{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,15]],"date-time":"2026-05-15T15:21:52Z","timestamp":1778858512316,"version":"3.51.4"},"reference-count":25,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2022,9,8]],"date-time":"2022-09-08T00:00:00Z","timestamp":1662595200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Ministry of Education, Youth and Sports of the Czech Republic","award":["90140"],"award-info":[{"award-number":["90140"]}]},{"name":"Ministry of Education, Youth and Sports of the Czech Republic","award":["TK03020027"],"award-info":[{"award-number":["TK03020027"]}]},{"name":"Ministry of Education, Youth and Sports of the Czech Republic","award":["SP2022\/42"],"award-info":[{"award-number":["SP2022\/42"]}]},{"DOI":"10.13039\/501100002969","name":"Technology Agency of the Czech Republic","doi-asserted-by":"publisher","award":["90140"],"award-info":[{"award-number":["90140"]}],"id":[{"id":"10.13039\/501100002969","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002969","name":"Technology Agency of the Czech Republic","doi-asserted-by":"publisher","award":["TK03020027"],"award-info":[{"award-number":["TK03020027"]}],"id":[{"id":"10.13039\/501100002969","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002969","name":"Technology Agency of the Czech Republic","doi-asserted-by":"publisher","award":["SP2022\/42"],"award-info":[{"award-number":["SP2022\/42"]}],"id":[{"id":"10.13039\/501100002969","id-type":"DOI","asserted-by":"publisher"}]},{"name":"VSB\u2014Technical University of Ostrava","award":["90140"],"award-info":[{"award-number":["90140"]}]},{"name":"VSB\u2014Technical University of Ostrava","award":["TK03020027"],"award-info":[{"award-number":["TK03020027"]}]},{"name":"VSB\u2014Technical University of Ostrava","award":["SP2022\/42"],"award-info":[{"award-number":["SP2022\/42"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we analyze the interpretable models from real gasification datasets of the project \u201cCentre for Energy and Environmental Technologies\u201d (CEET) discovered by symbolic regression. To evaluate CEET models based on input data, two different statistical metrics to quantify their accuracy are usually used: Mean Square Error (MSE) and the Pearson Correlation Coefficient (PCC). However, if the testing points and the points used to construct the models are not chosen randomly from the continuum of the input variable, but instead from the limited number of discrete input points, the behavior of the model between such points very possibly will not fit well the physical essence of the modelled phenomenon. For example, the developed model can have unexpected oscillatory tendencies between the used points, while the usually used statistical metrics cannot detect these anomalies. However, using dynamic system criteria in addition to statistical metrics, such suspicious models that do fit well-expected behavior can be automatically detected and abandoned. This communication will show the universal method based on dynamic system criteria which can detect suitable models among all those which have good properties following statistical metrics. The dynamic system criteria measure the complexity of the candidate models using approximate and sample entropy. The examples are given for waste gasification where the output data (percentage of each particular gas in the produced mixture) is given only for six values of the input data (temperature in the chamber in which the process takes place). In such cases instead, to produce expected simple spline-like curves, artificial intelligence tools can produce inappropriate oscillatory curves with sharp picks due to the known tendency of symbolic regression to produce overfitted and relatively more complex models if the nature of the physical model is simple.