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== Applications ==
== Applications ==

=== Ecology and evolution ===
Unlike theoretical [[Fractal Curves|fractal curves]] which can be easily measured and the underlying [[Property (mathematics)|mathematical properties]] calculated; [[Nature|natural]] systems are sources of heterogeneity and generate complex space-time structures that may only demonstrate partial [[self-similarity]].<ref name=":3">{{Citation|last=Frontier|first=Serge|title=Applications of Fractal Theory to Ecology|date=1987|url=http://dx.doi.org/10.1007/978-3-642-70880-0_9|work=Develoments in Numerical Ecology|pages=335–378|publisher=Springer Berlin Heidelberg|isbn=9783642708824|access-date=2019-03-26}}</ref><ref>{{Cite journal|last=Scheuring|first=István|last2=Riedi|first2=Rudolf H.|date=1994-8|title=Application of multifractals to the analysis of vegetation pattern|url=http://doi.wiley.com/10.2307/3235975|journal=Journal of Vegetation Science|language=en|volume=5|issue=4|pages=489–496|doi=10.2307/3235975}}</ref><ref>{{Cite journal|last=Seuront|first=Laurent|last2=Lagadeuc|first2=Yvan|date=1998|title=Spatio-temporal structure of tidally mixed coastal waters: variability and heterogeneity|url=https://academic.oup.com/plankt/article-lookup/doi/10.1093/plankt/20.7.1387|journal=Journal of Plankton Research|language=en|volume=20|issue=7|pages=1387–1401|doi=10.1093/plankt/20.7.1387|issn=0142-7873}}</ref> Using fractal analysis, it is possible to analyze and recognize when features of complex [[Ecology|ecological]] systems are altered since fractals are able to characterize the natural complexity in such systems.<ref name=":4">{{Cite journal|last=Rutherford|first=Kenneth M.D.|last2=Haskell|first2=Marie J.|last3=Glasbey|first3=Chris|last4=Jones|first4=R.Bryan|last5=Lawrence|first5=Alistair B.|date=2003-9|title=Detrended fluctuation analysis of behavioural responses to mild acute stressors in domestic hens|url=https://linkinghub.elsevier.com/retrieve/pii/S0168159103001151|journal=Applied Animal Behaviour Science|language=en|volume=83|issue=2|pages=125–139|doi=10.1016/S0168-1591(03)00115-1}}</ref> 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 [[Ecosystem|ecosystems]] and to reveal some of the underlying structural mechanisms of nature.<ref name=":02">{{Cite journal|last=Mandelbrot|first=B.|date=1967-05-05|title=How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension|url=http://www.sciencemag.org/cgi/doi/10.1126/science.156.3775.636|journal=Science|language=en|volume=156|issue=3775|pages=636–638|doi=10.1126/science.156.3775.636|issn=0036-8075}}</ref><ref>{{Cite journal|last=Bradbury|first=Rh|last2=Reichelt|first2=Re|date=1983|title=Fractal Dimension of a Coral Reef at Ecological Scales|url=http://www.int-res.com/articles/meps/10/m010p169.pdf|journal=Marine Ecology Progress Series|language=en|volume=10|pages=169–171|doi=10.3354/meps010169|issn=0171-8630}}</ref><ref>{{Cite journal|last=Hastings|first=Harold M.|last2=Pekelney|first2=Richard|last3=Monticciolo|first3=Richard|last4=Vun Kannon|first4=David|last5=Del Monte|first5=Diane|date=1982-01|title=Time scales, persistence and patchiness|url=http://dx.doi.org/10.1016/0303-2647(82)90043-0|journal=Biosystems|volume=15|issue=4|pages=281–289|doi=10.1016/0303-2647(82)90043-0|issn=0303-2647}}</ref> 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.<ref>{{Cite journal|last=West|first=G. B.