{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T04:25:39Z","timestamp":1772252739422,"version":"3.50.1"},"reference-count":46,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2022,11,18]],"date-time":"2022-11-18T00:00:00Z","timestamp":1668729600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001659","name":"German Research Foundation","doi-asserted-by":"publisher","award":["MA4759\/8"],"award-info":[{"award-number":["MA4759\/8"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001659","name":"German Research Foundation","doi-asserted-by":"publisher","award":["MA4759\/9"],"award-info":[{"award-number":["MA4759\/9"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In recurrence analysis, the \u03c4-recurrence rate encodes the periods of the cycles of the underlying high-dimensional time series. It, thus, plays a similar role to the autocorrelation for scalar time-series in encoding temporal correlations. However, its Fourier decomposition does not have a clean interpretation. Thus, there is no satisfactory analogue to the power spectrum in recurrence analysis. We introduce a novel method to decompose the \u03c4-recurrence rate using an over-complete basis of Dirac combs together with sparsity regularization. We show that this decomposition, the inter-spike spectrum, naturally provides an analogue to the power spectrum for recurrence analysis in the sense that it reveals the dominant periodicities of the underlying time series. We show that the inter-spike spectrum correctly identifies patterns and transitions in the underlying system in a wide variety of examples and is robust to measurement noise.<\/jats:p>","DOI":"10.3390\/e24111689","type":"journal-article","created":{"date-parts":[[2022,11,21]],"date-time":"2022-11-21T03:07:23Z","timestamp":1669000043000},"page":"1689","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Spike Spectra for Recurrences"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9943-5391","authenticated-orcid":false,"given":"K. Hauke","family":"Kraemer","sequence":"first","affiliation":[{"name":"Potsdam Institute for Climate Impact Research (PIK), Member of the Leibniz Association, 14473 Potsdam, Germany"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5635-4949","authenticated-orcid":false,"given":"Frank","family":"Hellmann","sequence":"additional","affiliation":[{"name":"Potsdam Institute for Climate Impact Research (PIK), Member of the Leibniz Association, 14473 Potsdam, Germany"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6643-9508","authenticated-orcid":false,"given":"Mehrnaz","family":"Anvari","sequence":"additional","affiliation":[{"name":"Potsdam Institute for Climate Impact Research (PIK), Member of the Leibniz Association, 14473 Potsdam, Germany"}]},{"given":"J\u00fcrgen","family":"Kurths","sequence":"additional","affiliation":[{"name":"Potsdam Institute for Climate Impact Research (PIK), Member of the Leibniz Association, 14473 Potsdam, Germany"},{"name":"Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam, Germany"},{"name":"Institute of Physics, Humboldt Universit\u00e4t zu Berlin, 12489 Berlin, Germany"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1437-7039","authenticated-orcid":false,"given":"Norbert","family":"Marwan","sequence":"additional","affiliation":[{"name":"Potsdam Institute for Climate Impact Research (PIK), Member of the Leibniz Association, 14473 Potsdam, Germany"},{"name":"Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam, Germany"},{"name":"Institute of Geosciences, University of Potsdam, 14476 Potsdam, Germany"}]}],"member":"1968","published-online":{"date-parts":[[2022,11,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1016\/j.physrep.2006.11.001","article-title":"Recurrence Plots for the Analysis of Complex Systems","volume":"438","author":"Marwan","year":"2007","journal-title":"Phys. Rep."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1140\/epjst\/e2008-00829-1","article-title":"A Historical Review of Recurrence Plots","volume":"164","author":"Marwan","year":"2008","journal-title":"Eur. Phys. J. Spec. Top."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Webber, C.L., and Marwan, N. (2015). Recurrence Quantification Analysis\u2014Theory and Best Practices, Springer.","DOI":"10.1007\/978-3-319-07155-8"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"104","DOI":"10.3389\/fncom.2014.00104","article-title":"Self-consistent determination of the spike-train power spectrum in a neural network with sparse connectivity","volume":"8","author":"Dummer","year":"2014","journal-title":"Front. Comput. Neurosci."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Orcioni, S., Paffi, A., Apollonio, F., and Liberti, M. (2020). Revealing Spectrum Features of Stochastic Neuron Spike Trains. Mathematics, 8.","DOI":"10.3390\/math8061011"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"026702","DOI":"10.1103\/PhysRevE.66.026702","article-title":"Recurrence Plot Based Measures of Complexity and its Application to Heart Rate Variability Data","volume":"66","author":"Marwan","year":"2002","journal-title":"Phys. Rev. E"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"213","DOI":"10.5194\/npg-28-213-2021","article-title":"Recurrence analysis of extreme event-like data","volume":"28","author":"Banerjee","year":"2021","journal-title":"Nonlinear Process. Geophys."