Fractal analysis

From Wikipedia

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, ecology and aquatic sciences, [1] sound, market fluctuations, [2] [3] [4] heart rates, [5] frequency domain in electroencephalography signals, [6] [7] digital images, [8] molecular motion, and data science. Fractal analysis is now widely used in all areas of science. [9] 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. [10] 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.

Underlying principles

Fractals have fractional dimensions, which are a measure of complexity that indicates the degree to which the objects fill the available space. [10] [11] The fractal dimension measures the change in "size" of a fractal set with the changing observational scale, and is not limited by integer values. [1] This is possible given that a smaller section of the fractal resembles the entirety, showing the same statistical properties at different scales. [10] 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). [1] Whether the view of the fractal is expanding or contracting, the structure remains the same and appears equivalently complex. [10] [11] 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. [12]

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. [2] [10] A common feature of all types of fractal analysis is the need for benchmark patterns against which to assess outputs. [13] 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. [14] It is suggested to use more than one approach in order to compare results and increase the robustness of one's findings.


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. [15] [16] [17] 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. [18] 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. [11] [19] [20] 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. [21] [22] 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. [23] [24] [25] [26] Despite its extensive use, it still receives some criticism. [27] [28]

Animal behaviour

Patterns in animal behaviour exhibit fractal properties on spatial and temporal scales. [14] Fractal analysis helps in understanding the behaviour of animals and how they interact with their environments on multiple scales in space and time. [1] Various animal movement signatures in their respective environments have been found to demonstrate spatially non-linear fractal patterns. [29] [30] 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. [31] [32] [33]

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. [12] [34] 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. [35] [36] For example, it has been used to assess welfare of domestic hens, [18] stress in bottlenose dolphins in response to human disturbance, [37] and parasitic infection in Japanese macaques [36] and sheep. [35] The research is furthering the field of behavioural ecology by simplifying and quantifying very complex relationships. [38] 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. [18] [39] [40]

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. [34] 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: [41]

