscript, the author summarizes his own story of recurrence plots. If we make the purpose of this review manuscript narrow, it is well written and easy to understand. However, if we look at the manuscript as an introduction to the current state-of-the art of recurrence plots, the manuscript is too biased to the author’s work and his colleagues’. 

Response 1: I apologise for the bias in the content of the manuscript. The review by the Editorial Board Member also raised similar concerns. To this end I have added several paragraphs in Sec 1 which give a broader overview of the potentials of recurrence plot applications in the last twenty years. I have also added a new section (Sec 2) on the various applications of recurrence plot based methods in engineering research.

Point 2: Therefore, if the author intends this review manuscript for his own story of recurrence plots, he should state this fact more explicitly and honestly in the abstract and introduction, and can keep the other contents as they are. In this case, the values for the manuscript would be limited. 

Response 2: I have modified the abstract to hopefully make it clearer that this tutorial review presents only a few of of the existing recurrence plot approaches. I have also tried to broaden the narrative of the review by including a broader literature survey in the introduction, a new section on recurrence plots in engineering, and an additional section on other aspects of recurrence plots such as the ones mentioned by the reviewer below. I have also included a new section “About the Review” which tries to circumscribe the boundaries of the manuscript more clearly and I have stated there that the “the review is biased by my own areas of expertise.”

Point 3: If the author would like to gain the values for the manuscript more, he should cite the others’ work, and discuss or mention, for example, but not limited to, how much information for the original time series a recurrence plot retains, the fact that we can infer dynamical invariants such as correlation dimensions and correlation entropy from recurrence plots, cross-recurrence plots, order recurrence plots, relationship of recurrence plots with unstable periodic orbits, how we can interpret the nature of deterministic chaos from a recurrence plot, how we can infer driving forces using recurrence plots, and applications of recurrence plots for more exotic data such as point processes, irregularly sampled data, and network data.

Response 3: I thank the reviewer for pointing out these topics and the fact that they were not covered in the review. As the manuscript is intended as a tutorial-plus-review for a beginner-to-intermediate level recurrence plot and nonlinear time series analysis reader, I have only touched upon most of the above listed topics without going too much into their details. The details, I feel, are outside of the purview of the current manuscript. That said, I have now included the following new references at appropriate locations in the revised version:

Casdagli, M. C. (1997). Recurrence plots revisited. Physica D, 108(1–2), 12–44. Iwanski, J. S., & Bradley, E. (1998). Recurrence plots of experimental data: To embed or not to embed? Chaos, 8(4), 861–871. Bradley, E., & Mantilla, R. (2002). Recurrence plots and unstable periodic orbits. Chaos, 12(3), 596–600. Thiel, M., Romano, M. C., Kurths, J., Meucci, R., Allaria, E., & Arecchi, F. T. (2002). Influence of observational noise on the recurrence quantification analysis. Physica D, 171(3), 138–152. Romano, M. C., Thiel, M., Kurths, J., & von Bloh, W. (2004). Multivariate recurrence plots. Physics Letters A, 330(3–4), 214–223. Thiel, M., Romano, M. C., Read, P. L., & Kurths, J. (2004). Estimation of dynamical invariants without embedding by recurrence plots. Chaos, 14(2), 234–243. Thiel, M., Romano, M. C., & Kurths, J. (2004). How much information is contained in a recurrence plot? Physics Letters A, 330(5), 343–349. March, T. K., Chapman, S. C., & Dendy, R. O. (2005). Recurrence plot statistics and the effect of embedding. Physica D, 200(1–2), 171–184. Facchini, A., Kantz, H., & Tiezzi, E. (2005). Recurrence plot analysis of nonstationary data: The understanding of curved patterns. Physical Review E, 72(2), 021915. Groth, A. (2005). Visualization of coupling in time series by order recurrence plots. Physical Review E, 72(4), 046220. Thiel, M., Romano, M. C., & Kurths, J. (2006). Spurious Structures in Recurrence Plots Induced by Embedding. Nonlinear Dynamics, 44(1–4), 299–305. Zou, Y., Thiel, M., Romano, M. C., & Kurths, J. (2007). Characterization of stickiness by means of recurrence. Chaos, 17(4), 043101. Schinkel, S., Marwan, N., & Kurths, J. (2007). Order patterns recurrence plots in the analysis of ERP data. Cognitive Neurodynamics, 1(4), 317–325. Hirata, Y., Horai, S., & Aihara, K. (2008). Reproduction of distance matrices and original time series from recurrence plots and their applications. European Physical Journal Special Topics, 164(1), 13–22. Tanio, M., Hirata, Y., & Suzuki, H. (2009). Reconstruction of driving forces through recurrence plots. Physics Letters A, 373(23–24), 2031–2040. Thiel, M., Kurths, J., Mergenthaler, K., & Engbert, R. (2009). Hypothesis test for synchronization: Twin surrogates revisited. Chaos, 19(1), 015108. Robinson, G., & Thiel, M. (2009). Recurrences determine the dynamics. Chaos, 19(2), 023104. Hirata, Y., & Aihara, K. (2010). Identifying hidden common causes from bivariate time series: A method using recurrence plots. Physical Review E, 81(1), 016203. Graben, P. B., & Hutt, A. (2013). Detecting Recurrence Domains of Dynamical Systems by Symbolic Dynamics. Physical Review Letters, 110(15), 154101. Iwayama, K., Hirata, Y., Suzuki, H., & Aihara, K. (2013). Change-point detection with recurrence networks. Nonlinear Theory and Its Applications, IEICE, 4(2), 160–171. Rapp, P. E., Darmon, D. M., & Cellucci, C. J. (2014). Hierarchical Transition Chronometries in the Human Central Nervous System. IEICE Proceeding Series, 2, 286–289. Eroglu, D., Peron, T. K. D., Marwan, N., Rodrigues, F. A., Costa, L. D. F., Sebek, M., … Kurths, J. (2014). Entropy of weighted recurrence plots. Physical Review E, 90(4), 042919. Fukino, M., Hirata, Y., & Aihara, K. (2016). Coarse-graining time series data: Recurrence plot of recurrence plots and its application for music. Chaos, 26(2), 023116. beim Graben, P., Sellers, K. K., Fröhlich, F., & Hutt, A. (2016). Optimal estimation of recurrence structures from time series. Europhysics Letters, 114(3), 38003. Pham, T. D. (2016). Fuzzy recurrence plots. Europhysics Letters, 116(5), 50008. Hutt, A., & beim Graben, P. (2017). Sequences by Metastable Attractors: Interweaving Dynamical Systems and Experimental Data. Frontiers in Applied Mathematics and Statistics, 3(May), 1–14.

