{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,9]],"date-time":"2026-01-09T17:18:25Z","timestamp":1767979105463,"version":"3.49.0"},"reference-count":31,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2018,1,19]],"date-time":"2018-01-19T00:00:00Z","timestamp":1516320000000},"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>This work is focused on latent-variable graphical models for multivariate time series. We show how an algorithm which was originally used for finding zeros in the inverse of the covariance matrix can be generalized such that to identify the sparsity pattern of the inverse of spectral density matrix. When applied to a given time series, the algorithm produces a set of candidate models. Various information theoretic (IT) criteria are employed for deciding the winner. A novel IT criterion, which is tailored to our model selection problem, is introduced. Some options for reducing the computational burden are proposed and tested via numerical examples. We conduct an empirical study in which the algorithm is compared with the state-of-the-art. The results are good, and the major advantage is that the subjective choices made by the user are less important than in the case of other methods.<\/jats:p>","DOI":"10.3390\/e20010076","type":"journal-article","created":{"date-parts":[[2018,1,22]],"date-time":"2018-01-22T04:51:13Z","timestamp":1516596673000},"page":"76","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Maximum Entropy Expectation-Maximization Algorithm for Fitting Latent-Variable Graphical Models to Multivariate Time Series"],"prefix":"10.3390","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9365-7586","authenticated-orcid":false,"given":"Sa\u00efd","family":"Maanan","sequence":"first","affiliation":[{"name":"Department of Statistics, University of Auckland, Auckland 1142, New Zealand"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bogdan","family":"Dumitrescu","sequence":"additional","affiliation":[{"name":"Department of Automatic Control and Computers, University Politehnica of Bucharest, 060042 Bucharest, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5512-0868","authenticated-orcid":false,"given":"Ciprian","family":"Giurc\u0103neanu","sequence":"additional","affiliation":[{"name":"Department of Statistics, University of Auckland, Auckland 1142, New Zealand"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,1,19]]},"reference":[{"key":"ref_1","first-page":"1","article-title":"Remarks concerning graphical models for time series and point processes","volume":"16","author":"Brillinger","year":"1996","journal-title":"Rev. Econom."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1016\/S0165-0270(97)00100-3","article-title":"Identification of synaptic connections in neural ensembles by graphical models","volume":"77","author":"Dahlhaus","year":"1997","journal-title":"J. Neurosci. Methods"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"138","DOI":"10.1214\/aos\/1176349846","article-title":"Gaussian Markov distributions over finite graphs","volume":"14","author":"Speed","year":"1986","journal-title":"Ann. Stat."},{"key":"ref_4","first-page":"1935","article-title":"Latent variable graphical model selection via convex optimization","volume":"40","author":"Chandrasekaran","year":"2012","journal-title":"Ann. Stat."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1973","DOI":"10.1214\/12-AOS980","article-title":"Discussion: Latent variable graphical model selection via convex optimization","volume":"40","author":"Lauritzen","year":"2012","journal-title":"Ann. Stat."},{"key":"ref_6","unstructured":"Palomar, D., and Eldar, Y. (2010). Graphical models of autoregressive processes. Convex Optimization in Signal Processing and Communications, Cambridge University Press."},{"key":"ref_7","first-page":"2671","article-title":"Topology selection in graphical models of autoregressive processes","volume":"11","author":"Songsiri","year":"2010","journal-title":"J. Mach. Learn. Res."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"122","DOI":"10.1016\/j.sigpro.2016.10.023","article-title":"Conditional independence graphs for multivariate autoregressive models by convex optimization: Efficient Algorithms","volume":"133","author":"Maanan","year":"2017","journal-title":"Signal Process."