{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,13]],"date-time":"2026-03-13T08:00:21Z","timestamp":1773388821776,"version":"3.50.1"},"reference-count":40,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2025,3,3]],"date-time":"2025-03-03T00:00:00Z","timestamp":1740960000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,3,3]],"date-time":"2025-03-03T00:00:00Z","timestamp":1740960000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["LA\/P\/0063\/2020"],"award-info":[{"award-number":["LA\/P\/0063\/2020"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["LA\/P\/0063\/2020"],"award-info":[{"award-number":["LA\/P\/0063\/2020"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["LA\/P\/0063\/2020"],"award-info":[{"award-number":["LA\/P\/0063\/2020"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["LA\/P\/0063\/2020"],"award-info":[{"award-number":["LA\/P\/0063\/2020"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100006752","name":"Universidade do Porto","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100006752","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Data Min Knowl Disc"],"published-print":{"date-parts":[[2025,5]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>Multivariate time series analysis is a vital but challenging task, with multidisciplinary applicability, tackling the characterization of multiple interconnected variables over time and their dependencies. Traditional methodologies often adapt univariate approaches or rely on assumptions specific to certain domains or problems, presenting limitations. A recent promising alternative is to map multivariate time series into high-level network structures such as multiplex networks, with past work relying on connecting successive time series components with interconnections between contemporary timestamps. In this work, we first define a novel cross-horizontal visibility mapping between lagged timestamps of different time series and then introduce the concept of multilayer horizontal visibility graphs. This allows describing cross-dimension dependencies via inter-layer edges, leveraging the entire structure of multilayer networks. To this end, a novel parameter-free topological measure is proposed and common measures are extended for the multilayer setting. Our approach is general and applicable to any kind of multivariate time series data. We provide an extensive experimental evaluation with both synthetic and real-world datasets. We first explore the proposed methodology and the data properties highlighted by each measure, showing that inter-layer edges based on cross-horizontal visibility preserve more information than previous mappings, while also complementing the information captured by commonly used intra-layer edges. We then illustrate the applicability and validity of our approach in multivariate time series mining tasks, showcasing its potential for enhanced data analysis and insights.<\/jats:p>","DOI":"10.1007\/s10618-025-01089-4","type":"journal-article","created":{"date-parts":[[2025,3,3]],"date-time":"2025-03-03T17:01:27Z","timestamp":1741021287000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Multilayer horizontal visibility graphs for multivariate time series analysis"],"prefix":"10.1007","volume":"39","author":[{"given":"Vanessa","family":"Freitas Silva","sequence":"first","affiliation":[]},{"given":"Maria Eduarda","family":"Silva","sequence":"additional","affiliation":[]},{"given":"Pedro","family":"Ribeiro","sequence":"additional","affiliation":[]},{"given":"Fernando","family":"Silva","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,3,3]]},"reference":[{"issue":"1","key":"1089_CR1","doi-asserted-by":"publisher","first-page":"47","DOI":"10.1103\/RevModPhys.74.47","volume":"74","author":"R