{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,16]],"date-time":"2026-05-16T16:12:59Z","timestamp":1778947979816,"version":"3.51.4"},"reference-count":45,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2013,5,24]],"date-time":"2013-05-24T00:00:00Z","timestamp":1369353600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Across geosciences, many investigated phenomena relate to specific complex systems consisting of intricately intertwined interacting subsystems. Such dynamical complex systems can be represented by a directed graph, where each link denotes an existence of a causal relation, or information exchange between the nodes. For geophysical systems such as global climate, these relations are commonly not theoretically known but estimated from recorded data using causality analysis methods. These include bivariate nonlinear methods based on information theory and their linear counterpart. The trade-off between the valuable sensitivity of nonlinear methods to more general interactions and the potentially higher numerical reliability of linear methods may affect inference regarding structure and variability of climate networks. We investigate the reliability of directed climate networks detected by selected methods and parameter settings, using a stationarized model of dimensionality-reduced surface air temperature data from reanalysis of 60-year global climate records. Overall, all studied bivariate causality methods provided reproducible estimates of climate causality networks, with the linear approximation showing higher reliability than the investigated nonlinear methods. On the example dataset, optimizing the investigated nonlinear methods with respect to reliability increased the similarity of the detected networks to their linear counterparts, supporting the particular hypothesis of the near-linearity of the surface air temperature reanalysis data.<\/jats:p>","DOI":"10.3390\/e15062023","type":"journal-article","created":{"date-parts":[[2013,5,24]],"date-time":"2013-05-24T13:02:03Z","timestamp":1369400523000},"page":"2023-2045","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":109,"title":["Reliability of Inference of Directed Climate Networks Using Conditional Mutual Information"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1402-1470","authenticated-orcid":false,"given":"Jaroslav","family":"Hlinka","sequence":"first","affiliation":[{"name":"Institute of Computer Science, Academy of Sciences of the Czech Republic, Pod vodarenskou vezi 2, 182 07, Prague 8, Czech Republic"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"David","family":"Hartman","sequence":"additional","affiliation":[{"name":"Institute of Computer Science, Academy of Sciences of the Czech Republic, Pod vodarenskou vezi 2, 182 07, Prague 8, Czech Republic"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Martin","family":"Vejmelka","sequence":"additional","affiliation":[{"name":"Institute of Computer Science, Academy of Sciences of the Czech Republic, Pod vodarenskou vezi 2, 182 07, Prague 8, Czech Republic"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jakob","family":"Runge","sequence":"additional","affiliation":[{"name":"Potsdam Institute for Climate Impact Research (PIK), 14473 Potsdam, Germany"},{"name":"Department of Physics, Humboldt University, 12489 Berlin, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Norbert","family":"Marwan","sequence":"additional","affiliation":[{"name":"Potsdam Institute for Climate Impact Research (PIK), 14473 Potsdam, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"J\u00fcrgen","family":"Kurths","sequence":"additional","affiliation":[{"name":"Potsdam Institute for Climate Impact Research (PIK), 14473 Potsdam, Germany"},{"name":"Department of Physics, Humboldt University, 12489 Berlin, Germany"},{"name":"Institute for Complex Systems and Mathematical Biology, University of Aberdeen, Aberdeen AB24 3UE, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Milan","family":"Palu\u0161","sequence":"additional","affiliation":[{"name":"Institute of Computer Science, Academy of Sciences of the Czech Republic, Pod vodarenskou vezi 2, 182 07, Prague 8, Czech Republic"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2013,5,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1137\/S003614450342480","article-title":"The structure and function of complex networks","volume":"45","author":"Newman","year":"2003","journal-title":"SIAM Rev."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1016\/j.physrep.2005.10.009","article-title":"Complex networks: Structure and dynamics","volume":"424","author":"Boccaletti","year":"2006","journal-title":"Phys. Rep."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"497","DOI":"10.1016\/j.physa.2003.10.045","article-title":"The architecture of the climate network","volume":"333","author":"Tsonis","year":"2004","journal-title":"Physica A"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Tsonis, A.A., Swanson, K.L., and Roebber, P.J. (2006). What do networks have to do with climate?. Bull. Am. Meteorol. Soc.","DOI":"10.1175\/BAMS-87-5-585"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Yamasaki, K., Gozolchiani, A., and Havlin, S. (2008). Climate networks around the globe are significantly affected by El Nino. Phys. Rev. Lett.","DOI":"10.1103\/PhysRevLett.100.228501"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"971","DOI":"10.1007\/s00382-011-1156-4","article-title":"Analysis of spatial and temporal extreme monsoonal rainfall over South Asia using complex networks","volume":"39","author":"Malik","year":"2012","journal-title":"Clim. Dyn."