Fractal analysis

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Short description: Mathematical technique in data science


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]

Underlying principles

Fractals have fractional dimensions, which are a measure of complexity that indicates the degree to which the objects fill the available space.[11][13] The fractal dimension measures the change in "size" of a fractal set with the changing observational scale, and is not limited by integer values.[2] This is possible given that a smaller section of the fractal resembles the entirety, showing the same statistical properties at different scales.[11] This characteristic is termed scale invariance, and can be further categorized as self-similarity or self-affinity, the latter scaled anisotropically (depending on the direction).[2] Whether the view of the fractal is expanding or contracting, the structure remains the same and appears equivalently complex.[11][13] Fractal analysis uses these underlying properties to help in the understanding and characterization of complex systems. It is also possible to expand the use of fractals to the lack of a single characteristic time scale, or pattern.[14]

Further information on the Origins: Fractal Geometry

Types of fractal analysis

There are various types of fractal analysis, including box counting, lacunarity analysis, mass methods, and multifractal analysis.[1][3][11] A common feature of all types of fractal analysis is the need for benchmark patterns against which to assess outputs.[15] 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. Other types include detrended fluctuation analysis and the Hurst absolute value method, which estimate the hurst exponent.[16] It is suggested to use more than one approach in order to compare results and increase the robustness of one's findings.

Applications

Ecology and evolution

Unlike theoretical fractal curves which can be easily measured and the underlying mathematical properties calculated; natural systems are sources of heterogeneity and generate complex space-time structures that may only demonstrate partial self-similarity.[17][18][19] Using fractal analysis, it is possible to analyze and recognize when features of complex ecological systems are altered since fractals are able to characterize the natural complexity in such systems.[20] Thus, fractal analysis can help to quantify patterns in nature and to identify deviations from these natural sequences. It helps to improve our overall understanding of ecosystems and to reveal some of the underlying structural mechanisms of nature.[13][21][22] For example, it was found that the structure of an individual tree’s xylem follows the same architecture as the spatial distribution of the trees in the forest, and that the distribution of the trees in the forest shared the same underlying fractal structure as the branches, scaling identically to the point of being able to use the pattern of the trees’ branches mathematically to determine the structure of the forest stand.[23][24] The use of fractal analysis for understanding structures, and spatial and temporal complexity in biological systems has already been well studied and its use continues to increase in ecological research.[25][26][27][28] Despite its extensive use, it still receives some criticism.[29][30]

Animal behaviour

Patterns in animal behaviour exhibit fractal properties on spatial and temporal scales.[16] Fractal analysis helps in understanding the behaviour of animals and how they interact with their environments on multiple scales in space and time.[2] Various animal movement signatures in their respective environments have been found to demonstrate spatially non-linear fractal patterns.[31][32] This has generated ecological interpretations such as the Lévy Flight Foraging hypothesis, which has proven to be a more accurate description of animal movement for some species.[33][34][35]

Spatial patterns and animal behaviour sequences in fractal time have an optimal complexity range, which can be thought of as the homeostatic state on the spectrum where the complexity sequence should regularly fall. An increase or a loss in complexity, either becoming more stereotypical or conversely more random in their behaviour patterns, indicates that there has been an alteration in the functionality of the individual.[14][36] Using fractal analysis, it is possible to examine the movement sequential complexity of animal behaviour and to determine whether individuals are experiencing deviations from their optimal range, suggesting a change in condition.[37][38] For example, it has been used to assess welfare of domestic hens,[20] stress in bottlenose dolphins in response to human disturbance,[39] and parasitic infection in Japanese macaques[38] and sheep.[37] The research is furthering the field of behavioural ecology by simplifying and quantifying very complex relationships.[40] When it comes to animal welfare and conservation, fractal analysis makes it possible to identify potential sources of stress on animal behaviour, stressors that may not always be discernible through classical behaviour research.[20][41][42]

This approach is more objective than classical behaviour measurements, such as frequency-based observations that are limited by the counts of behaviours, but is able to delve into the underlying reason for the behaviour.[36] Another important advantage of fractal analysis is the ability to monitor the health of wild and free-ranging animal populations in their natural habitats without invasive measurements.

