{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,29]],"date-time":"2025-10-29T19:41:28Z","timestamp":1761766888088,"version":"build-2065373602"},"reference-count":47,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2021,2,23]],"date-time":"2021-02-23T00:00:00Z","timestamp":1614038400000},"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":["UIDB\/50008\/2020","PTDC\/CCI-CIF\/29877\/2017","DSAIPA\/DS\/0026\/2019","UIDB\/50021\/2020"],"award-info":[{"award-number":["UIDB\/50008\/2020","PTDC\/CCI-CIF\/29877\/2017","DSAIPA\/DS\/0026\/2019","UIDB\/50021\/2020"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Applied Sciences"],"abstract":"<jats:p>Outliers are observations suspected of not having been generated by the underlying process of the remaining data. Many applications require a way of identifying interesting or unusual patterns in multivariate time series (MTS), now ubiquitous in many applications; however, most outlier detection methods focus solely on univariate series. We propose a complete and automatic outlier detection system covering the pre-processing of MTS data that adopts a dynamic Bayesian network (DBN) modeling algorithm. The latter encodes optimal inter and intra-time slice connectivity of transition networks capable of capturing conditional dependencies in MTS datasets. A sliding window mechanism is employed to score each MTS transition gradually, given the DBN model. Two score-analysis strategies are studied to assure an automatic classification of anomalous data. The proposed approach is first validated in simulated data, demonstrating the performance of the system. Further experiments are made on real data, by uncovering anomalies in distinct scenarios such as electrocardiogram series, mortality rate data, and written pen digits. The developed system proved beneficial in capturing unusual data resulting from temporal contexts, being suitable for any MTS scenario. A widely accessible web application employing the complete system is publicly available jointly with a tutorial.<\/jats:p>","DOI":"10.3390\/app11041955","type":"journal-article","created":{"date-parts":[[2021,2,23]],"date-time":"2021-02-23T12:40:16Z","timestamp":1614084016000},"page":"1955","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Outlier Detection for Multivariate Time Series Using Dynamic Bayesian Networks"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9049-829X","authenticated-orcid":false,"given":"Jorge L.","family":"Serras","sequence":"first","affiliation":[{"name":"Instituto de Telecomunica\u00e7\u00f5es, Instituto Superior T\u00e9cnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1954-5487","authenticated-orcid":false,"given":"Susana","family":"Vinga","sequence":"additional","affiliation":[{"name":"INESC-ID, Instituto Superior T\u00e9cnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal"},{"name":"Lisbon ELLIS Unit (LUMLIS\u2014Lisbon Unit for Learning and Intelligent Systems), Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal"}]},{"given":"Alexandra M.","family":"Carvalho","sequence":"additional","affiliation":[{"name":"Instituto de Telecomunica\u00e7\u00f5es, Instituto Superior T\u00e9cnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal"},{"name":"Lisbon ELLIS Unit (LUMLIS\u2014Lisbon Unit for Learning and Intelligent Systems), Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2021,2,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1080\/00401706.1969.10490657","article-title":"Procedures for detecting outlying observations in samples","volume":"11","author":"Grubbs","year":"1969","journal-title":"Technometrics"},{"key":"ref_2","unstructured":"L\u00f3pez-de Lacalle, J. (2017). tsoutliers: Detection of Outliers in Time Series, The Comprehensive R Archive Network (CRAN). R Package Version 0.6-6."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Matt Dancho, D.V. (2018). anomalize: Tidy Anomaly Detection, The Comprehensive R Archive Network (CRAN). R Package Version 0.1.1.","DOI":"10.32614\/CRAN.package.anomalize"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1145\/1541880.1541882","article-title":"Anomaly detection: A survey","volume":"41","author":"Chandola","year":"2009","journal-title":"ACM Comput. Surv. (CSUR)"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Aggarwal, C.C. (2017). Outlier Analysis, Springer.","DOI":"10.1007\/978-3-319-47578-3"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"2250","DOI":"10.1109\/TKDE.2013.184","article-title":"Outlier Detection for Temporal Data: A Survey","volume":"26","author":"Gupta","year":"2014","journal-title":"IEEE Trans. Knowl. Data Eng."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"654","DOI":"10.1198\/016214505000001131","article-title":"Outlier detection in multivariate time series by projection pursuit","volume":"101","author":"Galeano","year":"2006","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"823","DOI":"10.1109\/TKDE.2010.235","article-title":"Anomaly Detection for Discrete Sequences: A Survey","volume":"24","author":"Chandola","year":"2012","journal-title":"IEEE Trans. Knowl. Data Eng."