<\/jats:p>","DOI":"10.3390\/axioms11090463","type":"journal-article","created":{"date-parts":[[2022,9,8]],"date-time":"2022-09-08T04:18:32Z","timestamp":1662610712000},"page":"463","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Selection of Appropriate Symbolic Regression Models Using Statistical and Dynamic System Criteria: Example of Waste Gasification"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3913-7800","authenticated-orcid":false,"given":"Pavel","family":"Praks","sequence":"first","affiliation":[{"name":"IT4Innovations, VSB\u2014Technical University of Ostrava, 708 00 Ostrava, Czech Republic"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6349-8553","authenticated-orcid":false,"given":"Marek","family":"Lampart","sequence":"additional","affiliation":[{"name":"IT4Innovations, VSB\u2014Technical University of Ostrava, 708 00 Ostrava, Czech Republic"},{"name":"Department of Applied Mathematics, VSB\u2014Technical University of Ostrava, 708 00 Ostrava, Czech Republic"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7510-4723","authenticated-orcid":false,"given":"Ren\u00e1ta","family":"Praksov\u00e1","sequence":"additional","affiliation":[{"name":"IT4Innovations, VSB\u2014Technical University of Ostrava, 708 00 Ostrava, Czech Republic"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2502-0601","authenticated-orcid":false,"given":"Dejan","family":"Brki\u0107","sequence":"additional","affiliation":[{"name":"IT4Innovations, VSB\u2014Technical University of Ostrava, 708 00 Ostrava, Czech Republic"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tom\u00e1\u0161","family":"Kozubek","sequence":"additional","affiliation":[{"name":"IT4Innovations, VSB\u2014Technical University of Ostrava, 708 00 Ostrava, Czech Republic"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jan","family":"Najser","sequence":"additional","affiliation":[{"name":"ENET Centre, VSB\u2014Technical University of Ostrava, 708 00 Ostrava, Czech Republic"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,9,8]]},"reference":[{"key":"ref_1","unstructured":"Praks, P., Brki\u0107, D., Najser, J., Najser, T., Praksov\u00e1, R., and Staji\u0107, Z. (June, January 31). Methods of Artificial Intelligence for Simulation of Gasification of Biomass and Communal Waste. Proceedings of the 22nd International Carpathian Control Conference (ICCC), Velk\u00e9 Karlovice, Czech Republic."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Praks, P., and Brki\u0107, D. (2018). Symbolic Regression-Based Genetic Approximations of the Colebrook Equation for Flow Friction. Water, 10.","DOI":"10.20944\/preprints201808.0510.v1"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Dresp-Langley, B., Ekseth, O.K., Fesl, J., Gohshi, S., Kurz, M., and Sehring, H.-W. (2019). Occam\u2019s Razor for Big Data? On Detecting Quality in Large Unstructured Datasets. Appl. Sci., 9.","DOI":"10.3390\/app9153065"},{"key":"ref_4","first-page":"7","article-title":"The Effect of Temperature on the Gasification Process","volume":"52","year":"2012","journal-title":"Acta Polytech."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"eaay2631","DOI":"10.1126\/sciadv.aay2631","article-title":"AI Feynman: A physics-inspired method for symbolic regression","volume":"6","author":"Udrescu","year":"2020","journal-title":"Sci. Adv."},{"key":"ref_6","first-page":"17429","article-title":"Discovering symbolic models from deep learning with inductive biases","volume":"33","author":"Cranmer","year":"2020","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2297","DOI":"10.1073\/pnas.88.6.2297","article-title":"Approximate entropy as a measure of system complexity","volume":"88","author":"Pincus","year":"1991","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"18","DOI":"10.1109\/MEMB.2009.934629","article-title":"Approximate entropy for all signals","volume":"28","author":"Chon","year":"2009","journal-title":"IEEE Eng. Med. Biol. Mag."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Delgado-Bonal, A., and Marshak, A. (2019). Approximate Entropy and Sample Entropy: A Comprehensive Tutorial. Entropy, 21.","DOI":"10.3390\/e21060541"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"H2039","DOI":"10.1152\/ajpheart.2000.278.6.H2039","article-title":"Physiological time-series analysis using approximate entropy and sample entropy","volume":"278","author":"Richman","year":"2000","journal-title":"Am. J. Physiol. Heart Circ. Physiol."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"349","DOI":"10.1007\/s10439-012-0668-3","article-title":"The appropriate use of approximate entropy and sample entropy with short data sets","volume":"41","author":"Yentes","year":"2013","journal-title":"Ann. Biomed. Eng."