Applications include

Applications of fractal analysis include:[43]

  • Fractal calculus[12]

See also

References

  1. ↑ 1.0 1.1 Gerges, Firas; Geng, Xiaolong; Nassif, Hani; Boufadel, Michel C. (2021). "Anisotropic Multifractal Scaling of Mount Lebanon Topography: Approximate Conditioning" (in en). Fractals 29 (5): 2150112–2153322. doi:10.1142/S0218348X21501127. ISSN 0218-348X. Bibcode: 2021Fract..2950112G. https://www.worldscientific.com/doi/abs/10.1142/S0218348X21501127. 
  2. ↑ 2.0 2.1 2.2 2.3 Seuront, Laurent (2009-10-12). Fractals and Multifractals in Ecology and Aquatic Science. CRC Press. doi:10.1201/9781420004243. ISBN 9780849327827. 
  3. ↑ 3.0 3.1 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. 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. doi:10.1371/journal.pone.0100199. PMID 24967904. Bibcode: 2014PLoSO...9j0199Z. 
  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. 2008. pp. 3880–3. doi:10.1109/IEMBS.2008.4650057. ISBN 978-1-4244-1814-5. 
  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. http://www.worldscinet.com/fractals/fractals.shtml. 
  11. ↑ 11.0 11.1 11.2 11.3 11.4 11.5 Benoît B. Mandelbrot (1983). The fractal geometry of nature. Macmillan. ISBN 978-0-7167-1186-5. https://books.google.com/books?id=0R2LkE3N7-oC. Retrieved 1 February 2012. 
  12. ↑ 12.0 12.1 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. https://worldscientific.com/worldscibooks/10.1142/12988#t=aboutBook. 
  13. ↑ 13.0 13.1 13.2 Mandelbrot, B. (1967-05-05). "How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension". Science 156 (3775): 636–638. doi:10.1126/science.156.3775.636. ISSN 0036-8075. PMID 17837158. Bibcode: 1967Sci...156..636M. http://ena.lp.edu.ua:8080/handle/ntb/52473. 
  14. ↑ 14.0 14.1 Goldberger, Ary L; Peng, C.-K; Lipsitz, Lewis A (January 2002). "What is physiologic complexity and how does it change with aging and disease?". Neurobiology of Aging 23 (1): 23–26. doi:10.1016/S0197-4580(01)00266-4. PMID 11755014. 
  15. ↑ "Digital Images in FracLac". ImageJ. http://rsbweb.nih.gov/ij/plugins/fraclac/FLHelp/Images.htm. 
  16. ↑ 16.0 16.1 MacIntosh, Andrew J. J.; Pelletier, Laure; Chiaradia, Andre; Kato, Akiko; Ropert-Coudert, Yan (December 2013). "Temporal fractals in seabird foraging behaviour: diving through the scales of time". Scientific Reports 3 (1): 1884. doi:10.1038/srep01884. ISSN 2045-2322. PMID 23703258. Bibcode: 2013NatSR...3E1884M. 
  17. ↑ Frontier, Serge (1987), "Applications of Fractal Theory to Ecology", Develoments in Numerical Ecology, Springer Berlin Heidelberg, pp. 335–378, doi:10.1007/978-3-642-70880-0_9, ISBN 9783642708824 
  18. ↑ Scheuring, István; Riedi, Rudolf H. (August 1994). "Application of multifractals to the analysis of vegetation pattern". Journal of Vegetation Science 5 (4): 489–496. doi:10.2307/3235975. 
  19. ↑ Seuront, Laurent; Lagadeuc, Yvan (1998). "Spatio-temporal structure of tidally mixed coastal waters: variability and heterogeneity". Journal of Plankton Research 20 (7): 1387–1401. doi:10.1093/plankt/20.7.1387. ISSN 0142-7873. 
  20. ↑ 20.0 20.1 20.2 Rutherford, Kenneth M.D.; Haskell, Marie J.; Glasbey, Chris; Jones, R.Bryan; Lawrence, Alistair B. (September 2003). "Detrended fluctuation analysis of behavioural responses to mild acute stressors in domestic hens". Applied Animal Behaviour Science 83 (2): 125–139. doi:10.1016/S0168-1591(03)00115-1. 