},{"key":"ref_9","unstructured":"Ma, J., and Perkins, S. (2003, January 20\u201324). Time-Series Novelty Detection Using One-Class Support Vector Machines. Proceedings of the International Joint Conference on Neural Networks, Portland, OR, USA."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1007\/s00190-012-0582-3","article-title":"Robust estimation by expectation maximization algorithm","volume":"87","author":"Koch","year":"2013","journal-title":"J. Geod."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1806","DOI":"10.1007\/s10618-018-0569-7","article-title":"Exact variable-length anomaly detection algorithm for univariate and multivariate time series","volume":"32","author":"Wang","year":"2018","journal-title":"Data Min. Knowl. Discov."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Ding, N., Gao, H., Bu, H., Ma, H., and Si, H. (2018). Multivariate-Time-Series-Driven Real-time Anomaly Detection Based on Bayesian Network. Sensors, 18.","DOI":"10.3390\/s18103367"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"8846608","DOI":"10.1155\/2020\/8846608","article-title":"MTAD-TF: Multivariate Time Series Anomaly Detection Using the Combination of Temporal Pattern and Feature Pattern","volume":"2020","author":"He","year":"2020","journal-title":"Complexity"},{"key":"ref_14","unstructured":"Monteiro, J.L., Vinga, S., and Carvalho, A.M. (2015, January 12\u201316). Polynomial-Time Algorithm for Learning Optimal Tree-Augmented Dynamic Bayesian Networks. Proceedings of the Polynomial-Time Algorithm for Learning Optimal Tree-Augmented Dynamic Bayesian Networks (UAI 2015), Amsterdam, The Netherlands."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"W00D28","DOI":"10.1029\/2008WR006956","article-title":"Real-time Bayesian anomaly detection in streaming environmental data","volume":"45","author":"Hill","year":"2009","journal-title":"Water Resour. Res."},{"key":"ref_16","unstructured":"Murphy, K., and Mian, S. (1999). Modelling Gene Expression Data Using Dynamic Bayesian Networks, Computer Science Division, University of California. Technical Report."},{"key":"ref_17","unstructured":"Tukey, J.W. (1977). Exploratory Data Analysis, Pearson."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Hoaglin, D.C., and John, W. (2003). Tukey and data analysis. Stat. Sci., 311\u2013318.","DOI":"10.1214\/ss\/1076102418"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1146\/annurev-statistics-031017-100325","article-title":"Finite mixture models","volume":"5","author":"McLachlan","year":"2019","journal-title":"Annu. Rev. Stat. Appl."},{"key":"ref_20","unstructured":"Serras, J.L., Vinga, S., and Carvalho, A.M. (2021, February 23). METEOR\u2014Dynamic Bayesian Outlier Detection. Available online: https:\/\/meteor.jorgeserras.com\/."},{"key":"ref_21","unstructured":"Friedman, N. (1998). The Bayesian Structural EM Algorithm, Morgan Kaufmann."},{"key":"ref_22","first-page":"2181","article-title":"Discriminative Learning of Bayesian Networks via Factorized Conditional Log-Likelihood","volume":"12","author":"Carvalho","year":"2011","journal-title":"J. Mach. Learn. Res."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"2176","DOI":"10.3390\/e15072716","article-title":"Efficient Approximation of the Conditional Relative Entropy with Applications to Discriminative Learning of Bayesian Network Classifiers","volume":"15","author":"Carvalho","year":"2013","journal-title":"Entropy"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"3438","DOI":"10.1016\/j.patcog.2014.03.019","article-title":"Hybrid learning of Bayesian multinets for binary classification","volume":"47","author":"Carvalho","year":"2014","journal-title":"Pattern Recognit."},{"key":"ref_25","unstructured":"Carvalho, A.M. (2009). Scoring Functions for Learning Bayesian Networks, INESC.ID. INESC-ID Tech. Rep."},{"key":"ref_26","unstructured":"Friedman, N., Murphy, K.P., and Russell, S.J. (1998). Learning the Structure of Dynamic Probabilistic Networks, Morgan Kaufmann."},{"key":"ref_27","unstructured":"Chickering, D., Geiger, D., and Heckerman, D. (1995, January 20\u201325). Learning Bayesian Networks: Search Methods and Experimental Results. Proceedings of the Fifth Conference on Artificial Intelligence and Statistics, Montreal, QC, Canada."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Dojer, N. (2006). Learning Bayesian Networks Does Not Have to Be NP-Hard, Springer.","DOI":"10.1007\/11821069_27"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1023\/A:1007465528199","article-title":"Bayesian network classifiers","volume":"29","author":"Friedman","year":"1997","journal-title":"Mach. Learn."