Point 4: This is the author’s choice whether he keeps most parts of the current manuscript as they are by stating that this review is his personal story of recurrence plots, or he adds the contents of the others’ work to increase the values of the manuscript. In either case, I will support the publication of the manuscript after an appropriate revision is made. 

Response 4: I have tried my best to broaden the scope of the methods discussed in the manuscript. I have, however, not included any new model examples or real-world examples. The choice of the methods discussed in greater detail is biased by my own expertise in these approaches (I have noted this point in my response to the Editorial Board Member’s comments as well). Nevertheless, I have extensively included recurrence plot based research carried out by groups other than my own and in disciplines other than my own. I am hopeful that the reviewer agrees to the changes made to the manuscript following the comments raised above.

Reviewer 3 Report

This is an outstanding and excellent manuscript. The paper summarises in excellently balanced brevity today’s state of the art in nonlinear time-series and recurrence plot analysis.  I expect the work to be of utmost interest and relevance for the readers of Vibration, since without any doubt the future of complex systems vibration analysis will follow the route of methods and approaches described here. I thus strongly recommend without any reservation the publication of the manuscript in its present form.

In more detail: The paper is very well structured and language-wise absolutely flaw-less. The figures presented are graphically appealing, and the overall story told is highly convincing. The author succeeds in breaking down the timely and challenging topics of nonlinear time-series analysis and recurrence plot analysis to the essential parts. I expect the paper to serve as a highly valuable short-hand introductory text to the topic for beginning researchers in engineering and the sciences. Compared to fuller presentations of the topic in textbooks, it is the very focus on brevity and key aspects which is the special merit of the present manuscript.

In terms of content the logic of the presentation allows good reading, starting from the historical background, summarising the main aspects of nonlinear dynamics and its tools. Phase space reconstruction and embedding questions are well addressed, and recurrence analysis is laid out. Also the rather recent link to networks by using recurrence plots as adjacency matrices is presented. Some of the key methods to quantify recurrence plots are shown too. An example application towards climate data is also contained, and a summary and an outlook close the paper.

Altogether all formal aspects are excellent, the tutorial style introduction succeeds in full, and the reader is left with a terrific short introduction to the topic and its future potential. 

Author Response

Response to Reviewer 3 Comments

Point 1: This is an outstanding and excellent manuscript. The paper summarises in excellently balanced brevity today’s state of the art in nonlinear time-series and recurrence plot analysis. I expect the work to be of utmost interest and relevance for the readers of Vibration, since without any doubt the future of complex systems vibration analysis will follow the route of methods and approaches described here. I thus strongly recommend without any reservation the publication of the manuscript in its present form.
In more detail: The paper is very well structured and language-wise absolutely flaw-less. The figures presented are graphically appealing, and the overall story told is highly convincing. The author succeeds in breaking down the timely and challenging topics of nonlinear time-series analysis and recurrence plot analysis to the essential parts. I expect the paper to serve as a highly valuable short-hand introductory text to the topic for beginning researchers in engineering and the sciences. Compared to fuller presentations of the topic in textbooks, it is the very focus on brevity and key aspects which is the special merit of the present manuscript.
In terms of content the logic of the presentation allows good reading, starting from the historical background, summarising the main aspects of nonlinear dynamics and its tools. Phase space reconstruction and embedding questions are well addressed, and recurrence analysis is laid out. Also the rather recent link to networks by using recurrence plots as adjacency matrices is presented. Some of the key methods to quantify recurrence plots are shown too. An example application towards climate data is also contained, and a summary and an outlook close the paper. Altogether all formal aspects are excellent, the tutorial style introduction succeeds in full, and the reader is left with a terrific short introduction to the topic and its future potential. 

Response 1: I thank the reviewer for this extremely positive review of my work. I hope that the revised version of the manuscript is equally appreciated and that it manages to maintain the standard set by the first version.

Reviewer 4 Report

The paper presents an interesting overview of nonlinear time series analysis and recurrence plots. Even if no original material is presented, the review is clear and interesting, and can be a valid support for readers approaching such methods for a first time.

Author Response

Response to Reviewer 4 Comments

Point 1: The paper presents an interesting overview of nonlinear time series analysis and recurrence plots. Even if no original material is presented, the review is clear and interesting, and can be a valid support for readers approaching such methods for a first time. 

Response 1: I thank the reviewer for the positive review of the manuscript. I hope the revised version of the manuscript is positively reviewed as well.