  21. ↑ Bradbury, Rh; Reichelt, Re (1983). "Fractal Dimension of a Coral Reef at Ecological Scales". Marine Ecology Progress Series 10: 169–171. doi:10.3354/meps010169. ISSN 0171-8630. Bibcode: 1983MEPS...10..169B. 
  22. ↑ Hastings, Harold M.; Pekelney, Richard; Monticciolo, Richard; Vun Kannon, David; Del Monte, Diane (January 1982). "Time scales, persistence and patchiness". Biosystems 15 (4): 281–289. doi:10.1016/0303-2647(82)90043-0. ISSN 0303-2647. PMID 7165795. 
  23. ↑ West, G. B. (1997-04-04). "A General Model for the Origin of Allometric Scaling Laws in Biology". Science 276 (5309): 122–126. doi:10.1126/science.276.5309.122. PMID 9082983. 
  24. ↑ West, G. B.; Enquist, B. J.; Brown, J. H. (2009-04-28). "A general quantitative theory of forest structure and dynamics". Proceedings of the National Academy of Sciences 106 (17): 7040–7045. doi:10.1073/pnas.0812294106. ISSN 0027-8424. PMID 19363160. Bibcode: 2009PNAS..106.7040W. 
  25. ↑ Rieu, Michel; Sposito, Garrison (1991). "Fractal Fragmentation, Soil Porosity, and Soil Water Properties: II. Applications". Soil Science Society of America Journal 55 (5): 1239. doi:10.2136/sssaj1991.03615995005500050007x. ISSN 0361-5995. Bibcode: 1991SSASJ..55.1239R. 
  26. ↑ Morse, D. R.; Lawton, J. H.; Dodson, M. M.; Williamson, M. H. (April 1985). "Fractal dimension of vegetation and the distribution of arthropod body lengths". Nature 314 (6013): 731–733. doi:10.1038/314731a0. ISSN 0028-0836. Bibcode: 1985Natur.314..731M. 
  27. ↑ Li, Xiaoyan; Passow, Uta; Logan, Bruce E (January 1998). "Fractal dimensions of small (15–200 μm) particles in Eastern Pacific coastal waters". Deep Sea Research Part I: Oceanographic Research Papers 45 (1): 115–131. doi:10.1016/s0967-0637(97)00058-7. ISSN 0967-0637. 
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  29. ↑ Halley, J. M.; Hartley, S.; Kallimanis, A. S.; Kunin, W. E.; Lennon, J. J.; Sgardelis, S. P. (2004-02-24). "Uses and abuses of fractal methodology in ecology". Ecology Letters 7 (3): 254–271. doi:10.1111/j.1461-0248.2004.00568.x. ISSN 1461-023X. 
  30. ↑ Bryce, R. M.; Sprague, K. B. (December 2012). "Revisiting detrended fluctuation analysis". Scientific Reports 2 (1): 315. doi:10.1038/srep00315. ISSN 2045-2322. PMID 22419991. Bibcode: 2012NatSR...2E.315B. 
  31. ↑ Catalan, Jordi; Marrasé, Cèlia; Pueyo, Salvador; Peters, Francesc; Bartumeus, Frederic (2003-10-28). "Helical Lévy walks: Adjusting searching statistics to resource availability in microzooplankton". Proceedings of the National Academy of Sciences 100 (22): 12771–12775. doi:10.1073/pnas.2137243100. ISSN 0027-8424. PMID 14566048. Bibcode: 2003PNAS..10012771B. 
  32. ↑ Garcia, F.; Carrère, P.; Soussana, J.F.; Baumont, R. (September 2005). "Characterisation by fractal analysis of foraging paths of ewes grazing heterogeneous swards". Applied Animal Behaviour Science 93 (1–2): 19–37. doi:10.1016/j.applanim.2005.01.001. 
  33. ↑ Humphries, N. E.; Weimerskirch, H.; Queiroz, N.; Southall, E. J.; Sims, D. W. (2012-05-08). "Foraging success of biological Levy flights recorded in situ". Proceedings of the National Academy of Sciences 109 (19): 7169–7174. doi:10.1073/pnas.1121201109. ISSN 0027-8424. PMID 22529349. Bibcode: 2012PNAS..109.7169H. 
