{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:50:39Z","timestamp":1760161839754,"version":"build-2065373602"},"reference-count":33,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2021,2,8]],"date-time":"2021-02-08T00:00:00Z","timestamp":1612742400000},"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>Many methods of Granger causality, or broadly termed connectivity, have been developed to assess the causal relationships between the system variables based only on the information extracted from the time series. The power of these methods to capture the true underlying connectivity structure has been assessed using simulated dynamical systems where the ground truth is known. Here, we consider the presence of an unobserved variable that acts as a hidden source for the observed high-dimensional dynamical system and study the effect of the hidden source on the estimation of the connectivity structure. In particular, the focus is on estimating the direct causality effects in high-dimensional time series (not including the hidden source) of relatively short length. We examine the performance of a linear and a nonlinear connectivity measure using dimension reduction and compare them to a linear measure designed for latent variables. For the simulations, four systems are considered, the coupled H\u00e9non maps system, the coupled Mackey\u2013Glass system, the neural mass model and the vector autoregressive (VAR) process, each comprising 25 subsystems (variables for VAR) at close chain coupling structure and another subsystem (variable for VAR) driving all others acting as the hidden source. The results show that the direct causality measures estimate, in general terms, correctly the existing connectivity in the absence of the source when its driving is zero or weak, yet fail to detect the actual relationships when the driving is strong, with the nonlinear measure of dimension reduction performing best. An example from finance including and excluding the USA index in the global market indices highlights the different performance of the connectivity measures in the presence of hidden source.<\/jats:p>","DOI":"10.3390\/e23020208","type":"journal-article","created":{"date-parts":[[2021,2,9]],"date-time":"2021-02-09T23:43:16Z","timestamp":1612914196000},"page":"208","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["The Effect of a Hidden Source on the Estimation of Connectivity Networks from Multivariate Time Series"],"prefix":"10.3390","volume":"23","author":[{"given":"Christos","family":"Koutlis","sequence":"first","affiliation":[{"name":"Information Technologies Institute, Centre of Research and Technology Hellas, 57001 Thessaloniki, Greece"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5337-5688","authenticated-orcid":false,"given":"Dimitris","family":"Kugiumtzis","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, Aristotle University of Thessaloniki, University Campus, 54124 Thessaloniki, Greece"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,2,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1080\/00018732.2011.572452","article-title":"Analyzing and Modeling Real-World Phenomena with Complex Networks: A Survey of Applications","volume":"60","author":"Costa","year":"2011","journal-title":"Adv. Phys."},{"key":"ref_2","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_3","doi-asserted-by":"crossref","unstructured":"Siggiridou, E., Koutlis, C., Tsimpiris, A., and Kugiumtzis, D. (2019). Evaluation of Granger Causality Measures for Constructing Networks from Multivariate Time Series. Entropy, 21.","DOI":"10.3390\/e21111080"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1553","DOI":"10.3390\/e14081553","article-title":"Permutation Entropy and Its Main Biomedical and Econophysics Applications: A Review","volume":"14","author":"Zanin","year":"2012","journal-title":"Entropy"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"492902","DOI":"10.1155\/2012\/492902","article-title":"Editorial: Methodological Advances in Brain Connectivity","volume":"2012","author":"Faes","year":"2012","journal-title":"Comput. Math. Methods Med."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"282","DOI":"10.1109\/JPROC.2015.2476824","article-title":"Wiener-Granger Causality in Network Physiology with Applications to Cardiovascular Control and Neuroscience","volume":"104","author":"Porta","year":"2016","journal-title":"Proc. IEEE"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1016\/j.jeconom.2008.12.013","article-title":"Granger Causality in Risk and Detection of Extreme Risk Spillover between Financial Markets","volume":"150","author":"Hong","year":"2009","journal-title":"J. Econom."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"535","DOI":"10.1016\/j.jfineco.2011.12.010","article-title":"Econometric measures of connectedness and systemic risk in the finance and insurance sectors","volume":"104","author":"Billio","year":"2012","journal-title":"J. Financ. Econ."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"026222","DOI":"10.1103\/PhysRevE.72.026222","article-title":"Assessing Causality from Multivariate Time Series","volume":"72","author":"Verdes","year":"2005","journal-title":"Phys. Rev. E"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1038\/ncomms9502","article-title":"Identifying Causal Gateways and Mediators in Complex Spatio-Temporal Systems","volume":"6","author":"Runge","year":"2015","journal-title":"Nat. Commun."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1","DOI":"10.2307\/1912017","article-title":"Macroeconomics and Reality","volume":"48","author":"Sims","year":"1980","journal-title":"Econometrica"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"447","DOI":"10.1007\/s00181-011-0484-x","article-title":"Asymmetric causality tests with an application","volume":"43","author":"Hatemi","year":"2012","journal-title":"Empir. Econ."