Fractal analysis

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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 natural geometric objects, sound, market fluctuations,[1][2][3] heart rates,[4] digital images,[5] molecular motion, networks, etc. Fractal analysis is now widely used in all areas of science.[6] 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.[7]

Types of fractal analysis[edit]

Several types of fractal analysis are done, including box counting, lacunarity analysis, mass methods, and multifractal analysis.[1][7] A common feature of all types of fractal analysis is the need for benchmark patterns against which to assess outputs.[8] These can be acquired with various types of fractal generating software capable of generating benchmark patterns suitable for this purpose, which generally differ from software designed to render fractal art.

Applications[edit]

Applications of fractal analysis include:[9]

See also[edit]

References[edit]

  1. ^ a b Peters, Edgar (1996). Chaos and order in the capital markets : a new view of cycles, prices, and market volatility. New York: Wiley. ISBN 0-471-13938-6. 
  2. ^ 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. 
  3. ^ 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. 
  4. ^ a b 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. PMC 2746620Freely accessible. PMID 19528254. doi:10.1113/jphysiol.2009.169219. 
  5. ^ Fractal Analysis of Digital Images http://rsbweb.nih.gov/ij/plugins/fraclac/FLHelp/Fractals.htm
  6. ^ "Fractals: Complex Geometry, Patterns, and Scaling in Nature and Society". ISSN 1793-6543. 
  7. ^ a b c Benoît B. Mandelbrot (1983). The fractal geometry of nature. Macmillan. ISBN 978-0-7167-1186-5. Retrieved 1 February 2012. 
  8. ^ "Digital Images in FracLac". ImageJ. Archived from the original on 2012-02-08. Retrieved 2012-02-08 
  9. ^ "Applications". Retrieved 2007-10-21. 
  10. ^ a b Karperien, Audrey; Jelinek, Herbert F.; Leandro, Jorge de Jesus Gomes; Soares, João V. B.; Cesar Jr, Roberto M.; Luckie, Alan (2008). "Automated detection of proliferative retinopathy in clinical practice". Clinical ophthalmology (Auckland, N.Z.). 2 (1): 109–122. PMC 2698675Freely accessible. PMID 19668394. doi:10.2147/OPTH.S1579. 
  11. ^ Kam, Y.; Karperien, A.; Weidow, B.; Estrada, L.; Anderson, A. R.; Quaranta, V. (2009). "Nest expansion assay: A cancer systems biology approach to in vitro invasion measurements". BMC Research Notes. 2: 130. PMC 2716356Freely accessible. PMID 19594934. doi:10.1186/1756-0500-2-130. 
  12. ^ Losa, Gabriele A.; Nonnenmacher, Theo F., eds. (2005). Fractals in biology and medicine. Springer. ISBN 978-3-7643-7172-2. Retrieved 1 February 2012. 
  13. ^ Mandelbrot, B. (1967). "How Long is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension". Science. 156 (3775): 636–638. Bibcode:1967Sci...156..636M. PMID 17837158. doi:10.1126/science.156.3775.636. 
  14. ^ Li, H. (2013). "Fractal analysis of side channels for breakdown structures in XLPE cable insulation" (PDF). J Mater Sci: Mater Electron. Springer Science. 24: 1640–1643. doi:10.1007/s10854-012-0988-y. 
  15. ^ Reuveni, Shlomi; Granek, Rony; Klafter, Joseph (2008). "Proteins: Coexistence of Stability and Flexibility". Physical Review Letters. 100 (20). Bibcode:2008PhRvL.100t8101R. ISSN 0031-9007. doi:10.1103/PhysRevLett.100.208101. 