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Sousa, M., and Carvalho, A.M. (2018). Polynomial-Time Algorithm for Learning Optimal BFS-Consistent Dynamic Bayesian Networks. Entropy, 20.","DOI":"10.3390\/e20040274"},{"key":"ref_31","first-page":"179","article-title":"Learning Consistent Tree-Augmented Dynamic Bayesian Networks","volume":"Volume 11331","author":"Nicosia","year":"2019","journal-title":"Machine Learning, Optimization, and Data Science, Proceedings of the 4th International Conference, Volterra, Tuscany, Italy, 13\u201316 September 2018\u2014Revised Selected Papers"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Lin, J., Keogh, E.J., Lonardi, S., and Chiu, B.Y. (2003, January 13). A Symbolic Representation of Time Series, with Implications for Streaming Algorithms. Proceedings of the 8th ACM SIGMOD Workshop on Research Issues in Data Mining and Knowledge Discovery (DMKD 2003), San Diego, CA, USA.","DOI":"10.1145\/882082.882086"},{"key":"ref_33","unstructured":"Keogh, E., Lin, J., and Fu, A. (2004, January 1\u20134). HOT SAX: Finding the Most Unusual Time Series Subsequence: Algorithms and Applications. Proceedings of the Sixth International Conference on Data Mining (ICDM), Brighton, UK."},{"key":"ref_34","unstructured":"Larsen, R.J., and Marx, M.L. (1986). An Introduction to Mathematical Statistics and Its Applications, Prentice-Hall."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"233","DOI":"10.6028\/jres.071B.032","article-title":"Optimum branchings","volume":"71","author":"Edmonds","year":"1967","journal-title":"J. Res. Natl. Bur. Stand."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Manning, C.D., Raghavan, P., and Sch\u00fctze, H. (2008). Introduction to Information Retrieval, Cambridge University Press.","DOI":"10.1017\/CBO9780511809071"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1273","DOI":"10.1080\/01621459.1993.10476408","article-title":"Alternatives to the median absolute deviation","volume":"88","author":"Rousseeuw","year":"1993","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1189","DOI":"10.3758\/s13414-019-01726-3","article-title":"A note on detecting statistical outliers in psychophysical data","volume":"81","author":"Jones","year":"2019","journal-title":"Atten. Percept. Psychophys."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1109\/34.990138","article-title":"Unsupervised learning of finite mixture models","volume":"24","author":"Figueiredo","year":"2002","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_40","unstructured":"Fraley, C., Raftery, A., Scrucca, L., Murphy, T.B., Fop, M., and Scrucca, M.L. (2017). mclust: Gaussian Mixture Modelling for Model-Based Clustering, Classification, and Density Estimation, The Comprehensive R Archive Network (CRAN). R Package Version 5."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1023\/A:1026490906255","article-title":"The power of amnesia: Learning probabilistic automata with variable memory length","volume":"25","author":"Ron","year":"1996","journal-title":"Mach. Learn."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1","DOI":"10.18637\/jss.v072.i03","article-title":"Analyzing state sequences with probabilistic suffix trees: the PST R package","volume":"72","author":"Gabadinho","year":"2016","journal-title":"J. Stat. Softw."},{"key":"ref_43","unstructured":"Dau, H.A., Keogh, E., Kamgar, K., Yeh, C.C.M., Zhu, Y., Gharghabi, S., Ratanamahatana, C.A., Hu, B., Begum, N., and Bagnall, A. (2018, September 18). The UCR Time Series Classification Archive. Available online: https:\/\/www.cs.ucr.edu\/~eamonn\/time_series_data_2018\/."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1080\/00401706.2017.1340909","article-title":"Detecting deviating data cells","volume":"60","author":"Rousseeuw","year":"2018","journal-title":"Technometrics"},{"key":"ref_45","unstructured":"University of California, and Max Planck Institute for Demographic Research (Germany) (2018, September 18). Human Mortality Database. Available online: www.humanmortality.de."},{"key":"ref_46","unstructured":"Dheeru, D., and Karra Taniskidou, E. (2018, September 18). UCI Machine Learning Repository. Available online: http:\/\/archive.ics.uci.edu\/ml."},{"key":"ref_47","unstructured":"Alimoglu, F., and Alpaydin, E. (1996, January 27\u201328). Methods of Combining Multiple Classifiers Based on Different Representations for Pen-based Handwritten Digit Recognition. Proceedings of the Fifth Turkish Artificial Intelligence and Artificial Neural Networks Symposium (TAINN), Istanbul, Turkey."}],"container-title":["Applied Sciences"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2076-3417\/11\/4\/1955\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:26:55Z","timestamp":1760160415000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2076-3417\/11\/4\/1955"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,2,23]]},"references-count":47,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2021,2]]}},"alternative-id":["app11041955"],"URL":"https:\/\/doi.org\/10.3390\/app11041955","relation":{},"ISSN":["2076-3417"],"issn-type":[{"type":"electronic","value":"2076-3417"}],"subject":[],"published":{"date-parts":[[2021,2,23]]}}}