{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,20]],"date-time":"2026-03-20T19:55:17Z","timestamp":1774036517910,"version":"3.50.1"},"reference-count":39,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2020,10,3]],"date-time":"2020-10-03T00:00:00Z","timestamp":1601683200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We propose a new estimator to measure directed dependencies in time series. The dimensionality of data is first reduced using a new non-uniform embedding technique, where the variables are ranked according to a weighted sum of the amount of new information and improvement of the prediction accuracy provided by the variables. Then, using a greedy approach, the most informative subsets are selected in an iterative way. The algorithm terminates, when the highest ranked variable is not able to significantly improve the accuracy of the prediction as compared to that obtained using the existing selected subsets. In a simulation study, we compare our estimator to existing state-of-the-art methods at different data lengths and directed dependencies strengths. It is demonstrated that the proposed estimator has a significantly higher accuracy than that of existing methods, especially for the difficult case, where the data are highly correlated and coupled. Moreover, we show its false detection of directed dependencies due to instantaneous couplings effect is lower than that of existing measures. We also show applicability of the proposed estimator on real intracranial electroencephalography data.<\/jats:p>","DOI":"10.3390\/e22101124","type":"journal-article","created":{"date-parts":[[2020,10,3]],"date-time":"2020-10-03T07:22:16Z","timestamp":1601709736000},"page":"1124","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":17,"title":["Estimating Conditional Transfer Entropy in Time Series Using Mutual Information and Nonlinear Prediction"],"prefix":"10.3390","volume":"22","author":[{"given":"Payam","family":"Shahsavari Baboukani","sequence":"first","affiliation":[{"name":"Department of Electronic Systems, Aalborg University, 9220 Aalborg, Denmark"}]},{"given":"Carina","family":"Graversen","sequence":"additional","affiliation":[{"name":"Eriksholm Research Centre, Oticon A\/S, 3070 Snekkersten, Denmark"}]},{"given":"Emina","family":"Alickovic","sequence":"additional","affiliation":[{"name":"Eriksholm Research Centre, Oticon A\/S, 3070 Snekkersten, Denmark"},{"name":"Department of Electrical Engineering, Link\u00f6ping University, 581 83 Link\u00f6ping, Sweden"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3724-6114","authenticated-orcid":false,"given":"Jan","family":"\u00d8stergaard","sequence":"additional","affiliation":[{"name":"Department of Electronic Systems, Aalborg University, 9220 Aalborg, Denmark"}]}],"member":"1968","published-online":{"date-parts":[[2020,10,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"680","DOI":"10.1109\/TBME.2013.2286394","article-title":"Measuring time-varying information flow in scalp EEG signals: Orthogonalized partial directed coherence","volume":"61","author":"Omidvarnia","year":"2013","journal-title":"IEEE Trans. Biomed. Eng."},{"key":"ref_2","unstructured":"Cover, T.M., and Thomas, J.A. (2012). Elements of Information Theory, John Wiley & Sons."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1016\/j.dsp.2018.08.019","article-title":"A novel multivariate phase synchrony measure: Application to multichannel newborn EEG analysis","volume":"84","author":"Baboukani","year":"2019","journal-title":"Digit. Signal Process."},{"key":"ref_4","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_5","unstructured":"Baboukani, P.S., Mohammadi, S., and Azemi, G. (December, January 30). Classifying Single-Trial EEG During Motor Imagery Using a Multivariate Mutual Information Based Phase Synchrony Measure. Proceedings of the 2017 24th National and 2nd International Iranian Conference on Biomedical Engineering (ICBME), Tehran, Iran."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Gen\u00e7a\u011fa, D. (2018). Transfer Entropy. Entropy, 20.","DOI":"10.3390\/e20040288"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Faes, L., Marinazzo, D., and Stramaglia, S. (2017). Multiscale information decomposition: Exact computation for multivariate Gaussian processes. Entropy, 19.","DOI":"10.3390\/e19080408"},{"key":"ref_8","unstructured":"Derpich, M.S., Silva, E.I., and \u00d8stergaard, J. (2013). Fundamental inequalities and identities involving mutual and directed informations in closed-loop systems. arXiv."},{"key":"ref_9","unstructured":"Massey, J. (1990, January 27\u201330). Causality, feedback and directed information. Proceedings of the 1990 International Symposium on Information Theory and its Applications (ISITA-90), Waikiki, HI, USA."},{"key":"ref_10","unstructured":"Wiener, N. (1956). The Theory of Prediction. Modern Mathematics for Engineers, McGraw-Hill."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"238701","DOI":"10.1103\/PhysRevLett.116.238701","article-title":"Information flows? A critique of transfer entropies","volume":"116","author":"James","year":"2016","journal-title":"Phys. Rev. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"605","DOI":"10.1140\/epjb\/e2010-00034-5","article-title":"Differentiating information transfer and causal effect","volume":"73","author":"Lizier","year":"2010","journal-title":"Eur. Phys. J. B"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Montalto, A., Faes, L., and Marinazzo, D. (2014). MuTE: A MATLAB toolbox to compare established and novel estimators of the multivariate transfer entropy. PLoS ONE, 9.","DOI":"10.1371\/journal.pone.0109462"},{"key":"ref_14","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_15","doi-asserted-by":"crossref","unstructured":"Lindner, M., Vicente, R., Priesemann, V., and Wibral, M. (2011). TRENTOOL: A Matlab open source toolbox to analyse information flow in time series data with transfer entropy. BMC Neurosci., 12.","DOI":"10.1186\/1471-2202-12-119"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Wibral, M., Pampu, N., Priesemann, V., Siebenh\u00fchner, F., Seiwert, H., Lindner, M., Lizier, J.T., and Vicente, R. (2013). Measuring information-transfer delays. PLoS ONE, 8.","DOI":"10.1371\/journal.pone.0055809"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Bossomaier, T., Barnett, L., Harr\u00e9, M., and Lizier, J.T. (2016). An Introduction to Transfer Entropy, Springer International Publishing.","DOI":"10.1007\/978-3-319-43222-9"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"066019","DOI":"10.1088\/1741-2552\/ab4024","article-title":"Computational modeling of the effects of EEG volume conduction on functional connectivity metrics. Application to Alzheimer\u2019s disease continuum","volume":"16","author":"Hornero","year":"2019","journal-title":"J. Neural Eng."