{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,18]],"date-time":"2025-10-18T15:12:52Z","timestamp":1760800372337,"version":"build-2065373602"},"reference-count":38,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2020,8,6]],"date-time":"2020-08-06T00:00:00Z","timestamp":1596672000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Romanian Ministry of Research and Innovation","award":["1EU- 4\/2"],"award-info":[{"award-number":["1EU- 4\/2"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>This article describes a refinement of recurrence analysis to determine the delay in the causal influence between a driver and a target, in the presence of additional perturbations affecting the time series of the response observable. The methodology is based on the definition of a new type of recurrence plots, the Conditional Joint Recurrence plot. The potential of the proposed approach resides in the great flexibility of recurrence plots themselves, which allows extending the technique to more than three quantities. Autoregressive time series, both linear and nonlinear, with different couplings and percentage of additive Gaussian noise have been investigated in detail, with and without outliers. The approach has also been applied to the case of synthetic periodic signals, representing realistic situations of synchronization experiments in thermonuclear fusion. The results obtained have been very positive; the proposed Conditional Joint Recurrence plots have always managed to identify the right interval of the causal influences and are very competitive with alternative techniques such as the Conditional Transfer Entropy.<\/jats:p>","DOI":"10.3390\/e22080865","type":"journal-article","created":{"date-parts":[[2020,8,6]],"date-time":"2020-08-06T09:41:21Z","timestamp":1596706881000},"page":"865","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["A Refinement of Recurrence Analysis to Determine the Time Delay of Causality in Presence of External Perturbations"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6829-2180","authenticated-orcid":false,"given":"Emmanuele","family":"Peluso","sequence":"first","affiliation":[{"name":"Department of Industrial Engineering, University of Rome \u201cTor Vergata\u201d, via del Politecnico 1, 00133 Roma, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0012-4260","authenticated-orcid":false,"given":"Teddy","family":"Craciunescu","sequence":"additional","affiliation":[{"name":"National Institute for Laser, Plasma and Radiation Physics, RO-077125 Magurele-Bucharest, Romania"}]},{"given":"Andrea","family":"Murari","sequence":"additional","affiliation":[{"name":"Consorzio RFX (CNR, ENEA, INFN, Universita di Padova, Acciaierie Venete SpA), Corso Stati Uniti 4, 35127 Padova, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2020,8,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1901","DOI":"10.1016\/j.cnsns.2008.07.002","article-title":"A theory for synchronization of dynamical systems","volume":"14","author":"Luo","year":"2009","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"046211","DOI":"10.1103\/PhysRevE.63.046211","article-title":"Synchronization as adjustment of information rates: Detection from bivariate time series","volume":"63","author":"Palus","year":"2001","journal-title":"Phys. Rev. E"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"214","DOI":"10.1016\/j.physleta.2004.07.066","article-title":"Multivariate recurrence plots","volume":"330","author":"Romano","year":"2004","journal-title":"Phys. Lett. A"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1099","DOI":"10.1142\/S0218127411029033","article-title":"Inferring indirect coupling by means of recurrences","volume":"21","author":"Zou","year":"2011","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"122586","DOI":"10.1016\/j.physa.2019.122586","article-title":"Scaling features of intermittent dynamics: Differences of characterizing correlated and anti-correlated data sets","volume":"536","author":"Pavlova","year":"2019","journal-title":"Physica  A"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"663","DOI":"10.1016\/j.physa.2016.09.058","article-title":"Complex network analysis of brain functional connectivity under a multi-step cognitive task","volume":"466","author":"Chai","year":"2017","journal-title":"Physica  A"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"063116","DOI":"10.1063\/5.0007670","article-title":"Detecting causality from time series in a machine learning framework","volume":"30","author":"Huang","year":"2020","journal-title":"Chaos"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"14750","DOI":"10.1038\/srep14750","article-title":"Distinguishing time delayed causal interactions using convergent cross mapping","volume":"5","author":"Ye","year":"2015","journal-title":"Sci. Rep."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1038\/srep10829","article-title":"Inferring the connectivity of coupled oscillators from time series statistical similarity analysis","volume":"5","author":"Tirabassi","year":"2015","journal-title":"Sci. Rep."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"013137","DOI":"10.1063\/1.5112799","article-title":"Reciprocal characterization from multivariate time series to multilayer complex networks","volume":"30","author":"Zhao","year":"2020","journal-title":"Chaos"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Cuautle, E.T., Valencia-Ponce, M.A., and De la Fraga, L.G. (2020). Sizing CMOS Amplifiers by PSO and MOL to Improve DC Operating Point Conditions. Electronics, 9.","DOI":"10.3390\/electronics9061027"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"116643","DOI":"10.1109\/ACCESS.2019.2936535","article-title":"On a Simple Single-Transistor-Based Chaotic Snap Circuit: A Maximized Attractor Dimension at Minimized Damping and a Stable Equilibrium","volume":"7","author":"Srisuchinwong","year":"2019","journal-title":"IEEE Access"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"042207","DOI":"10.1103\/PhysRevE.97.042207","article-title":"Comparison of six methods for the detection of causality in a bivariate time series","volume":"97","author":"Krakovska","year":"2018","journal-title":"Phys. Rev. E"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"083109","DOI":"10.1063\/1.4997757","article-title":"Detection of coupling delay: A problem not yet