{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,12]],"date-time":"2026-05-12T23:24:39Z","timestamp":1778628279584,"version":"3.51.4"},"reference-count":35,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2017,3,31]],"date-time":"2017-03-31T00:00:00Z","timestamp":1490918400000},"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 paper formulates an unsupervised algorithm for symbolization of signal time series to capture the embedded dynamic behavior. The key idea is to convert time series of the digital signal into a string of (spatially discrete) symbols from which the embedded dynamic information can be extracted in an unsupervised manner (i.e., no requirement for labeling of time series). The main challenges here are: (1) definition of the symbol assignment for the time series; (2) identification of the partitioning segment locations in the signal space of time series; and (3) construction of probabilistic finite-state automata (PFSA) from the symbol strings that contain temporal patterns. The reported work addresses these challenges by maximizing the mutual information measures between symbol strings and PFSA states. The proposed symbolization method has been validated by numerical simulation as well as by experimentation in a laboratory environment. Performance of the proposed algorithm has been compared to that of two commonly used algorithms of time series partitioning.<\/jats:p>","DOI":"10.3390\/e19040148","type":"journal-article","created":{"date-parts":[[2017,3,31]],"date-time":"2017-03-31T10:35:49Z","timestamp":1490956549000},"page":"148","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":17,"title":["Unsupervised Symbolization of Signal Time Series for Extraction of the Embedded Information"],"prefix":"10.3390","volume":"19","author":[{"given":"Yue","family":"Li","sequence":"first","affiliation":[{"name":"Department of Mechanical &amp; Nuclear Engineering, Pennsylvania State University, University Park, PA 16802, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Asok","family":"Ray","sequence":"additional","affiliation":[{"name":"Department of Mechanical &amp; Nuclear Engineering, Pennsylvania State University, University Park, PA 16802, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2017,3,31]]},"reference":[{"key":"ref_1","first-page":"27","article-title":"Les surfaces \u00e0 courbures oppos\u00e9es et leurs lignes g\u00e9od\u00e9sique","volume":"4","author":"Hadamard","year":"1898","journal-title":"J. Math. Pures Appl."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1145\/584091.584093","article-title":"A mathematical theory of communication","volume":"5","author":"Shannon","year":"2001","journal-title":"ACM SIGMOBILE Mob. Comput. Commun. Rev."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Lin, J., Keogh, E., Lonardi, S., and Chiu, B. (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, San Diego, CA, USA.","DOI":"10.1145\/882082.882086"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"915","DOI":"10.1063\/1.1531823","article-title":"A review of symbolic analysis of experimental data","volume":"74","author":"Daw","year":"2003","journal-title":"Rev. Sci. Instrum."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1115","DOI":"10.1016\/j.sigpro.2004.03.011","article-title":"Symbolic dynamic analysis of complex systems for anomaly detection","volume":"84","author":"Ray","year":"2004","journal-title":"Signal Process."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1016\/j.sigpro.2014.03.045","article-title":"State splitting and merging in probabilistic finite state automata for signal representation and analysis","volume":"104","author":"Mukherjee","year":"2014","journal-title":"Signal Process."},{"key":"ref_7","unstructured":"Martin, L., and Puterman, M.L. (2014). Markov Decision Processes: Discrete Stochastic Dynamic Programming, Wiley."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Lind, D., and Marcus, B. (1995). An Introduction to Symbolic Dynamics and Coding, Cambridge University Press.","DOI":"10.1017\/CBO9780511626302"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"3309","DOI":"10.1016\/j.sigpro.2006.01.014","article-title":"Symbolic time series analysis via wavelet-based partitioning","volume":"86","author":"Rajagopalan","year":"2006","journal-title":"Signal Process."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"051104","DOI":"10.1103\/PhysRevE.64.051104","article-title":"Estimating and improving the signal-to-noise ratio of time series by symbolic dynamics","volume":"64","author":"Graben","year":"2001","journal-title":"Phys. Rev. E"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1026","DOI":"10.1063\/1.1796071","article-title":"On the control of chaotic systems via symbolic time series analysis","volume":"14","author":"Piccardi","year":"2004","journal-title":"Chaos"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"374","DOI":"10.1002\/stc.1683","article-title":"Damage localization based on symbolic time series analysis","volume":"22","author":"Alamdari","year":"2015","journal-title":"Struct. Control Health Monit."