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2488","DOI":"10.1109\/TBME.2016.2569823","article-title":"An information-theoretic framework to map the spatiotemporal dynamics of the scalp electroencephalogram","volume":"63","author":"Faes","year":"2016","journal-title":"IEEE Trans. Biomed. Eng."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"150","DOI":"10.1109\/TMBMC.2018.2806454","article-title":"Directional and Causal Information Flow in EEG for Assessing Perceived Audio Quality","volume":"3","author":"Mehta","year":"2017","journal-title":"IEEE Trans. Mol. Biol. Multi-Scale Commun."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Zhang, J. (2018). Low-dimensional approximation searching strategy for transfer entropy from non-uniform embedding. PLoS ONE, 13.","DOI":"10.1371\/journal.pone.0194382"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"062114","DOI":"10.1103\/PhysRevE.95.062114","article-title":"Entropy measures, entropy estimators, and their performance in quantifying complex dynamics: Effects of artifacts, nonstationarity, and long-range correlations","volume":"95","author":"Xiong","year":"2017","journal-title":"Phys. Rev. E"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Jia, Z., Lin, Y., Jiao, Z., Ma, Y., and Wang, J. (2019). Detecting causality in multivariate time series via non-uniform embedding. Entropy, 21.","DOI":"10.3390\/e21121233"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"062918","DOI":"10.1103\/PhysRevE.87.062918","article-title":"Direct-coupling information measure from nonuniform embedding","volume":"87","author":"Kugiumtzis","year":"2013","journal-title":"Phys. Rev. E"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"036017","DOI":"10.1088\/1741-2552\/aa6401","article-title":"Comparison of connectivity analyses for resting state EEG data","volume":"14","author":"Olejarczyk","year":"2017","journal-title":"J. Neural Eng."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"827","DOI":"10.1162\/netn_a_00092","article-title":"Large-scale directed network inference with multivariate transfer entropy and hierarchical statistical testing","volume":"3","author":"Novelli","year":"2019","journal-title":"Netw. Neurosci."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1312","DOI":"10.1016\/j.envsoft.2008.03.007","article-title":"Non-linear variable selection for artificial neural networks using partial mutual information","volume":"23","author":"May","year":"2008","journal-title":"Environ. Model. Softw."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1016\/j.envsoft.2014.11.028","article-title":"Improved PMI-based input variable selection approach for artificial neural network and other data driven environmental and water resource models","volume":"65","author":"Li","year":"2015","journal-title":"Environ. Model. Softw."},{"key":"ref_29","unstructured":"Baboukani, P.S., Graversen, C., and \u00d8stergaard, J. Estimation of Directed Dependencies in Time Series Using Conditional Mutual Information and Non-linear Prediction. Proceedings of the European Signal Processing Conference (EUSIPCO), European Association for Signal Processing (EURASIP), in press."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1080\/00031305.1992.10475879","article-title":"An introduction to kernel and nearest-neighbor nonparametric regression","volume":"46","author":"Altman","year":"1992","journal-title":"Am. Stat."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"198","DOI":"10.3390\/e15010198","article-title":"Compensated transfer entropy as a tool for reliably estimating information transfer in physiological time series","volume":"15","author":"Faes","year":"2013","journal-title":"Entropy"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"032904","DOI":"10.1103\/PhysRevE.91.032904","article-title":"Estimating the decomposition of predictive information in multivariate systems","volume":"91","author":"Faes","year":"2015","journal-title":"Phys. Rev. E"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Danafar, S., Fukumizu, K., and Gomez, F. (2014). Kernel-based Information Criterion. arXiv.","DOI":"10.5539\/cis.v8n1p10"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"2556","DOI":"10.1109\/TBME.2014.2323131","article-title":"Lag-specific transfer entropy as a tool to assess cardiovascular and cardiorespiratory information transfer","volume":"61","author":"Faes","year":"2014","journal-title":"IEEE Trans. Biomed. Eng."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1016\/j.eplepsyres.2008.02.002","article-title":"Emergent network topology at seizure onset in humans","volume":"79","author":"Kramer","year":"2008","journal-title":"Epilepsy Res."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Wibral, M., Vicente, R., and Lizier, J.T. (2014). Directed Information Measures in Neuroscience, Springer.","DOI":"10.1007\/978-3-642-54474-3"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"238701","DOI":"10.1103\/PhysRevLett.103.238701","article-title":"Granger causality and transfer entropy are equivalent for Gaussian variables","volume":"103","author":"Barnett","year":"2009","journal-title":"Phys. Rev. Lett."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1093\/ije\/dyp334","article-title":"Illustrating bias due to conditioning on a collider","volume":"39","author":"Cole","year":"2010","journal-title":"Int. J. Epidemiol."},{"key":"ref_39","unstructured":"Williams, P.L., and Beer, R.D. (2011). Generalized measures of information transfer. arXiv."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/10\/1124\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:16:16Z","timestamp":1760177776000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/10\/1124"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,10,3]]},"references-count":39,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2020,10]]}},"alternative-id":["e22101124"],"URL":"https:\/\/doi.org\/10.3390\/e22101124","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,10,3]]}}}