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"90","DOI":"10.1016\/j.jneumeth.2005.05.011","article-title":"Entropy analysis of neuronal spike train synchrony","volume":"149","author":"Kajikawa","year":"2005","journal-title":"J. Neurosci. Methods"},{"key":"ref_9","unstructured":"Canale, A., Lijoi, A., Nipoti, B., and Pr\u00fcnster, I. (2021). Inner spike and slab Bayesian nonparametric models. Econom. Stat."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"085720","DOI":"10.1063\/1.5024914","article-title":"Recurrence threshold selection for obtaining robust recurrence characteristics in different embedding dimensions","volume":"28","author":"Kraemer","year":"2018","journal-title":"Chaos: Interdiscip. J. Nonlinear Sci."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1016\/j.physd.2004.11.002","article-title":"Recurrence plot statistics and the effect of embedding","volume":"200","author":"March","year":"2005","journal-title":"Phys. D"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1135","DOI":"10.1016\/j.physleta.2009.12.057","article-title":"Kolmogorov-Sinai entropy from recurrence times","volume":"374","author":"Baptista","year":"2010","journal-title":"Phys. Lett. A"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"125977","DOI":"10.1016\/j.physleta.2019.125977","article-title":"Border effect corrections for diagonal line based recurrence quantification analysis measures","volume":"383","author":"Kraemer","year":"2019","journal-title":"Phys. Lett. A"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1016\/S0375-9601(02)01170-2","article-title":"Nonlinear analysis of bivariate data with cross recurrence plots","volume":"302","author":"Marwan","year":"2002","journal-title":"Phys. Lett. A"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"6622","DOI":"10.1016\/j.physleta.2008.09.027","article-title":"The Wiener\u2013Khinchin theorem and recurrence quantification","volume":"372","author":"Zbilut","year":"2008","journal-title":"Phys. Lett. A"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"033017","DOI":"10.1088\/1367-2630\/abe336","article-title":"A unified and automated approach to attractor reconstruction","volume":"23","author":"Kraemer","year":"2021","journal-title":"New J. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"179","DOI":"10.3109\/00207458208985921","article-title":"An Efficient Method for the Fourier Transform of a Neuronal Spike Train","volume":"17","author":"Schild","year":"1982","journal-title":"Int. J. Neurosci."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1007\/BF00401584","article-title":"Dirac combs","volume":"17","year":"1989","journal-title":"Lett. Math. Phys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"903","DOI":"10.1098\/rspa.1998.0193","article-title":"The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis","volume":"454","author":"Huang","year":"1998","journal-title":"Proc. R. Soc. London. Ser. Math. Phys. Eng. Sci."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"328","DOI":"10.1109\/JBHI.2016.2530943","article-title":"Homomorphic Deconvolution for MUAP Estimation From Surface EMG Signals","volume":"21","author":"Biagetti","year":"2017","journal-title":"IEEE J. Biomed. Health Inform."},{"key":"ref_21","unstructured":"Bracewell, R.N., and Bracewell, R.N. (1986). The Fourier Transform and its Applications, McGraw-Hill."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1037\/h0071325","article-title":"Analysis of a complex of statistical variables into principal components","volume":"24","author":"Hotelling","year":"1933","journal-title":"J. Educ. Psychol."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Brunton, S.L., and Kutz, J.N. (2019). Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control, Cambridge University Press.","DOI":"10.1017\/9781108380690"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1111\/j.2517-6161.1996.tb02080.x","article-title":"Regression Shrinkage and Selection Via the Lasso","volume":"58","author":"Tibshirani","year":"1996","journal-title":"J. R. Stat. Soc. Ser. B"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"3932","DOI":"10.1073\/pnas.1517384113","article-title":"Discovering governing equations from data by sparse identification of nonlinear dynamical systems","volume":"113","author":"Brunton","year":"2016","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"635","DOI":"10.1103\/PhysRevLett.77.635","article-title":"Improved Surrogate Data for Nonlinearity Tests","volume":"77","author":"Schreiber","year":"1996","journal-title":"Phys. Rev. Lett."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"346","DOI":"10.1016\/S0167-2789(00)00043-9","article-title":"Surrogate time series","volume":"142","author":"Schreiber","year":"2000","journal-title":"Phys. D: Nonlinear Phenom."},{"key":"ref_28","unstructured":"Kundur, P., Balu, N.J., and Lauby, M.G. (1994). Power System Stability and Control, McGraw-Hill."