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"889","DOI":"10.1007\/s00382-011-1135-9","article-title":"Multivariate and multiscale dependence in the global climate system revealed through complex networks","volume":"39","author":"Steinhaeuser","year":"2012","journal-title":"Clim. Dyn."},{"key":"ref_8","unstructured":"Steinhaeuser, K., Chawla, N.V., and Ganguly, A.R. (2013, January 5\u20136). Complex Networks in Climate Science: Progress, Opportunities and Challenges. Proceedings of the 2010 Conference on Intelligent Data Understanding, CIDU 2010, Mountain View, CA, USA."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Donges, J.F., Zou, Y., Marwan, N., and Kurths, J. (2009). The backbone of the climate network. EPL.","DOI":"10.1209\/0295-5075\/87\/48007"},{"key":"ref_10","first-page":"157","article-title":"Complex networks in climate dynamics","volume":"174","author":"Donges","year":"2009","journal-title":"Eur. Phys. J."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Hlinka, J., Hartman, D., Vejmelka, M., Novotna, D., and Palu\u0161, M. (2012). Non-linear dependence and teleconnections in climate data: Sources, relevance, nonstationarity. Clim. Dyn.","DOI":"10.1007\/s00382-013-1780-2"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"5648","DOI":"10.1175\/JCLI-D-11-00387.1","article-title":"Causal discovery for climate research using graphical models","volume":"25","author":"Deng","year":"2012","journal-title":"J. Clim."},{"key":"ref_13","first-page":"5648","article-title":"A new type of climate network based on probabilistic graphical models: Results of boreal winter versus summer","volume":"39","author":"Deng","year":"2012","journal-title":"Geophys. Res. Lett."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"424","DOI":"10.2307\/1912791","article-title":"Investigating causal relations by econometric model and cross-spectral methods","volume":"37","author":"Granger","year":"1969","journal-title":"Econometrica"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Vejmelka, M., and Palus, M. (2008). Inferring the directionality of coupling with conditional mutual information. Phys. Rev. E.","DOI":"10.1103\/PhysRevE.77.026214"},{"key":"ref_16","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_17","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1175\/1520-0477(2001)082<0247:TNNYRM>2.3.CO;2","article-title":"The NCEP-NCAR 50-year reanalysis: Monthly means CD-ROM and documentation","volume":"82","author":"Kistler","year":"2001","journal-title":"Bull. Am. Meteorol. Soc."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"437","DOI":"10.1175\/1520-0477(1996)077<0437:TNYRP>2.0.CO;2","article-title":"The NCEP\/NCAR 40-year reanalysis project","volume":"77","author":"Kalnay","year":"1996","journal-title":"Bull. Am. Meteorol.Soc."},{"key":"ref_19","unstructured":"Wiener, N. (1956). Modern Mathermatics for Engineers, McGraw-Hill. chapter The theory of prediction."},{"key":"ref_20","unstructured":"Ding, M., Chen, Y., and Bressler, S.L. (2006). Handbook of Time Series Analysis, Wiley-VCH Verlag GmbH & Co. KGaA."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"907","DOI":"10.1080\/01621459.1984.10477110","article-title":"Measurement of linear dependence and feedback between multiple time series","volume":"79","author":"Geweke","year":"1982","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"907","DOI":"10.1080\/01621459.1984.10477110","article-title":"Measures of conditional linear dependence and feedback between time series","volume":"79","author":"Geweke","year":"1984","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Barnett, L., Barrett, A.B., and Seth, A.K. (2009). Granger causality and transfer entropy are equivalent for gaussian variables. Phys. Rev. Lett., 103.","DOI":"10.1103\/PhysRevLett.103.238701"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1016\/0375-9601(93)90827-M","article-title":"Information theoretic test for nonlinearity in time series","volume":"175","author":"Palus","year":"1993","journal-title":"Phys. Lett."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Frenzel, S., and Pompe, B. (2007). Partial mutual information for coupling analysis of multivariate time series. Phys. Rev. Lett., 99.","DOI":"10.1103\/PhysRevLett.99.204101"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1007\/BF02289233","article-title":"The varimax criterion for analytic rotation in factor analysis","volume":"23","author":"Kaiser","year":"1958","journal-title":"Psychometrika"},{"key":"ref_27","first-page":"461","article-title":"Estimating the dimension of a model","volume":"5","author":"Schwarz","year":"1978","journal-title":"Ann. Stat."