|date=1997-04-04|title=A General Model for the Origin of Allometric Scaling Laws in Biology|url=http://www.sciencemag.org/cgi/doi/10.1126/science.276.5309.122|journal=Science|volume=276|issue=5309|pages=122–126|doi=10.1126/science.276.5309.122}}</ref><ref>{{Cite journal|last=West|first=G. B.|last2=Enquist|first2=B. J.|last3=Brown|first3=J. H.|date=2009-04-28|title=A general quantitative theory of forest structure and dynamics|url=http://www.pnas.org/cgi/doi/10.1073/pnas.0812294106|journal=Proceedings of the National Academy of Sciences|language=en|volume=106|issue=17|pages=7040–7045|doi=10.1073/pnas.0812294106|issn=0027-8424|pmc=PMC2678466|pmid=19363160}}</ref> 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.<ref>{{Cite journal|last=Rieu|first=Michel|last2=Sposito|first2=Garrison|date=1991|title=Fractal Fragmentation, Soil Porosity, and Soil Water Properties: II. Applications|url=https://www.soils.org/publications/sssaj/abstracts/55/5/SS0550051239|journal=Soil Science Society of America Journal|language=en|volume=55|issue=5|pages=1239|doi=10.2136/sssaj1991.03615995005500050007x|issn=0361-5995}}</ref><ref>{{Cite journal|last=Morse|first=D. R.|last2=Lawton|first2=J. H.|last3=Dodson|first3=M. M.|last4=Williamson|first4=M. H.|date=1985-4|title=Fractal dimension of vegetation and the distribution of arthropod body lengths|url=http://www.nature.com/articles/314731a0|journal=Nature|language=en|volume=314|issue=6013|pages=731–733|doi=10.1038/314731a0|issn=0028-0836}}</ref><ref>{{Cite journal|last=Li|first=Xiaoyan|last2=Passow|first2=Uta|last3=Logan|first3=Bruce E|date=1998-01|title=Fractal dimensions of small (15–200 μm) particles in Eastern Pacific coastal waters|url=http://dx.doi.org/10.1016/s0967-0637(97)00058-7|journal=Deep Sea Research Part I: Oceanographic Research Papers|volume=45|issue=1|pages=115–131|doi=10.1016/s0967-0637(97)00058-7|issn=0967-0637}}</ref><ref>{{Cite journal|last=Lovejoy|first=S.|last2=Schertzer|first2=D.|date=2006-5|title=Multifractals, cloud radiances and rain|url=https://linkinghub.elsevier.com/retrieve/pii/S002216940500106X|journal=Journal of Hydrology|language=en|volume=322|issue=1-4|pages=59–88|doi=10.1016/j.jhydrol.2005.02.042}}</ref> Despite its extensive use, it still receives some [[criticism]].<ref>{{Cite journal|last=Halley|first=J. M.|last2=Hartley|first2=S.|last3=Kallimanis|first3=A. S.|last4=Kunin|first4=W. E.|last5=Lennon|first5=J. J.|last6=Sgardelis|first6=S. P.|date=2004-02-24|title=Uses and abuses of fractal methodology in ecology|url=http://dx.doi.org/10.1111/j.1461-0248.2004.00568.x|journal=Ecology Letters|volume=7|issue=3|pages=254–271|doi=10.1111/j.1461-0248.2004.00568.x|issn=1461-023X}}</ref><ref>{{Cite journal|last=Bryce|first=R. M.|last2=Sprague|first2=K. B.|date=2012-12|title=Revisiting detrended fluctuation analysis|url=http://www.nature.com/articles/srep00315|journal=Scientific Reports|language=en|volume=2|issue=1|doi=10.1038/srep00315|issn=2045-2322}}</ref>

==== Animal behaviour ====