  16. ^ Panteha Saeedi, and Soren A. Sorensen. "An Algorithmic Approach to Generate After-disaster Test Fields for Search and Rescue Agents" (PDF). Proceedings of the World Congress on Engineering 2009: 93–98. ISBN 978-988-17-0125-1. 
  17. ^ a b Chen, Yanguang (2011). "Modeling Fractal Structure of City-Size Distributions Using Correlation Functions". PLoS ONE. 6 (9): e24791. Bibcode:2011PLoSO...624791C. PMC 3176775Freely accessible. PMID 21949753. arXiv:1104.4682Freely accessible. doi:10.1371/journal.pone.0024791. 
  18. ^ Karperien, Audrey L.; Jelinek, Herbert F.; Buchan, Alastair M. (2008). "Box-Counting Analysis of Microglia Form in Schizophrenia, Alzheimer's Disease and Affective Disorder". Fractals. 16 (2): 103–107. doi:10.1142/S0218348X08003880. 
  19. ^ Liu, Jing Z.; Zhang, Lu D.; Yue, Guang H. (2003). "Fractal Dimension in Human Cerebellum Measured by Magnetic Resonance Imaging". Biophysical Journal. 85 (6): 4041–4046. Bibcode:2003BpJ....85.4041L. PMC 1303704Freely accessible. PMID 14645092. doi:10.1016/S0006-3495(03)74817-6. 
  20. ^ Nikolić, D.; Moca, V.V.; Singer, W.; Mureşan, R.C. (2008). "Properties of multivariate data investigated by fractal dimensionality". Journal of Neuroscience Methods. 172 (1): 27–33. doi:10.1016/j.jneumeth.2008.04.007. 
  21. ^ Smith, Robert F.; Mohr, David N.; Torres, Vicente E.; Offord, Kenneth P.; Melton III, L. Joseph (1989). "Renal insufficiency in community patients with mild asymptomatic microhematuria". Mayo Clinic proceedings. Mayo Clinic. 64 (4): 409–414. PMID 2716356. doi:10.1016/s0025-6196(12)65730-9. 
  22. ^ Al-Kadi O.S, Watson D. (2008). "Texture Analysis of Aggressive and non-Aggressive Lung Tumor CE CT Images" (PDF). IEEE Transactions on Biomedical Engineering. 55 (7): 1822–1830. PMID 18595800. doi:10.1109/tbme.2008.919735. 
  23. ^ Landini, Gabriel (2011). "Fractals in microscopy". Journal of Microscopy. 241 (1): 1–8. PMID 21118245. doi:10.1111/j.1365-2818.2010.03454.x. 
  24. ^ Cheng, Qiuming (1997). "Multifractal Modeling and Lacunarity Analysis". Mathematical Geology. 29 (7): 919–932. doi:10.1023/A:1022355723781. 
  25. ^ Burkle-Elizondo, Gerardo; Valdéz-Cepeda, Ricardo David (2006). "Fractal analysis of Mesoamerican pyramids". Nonlinear dynamics, psychology, and life sciences. 10 (1): 105–122. PMID 16393505. 
  26. ^ Brown, Clifford T.; Witschey, Walter R. T.; Liebovitch, Larry S. (2005). "The Broken Past: Fractals in Archaeology". Journal of Archaeological Method and Theory. 12: 37–78. doi:10.1007/s10816-005-2396-6. 
  27. ^ Vannucchi, Paola; Leoni, Lorenzo (2007). "Structural characterization of the Costa Rica décollement: Evidence for seismically-induced fluid pulsing". Earth and Planetary Science Letters. 262 (3–4): 413–428. Bibcode:2007E&PSL.262..413V. doi:10.1016/j.epsl.2007.07.056. 
  28. ^ Didier Sornette (2004). Critical phenomena in natural sciences: chaos, fractals, self-organization, and disorder : concepts and tools. Springer. pp. 128–140. ISBN 978-3-540-40754-6. 
  29. ^ Hu, Shougeng; Cheng, Qiuming; Wang, Le; Xie, Shuyun (2012). "Multifractal characterization of urban residential land price in space and time". Applied Geography. 34: 161–170. doi:10.1016/j.apgeog.2011.10.016. 
  30. ^ Brothers, Harlan J. (2007). "Structural Scaling in Bach's Cello Suite No. 3". Fractals. 15: 89–95. doi:10.1142/S0218348X0700337X. 

Further reading[edit]