{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,4]],"date-time":"2026-03-04T05:59:47Z","timestamp":1772603987087,"version":"3.50.1"},"reference-count":43,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2020,3,13]],"date-time":"2020-03-13T00:00:00Z","timestamp":1584057600000},"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>Financial markets give a large number of trading opportunities. However, over-complicated systems make it very difficult to be effectively used by decision-makers. Volatility and noise present in the markets evoke a need to simplify the market picture derived for the decision-makers. Symbolic representation fits in this concept and greatly reduces data complexity. However, at the same time, some information from the market is lost. Our motivation is to answer the question: What is the impact of introducing different data representation on the overall amount of information derived for the decision-maker? We concentrate on the possibility of using entropy as a measure of the information gain\/loss for the financial data, and as a basic form, we assume permutation entropy with later modifications. We investigate different symbolic representations and compare them with classical data representation in terms of entropy. The real-world data covering the time span of 10 years are used in the experiments. The results and the statistical verification show that extending the symbolic description of the time series does not affect the permutation entropy values.<\/jats:p>","DOI":"10.3390\/e22030330","type":"journal-article","created":{"date-parts":[[2020,3,17]],"date-time":"2020-03-17T09:27:41Z","timestamp":1584437261000},"page":"330","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["Permutation Entropy as a Measure of Information Gain\/Loss in the Different Symbolic Descriptions of Financial Data"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2128-6998","authenticated-orcid":false,"given":"Jan","family":"Kozak","sequence":"first","affiliation":[{"name":"Faculty of Informatics and Communication; Department of Knowledge Engineering, University of Economics, 1 Maja 50, 40-287 Katowice, Poland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7123-6972","authenticated-orcid":false,"given":"Krzysztof","family":"Kania","sequence":"additional","affiliation":[{"name":"Faculty of Informatics and Communication; Department of Knowledge Engineering, University of Economics, 1 Maja 50, 40-287 Katowice, Poland"}]},{"given":"Przemys\u0142aw","family":"Juszczuk","sequence":"additional","affiliation":[{"name":"Faculty of Informatics and Communication; Department of Knowledge Engineering, University of Economics, 1 Maja 50, 40-287 Katowice, Poland"}]}],"member":"1968","published-online":{"date-parts":[[2020,3,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.finmar.2019.05.001","article-title":"Make-take decisions under high-frequency trading competition","volume":"45","author":"Bernales","year":"2019","journal-title":"J. Financ. Mark."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"158","DOI":"10.1016\/j.ribaf.2017.04.031","article-title":"Is high-frequency trading tiering the financial markets?","volume":"41","author":"Virgilio","year":"2017","journal-title":"Res. Int. Bus. Financ."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"322","DOI":"10.1016\/j.ribaf.2019.06.004","article-title":"An empirical investigation of volatility dynamics in the cryptocurrency market","volume":"50","author":"Katsiampa","year":"2019","journal-title":"Res. Int. Bus. Financ."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1016\/j.qref.2017.01.014","article-title":"A literature review of technical analysis on stock markets","volume":"66","author":"Silva","year":"2017","journal-title":"Q. Rev. Econ. Financ."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"939","DOI":"10.1016\/S2212-5671(15)01344-1","article-title":"Fundamental Analysis Models in Financial Markets\u2014Review Study","volume":"30","author":"Wafi","year":"2015","journal-title":"Proced. Econ. Financ."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"160","DOI":"10.1016\/j.qref.2018.04.009","article-title":"To follow or not to follow\u2014An empirical analysis of the returns of actors on social trading platforms","volume":"70","author":"Dorfleitner","year":"2018","journal-title":"Quart. Rev. Econ. Financ."},{"key":"ref_7","first-page":"22","article-title":"Ichimoku charts: Technical analysis of stocks and commodities","volume":"18","author":"Muranaka","year":"2000","journal-title":"Stocks Commod."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Bandt, C., and Pompe, B. (2002). Permutation Entropy: A Natural Complexity Measure for Time Series. Phys. Rev. Lett., 88.","DOI":"10.1103\/PhysRevLett.88.174102"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"3344","DOI":"10.1016\/j.physa.2013.03.041","article-title":"Time-series analysis of foreign exchange rates using time-dependent pattern entropy","volume":"392","author":"Ishizaki","year":"2013","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Gencay, R., and Gradojevic, N. (2017). The tale of two financial crises: An entropic perspective. Entropy, 19.","DOI":"10.3390\/e19060244"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Henry, M., and Judge, G. (2019). Permutation Entropy and Information Recovery in Nonlinear Dynamic Economic Time Series. Econometrics, 7.","DOI":"10.3390\/econometrics7010010"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Rojas, I., and Pomares, H. (2016). Permutation Entropy and Order Patterns in Long Time Series. Time Series Analysis and Forecasting, Springer. [2016 ed.].","DOI":"10.1007\/978-3-319-28725-6"},{"key":"ref_13","unstructured":"Dylee, D.L. (2020, March 11). Permutation Entropies (PEs) of International Short-Term Interest Rates and Interest Rate Spreads before the Financial Crisis of 2007\u201309. Available online: https:\/\/pdfs.semanticscholar.org\/ef74\/868ae488675fd4a6ab350868bc0e0015682a.pdf."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1142\/S0219493707002025","article-title":"Time series from the ordinal viewpoint","volume":"7","author":"Keller","year":"2007","journal-title":"Stoch. Dyn."