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1111\/j.2517-6161.1995.tb02031.x","article-title":"Controlling the false discovery rate: A practical and powerful approach to multiple testing","volume":"57","author":"Benjamini","year":"1995","journal-title":"J. R. Stat. Soc. Series B-Methodol."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"951","DOI":"10.1103\/PhysRevLett.73.951","article-title":"Generating surrogate data for time series with several simultaneously measured variables","volume":"73","author":"Prichard","year":"1994","journal-title":"Phys. Rev. Lett."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1016\/S0375-9601(97)00635-X","article-title":"Detecting phase synchronization in noisy systems","volume":"235","author":"Palus","year":"1997","journal-title":"Phys. Lett."},{"key":"ref_31","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_32","doi-asserted-by":"crossref","unstructured":"Runge, J., Heitzig, J., Petoukhov, V., and Kurths, J. (2012). Escaping the curse of dimensionality in estimating multivariate transfer entropy. Phys. Rev. Lett.","DOI":"10.1103\/PhysRevLett.108.258701"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Runge, J., Heitzig, J., Marwan, N., and Kurths, J. (2012). Quantifying causal coupling strength: A lag-specific measure for multivariate time series related to transfer entropy. Phys. Rev.","DOI":"10.1103\/PhysRevE.86.061121"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"463","DOI":"10.1007\/PL00007990","article-title":"Partial directed coherence: A new concept in neural structure determination","volume":"84","author":"Baccala","year":"2001","journal-title":"Biol. Cybern."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1007\/s004220000235","article-title":"Evaluating causal relations in neural systems: Granger causality, directed transfer function and statistical assessment of significance","volume":"85","author":"Kaminski","year":"2001","journal-title":"Biol. Cybern."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"2218","DOI":"10.1016\/j.neuroimage.2010.08.042","article-title":"Functional connectivity in resting-state fMRI: Is linear correlation sufficient?","volume":"54","author":"Hlinka","year":"2011","journal-title":"NeuroImage"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1016\/0375-9601(94)91002-2","article-title":"Testing for nonlinearity in weather records","volume":"193","author":"Palus","year":"1994","journal-title":"Phys. Lett."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"721","DOI":"10.5194\/npg-11-721-2004","article-title":"Enhanced Monte Carlo Singular System Analysis and detection of period 7.8 years oscillatory modes in the monthly NAO index and temperature records","volume":"11","author":"Palus","year":"2004","journal-title":"Nonlinear Process. Geophys."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"287","DOI":"10.5194\/npg-13-287-2006","article-title":"Quasi-biennial oscillations extracted from the monthly NAO index and temperature records are phase-synchronized","volume":"13","author":"Palus","year":"2006","journal-title":"Nonlinear Process. Geophys."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"923","DOI":"10.1016\/j.jastp.2009.03.012","article-title":"Phase-coherent oscillatory modes in solar and geomagnetic activity and climate variability","volume":"71","author":"Palus","year":"2009","journal-title":"J. Atmos. Solar-terr. Phys."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"751","DOI":"10.5194\/npg-18-751-2011","article-title":"Discerning connectivity from dynamics in climate networks","volume":"18","author":"Palus","year":"2011","journal-title":"Nonlinear Process. Geophys."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"4060","DOI":"10.1175\/2010JCLI3181.1","article-title":"Oscillatory climate modes in the eastern mediterranean and their synchronization with the north atlantic oscillation","volume":"23","author":"Feliks","year":"2010","journal-title":"J. Clim."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"2045","DOI":"10.1007\/s00382-011-1119-9","article-title":"ENSO nonlinearity in a warming climate","volume":"37","author":"Boucharel","year":"2011","journal-title":"Clim. Dyn."},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Osprey, S.M., and Ambaum, M.H.P. (2011). Evidence for the chaotic origin of northern annular mode variability. Geophys. Res. Lett.","DOI":"10.1029\/2011GL048181"},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Mokhov, I.I., Smirnov, D.A., Nakonechny, P.I., Kozlenko, S.S., Seleznev, E.P., and Kurths, J. (2011). Alternating mutual influence of El-Nino\/Southern Oscillation and Indian monsoon. Geophys. Res. Lett.","DOI":"10.1029\/2010GL045932"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/15\/6\/2023\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:46:59Z","timestamp":1760219219000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/15\/6\/2023"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,5,24]]},"references-count":45,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2013,6]]}},"alternative-id":["e15062023"],"URL":"https:\/\/doi.org\/10.3390\/e15062023","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,5,24]]}}}