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"43082","DOI":"10.1109\/ACCESS.2020.2967834","article-title":"Data-driven model of the power-grid frequency dynamics","volume":"8","author":"Anvari","year":"2020","journal-title":"IEEE Access"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"013339","DOI":"10.1103\/PhysRevResearch.2.013339","article-title":"Stochastic properties of the frequency dynamics in real and synthetic power grids","volume":"2","author":"Anvari","year":"2020","journal-title":"Phys. Rev. Res."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"013130","DOI":"10.1063\/1.5123778","article-title":"Identifying characteristic time scales in power grid frequency fluctuations with DFA","volume":"30","author":"Meyer","year":"2020","journal-title":"Chaos: Interdiscip. J. Nonlinear Sci."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Wei\u00dfbach, T., and Welfonder, E. (2009, January 15\u201318). High frequency deviations within the European power system: Origins and proposals for improvement. Proceedings of the 2009 IEEE\/PES Power Systems Conference and Exposition, Seattle, WA, USA.","DOI":"10.1109\/PSCE.2009.4840180"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"A89","DOI":"10.1051\/0004-6361\/201116836","article-title":"La2010: A new orbital solution for the long-term motion of the Earth","volume":"532","author":"Laskar","year":"2011","journal-title":"Astron. Astrophys."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1051\/0004-6361:20041335","article-title":"A long-term numerical solution for the insolation quantities of the Earth","volume":"428","author":"Laskar","year":"2004","journal-title":"Astron. Astrophys."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1383","DOI":"10.1126\/science.aba6853","article-title":"An astronomically dated record of Earth\u2019s climate and its predictability over the last 66 million years","volume":"369","author":"Westerhold","year":"2020","journal-title":"Science"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1525","DOI":"10.1007\/s11071-022-07280-2","article-title":"Optimal state space reconstruction via Monte Carlo decision tree search","volume":"108","author":"Kraemer","year":"2022","journal-title":"Nonlinear Dyn."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"4970","DOI":"10.1103\/PhysRevE.60.4970","article-title":"Improved false nearest neighbor method to detect determinism in time series data","volume":"60","author":"Hegger","year":"1999","journal-title":"Phys. Rev. E"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"013110","DOI":"10.1063\/1.2430294","article-title":"A unified approach to attractor reconstruction","volume":"17","author":"Pecora","year":"2007","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"598","DOI":"10.21105\/joss.00598","article-title":"DynamicalSystems.jl: A Julia software library for chaos and nonlinear dynamics","volume":"3","author":"Datseris","year":"2018","journal-title":"J. Open Source Softw."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"15","DOI":"10.5334\/jors.151","article-title":"Differentialequations.jl\u2013a performant and feature-rich ecosystem for solving differential equations in julia","volume":"5","author":"Rackauckas","year":"2017","journal-title":"J. Open Res. Softw."},{"key":"ref_41","unstructured":"Zhang, S., Widmann, D., and Barreira, D.S. (2022, November 16). JuliaOptimalTransport\/OptimalTransport.jl: v0.3.19, Zenodo. Available online: https:\/\/zenodo.org\/record\/5920148#.Y3SbbORByUk."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1175\/1520-0469(1963)020<0130:DNF>2.0.CO;2","article-title":"Deterministic Nonperiodic Flow","volume":"20","author":"Lorenz","year":"1963","journal-title":"J. Atmos. Sci."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1016\/0375-9601(76)90101-8","article-title":"An equation for continuous chaos","volume":"57","year":"1976","journal-title":"Phys. Lett. A"},{"key":"ref_44","unstructured":"(2022, July 25). National Grid, Frequency Data (2014\u20132018). Available online: https:\/\/power-grid-frequency.org\/database\/."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"30001","DOI":"10.1209\/0295-5075\/121\/30001","article-title":"The footprint of atmospheric turbulence in power grid frequency measurements","volume":"121","author":"Haehne","year":"2018","journal-title":"EPL (Europhys. Lett.)"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"1134","DOI":"10.1103\/PhysRevA.33.1134","article-title":"Independent coordinates for strange attractors from mutual information","volume":"33","author":"Fraser","year":"1986","journal-title":"Phys. Rev. A"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/11\/1689\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:21:39Z","timestamp":1760145699000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/11\/1689"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,11,18]]},"references-count":46,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2022,11]]}},"alternative-id":["e24111689"],"URL":"https:\/\/doi.org\/10.3390\/e24111689","relation":{"has-preprint":[{"id-type":"doi","id":"10.20944\/preprints202209.0054.v1","asserted-by":"object"}]},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,11,18]]}}}