{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,20]],"date-time":"2026-10-20T11:36:49Z","timestamp":1792496209168,"version":"4.5.0"},"reference-count":21,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2017,3,31]],"date-time":"2017-03-31T00:00:00Z","timestamp":1490918400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"S\u00e3o Paulo Research Foundation FAPESP","award":["2015\/50122-0"],"award-info":[{"award-number":["2015\/50122-0"]}]},{"name":"S\u00e3o Paulo Research Foundation FAPESP","award":["2014\/14229-2"],"award-info":[{"award-number":["2014\/14229-2"]}]},{"name":"S\u00e3o Paulo Research Foundation FAPESP","award":["2015\/07373-2"],"award-info":[{"award-number":["2015\/07373-2"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Transfer Entropy has been applied to experimental datasets to unveil causality between variables. In particular, its application to non-stationary systems has posed a great challenge due to restrictions on the sample size. Here, we have investigated the minimum sample size that produces a reliable causal inference. The methodology has been applied to two prototypical models: the linear model autoregressive-moving average and the non-linear logistic map. The relationship between the Transfer Entropy value and the sample size has been systematically examined. Additionally, we have shown the dependence of the reliable sample size and the strength of coupling between the variables. Our methodology offers a realistic lower bound for the sample size to produce a reliable outcome.<\/jats:p>","DOI":"10.3390\/e19040150","type":"journal-article","created":{"date-parts":[[2017,3,31]],"date-time":"2017-03-31T10:35:49Z","timestamp":1490956549000},"page":"150","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":17,"title":["Minimum Sample Size for Reliable Causal Inference Using Transfer Entropy"],"prefix":"10.3390","volume":"19","author":[{"given":"Ant\u00f4nio","family":"Ramos","sequence":"first","affiliation":[{"name":"National Institute for Space Research, S\u00e3o Jos\u00e9 dos Campos 12227-010, Brazil"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Elbert","family":"Macau","sequence":"additional","affiliation":[{"name":"National Institute for Space Research, S\u00e3o Jos\u00e9 dos Campos 12227-010, Brazil"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2017,3,31]]},"reference":[{"key":"ref_1","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":"Phys. Rev. Lett."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"4844","DOI":"10.3390\/e15114844","article-title":"Statistical mechanics and information-theoretic perspectives on complexity in the earth system","volume":"15","author":"Balasis","year":"2013","journal-title":"Entropy"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Li, Z., and Li, X. (2013). Estimating temporal causal interaction between spike trains with permutation and transfer entropy. PLoS ONE, 8.","DOI":"10.1371\/journal.pone.0070894"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"813","DOI":"10.1088\/0967-3334\/36\/4\/813","article-title":"Quantifying the causal strength of multivariate cardiovascular couplings with momentary information transfer","volume":"36","author":"Runge","year":"2015","journal-title":"Physiol. Meas."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"022114","DOI":"10.1103\/PhysRevE.93.022114","article-title":"Thermodynamic aspects of information transfer in complex dynamical systems","volume":"93","author":"Cafaro","year":"2016","journal-title":"Phys. Rev. E"},{"key":"ref_6","unstructured":"Miller, G.A. (1955). Information Theory in Psychology: Problems and Methods II\u2013B, Free Press."},{"key":"ref_7","unstructured":"Green, B.F., and Rogers, M.S. (1955). Information Theory in Psychology: Problems and Methods II\u2013B, Free Press."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"3416","DOI":"10.3390\/e16063416","article-title":"Identifying the coupling structure in complex systems through the optimal causation entropy principle","volume":"16","author":"Sun","year":"2014","journal-title":"Entropy"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"258701","DOI":"10.1103\/PhysRevLett.108.258701","article-title":"Escaping the curse of dimensionality in estimating multivariate transfer entropy","volume":"108","author":"Runge","year":"2012","journal-title":"Phys. Rev. Lett."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"6841","DOI":"10.1103\/PhysRevE.52.6841","article-title":"Estimating functions of probability distributions from a finite set of samples","volume":"52","author":"Wolpert","year":"1995","journal-title":"Phys. Rev. E"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2551","DOI":"10.1088\/0305-4470\/31\/11\/007","article-title":"Bayes\u2019 estimators of generalized entropies","volume":"31","author":"Holste","year":"1998","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"202001","DOI":"10.1088\/1751-8113\/41\/20\/202001","article-title":"Entropy estimates of small data sets","volume":"41","author":"Bonachela","year":"2008","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"040101","DOI":"10.1103\/PhysRevE.91.040101","article-title":"Granger causality for state-space models","volume":"91","author":"Barnett","year":"2015","journal-title":"Phys. Rev. E"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1016\/0025-5564(93)90032-6","article-title":"Does migration stabilize local population dynamics? Analysis of a discrete metapopulation model","volume":"118","author":"Gyllenberg","year":"1993","journal-title":"Math. Biosci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1362","DOI":"10.2307\/1940066","article-title":"Complex interactions between dispersal and dynamics: Lessons from coupled logistic equations","volume":"74","author":"Hastings","year":"1993","journal-title":"Ecology"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1016\/S0167-2789(98)00269-3","article-title":"Estimating the errors on measured entropy and mutual information","volume":"125","author":"Roulston","year":"1999","journal-title":"Physica D"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1016\/j.physd.2013.07.001","article-title":"Causation entropy identifies indirect influences, dominance of neighbors and anticipatory couplings","volume":"267","author":"Sun","year":"2014","journal-title":"Physica D"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"174102","DOI":"10.1103\/PhysRevLett.88.174102","article-title":"Permutation entropy: A natural complexity measure for time series","volume":"88","author":"Bandt","year":"2002","journal-title":"Phys. Rev. Lett."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"646","DOI":"10.1111\/j.1467-9892.2007.00528.x","article-title":"Order patterns in time series","volume":"28","author":"Bandt","year":"2007","journal-title":"J. Time Ser. Anal."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Takens, F. (1981). Detecting Strange Attractors in Turbulence, Springer.","DOI":"10.1007\/BFb0091924"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"712","DOI":"10.1103\/PhysRevLett.45.712","article-title":"Geometry from a time series","volume":"45","author":"Packard","year":"1980","journal-title":"Phys. Rev. Lett."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/4\/150\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:31:43Z","timestamp":1760207503000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/4\/150"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,3,31]]},"references-count":21,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2017,4]]}},"alternative-id":["e19040150"],"URL":"https:\/\/doi.org\/10.3390\/e19040150","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,3,31]]}}}