Albert","year":"2002","unstructured":"Albert R, Barab\u00e1si AL (2002) Statistical mechanics of complex networks. Rev Mod Phys 74(1):47. https:\/\/doi.org\/10.1103\/RevModPhys.74.47","journal-title":"Rev Mod Phys"},{"key":"1089_CR2","unstructured":"Bagnall AJ, Dau HA, Lines J, et\u00a0al (2018) The UEA multivariate time series classification archive, 2018. CoRR abs\/1811.00075. http:\/\/arxiv.org\/abs\/1811.00075,"},{"key":"1089_CR3","volume-title":"Network Science","author":"AL Barab\u00e1si","year":"2016","unstructured":"Barab\u00e1si AL (2016) Network Science. Cambridge University Press, Cambridge, United Kingdom"},{"key":"1089_CR4","unstructured":"Barbosa SM (2012) mAr: Multivariate AutoRegressive analysis. https:\/\/CRAN.R-project.org\/package=mAr, r package version 1.1-2"},{"key":"1089_CR5","doi-asserted-by":"publisher","first-page":"P1000","DOI":"10.1088\/1742-5468\/2008\/10\/P10008","volume":"10","author":"VD Blondel","year":"2008","unstructured":"Blondel VD, Guillaume JL, Lambiotte R et al (2008) (2008) Fast unfolding of communities in large networks. J Stat Mech: Theory Exp 10:P1000. https:\/\/doi.org\/10.1088\/1742-5468\/2008\/10\/P10008","journal-title":"J Stat Mech: Theory Exp"},{"issue":"1","key":"1089_CR6","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.physrep.2014.07.001","volume":"544","author":"S Boccaletti","year":"2014","unstructured":"Boccaletti S, Bianconi G, Criado R et al (2014) The structure and dynamics of multilayer networks. Phys Rep 544(1):1\u201312. https:\/\/doi.org\/10.1016\/j.physrep.2014.07.001","journal-title":"Phys Rep"},{"key":"1089_CR7","doi-asserted-by":"publisher","DOI":"10.14778\/3565816.3565822","author":"A Bonifati","year":"2022","unstructured":"Bonifati A, Buono FD, Guerra F et al (2022) Time2Feat: learning interpretable representations for multivariate time series clustering. Proc VLDB Endowment. https:\/\/doi.org\/10.14778\/3565816.3565822","journal-title":"Proc VLDB Endowment"},{"key":"1089_CR8","doi-asserted-by":"publisher","unstructured":"Cipra T (2020) Time series in economics and finance. Springer, Wiesbaden, Deutschlan. https:\/\/doi.org\/10.1007\/978-3-030-46347-2","DOI":"10.1007\/978-3-030-46347-2"},{"issue":"1","key":"1089_CR9","doi-asserted-by":"publisher","first-page":"167","DOI":"10.1080\/00018730601170527","volume":"56","author":"LdF Costa","year":"2007","unstructured":"Costa LdF, Rodrigues FA, Travieso G et al (2007) Characterization of complex networks: a survey of measurements. Adv Phys 56(1):167\u2013242. https:\/\/doi.org\/10.1080\/00018730601170527","journal-title":"Adv Phys"},{"issue":"1","key":"1089_CR10","doi-asserted-by":"publisher","first-page":"01231","DOI":"10.1103\/physreve.97.012312","volume":"97","author":"D Eroglu","year":"2018","unstructured":"Eroglu D, Marwan N, Stebich M et al (2018) Multiplex recurrence networks. Phys Rev E 97(1):01231. https:\/\/doi.org\/10.1103\/physreve.97.012312","journal-title":"Phys Rev E"},{"key":"1089_CR11","doi-asserted-by":"publisher","unstructured":"Fulcher BD (2018) Feature-based time-series analysis. In: Feature engineering for machine learning and data analytics. CRC Press, Boca Raton, Florida, p 87\u201311https:\/\/doi.org\/10.1201\/9781315181080","DOI":"10.1201\/9781315181080"},{"issue":"12","key":"1089_CR12","doi-asserted-by":"publisher","first-page":"3026","DOI":"10.1080\/00018730601170527","volume":"26","author":"BD Fulcher","year":"2014","unstructured":"Fulcher BD, Jones NS (2014) Highly comparative feature-based time-series classification. IEEE Trans Knowledge Data Eng 