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Sarkar, S., Chattopadhyay, P., and Ray, A. (2016). Symbolization of dynamic data-driven systems for signal representation. Signal Image Video Process.","DOI":"10.1007\/s11760-016-0967-5"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"056209","DOI":"10.1103\/PhysRevE.66.056209","article-title":"Context-tree modeling of observed symbolic dynamics","volume":"66","author":"Kennel","year":"2002","journal-title":"Phys. Rev. E"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Takens, F. (1981). Detecting Strange Attractors in Turbulence, Springer.","DOI":"10.1007\/BFb0091924"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"656","DOI":"10.1109\/TIT.1983.1056741","article-title":"A universal data compression system","volume":"29","author":"Rissanen","year":"1983","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Ahmed, A.M., Bakar, A.A., and Hamdan, A.R. (2009, January 27\u201328). Dynamic data discretization technique based on frequency and k-nearest neighbour algorithm. Proceedings of the 2nd Conference on Data Mining and Optimization, DMO\u201909, Pekan Bangi, Malaysia.","DOI":"10.1109\/DMO.2009.5341919"},{"key":"ref_18","unstructured":"Bishop, C.M. (2006). Pattern Recognition and Machine Learning (Information Science and Statistics), Springer."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"54","DOI":"10.1103\/PhysRevLett.78.54","article-title":"Symbolic analysis of chaotic signals and turbulent fluctuations","volume":"78","author":"Lehrman","year":"1997","journal-title":"Phys. Rev. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"M\u00f6rchen, F., and Ultsch, A. (2005, January 21\u201324). Optimizing time series discretization for knowledge discovery. Proceedings of the Eleventh ACM SIGKDD International Conference on Knowledge Discovery in Data Mining, Chicago, IL, USA.","DOI":"10.1145\/1081870.1081953"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1214\/aoms\/1177729694","article-title":"On information and sufficiency","volume":"22","author":"Kullback","year":"1951","journal-title":"Ann. Math. Stat."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2743","DOI":"10.1109\/18.720554","article-title":"The minimum description length principle in coding and modeling","volume":"44","author":"Barron","year":"1998","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"653","DOI":"10.1109\/18.382012","article-title":"The context-tree weighting method: basic properties","volume":"41","author":"Willems","year":"1995","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Darema, F. (2005, January 22\u201325). Dynamic data driven applications systems: New capabilities for application simulations and measurements. Proceedings of the 5th International Conference on Computational Science\u2014ICCS 2005, Atlanta, GA, USA.","DOI":"10.1007\/11428848_79"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"399","DOI":"10.1007\/978-3-662-07675-0_9","article-title":"Automata for matching patterns","volume":"Volume 2","author":"Crochemore","year":"1997","journal-title":"Handbook of Formal Languages"},{"key":"ref_26","first-page":"463","article-title":"Symbolic dynamics and finite automata","volume":"Volume 2","author":"Perrin","year":"1997","journal-title":"Handbook of Formal Languages"},{"key":"ref_27","unstructured":"Sipser, M. (2012). Introduction to the Theory of Computation, Cengage Learning."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"18","DOI":"10.1016\/j.tcs.2007.07.023","article-title":"Pac-learnability of probabilistic deterministic finite state automata in terms of variation distance","volume":"387","author":"Palmer","year":"2007","journal-title":"Theor. Comput. Sci."},{"key":"ref_29","unstructured":"Cover, T.M., and Thomas, J.A. (2012). Elements of Information Theory, Wiley."},{"key":"ref_30","unstructured":"Feldman, D.P. (2002). A Brief Introduction to: Information Theory, Excess Entropy and Computational Mechanics, College of the Atlantic."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1349","DOI":"10.1016\/j.patcog.2004.03.020","article-title":"Links between probabilistic automata and hidden markov models, probability distributions, learning models and induction algorithms","volume":"38","author":"Dupont","year":"2005","journal-title":"Pattern Recognit."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1109\/TCYB.2015.2508024","article-title":"Information fusion of passive sensors for detection of moving targets in dynamic environments","volume":"47","author":"Li","year":"2017","journal-title":"IEEE Trans. Cybern."},{"key":"ref_33","unstructured":"Robot, K. (2017, March 30). User Manual, version 2.1. Available online: https:\/\/www.k-team.com\/mobile-robotics-products\/khepera-iii."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"073114","DOI":"10.1063\/1.4959537","article-title":"Using ordinal partition transition networks to analyze ecg data","volume":"26","author":"Kulp","year":"2016","journal-title":"Chaos"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Malinowski, S., Guyet, T., Quiniou, R., and Tavenard, R. (2013, January 17\u201319). 1d-sax: A novel symbolic representation for time series. Proceedings of the International Symposium on Intelligent Data Analysis, IDA2013: Advances in Intelligent Data Analysis XII, London, UK.","DOI":"10.1007\/978-3-642-41398-8_24"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/4\/148\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:31:42Z","timestamp":1760207502000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/4\/148"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,3,31]]},"references-count":35,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2017,4]]}},"alternative-id":["e19040148"],"URL":"https:\/\/doi.org\/10.3390\/e19040148","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,3,31]]}}}