solved","volume":"27","author":"Coufal","year":"2017","journal-title":"Chaos"},{"key":"ref_15","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_16","doi-asserted-by":"crossref","unstructured":"Bossomaier, T., Barnett, L., Harr\u00e9, M., and Lizier, J.T. (2016). An Introduction to Transfer Entropy, Springer Nature. ebook.","DOI":"10.1007\/978-3-319-43222-9"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Akay, M. (2006). Recurrence Quantification Analysis. Wiley Encyclopedia of Biomedical Engineering, John Wiley & Sons.","DOI":"10.1002\/9780471740360"},{"key":"ref_18","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_19","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1103\/PhysRevLett.85.461","article-title":"Measuring Information Transfer","volume":"85","author":"Schreiber","year":"2000","journal-title":"PRL"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"424","DOI":"10.2307\/1912791","article-title":"Investigating causal relations by econometric models and cross-spectral methods","volume":"37","author":"Granger","year":"1969","journal-title":"Econometrica"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"066138","DOI":"10.1103\/PhysRevE.69.066138","article-title":"Estimating mutual information","volume":"69","author":"Kraskov","year":"2004","journal-title":"Phys. Rev. E"},{"key":"ref_22","first-page":"9","article-title":"A statistical estimate for the entropy of a random vector","volume":"23","author":"Kozachenko","year":"1987","journal-title":"Probl. Inf. Transm."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"11","DOI":"10.3389\/frobt.2014.00011","article-title":"JIDT: An information-theoretic toolkit for studying the dynamics of complex systems","volume":"1","author":"Lizier","year":"2014","journal-title":"Front. Robot. AI"},{"key":"ref_24","first-page":"463","article-title":"Partial directed coherence: A new concept in neural determination","volume":"84","author":"Sameshima","year":"2000","journal-title":"Biol. Cybern."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"3403","DOI":"10.1103\/PhysRevA.45.3403","article-title":"Determining embedding dimension for phase-space reconstruction using a geometrical construction","volume":"45","author":"Kennel","year":"1992","journal-title":"Phys. Rew. A"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Murari, A., Lungaroni, M., Peluso, E., Gaudio, P., Lerche, E., Garzotti, L., and Gelfusa, M. (2018). On the Use of Transfer Entropy to Investigate the Time Horizon of Causal Influences between Signals. Entropy, 20.","DOI":"10.3390\/e20090627"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"026006","DOI":"10.1088\/0029-5515\/56\/2\/026006","article-title":"Application of transfer entropy to causality detection and synchronization experiments in tokamaks","volume":"56","author":"Murari","year":"2016","journal-title":"Nucl. Fusion"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"076008","DOI":"10.1088\/0029-5515\/56\/7\/076008","article-title":"How to assess the efficiency of synchronization experiments in tokamaks","volume":"56","author":"Murari","year":"2016","journal-title":"Nucl. Fusion"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"126057","DOI":"10.1088\/1741-4326\/aa87e7","article-title":"On efficiency and interpretation of sawteeth pacing with on-axis ICRH modulation in JET","volume":"57","author":"Murari","year":"2017","journal-title":"Nucl. Fusion"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"036027","DOI":"10.1088\/1741-4326\/aa53b6","article-title":"Sawtooth pacing with on-axis ICRH modulation in JET-ILW","volume":"57","author":"Lerche","year":"2017","journal-title":"Nucl. Fusion"},{"key":"ref_31","unstructured":"Garzotti, L., Lang, P.T., Alonso, A., Alper, B., Belonohy, E., Boboc, A., Devaux, S., Eich, T., Frigione, D., and G\u00e1l, K. (2010, January 21\u201325). Investigating pellet ELM triggering physics using the new small size pellet launcher at JET. Proceedings of the 37th EPS Conference on Plasma Physics, Dublin, Ireland. Available online: http:\/\/ocs.ciemat.es\/EPS2010PAP\/pdf\/P2.131.pdf."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"056002","DOI":"10.1088\/1741-4326\/aaaf9c","article-title":"Adaptive predictors based on probabilistic SVM for real time disruption mitigation on JET","volume":"58","author":"Murari","year":"2018","journal-title":"Nucl. Fusion"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"056101","DOI":"10.1063\/1.4872017","article-title":"The influence of an ITER-like wall on disruptions at JET","volume":"21","year":"2014","journal-title":"Phys. Plasmas"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"579","DOI":"10.1111\/1467-9868.00194","article-title":"Statistical applications of the multivariate skew-normal distribution","volume":"61","author":"Azzalini","year":"1999","journal-title":"J. R. Stat. Soc."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"590","DOI":"10.1016\/j.fusengdes.2007.03.003","article-title":"The JET programme in support of ITER","volume":"82","author":"Pamela","year":"2007","journal-title":"Fusion Eng. Des."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"10D533","DOI":"10.1063\/1.3502039","article-title":"Mutual interaction of Faraday rotation and Cotton\u2014Mouton phase shift in JET polarimetric measurements","volume":"81","author":"Orsitto","year":"2010","journal-title":"Rev. Sci. Instrum."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"114007","DOI":"10.1088\/0741-3335\/56\/11\/114007","article-title":"Extensive statistical analysis of ELMs on JET with a carbon wall","volume":"56","author":"Murari","year":"2014","journal-title":"Plasma Phys. Control Fusion"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"373","DOI":"10.1007\/s00340-007-2607-9","article-title":"Application of a CO2 dial system for infrared detection of forest fire and reduction of false alarm","volume":"87","author":"Bellecci","year":"2007","journal-title":"Appl. Phys. B Lasers Opt."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/8\/865\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:57:13Z","timestamp":1760176633000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/8\/865"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,8,6]]},"references-count":38,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2020,8]]}},"alternative-id":["e22080865"],"URL":"https:\/\/doi.org\/10.3390\/e22080865","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2020,8,6]]}}}