26(12):3026\u20133037. https:\/\/doi.org\/10.1080\/00018730601170527","journal-title":"IEEE Trans Knowledge Data Eng"},{"issue":"5","key":"1089_CR13","doi-asserted-by":"publisher","first-page":"527","DOI":"10.1016\/j.cels.2017.10.001","volume":"5","author":"BD Fulcher","year":"2017","unstructured":"Fulcher BD, Jones NS (2017) hctsa: a computational framework for automated time-series phenotyping using massive feature extraction. Cell Syst 5(5):527\u2013531. https:\/\/doi.org\/10.1016\/j.cels.2017.10.001","journal-title":"Cell Syst"},{"issue":"1","key":"1089_CR14","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/S10618-020-00716-6","volume":"35","author":"X Huang","year":"2021","unstructured":"Huang X, Chen D, Ren T et al (2021) A survey of community detection methods in multilayer networks. Data Mining Knowledge Discovery 35(1):1\u20134. https:\/\/doi.org\/10.1007\/S10618-020-00716-6","journal-title":"Data Mining Knowledge Discovery"},{"key":"1089_CR15","doi-asserted-by":"publisher","unstructured":"Hyndman RJ, Wang E, Laptev N (2015) Large-scale unusual time series detection. In: 2015 IEEE international conference on data mining workshop (ICDMW), IEEE, pp 1616\u2013161https:\/\/doi.org\/10.1109\/ICDMW.2015.104","DOI":"10.1109\/ICDMW.2015.104"},{"key":"1089_CR16","doi-asserted-by":"publisher","unstructured":"Kang Y, Hyndman RJ, Li F (2020) GRATIS: generating time series with diverse and controllable characteristics. The ASA Data Science Journ, Statistical Analysis and Data Mining. https:\/\/doi.org\/10.1002\/SAM.11461","DOI":"10.1002\/SAM.11461"},{"issue":"3","key":"1089_CR17","doi-asserted-by":"publisher","first-page":"203","DOI":"10.2139\/ssrn.2341334","volume":"2","author":"M Kivel\u00e4","year":"2014","unstructured":"Kivel\u00e4 M, Arenas A, Barthelemy M et al (2014) Multilayer networks. J Complex Netw 2(3):203\u2013271. https:\/\/doi.org\/10.2139\/ssrn.2341334","journal-title":"J Complex Netw"},{"issue":"13","key":"1089_CR18","doi-asserted-by":"publisher","first-page":"4972","DOI":"10.1073\/pnas.0709247105","volume":"105","author":"L Lacasa","year":"2008","unstructured":"Lacasa L, Luque B, Ballesteros F et al (2008) From time series to complex networks: the visibility graph. Proc National Acad Sci 105(13):4972\u20134975. https:\/\/doi.org\/10.1073\/pnas.0709247105","journal-title":"Proc National Acad Sci"},{"issue":"1","key":"1089_CR19","doi-asserted-by":"publisher","first-page":"1550","DOI":"10.1038\/srep15508","volume":"5","author":"L Lacasa","year":"2015","unstructured":"Lacasa L, Nicosia V, Latora V (2015) Network structure of multivariate time series. Scientif Rep 5(1):1550. https:\/\/doi.org\/10.1038\/srep15508","journal-title":"Scientif Rep"},{"issue":"8","key":"1089_CR20","doi-asserted-by":"publisher","first-page":"08310","DOI":"10.1063\/1.4927835","volume":"25","author":"X Lan","year":"2015","unstructured":"Lan X, Mo H, Chen S et al (2015) Fast transformation from time series to visibility graphs. Chaos: An Interdisciplinary J Nonlinear Sci 25(8):08310. https:\/\/doi.org\/10.1063\/1.4927835","journal-title":"Chaos: An Interdisciplinary J Nonlinear Sci"},{"key":"1089_CR21","doi-asserted-by":"publisher","first-page":"10791","DOI":"10.1016\/j.patcog.2021.107919","volume":"115","author":"H Li","year":"2021","unstructured":"Li H, Liu Z (2021) Multivariate time series clustering based on complex network. Pattern Recognition 115:10791. https:\/\/doi.org\/10.1016\/j.patcog.2021.107919","journal-title":"Pattern Recognition"},{"issue":"4","key":"1089_CR22","doi-asserted-by":"publisher","first-page":"04610","DOI":"10.1103\/PhysRevE.80.046103","volume":"80","author":"B Luque","year":"2009","unstructured":"Luque B, Lacasa L, Ballesteros F et