{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,25]],"date-time":"2026-02-25T20:18:27Z","timestamp":1772050707418,"version":"3.50.1"},"reference-count":84,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2022,6,22]],"date-time":"2022-06-22T00:00:00Z","timestamp":1655856000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001824","name":"Czech Science Foundation","doi-asserted-by":"publisher","award":["19-16066S"],"award-info":[{"award-number":["19-16066S"]}],"id":[{"id":"10.13039\/501100001824","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001824","name":"Czech Science Foundation","doi-asserted-by":"publisher","award":["18696"],"award-info":[{"award-number":["18696"]}],"id":[{"id":"10.13039\/501100001824","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001824","name":"Czech Science Foundation","doi-asserted-by":"publisher","award":["W911NF-17-1-0108"],"award-info":[{"award-number":["W911NF-17-1-0108"]}],"id":[{"id":"10.13039\/501100001824","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Jubil\u00e4umsfonds der \u00d6sterreichischen Nationalbank Project","award":["19-16066S"],"award-info":[{"award-number":["19-16066S"]}]},{"name":"Jubil\u00e4umsfonds der \u00d6sterreichischen Nationalbank Project","award":["18696"],"award-info":[{"award-number":["18696"]}]},{"name":"Jubil\u00e4umsfonds der \u00d6sterreichischen Nationalbank Project","award":["W911NF-17-1-0108"],"award-info":[{"award-number":["W911NF-17-1-0108"]}]},{"name":"U.S. Army RDECOM\u2014Atlantic","award":["19-16066S"],"award-info":[{"award-number":["19-16066S"]}]},{"name":"U.S. Army RDECOM\u2014Atlantic","award":["18696"],"award-info":[{"award-number":["18696"]}]},{"name":"U.S. Army RDECOM\u2014Atlantic","award":["W911NF-17-1-0108"],"award-info":[{"award-number":["W911NF-17-1-0108"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Uncovering causal interdependencies from observational data is one of the great challenges of a nonlinear time series analysis. In this paper, we discuss this topic with the help of an information-theoretic concept known as R\u00e9nyi\u2019s information measure. In particular, we tackle the directional information flow between bivariate time series in terms of R\u00e9nyi\u2019s transfer entropy. We show that by choosing R\u00e9nyi\u2019s parameter \u03b1, we can appropriately control information that is transferred only between selected parts of the underlying distributions. This, in turn, is a particularly potent tool for quantifying causal interdependencies in time series, where the knowledge of \u201cblack swan\u201d events, such as spikes or sudden jumps, are of key importance. In this connection, we first prove that for Gaussian variables, Granger causality and R\u00e9nyi transfer entropy are entirely equivalent. Moreover, we also partially extend these results to heavy-tailed \u03b1-Gaussian variables. These results allow establishing a connection between autoregressive and R\u00e9nyi entropy-based information-theoretic approaches to data-driven causal inference. To aid our intuition, we employed the Leonenko et al. entropy estimator and analyzed R\u00e9nyi\u2019s information flow between bivariate time series generated from two unidirectionally coupled R\u00f6ssler systems. Notably, we find that R\u00e9nyi\u2019s transfer entropy not only allows us to detect a threshold of synchronization but it also provides non-trivial insight into the structure of a transient regime that exists between the region of chaotic correlations and synchronization threshold. In addition, from R\u00e9nyi\u2019s transfer entropy, we could reliably infer the direction of coupling and, hence, causality, only for coupling strengths smaller than the onset value of the transient regime, i.e., when two R\u00f6ssler systems are coupled but have not yet entered synchronization.<\/jats:p>","DOI":"10.3390\/e24070855","type":"journal-article","created":{"date-parts":[[2022,6,22]],"date-time":"2022-06-22T21:31:06Z","timestamp":1655933466000},"page":"855","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":19,"title":["Causal Inference in Time Series in Terms of R\u00e9nyi Transfer Entropy"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7940-204X","authenticated-orcid":false,"given":"Petr","family":"Jizba","sequence":"first","affiliation":[{"name":"Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, B\u0159ehov\u00e1 7, 115 19 Prague, Czech Republic"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7868-106X","authenticated-orcid":false,"given":"Hynek","family":"Lavi\u010dka","sequence":"additional","affiliation":[{"name":"Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, B\u0159ehov\u00e1 7, 115 19 Prague, Czech Republic"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8423-8574","authenticated-orcid":false,"given":"Zlata","family":"Tabachov\u00e1","sequence":"additional","affiliation":[{"name":"Complexity Science Hub Vienna, Josefst\u00e4dter Stra\u00dfe 39, 1080 Vienna, Austria"}]}],"member":"1968","published-online":{"date-parts":[[2022,6,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0370-1573(98)00035-0","article-title":"Interdisciplinary application of nonlinear time series methods","volume":"308","author":"Schreiber","year":"1999","journal-title":"Phys. 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