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1167","DOI":"10.1109\/TAC.2012.2231551","article-title":"ARMA identification of graphical models","volume":"58","author":"Avventi","year":"2013","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2327","DOI":"10.1109\/TAC.2015.2491678","article-title":"AR Identification of latent-variable graphical models","volume":"61","author":"Zorzi","year":"2016","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1007\/s001840000055","article-title":"Graphical interaction models for multivariate time series","volume":"51","author":"Dahlhaus","year":"2000","journal-title":"Metrika"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Liegeois, R., Mishra, B., Zorzi, M., and Sepulchre, R. (2015, January 15\u201318). Sparse plus low-rank autoregressive identification in neuroimaging time series. Proceedings of the 54th IEEE Conference on Decision and Control (CDC), Osaka, Japan.","DOI":"10.1109\/CDC.2015.7402835"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Brockwell, P.J., and Davis, R.A. (1991). Time Series: Theory and Methods, Springer.","DOI":"10.1007\/978-1-4419-0320-4"},{"key":"ref_14","unstructured":"Grant, M., and Boyd, S. (2018, January 17). CVX: Matlab Software for Disciplined Convex Programming. Available online: http:\/\/cvxr.com\/cvx."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1214\/aos\/1176344136","article-title":"Estimating the dimension of a model","volume":"6","author":"Schwarz","year":"1978","journal-title":"Ann. Stat."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"716","DOI":"10.1109\/TAC.1974.1100705","article-title":"A new look at the statistical model identification","volume":"19","author":"Akaike","year":"1974","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1111\/j.1467-9892.1993.tb00144.x","article-title":"A corrected Akaike information criterion for vector autoregressive model selection","volume":"14","author":"Hurvich","year":"1993","journal-title":"J. Time Ser. Anal."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1007\/BF02479221","article-title":"Autoregressive model fitting for control","volume":"23","author":"Akaike","year":"1971","journal-title":"Ann. Inst. Stat. Math."},{"key":"ref_19","first-page":"175","article-title":"Universal sequential coding of single messages","volume":"23","author":"Shtarkov","year":"1987","journal-title":"Probl. Inf. Transm."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2537","DOI":"10.1109\/18.887861","article-title":"MDL denoising","volume":"46","author":"Rissanen","year":"2000","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Rissanen, J. (2007). Information and Complexity in Statistical Modeling, Springer.","DOI":"10.1007\/978-0-387-68812-1"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Maanan, S., Dumitrescu, B., and Giurc\u0103neanu, C.D. (September, January 29). Renormalized maximum likelihood for multivariate autoregressive models. Proceedings of the 2016 European Signal Processing Conference (EUSIPCO 2016), Budapest, Hungary.","DOI":"10.1109\/EUSIPCO.2016.7760228"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"759","DOI":"10.1093\/biomet\/asn034","article-title":"Extended Bayesian information criteria for model selection with large model spaces","volume":"95","author":"Chen","year":"2008","journal-title":"Biometrika"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"3347","DOI":"10.1109\/TSP.2009.2021633","article-title":"MDL denoising revisited","volume":"57","author":"Roos","year":"2009","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_25","unstructured":"Foygel, R., and Drton, M. (2010, January 6\u20139). Extended Bayesian information criteria for Gaussian graphical models. Proceedings of the 24th Annual Conference on Neural Information Processing Systems, Vancouver, BC, Canada."},{"key":"ref_26","unstructured":"Stoica, P., and Moses, R. (2005). Spectral Analysis of Signals, Pearson Prentice Hall."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2189","DOI":"10.1109\/TSP.2004.831032","article-title":"Learning graphical models for stationary time series","volume":"52","author":"Bach","year":"2004","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2561","DOI":"10.1109\/TAC.2012.2190153","article-title":"Time and spectral domain relative entropy: A New approach to multivariate spectral estimation","volume":"57","author":"Ferrante","year":"2012","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1647","DOI":"10.1109\/TAC.2014.2359713","article-title":"Multivariate spectral estimation based on the concept of optimal prediction","volume":"60","author":"Zorzi","year":"2015","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/j.automatica.2015.09.023","article-title":"An interpretation of the dual problem of the THREE-like approaches","volume":"62","author":"Zorzi","year":"2015","journal-title":"Automatica"},{"key":"ref_31","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":"Ann. Stat."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/1\/76\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T14:51:53Z","timestamp":1760194313000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/1\/76"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,1,19]]},"references-count":31,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2018,1]]}},"alternative-id":["e20010076"],"URL":"https:\/\/doi.org\/10.3390\/e20010076","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,1,19]]}}}