Patterns in animal [[Behavior|behaviour]] exhibit fractal properties on spatial and temporal scales.<ref name=":82">{{Cite journal|last=MacIntosh|first=Andrew J. J.|last2=Pelletier|first2=Laure|last3=Chiaradia|first3=Andre|last4=Kato|first4=Akiko|last5=Ropert-Coudert|first5=Yan|date=2013-12|title=Temporal fractals in seabird foraging behaviour: diving through the scales of time|url=http://www.nature.com/articles/srep01884|journal=Scientific Reports|language=en|volume=3|issue=1|doi=10.1038/srep01884|issn=2045-2322|pmc=PMC3662970|pmid=23703258}}</ref> Fractal analysis helps in understanding the behaviour of animals and how they interact with their environments on multiple scales in space and time.<ref name=":23">{{Cite book|url=http://dx.doi.org/10.1201/9781420004243|title=Fractals and Multifractals in Ecology and Aquatic Science|last=Seuront|first=Laurent|date=2009-10-12|publisher=CRC Press|isbn=9780849327827}}</ref> Various animal movement signatures in their respective environments have been found to demonstrate spatially non-linear fractal patterns.<ref>{{Cite journal|last=Catalan|first=Jordi|last2=Marrasé|first2=Cèlia|last3=Pueyo|first3=Salvador|last4=Peters|first4=Francesc|last5=Bartumeus|first5=Frederic|date=2003-10-28|title=Helical Lévy walks: Adjusting searching statistics to resource availability in microzooplankton|url=https://www.pnas.org/content/100/22/12771|journal=Proceedings of the National Academy of Sciences|language=en|volume=100|issue=22|pages=12771–12775|doi=10.1073/pnas.2137243100|issn=0027-8424|pmc=PMC240693|pmid=14566048}}</ref><ref>{{Cite journal|last=Garcia|first=F.|last2=Carrère|first2=P.|last3=Soussana|first3=J.F.|last4=Baumont|first4=R.|date=2005-9|title=Characterisation by fractal analysis of foraging paths of ewes grazing heterogeneous swards|url=https://linkinghub.elsevier.com/retrieve/pii/S0168159105000031|journal=Applied Animal Behaviour Science|language=en|volume=93|issue=1-2|pages=19–37|doi=10.1016/j.applanim.2005.01.001}}</ref> This has generated ecological interpretations such as the [[Lévy flight foraging hypothesis|Lévy Flight Foraging hypothesis]], which has proven to be a more accurate description of animal movement for some species.<ref>{{Cite journal|last=Humphries|first=N. E.|last2=Weimerskirch|first2=H.|last3=Queiroz|first3=N.|last4=Southall|first4=E. J.|last5=Sims|first5=D. W.|date=2012-05-08|title=Foraging success of biological Levy flights recorded in situ|url=http://www.pnas.org/cgi/doi/10.1073/pnas.1121201109|journal=Proceedings of the National Academy of Sciences|language=en|volume=109|issue=19|pages=7169–7174|doi=10.1073/pnas.1121201109|issn=0027-8424|pmc=PMC3358854|pmid=22529349}}</ref><ref>{{Cite journal|last=Raposo|first=E P|last2=Buldyrev|first2=S V|last3=da Luz|first3=M G E|last4=Viswanathan|first4=G M|last5=Stanley|first5=H E|date=2009-10-30|title=Lévy flights and random searches|url=http://stacks.iop.org/1751-8121/42/i=43/a=434003?key=crossref.bd9ecd1b19a1e4362037d05351cf7384|journal=Journal of Physics A: Mathematical and Theoretical|volume=42|issue=43|pages=434003|doi=10.1088/1751-8113/42/43/434003|issn=1751-8113}}</ref><ref>{{Cite journal|last=Viswanathan|first=G.M|last2=Afanasyev|first2=V|last3=Buldyrev|first3=Sergey V|last4=Havlin|first4=Shlomo|last5=da Luz|first5=M.G.E|last6=Raposo|first6=E.P|last7=Stanley|first7=H.Eugene|date=2001-6|title=Lévy flights search patterns of biological organisms|url=http://linkinghub.elsevier.com/retrieve/pii/S0378437101000577|journal=Physica A: Statistical Mechanics and its Applications|language=en|volume=295|issue=1-2|pages=85–88|doi=10.1016/S0378-4371(01)00057-7}}</ref>