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Amigo, J., Keller, K., and Kurths, J. (2013). Recent Progress in Symbolic Dynamics and Permutation Complexity Ten Years of Permutation Entropy. Eur. Phys. J. Spec. Top., 222.","DOI":"10.1140\/epjst\/e2013-01840-1"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1140\/epjst\/e2013-01862-7","article-title":"Practical considerations of permutation entropy","volume":"222","author":"Riedl","year":"2013","journal-title":"Eur. Phys. J. Spec. Top."},{"key":"ref_17","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_18","doi-asserted-by":"crossref","unstructured":"Ji, A., and Shang, P. (2019). Analysis of financial time series through forbidden patterns. Phys. A Stat. Mech. Appl., 534.","DOI":"10.1016\/j.physa.2019.122038"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"McCullough, M., Sakellariou, K., Stemler, T., and Small, M. (2016). Counting forbidden patterns in irregularly sampled time series. I. The effects of under-sampling, random depletion, and timing jitter. Chaos Interdiscip. J. Nonlinear Sci., 26.","DOI":"10.1063\/1.4968551"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Zanin, M. (2008). Forbidden patterns in financial time series. Chaos, 18.","DOI":"10.1063\/1.2841197"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2854","DOI":"10.1016\/j.physa.2009.03.042","article-title":"Forbidden patterns, permutation entropy and stock market inefficiency","volume":"388","author":"Zunino","year":"2009","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Liu, X., and Yue, W. (2009). Fine-grained permutation entropy as a measure of natural complexity for time series. Chin. Phys. B, 18.","DOI":"10.1088\/1674-1056\/18\/7\/011"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Liu, X., Jiang, A., Xu, N., and Xue, J. (2016). Increment Entropy as a Measure of Complexity for Time Series. Entropy, 18.","DOI":"10.3390\/e18010022"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1016\/j.physa.2015.09.067","article-title":"Weighted permutation entropy based on different symbolic approaches for financial time series","volume":"443","author":"Yin","year":"2016","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Namdari, A., and Li, Z. (2019). A review of entropy measures for uncertainty quantification of stochastic processes. Adv. Mech. Eng., 11.","DOI":"10.1177\/1687814019857350"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Gradojevic, N., and Caric, M. (2016). Predicting Systemic Risk with Entropic Indicators. J. Forecast., 36.","DOI":"10.1002\/for.2411"},{"key":"ref_27","first-page":"161","article-title":"An econophysics approach to analyse uncertainty in financial markets: An application to the Portuguese stock market","volume":"50","author":"Menezes","year":"2006","journal-title":"Phys. Condens. Matter"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Leonarduzzi, R., Rochette, G., Bouchaud, J.P., and Mallat, S. (2019, January 12\u201317). Maximum-entropy Scattering Models for Financial Time Series. Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Brighton, UK.","DOI":"10.1109\/ICASSP.2019.8683734"},{"key":"ref_29","first-page":"733","article-title":"Local order, entropy and predictability of financial time series","volume":"15","author":"Molgedey","year":"2000","journal-title":"Phys. Condens. Matter"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1016\/j.sepro.2011.10.030","article-title":"Multi-Scale Approximate Entropy Analysis of Foreign Exchange Markets Efficiency","volume":"3","author":"Wang","year":"2012","journal-title":"Syst. Eng. Proced."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Bentes, S., and Menezes, R. (2012). Entropy: A new measure of stock market volatility?. J. Phys. Conf. Ser., 394.","DOI":"10.1088\/1742-6596\/394\/1\/012033"},{"key":"ref_32","unstructured":"Matos, O. (2006). Entropy Measures Applied to Financial Time Series\u2014An Econophysics Approach, Departamento de Matematica Aplicada, Universidade do Porto."},{"key":"ref_33","unstructured":"Schwill, S. (2018). Entropy Analysis of Financial Time Series. arXiv."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"4909","DOI":"10.3390\/e15114909","article-title":"Applications of Entropy in Finance: A Review","volume":"15","author":"Zhou","year":"2013","journal-title":"Entropy"},{"key":"ref_35","first-page":"7","article-title":"Financial Time Series and Their Features","volume":"9","author":"Arlt","year":"2001","journal-title":"Acta Oeconomica Pragensia V\u0160E Praha"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Tsay, R.S. (2002). Financial Econometrics. Analysis of Financial Time Series, John Wiley & Sons, Inc.. [2nd ed.].","DOI":"10.1002\/0471264105"},{"key":"ref_37","first-page":"345","article-title":"Investigating Patterns in the Financial Data with Enhanced Symbolic Description","volume":"11056","author":"Kania","year":"2018","journal-title":"Int. Conf. Comput. Collect. Intell."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Piek, A.B., Stolz, I., and Keller, K. (2019). Algorithmics, Possibilities and Limits of Ordinal Pattern Based Entropies. Entropy, 21.","DOI":"10.3390\/e21060547"},{"key":"ref_39","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_40","doi-asserted-by":"crossref","unstructured":"Traversaro, F., Redelico, F., Risk, M., Frery, A., and Rosso, O. (2018). Bandt-Pompe symbolization dynamics for time series with tied values: A data-driven approach. Chaos, 28.","DOI":"10.1063\/1.5022021"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1883","DOI":"10.1016\/j.physleta.2017.03.052","article-title":"Permutation entropy based time series analysis: Equalities in the input signal can lead to false conclusions","volume":"381","author":"Zunino","year":"2017","journal-title":"Phys. Lett. A"},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Stephen, M., Gu, C., and Yang, H. (2015). Visibility Graph Based Time Series Analysis. PLoS ONE.","DOI":"10.1371\/journal.pone.0143015"},{"key":"ref_43","first-page":"77","article-title":"Portfolio selection","volume":"7","author":"Markowitz","year":"1952","journal-title":"J. Financ."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/3\/330\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:06:46Z","timestamp":1760173606000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/3\/330"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,3,13]]},"references-count":43,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2020,3]]}},"alternative-id":["e22030330"],"URL":"https:\/\/doi.org\/10.3390\/e22030330","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,3,13]]}}}