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1402","DOI":"10.1080\/1351847X.2019.1599406","article-title":"Further Insights on the Relationship Between SP500, VIX and Volume: A New Asymmetric Causality Test","volume":"25","author":"Kyrtsou","year":"2019","journal-title":"Eur. J. Financ."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"105003","DOI":"10.1088\/1367-2630\/16\/10\/105003","article-title":"Synergy and redundancy in the Granger causal analysis of dynamical networks","volume":"16","author":"Stramaglia","year":"2014","journal-title":"New J. Phys."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1049\/iet-syb.2014.0013","article-title":"Identifying Latent Dynamic Components in Biological Systems","volume":"9","author":"Kondofersky","year":"2015","journal-title":"IET Syst. Biol."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"103792","DOI":"10.1016\/j.jedc.2019.103792","article-title":"Dynamic Interbank Network Analysis Using Latent Space Models","volume":"112","author":"Linardi","year":"2020","journal-title":"J. Econ. Dyn. Control"},{"key":"ref_17","unstructured":"Zhang, K., and Hyv\u00e4rinen, A. (2009). On the Identifiability of the Post-nonlinear Causal Model. Proceedings of the Twenty-Fifth Conference on Uncertainty in Artificial Intelligence, AUAI Press Arlington."},{"key":"ref_18","unstructured":"Peters, J., Janzing, D., and Sch\u00f6lkopf, B. (2014, January 5\u201310). Causal Inference on Time Series Using Restricted Structural Equation Models. Proceedings of the Advances in Neural Information Processing Systems 26, 27th Annual Conference on Neural Information Processing Systems 2013 Curran Associates, Lake Tahoe, NV, USA."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"075310","DOI":"10.1063\/1.5025050","article-title":"Causal Network Reconstruction from Time Series: From Theoretical Assumptions to Practical Estimation","volume":"28","author":"Runge","year":"2018","journal-title":"Chaos"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1016\/j.jneumeth.2008.04.011","article-title":"Partial Granger Causality\u2013Eliminating Exogenous Inputs and Latent Variables","volume":"172","author":"Guo","year":"2008","journal-title":"J. Neurosci. Methods"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Faes, L., Erla, S., Porta, A., and Nollo, G. (2013). A Framework for Assessing Frequency Domain Causality in Physiological Time Series with Instantaneous Effects. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci., 371.","DOI":"10.1098\/rsta.2011.0618"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1850051","DOI":"10.1142\/S012906571850051X","article-title":"Identification of Hidden Sources by Estimating Instantaneous Causality in High-Dimensional Biomedical Time Series","volume":"29","author":"Koutlis","year":"2019","journal-title":"Int. J. Neural Syst."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"20110612","DOI":"10.1098\/rsta.2011.0612","article-title":"The impact of latent confounders in directed network analysis in neuroscience","volume":"371","author":"Ramb","year":"2013","journal-title":"Philos. Trans. R. Soc. A"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1016\/j.jneumeth.2015.02.015","article-title":"Network inference in the presence of latent confounders: The role of instantaneous causalities","volume":"245","author":"Elsegai","year":"2015","journal-title":"J. Neurosci. Methods"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1007\/s12021-015-9281-6","article-title":"Temporal Information of Directed Causal Connectivity in Multi-Trial ERP Data using Partial Granger Causality","volume":"14","author":"Youssofzadeh","year":"2015","journal-title":"Neuroinformatics"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1759","DOI":"10.1109\/TSP.2015.2500893","article-title":"Granger Causality in Multivariate Time Series Using a Time-Ordered Restricted Vector Autoregressive Model","volume":"64","author":"Siggiridou","year":"2016","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_27","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_28","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_29","doi-asserted-by":"crossref","first-page":"1250222","DOI":"10.1142\/S0218127412502227","article-title":"Detection of Direct Causal Effects and Application in the Analysis of Electroencephalograms from Patients with Epilepsy","volume":"22","author":"Papana","year":"2012","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"023118","DOI":"10.1063\/1.2911541","article-title":"Transition from Phase to Generalized Synchronization in Time-Delay Systems","volume":"18","author":"Senthilkumar","year":"2008","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1007\/s004220000160","article-title":"Relevance of Nonlinear Lumped-Parameter Models in the Analysis of Depth-EEG Epileptic Signals","volume":"83","author":"Wendling","year":"2000","journal-title":"Biol. Cybern."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1535","DOI":"10.1214\/15-AOS1315","article-title":"Regularized Estimation in Sparse High-Dimensional Time Series Models","volume":"43","author":"Basu","year":"2015","journal-title":"The Annals of Statistics"},{"key":"ref_33","first-page":"893","article-title":"Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation","volume":"96","author":"Engle","year":"1988","journal-title":"Econometrica"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/2\/208\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:21:31Z","timestamp":1760160091000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/2\/208"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,2,8]]},"references-count":33,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2021,2]]}},"alternative-id":["e23020208"],"URL":"https:\/\/doi.org\/10.3390\/e23020208","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2021,2,8]]}}}