},{"key":"ref_12","first-page":"21031","article-title":"Detection of embedded dynamics in the Gy\u00f6rgyi-Field model","volume":"10","author":"Lampart","year":"2000","journal-title":"Sci. Rep."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"474","DOI":"10.1080\/17445760.2017.1324026","article-title":"Dynamical properties of partial-discharge patterns","volume":"33","author":"Lampart","year":"2018","journal-title":"Int. J. Parallel Emergent Distrib. Syst."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Lampart, M., and Zapom\u011bl, J. (2021). Motion of an Unbalanced Impact Body Colliding with a Moving Belt. Mathematics, 9.","DOI":"10.3390\/math9091071"},{"key":"ref_15","first-page":"304","article-title":"A survey of tools detecting the dynamical properties of one-dimensional families","volume":"15","author":"Lampart","year":"2017","journal-title":"Adv. Electr. Electron. Eng."},{"key":"ref_16","unstructured":"Akkaya, E., and Demir, A. (2009, January 13\u201315). Energy content estimation of municipal solid waste by multiple regression analysis. Proceedings of the 5th International Advanced Technologies Symposium IATS\u201909, Karabuk, Turkey. Available online: https:\/\/www.academia.edu\/download\/54979427\/IATS09_03-99_1292.pdf."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"650","DOI":"10.1080\/10473289.1996.10467499","article-title":"Modeling the energy content of municipal solid waste using multiple regression analysis","volume":"46","author":"Liu","year":"1996","journal-title":"J. Air Waste Manag. Assoc."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"112052","DOI":"10.1016\/j.rser.2021.112052","article-title":"Modeling the impact of some independent parameters on the syngas characteristics during plasma gasification of municipal solid waste using artificial neural network and stepwise linear regression methods","volume":"157","author":"Chu","year":"2022","journal-title":"Renew. Sustain. Energy Rev."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Mala\u0165\u00e1kov\u00e1, J., Jankovsk\u00fd, M., Mala\u0165\u00e1k, J., Velebil, J., Tamelov\u00e1, B., Gendek, A., and Aniszewska, M. (2021). Evaluation of Small-Scale Gasification for CHP for Wood from Salvage Logging in the Czech Republic. Forests, 12.","DOI":"10.3390\/f12111448"},{"key":"ref_20","first-page":"149","article-title":"Possibilities of gasification and pyrolysis technology in branch of energy recovery from waste","volume":"15","author":"Lapcik","year":"2014","journal-title":"In\u017cynieria Mineralna"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"110058","DOI":"10.1016\/j.rser.2020.110058","article-title":"Legislation-induced planning of waste processing infrastructure: A case study of the Czech Republic","volume":"132","year":"2020","journal-title":"Renew. Sustain. Energy Rev."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Ma, H., Narayanaswamy, A., Riley, P., and Li, L. (2022, July 19). Evolving symbolic density functionals. Science Advances. Available online: https:\/\/doi.org\/10.1126\/sciadv.abq0279.","DOI":"10.1126\/sciadv.abq0279"},{"key":"ref_23","first-page":"381","article-title":"Study towards the time-based MCDA ranking analysis\u2014A supplier selection case study","volume":"19","author":"Kizielewicz","year":"2021","journal-title":"Facta Univ. Ser. Mech. Eng."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Bogach, N., Boitsova, E., Chernonog, S., Lamtev, A., Lesnichaya, M., Lezhenin, I., Novopashenny, A., Svechnikov, R., Tsikach, D., and Vasiliev, K. (2021). Speech Processing for Language Learning: A Practical Approach to Computer-Assisted Pronunciation Teaching. Electronics, 10.","DOI":"10.3390\/electronics10030235"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Kantz, H., and Schreiber, T. (2003). Nonlinear Time Series Analysis, Cambridge University Press. [2nd ed.].","DOI":"10.1017\/CBO9780511755798"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/9\/463\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:25:29Z","timestamp":1760142329000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/9\/463"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,9,8]]},"references-count":25,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2022,9]]}},"alternative-id":["axioms11090463"],"URL":"https:\/\/doi.org\/10.3390\/axioms11090463","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,9,8]]}}}