Fractal analysis

Tree branches as seen from below. There are no leaves on the branches and they split many times.
Fractal branching of trees

Fractal analysis is assessing fractal characteristics of data. It consists of several methods to assign a fractal dimension and other fractal characteristics to a dataset which may be a theoretical dataset, or a pattern or signal extracted from phenomena including topography,[1] natural geometric objects, ecology and aquatic sciences,[2] sound, market fluctuations,[3][4][5] heart rates,[6] frequency domain in electroencephalography signals,[7][8] digital images,[9] molecular motion, and data science. Fractal analysis is now widely used in all areas of science.[10] An important limitation of fractal analysis is that arriving at an empirically determined fractal dimension does not necessarily prove that a pattern is fractal; rather, other essential characteristics have to be considered.[11] Fractal analysis is valuable in expanding our knowledge of the structure and function of various systems, and as a potential tool to mathematically assess novel areas of study. Fractal calculus was formulated which is a generalization of ordinary calculus. [12]

  1. ^ Gerges, Firas; Geng, Xiaolong; Nassif, Hani; Boufadel, Michel C. (2021). "Anisotropic Multifractal Scaling of Mount Lebanon Topography: Approximate Conditioning". Fractals. 29 (5): 2150112–2153322. Bibcode:2021Fract..2950112G. doi:10.1142/S0218348X21501127. ISSN 0218-348X. S2CID 234272453.
  2. ^ Seuront, Laurent (2009-10-12). Fractals and Multifractals in Ecology and Aquatic Science. CRC Press. doi:10.1201/9781420004243. ISBN 9780849327827.
  3. ^ Peters, Edgar (1996). Chaos and order in the capital markets: a new view of cycles, prices, and market volatility. New York: Wiley. ISBN 978-0-471-13938-6.
  4. ^ Mulligan, R. (2004). "Fractal analysis of highly volatile markets: an application to technology equities". The Quarterly Review of Economics and Finance. 44: 155–179. doi:10.1016/S1062-9769(03)00028-0.
  5. ^ Kamenshchikov, S. (2014). "Transport Catastrophe Analysis as an Alternative to a Monofractal Description: Theory and Application to Financial Crisis Time Series". Journal of Chaos. 2014: 1–8. doi:10.1155/2014/346743.
  6. ^ Tan, Can Ozan; Cohen, Michael A.; Eckberg, Dwain L.; Taylor, J. Andrew (2009). "Fractal properties of human heart period variability: Physiological and methodological implications". The Journal of Physiology. 587 (15): 3929–3941. doi:10.1113/jphysiol.2009.169219. PMC 2746620. PMID 19528254.
  7. ^ Zappasodi, Filippo; Olejarczyk, Elzbieta; Marzetti, Laura; Assenza, Giovanni (2014). "Fractal Dimension of EEG Activity Senses Neuronal Impairment in Acute Stroke". PLOS ONE. 9 (6): 3929–3941. Bibcode:2014PLoSO...9j0199Z. doi:10.1371/journal.pone.0100199. PMC 4072666. PMID 24967904.
  8. ^ Hisonothai, M.; Nakagawa, M. (2008). "EEG signal classification method based on fractal features and neural network". 2008 30th Annual International Conference of the IEEE Engineering in Medicine and Biology Society. Vol. 2008. pp. 3880–3. doi:10.1109/IEMBS.2008.4650057. ISBN 978-1-4244-1814-5. PMID 19163560. S2CID 22136019.
  9. ^ Fractal Analysis of Digital Images http://rsbweb.nih.gov/ij/plugins/fraclac/FLHelp/Fractals.htm
  10. ^ "Fractals: Complex Geometry, Patterns, and Scaling in Nature and Society". Fractals: An Interdiscipinary Journal on the Complex Geometry of Nature. ISSN 1793-6543.
  11. ^ Benoît B. Mandelbrot (1983). The fractal geometry of nature. Macmillan. ISBN 978-0-7167-1186-5. Retrieved 1 February 2012.
  12. ^ Khalili Golmankhaneh, Alireza (2022). Fractal Calculus and its Applications. Singapore: World Scientific Pub Co Inc. p. 328. doi:10.1142/12988. ISBN 978-981-126-110-7. S2CID 248575991.