  34. ↑ Raposo, E P; Buldyrev, S V; da Luz, M G E; Viswanathan, G M; Stanley, H E (2009-10-30). "Lévy flights and random searches". Journal of Physics A: Mathematical and Theoretical 42 (43): 434003. doi:10.1088/1751-8113/42/43/434003. ISSN 1751-8113. Bibcode: 2009JPhA...42Q4003R. 
  35. ↑ Viswanathan, G.M; Afanasyev, V; Buldyrev, Sergey V; Havlin, Shlomo; da Luz, M.G.E; Raposo, E.P; Stanley, H.Eugene (June 2001). "Lévy flights search patterns of biological organisms". Physica A: Statistical Mechanics and Its Applications 295 (1–2): 85–88. doi:10.1016/S0378-4371(01)00057-7. Bibcode: 2001PhyA..295...85V. 
  36. ↑ 36.0 36.1 MacIntosh, Andrew James Jonathan (2014). "The Fractal Primate". Primate Research 30 (1): 95–119. doi:10.2354/psj.30.011. ISSN 1880-2117. 
  37. ↑ 37.0 37.1 Burgunder, Jade; Petrželková, Klára J.; Modrý, David; Kato, Akiko; MacIntosh, Andrew J.J. (August 2018). "Fractal measures in activity patterns: Do gastrointestinal parasites affect the complexity of sheep behaviour?". Applied Animal Behaviour Science 205: 44–53. doi:10.1016/j.applanim.2018.05.014. 
  38. ↑ 38.0 38.1 MacIntosh, A. J. J.; Alados, C. L.; Huffman, M. A. (2011-10-07). "Fractal analysis of behaviour in a wild primate: behavioural complexity in health and disease". Journal of the Royal Society Interface 8 (63): 1497–1509. doi:10.1098/rsif.2011.0049. ISSN 1742-5689. PMID 21429908. 
  39. ↑ Cribb, Nardi; Seuront, Laurent (September 2016). "Changes in the behavioural complexity of bottlenose dolphins along a gradient of anthropogenically-impacted environments in South Australian coastal waters: Implications for conservation and management strategies". Journal of Experimental Marine Biology and Ecology 482: 118–127. doi:10.1016/j.jembe.2016.03.020. ISSN 0022-0981. 
  40. ↑ Bradbury, J. W.; Vehrencamp, S. L. (2014-05-01). "Complexity and behavioral ecology". Behavioral Ecology 25 (3): 435–442. doi:10.1093/beheco/aru014. ISSN 1045-2249. 
  41. ↑ Alados, C.L.; Escos, J.M.; Emlen, J.M. (February 1996). "Fractal structure of sequential behaviour patterns: an indicator of stress". Animal Behaviour 51 (2): 437–443. doi:10.1006/anbe.1996.0040. 
  42. ↑ Rutherford, K. M. D.; Haskell, M. J.; Glasbey, C.; Jones, R. B.; Lawrence, A. B. (February 2004). "Fractal analysis of animal behaviour as an indicator of animal welfare". Animal Welfare 13 (1): 99–103. doi:10.1017/S0962728600014433. https://www.ingentaconnect.com/contentone/ufaw/aw/2004/00000013/a00101s1/art00014. Retrieved 2019-03-27. 
  43. ↑ "Applications". http://library.thinkquest.org/26242/full/ap/ap.html. 
  44. ↑ 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. PMID 19528254. 
  45. ↑ Costa, Isis da Silva; Gamundí, Antoni; Miranda, José G. Vivas; França, Lucas G. Souza; Santana, De; Novaes, Charles; Montoya, Pedro (2017). "Altered Functional Performance in Patients with Fibromyalgia" (in en). Frontiers in Human Neuroscience 11: 14. doi:10.3389/fnhum.2017.00014. ISSN 1662-5161. PMID 28184193. 
  46. ↑ França, L. G. S.; Montoya, Pedro; Miranda, J. G. V. (2017). "On multifractals: a non-linear study of actigraphy data". Physica A: Statistical Mechanics and Its Applications 514: 612–619. doi:10.1016/j.physa.2018.09.122. 
  47. ↑ 47.0 47.1 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 2 (1): 109–122. doi:10.2147/OPTH.S1579. PMID 19668394. 
  48. ↑ 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. doi:10.1186/1756-0500-2-130. PMID 19594934. 
