{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,31]],"date-time":"2026-01-31T17:05:11Z","timestamp":1769879111641,"version":"3.49.0"},"reference-count":21,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2023,4,7]],"date-time":"2023-04-07T00:00:00Z","timestamp":1680825600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>This work was greatly influenced by the opinions of one of the authors (JS), who demonstrated in a recent book that it is important to distinguish between \u201cfractal models\u201d and \u201cfractal\u201d (power-law) behaviors. According to the self-similarity principle (SSP), the authors of this study completely distinguish between independent \u201cfractal\u201d (power-law) behavior and the \u201cfractal models\u201d, which result from the solution of equations incorporating non-integer differentiation\/integration operators. It is feasible to demonstrate how many random curves resemble one another and how they can be predicted by functions with real and complex-conjugated power-law exponents. Bellman\u2019s inequality can be used to demonstrate that the generalized geometric mean, not the arithmetic mean, which is typically recognized as the fundamental criterion in the signal processing field, corresponds to the global fitting minimum. To highlight the efficiency of the proposed algorithms, they are applied to two sets of data: one without a clearly expressed power-law behavior, the other containing clear power-law dependence.<\/jats:p>","DOI":"10.3390\/a16040199","type":"journal-article","created":{"date-parts":[[2023,4,10]],"date-time":"2023-04-10T03:19:54Z","timestamp":1681096794000},"page":"199","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Can Self-Similarity Processes Be Reflected by the Power-Law Dependencies?"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2931-4428","authenticated-orcid":false,"given":"Raoul R.","family":"Nigmatullin","sequence":"first","affiliation":[{"name":"Radioelectronics and Informative Measurements Techniques Department, Kazan National Research Technical University Named After A.N. Tupolev, Karl Marx Str.10, 420111 Kazan, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jocelyn","family":"Sabatier","sequence":"additional","affiliation":[{"name":"IMS Laboratory, Bordeaux University, UMR 5218 CNRS, 351 Cours de la Lib\u00e9ration, 33405 Talence, France"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,4,7]]},"reference":[{"key":"ref_1","unstructured":"Sabatier, J., Farges, C., and Tartaglione, V. (2022). Intelligent Systems, Control and Automation: Science and Engineering, Springer."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"716110","DOI":"10.3389\/fcteg.2021.716110","article-title":"Modelling Fractional Behaviours without Fractional Models","volume":"2","author":"Sabatier","year":"2021","journal-title":"Front. 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Fractals, Plenum Press.","DOI":"10.1007\/978-1-4899-2124-6"},{"key":"ref_8","unstructured":"Nigmatullin, R.R., and Vorobev, A.S. (2019). The Proceedings of the ICFDA19 Conference; Proceedings in Mathematics & Statistics, Springer."},{"key":"ref_9","first-page":"25","article-title":"Frequency band complex noninteger differentiator: Characterization and synthesis, IEEE Transactions on Circuits and Systems","volume":"47","author":"Oustaloup","year":"2000","journal-title":"IEEE Fundam. Theory Appl."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Gil\u2019mutdinov, A.K., Ushakov, P.A., and El-Khazali, R. (2017). Fractal Elements and Their Applications, Springer International Publishing.","DOI":"10.1007\/978-3-319-45249-4"},{"key":"ref_11","first-page":"724","article-title":"Nigmatullin1 Jos\u00e9 Tenreiro Machado, Rui Menezes. Self-similarity principle: The reduced description of randomness","volume":"11","author":"Raoul","year":"2013","journal-title":"Cent. Eur. J. 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Anal."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/16\/4\/199\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:12:04Z","timestamp":1760123524000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/16\/4\/199"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,4,7]]},"references-count":21,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2023,4]]}},"alternative-id":["a16040199"],"URL":"https:\/\/doi.org\/10.3390\/a16040199","relation":{},"ISSN":["1999-4893"],"issn-type":[{"value":"1999-4893","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,4,7]]}}}