al (2009) Horizontal visibility graphs: exact results for random time series. Phys Rev E 80(4):04610. https:\/\/doi.org\/10.1103\/PhysRevE.80.046103","journal-title":"Phys Rev E"},{"key":"1089_CR23","doi-asserted-by":"publisher","unstructured":"Maharaj EA, D\u2019Urso P, Caiado J (2019) Time series clustering and classification. CRC Press, Boca Raton, Florid. https:\/\/doi.org\/10.1201\/9780429058264","DOI":"10.1201\/9780429058264"},{"issue":"1","key":"1089_CR24","doi-asserted-by":"publisher","first-page":"86","DOI":"10.1016\/j.ijforecast.2019.02.011","volume":"36","author":"P Montero-Manso","year":"2020","unstructured":"Montero-Manso P, Athanasopoulos G, Hyndman RJ et al (2020) FFORMA: feature-based forecast model averaging. Int J Forecasting 36(1):86\u20139. https:\/\/doi.org\/10.1016\/j.ijforecast.2019.02.011","journal-title":"Int J Forecasting"},{"key":"1089_CR25","unstructured":"Nakatani T (2014) ccgarch: An R Package for Modelling Multivariate GARCH Models with Conditional Correlations. https:\/\/mran.microsoft.com\/snapshot\/2017-05-03\/web\/packages\/ccgarch\/ccgarch.pdf, r package version 0.2.3"},{"issue":"4","key":"1089_CR26","doi-asserted-by":"publisher","first-page":"10022","DOI":"10.1016\/j.patter.2021.100227","volume":"2","author":"RL Peach","year":"2021","unstructured":"Peach RL, Arnaudon A, Schmidt JA et al (2021) HCGA: highly comparative graph analysis for network phenotyping. Patterns 2(4):10022. https:\/\/doi.org\/10.1016\/j.patter.2021.100227","journal-title":"Patterns"},{"issue":"3","key":"1089_CR27","doi-asserted-by":"publisher","first-page":"208","DOI":"10.1162\/NETN_a_00012","volume":"1","author":"S Sannino","year":"2017","unstructured":"Sannino S, Stramaglia S, Lacasa L et al (2017) Visibility graphs for fMRI data: multiplex temporal graphs and their modulations across resting-state networks. Netw Neurosci 1(3):208\u2013221. https:\/\/doi.org\/10.1162\/NETN_a_00012","journal-title":"Netw Neurosci"},{"key":"1089_CR28","doi-asserted-by":"publisher","unstructured":"Shumway RH, Stoffer DS (2017) Time Series Analysis and its Applications: with R examples, 4th edn. 1431-875X, Springer, New York, United Stahttps:\/\/doi.org\/10.1007\/978-3-319-52452-8","DOI":"10.1007\/978-3-319-52452-8"},{"key":"1089_CR29","unstructured":"Silva VF (2018) Time series analysis based on complex networks. Msc thesis, University of Porto"},{"issue":"3","key":"1089_CR30","doi-asserted-by":"publisher","DOI":"10.1002\/widm.1404","volume":"11","author":"VF Silva","year":"2021","unstructured":"Silva VF, Silva ME, Ribeiro P et al (2021) Time series analysis via network science: concepts and algorithms. WIREs Data Mining and Knowledge Discovery 11(3):e140. https:\/\/doi.org\/10.1002\/widm.1404","journal-title":"WIREs Data Mining and Knowledge Discovery"},{"key":"1089_CR31","doi-asserted-by":"publisher","first-page":"1062","DOI":"10.1007\/s10618-022-00826-3","volume":"36","author":"VF Silva","year":"2022","unstructured":"Silva VF, Silva ME, Ribeiro P et al (2022) Novel features for time series analysis: a complex networks approach. Data Mining and Knowledge Discovery 36:1062\u2013110. https:\/\/doi.org\/10.1007\/s10618-022-00826-3","journal-title":"Data Mining and Knowledge Discovery"},{"key":"1089_CR32","unstructured":"Silva VF, Silva ME, Ribeiro P, et\u00a0al (2023) MHVG2MTS: Multilayer Horizontal Visibility Graphs for Multivariate Time Series Analysis. https:\/\/arxiv.org\/abs\/2301.02333"},{"key":"1089_CR33","doi-asserted-by":"crossref","unstructured":"Sucarrat G (2015) lgarch: Simulation and Estimation of Log-GARCH Models. https:\/\/CRAN.R-project.org\/package=lgarch, r package version 