  49. ↑ Xiao, Xiongye; Chen, Hanlong; Bogdan, Paul (25 November 2021). "Deciphering the generating rules and functionalities of complex networks". Scientific Reports 11 (1): 22964. doi:10.1038/s41598-021-02203-4. PMID 34824290. Bibcode: 2021NatSR..1122964X. 
  50. ↑ Losa, Gabriele A.; Nonnenmacher, Theo F., eds (2005). Fractals in biology and medicine. Springer. ISBN 978-3-7643-7172-2. https://books.google.com/books?id=t9l9GdAt95gC. Retrieved 1 February 2012. 
  51. ↑ Mandelbrot, B. (1967). "How Long is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension". Science 156 (3775): 636–638. doi:10.1126/science.156.3775.636. PMID 17837158. Bibcode: 1967Sci...156..636M. http://ena.lp.edu.ua:8080/handle/ntb/52473. 
  52. ↑ Li, H. (2013). "Fractal analysis of side channels for breakdown structures in XLPE cable insulation". J Mater Sci: Mater Electron 24 (5): 1640–1643. doi:10.1007/s10854-012-0988-y. 
  53. ↑ Reuveni, Shlomi; Granek, Rony; Klafter, Joseph (2008). "Proteins: Coexistence of Stability and Flexibility". Physical Review Letters 100 (20): 208101. doi:10.1103/PhysRevLett.100.208101. ISSN 0031-9007. PMID 18518581. Bibcode: 2008PhRvL.100t8101R. 
  54. ↑ Panteha Saeedi, and Soren A. Sorensen (2009). An Algorithmic Approach to Generate After-disaster Test Fields for Search and Rescue Agents. 93–98. ISBN 978-988-17-0125-1. http://www.iaeng.org/publication/WCE2009/WCE2009_pp93-98.pdf. 
  55. ↑ 55.0 55.1 Chen, Yanguang (2011). "Modeling Fractal Structure of City-Size Distributions Using Correlation Functions". PLOS ONE 6 (9): e24791. doi:10.1371/journal.pone.0024791. PMID 21949753. Bibcode: 2011PLoSO...624791C. 
  56. ↑ 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. 
  57. ↑ França, Lucas G. Souza; Miranda, José G. Vivas; Leite, Marco; Sharma, Niraj K.; Walker, Matthew C.; Lemieux, Louis; Wang, Yujiang (2018). "Fractal and Multifractal Properties of Electrographic Recordings of Human Brain Activity: Toward Its Use as a Signal Feature for Machine Learning in Clinical Applications" (in en). Frontiers in Physiology 9: 1767. doi:10.3389/fphys.2018.01767. ISSN 1664-042X. PMID 30618789. Bibcode: 2018arXiv180603889F. 
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  60. ↑ 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 64 (4): 409–414. doi:10.1016/s0025-6196(12)65730-9. PMID 2716356. 
  61. ↑ Al-Kadi O.S, Watson D. (2008). "Texture Analysis of Aggressive and non-Aggressive Lung Tumor CE CT Images". IEEE Transactions on Biomedical Engineering 55 (7): 1822–1830. doi:10.1109/tbme.2008.919735. PMID 18595800. http://sro.sussex.ac.uk/1919/1/tbme.pdf. Retrieved 2014-04-10. 
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  65. ↑ 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. 
  66. ↑ 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. doi:10.1016/j.epsl.2007.07.056. Bibcode: 2007E&PSL.262..413V. 
  67. ↑ 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. 
  68. ↑ 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. 
  69. ↑ Brothers, Harlan J. (2007). "Structural Scaling in Bach's Cello Suite No. 3". Fractals 15: 89–95. doi:10.1142/S0218348X0700337X. 
  70. ↑ Brothers, Harlan J. (2009). "Intervallic Scaling In The Bach Cello Suites". Fractals 17 (4): 537–545. doi:10.1142/S0218348X09004521. 

Further reading



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