See also


  1. ^ a b c d Seuront, Laurent (2009-10-12). Fractals and Multifractals in Ecology and Aquatic Science. CRC Press. doi: 10.1201/9781420004243. ISBN  9780849327827.
  2. ^ 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  978-0-471-13938-6.
  3. ^ 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.
  4. ^ 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.
  5. ^ 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.
  6. ^ 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.
  7. ^ 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. Annual International Conference of the IEEE Engineering in Medicine and Biology Society. IEEE Engineering in Medicine and Biology Society. Annual International Conference. 2008. pp. 3880–3. doi: 10.1109/IEMBS.2008.4650057. ISBN  978-1-4244-1814-5. PMID  19163560. S2CID  22136019.
  8. ^ Fractal Analysis of Digital Images
  9. ^ "Fractals: Complex Geometry, Patterns, and Scaling in Nature and Society". Fractals : An Interdiscipinary Journal on the Complex Geometry of Nature. ISSN  1793-6543.
  10. ^ a b c d e f Benoît B. Mandelbrot (1983). The fractal geometry of nature. Macmillan. ISBN  978-0-7167-1186-5. Retrieved 1 February 2012.
  11. ^ a b c Mandelbrot, B. (1967-05-05). "How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension". Science. 156 (3775): 636–638. Bibcode: 1967Sci...156..636M. doi: 10.1126/science.156.3775.636. ISSN  0036-8075. PMID  17837158. S2CID  15662830.
  12. ^ a b 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. S2CID  17022186.
  13. ^ "Digital Images in FracLac". ImageJ. Archived from the original on 2012-02-08. Retrieved 2012-02-08. Cite journal requires |journal= ( help)CS1 maint: bot: original URL status unknown ( link)
  14. ^ a b 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. Bibcode: 2013NatSR...3E1884M. doi: 10.1038/srep01884. ISSN  2045-2322. PMC  3662970. PMID  23703258.
  15. ^ 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
  16. ^ 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. JSTOR  3235975.
  17. ^ 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.
  18. ^ a b c 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.
  19. ^ Bradbury, Rh; Reichelt, Re (1983). "Fractal Dimension of a Coral Reef at Ecological Scales". Marine Ecology Progress Series. 10: 169–171. Bibcode: 1983MEPS...10..169B. doi: 10.3354/meps010169. ISSN  0171-8630.
  20. ^ 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.
  21. ^ 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. S2CID  3140271.
  22. ^ 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. Bibcode: 2009PNAS..106.7040W. doi: 10.1073/pnas.0812294106. ISSN  0027-8424. PMC  2678466. PMID  19363160.
  23. ^ Rieu, Michel; Sposito, Garrison (1991). "Fractal Fragmentation, Soil Porosity, and Soil Water Properties: II. Applications". Soil Science Society of America Journal. 55 (5): 1239. Bibcode: 1991SSASJ..55.1239R. doi: 10.2136/sssaj1991.03615995005500050007x. ISSN  0361-5995.
  24. ^ 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. Bibcode: 1985Natur.314..731M. doi: 10.1038/314731a0. ISSN  0028-0836. S2CID  4362382.
  25. ^ 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.
  26. ^ Lovejoy, S.; Schertzer, D. (May 2006). "Multifractals, cloud radiances and rain". Journal of Hydrology. 322 (1–4): 59–88. Bibcode: 2006JHyd..322...59L. doi: 10.1016/j.jhydrol.2005.02.042.
  27. ^ 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. S2CID  6059069.
  28. ^ Bryce, R. M.; Sprague, K. B. (December 2012). "Revisiting detrended fluctuation analysis". Scientific Reports. 2 (1): 315. Bibcode: 2012NatSR...2E.315B. doi: 10.1038/srep00315. ISSN  2045-2322. PMC  3303145. PMID  22419991.
  29. ^ 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. Bibcode: 2003PNAS..10012771B. doi: 10.1073/pnas.2137243100. ISSN  0027-8424. PMC  240693. PMID  14566048.
  30. ^ 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.
  31. ^ 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. Bibcode: 2012PNAS..109.7169H. doi: 10.1073/pnas.1121201109. ISSN  0027-8424. PMC  3358854. PMID  22529349.
  32. ^ 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. Bibcode: 2009JPhA...42Q4003R. doi: 10.1088/1751-8113/42/43/434003. ISSN  1751-8113.
  33. ^ 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. Bibcode: 2001PhyA..295...85V. doi: 10.1016/S0378-4371(01)00057-7.
  34. ^ a b MacIntosh, Andrew James Jonathan (2014). "The Fractal Primate". Primate Research. 30 (1): 95–119. doi: 10.2354/psj.30.011. ISSN  1880-2117.
  35. ^ a b 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.
  36. ^ a b 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. PMC  3163426. PMID  21429908.
  37. ^ 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.
  38. ^ 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.
  39. ^ 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. S2CID  53184132.
  40. ^ 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". Retrieved 2019-03-27.
  41. ^ "Applications". Archived from the original on 2007-10-12. Retrieved 2007-10-21.
  42. ^ 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.
  43. ^ 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". Frontiers in Human Neuroscience. 11: 14. doi: 10.3389/fnhum.2017.00014. ISSN  1662-5161. PMC  5266716. PMID  28184193.
  44. ^ 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. arXiv: 1702.03912. doi: 10.1016/j.physa.2018.09.122. S2CID  18259316.
  45. ^ 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. doi: 10.2147/OPTH.S1579. PMC  2698675. PMID  19668394.
  46. ^ 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. PMC  2716356. PMID  19594934.
  47. ^ Losa, Gabriele A.; Nonnenmacher, Theo F., eds. (2005). Fractals in biology and medicine. Springer. ISBN  978-3-7643-7172-2. Retrieved 1 February 2012.
  48. ^ 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. doi: 10.1126/science.156.3775.636. PMID  17837158. S2CID  15662830.
  49. ^ 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. S2CID  136564926.
  50. ^ Reuveni, Shlomi; Granek, Rony; Klafter, Joseph (2008). "Proteins: Coexistence of Stability and Flexibility". Physical Review Letters. 100 (20): 208101. Bibcode: 2008PhRvL.100t8101R. doi: 10.1103/PhysRevLett.100.208101. ISSN  0031-9007. PMID  18518581. S2CID  16203048.
  51. ^ Panteha Saeedi, and Soren A. Sorensen (2009). An Algorithmic Approach to Generate After-disaster Test Fields for Search and Rescue Agents (PDF). Proceedings of the World Congress on Engineering 2009. pp. 93–98. ISBN  978-988-17-0125-1.
  52. ^ a b Chen, Yanguang (2011). "Modeling Fractal Structure of City-Size Distributions Using Correlation Functions". PLOS ONE. 6 (9): e24791. arXiv: 1104.4682. Bibcode: 2011PLoSO...624791C. doi: 10.1371/journal.pone.0024791. PMC  3176775. PMID  21949753.
  53. ^ 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.
  54. ^ 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". Frontiers in Physiology. 9: 1767. arXiv: 1806.03889. Bibcode: 2018arXiv180603889F. doi: 10.3389/fphys.2018.01767. ISSN  1664-042X. PMC  6295567. PMID  30618789.
  55. ^ 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. doi: 10.1016/S0006-3495(03)74817-6. PMC  1303704. PMID  14645092.
  56. ^ 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. PMID  18495248. S2CID  12268410.
  57. ^ 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.
  58. ^ 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. doi: 10.1109/tbme.2008.919735. PMID  18595800. S2CID  14784161. Archived from the original (PDF) on 2014-04-13. Retrieved 2014-04-10.
  59. ^ Landini, Gabriel (2011). "Fractals in microscopy". Journal of Microscopy. 241 (1): 1–8. doi: 10.1111/j.1365-2818.2010.03454.x. PMID  21118245. S2CID  40311727.
  60. ^ Cheng, Qiuming (1997). "Multifractal Modeling and Lacunarity Analysis". Mathematical Geology. 29 (7): 919–932. doi: 10.1023/A:1022355723781. S2CID  118918429.
  61. ^ 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.
  62. ^ 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. S2CID  7481018.
  63. ^ 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.
  64. ^ 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.
  65. ^ 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.
  66. ^ Brothers, Harlan J. (2007). "Structural Scaling in Bach's Cello Suite No. 3". Fractals. 15: 89–95. doi: 10.1142/S0218348X0700337X.

Further reading