0.6-2","DOI":"10.32614\/CRAN.package.lgarch"},{"key":"1089_CR34","volume-title":"Multivariate time series analysis: with R and financial applications","author":"RS Tsay","year":"2013","unstructured":"Tsay RS (2013) Multivariate time series analysis: with R and financial applications. John Wiley & Sons, Hoboken, New Jersey"},{"key":"1089_CR35","doi-asserted-by":"publisher","DOI":"10.1038\/d41586-018-05444-y","author":"A Vespignani","year":"2018","unstructured":"Vespignani A (2018). Twenty years of network scienc. https:\/\/doi.org\/10.1038\/d41586-018-05444-y","journal-title":"Twenty years of network scienc"},{"issue":"3","key":"1089_CR36","doi-asserted-by":"publisher","first-page":"335","DOI":"10.1007\/S10618-005-0039-X","volume":"13","author":"X Wang","year":"2006","unstructured":"Wang X, Smith K, Hyndman RJ (2006) Characteristic-based clustering for time series data. Data mining and knowledge Discovery 13(3):335\u201336. https:\/\/doi.org\/10.1007\/S10618-005-0039-X","journal-title":"Data mining and knowledge Discovery"},{"key":"1089_CR37","doi-asserted-by":"publisher","unstructured":"Wei WW (2019) Multivariate Time Series Analysis and Applications. John Wiley & Sons, Hoboken, New Jerse. https:\/\/doi.org\/10.1002\/9781119502951","DOI":"10.1002\/9781119502951"},{"key":"1089_CR38","doi-asserted-by":"publisher","first-page":"239","DOI":"10.1016\/j.physa.2017.10.052","volume":"493","author":"R Zhang","year":"2018","unstructured":"Zhang R, Ashuri B, Shyr Y et al (2018) Forecasting construction cost index based on visibility graph: a network approach. Physica A: Stat Mech Appl 493:239\u2013252. https:\/\/doi.org\/10.1016\/j.physa.2017.10.052","journal-title":"Physica A: Stat Mech Appl"},{"issue":"3","key":"1089_CR39","doi-asserted-by":"publisher","first-page":"03050","DOI":"10.7498\/aps.61.030506","volume":"61","author":"TT Zhou","year":"2012","unstructured":"Zhou TT, Jin ND, Gao ZK et al (2012) Limited penetrable visibility graph for establishing complex network from time series. Acta Physica Sinica 61(3):03050. https:\/\/doi.org\/10.7498\/aps.61.030506","journal-title":"Acta Physica Sinica"},{"key":"1089_CR40","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.physrep.2018.10.005","volume":"787","author":"Y Zou","year":"2019","unstructured":"Zou Y, Donner RV, Marwan N et al (2019) Complex network approaches to nonlinear time series analysis. Phys Rep 787:1\u20139. https:\/\/doi.org\/10.1016\/j.physrep.2018.10.005","journal-title":"Phys Rep"}],"container-title":["Data Mining and Knowledge Discovery"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10618-025-01089-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10618-025-01089-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10618-025-01089-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,4,26]],"date-time":"2025-04-26T01:35:16Z","timestamp":1745631316000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10618-025-01089-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,3,3]]},"references-count":40,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2025,5]]}},"alternative-id":["1089"],"URL":"https:\/\/doi.org\/10.1007\/s10618-025-01089-4","relation":{},"ISSN":["1384-5810","1573-756X"],"issn-type":[{"value":"1384-5810","type":"print"},{"value":"1573-756X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,3,3]]},"assertion":[{"value":"6 February 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 January 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 March 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no Conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"Not applicable.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethics approval"}}],"article-number":"17"}}