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.<ref name=":72">{{Cite journal|last=Goldberger|first=Ary L|last2=Peng|first2=C.-K|last3=Lipsitz|first3=Lewis A|date=2002-1|title=What is physiologic complexity and how does it change with aging and disease?|url=http://linkinghub.elsevier.com/retrieve/pii/S0197458001002664|journal=Neurobiology of Aging|language=en|volume=23|issue=1|pages=23–26|doi=10.1016/S0197-4580(01)00266-4}}</ref><ref name=":5">{{Cite journal|last=MacIntosh|first=Andrew James Jonathan|date=2014|title=The Fractal Primate:|url=http://jlc.jst.go.jp/DN/JST.JSTAGE/psj/30.011?lang=en&from=CrossRef&type=abstract|journal=Primate Research|language=en|volume=30|issue=1|pages=95–119|doi=10.2354/psj.30.011|issn=1880-2117}}</ref> 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.<ref name=":1">{{Cite journal|last=Burgunder|first=Jade|last2=Petrželková|first2=Klára J.|last3=Modrý|first3=David|last4=Kato|first4=Akiko|last5=MacIntosh|first5=Andrew J.J.|date=2018-8|title=Fractal measures in activity patterns: Do gastrointestinal parasites affect the complexity of sheep behaviour?|url=https://linkinghub.elsevier.com/retrieve/pii/S0168159118302429|journal=Applied Animal Behaviour Science|language=en|volume=205|pages=44–53|doi=10.1016/j.applanim.2018.05.014}}</ref><ref>{{Cite journal|last=MacIntosh|first=A. J. J.|last2=Alados|first2=C. L.|last3=Huffman|first3=M. A.|date=2011-10-07|title=Fractal analysis of behaviour in a wild primate: behavioural complexity in health and disease|url=http://rsif.royalsocietypublishing.org/cgi/doi/10.1098/rsif.2011.0049|journal=Journal of The Royal Society Interface|language=en|volume=8|issue=63|pages=1497–1509|doi=10.1098/rsif.2011.0049|issn=1742-5689|pmc=PMC3163426|pmid=21429908}}</ref> For example, it has been used to assess welfare of domestic hens,<ref name=":4" /> stress in bottlenose dolphins in response to human disturbance,<ref>{{Cite journal|last=Cribb|first=Nardi|last2=Seuront|first2=Laurent|date=2016-09|title=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|url=http://dx.doi.org/10.1016/j.jembe.2016.03.020|journal=Journal of Experimental Marine Biology and Ecology|volume=482|pages=118–127|doi=10.1016/j.jembe.2016.03.020|issn=0022-0981}}</ref> and parasitic infection in Japanese macaques<ref>{{Cite journal|last=MacIntosh|first=A. J. J.|last2=Alados|first2=C. L.|last3=Huffman|first3=M. A.|date=2011-10-07|title=Fractal analysis of behaviour in a wild primate: behavioural complexity in health and disease|url=http://rsif.royalsocietypublishing.org/cgi/doi/10.1098/rsif.2011.0049|journal=Journal of The Royal Society Interface|language=en|volume=8|issue=63|pages=1497–1509|doi=10.1098/rsif.2011.0049|issn=1742-5689|pmc=PMC3163426|pmid=21429908}}</ref> and sheep.<ref name=":1" /> The research is furthering the field of behavioural ecology by simplifying and quantifying very complex relationships.<ref name=":6">{{Cite journal|last=Bradbury|first=J. W.|last2=Vehrencamp|first2=S. L.|date=2014-05-01|title=Complexity and behavioral ecology|url=https://academic.oup.com/beheco/article-lookup/doi/10.1093/beheco/aru014|journal=Behavioral Ecology|language=en|volume=25|issue=3|pages=435–442|doi=10.1093/beheco/aru014|issn=1045-2249}}</ref> When it comes to animal welfare and [[Conservation biology|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.<ref name=":4" /><ref>{{Cite journal|last=Alados|first=C.L.|last2=Escos|first2=J.M.|last3=Emlen|first3=J.M.|date=1996-2|title=Fractal structure of sequential behaviour patterns: an indicator of stress|url=https://linkinghub.elsevier.com/retrieve/pii/S0003347296900408|journal=Animal Behaviour|language=en|volume=51|issue=2|pages=437–443|doi=10.1006/anbe.1996.0040}}</ref><ref>{{Cite web|url=https://www.ingentaconnect.com/contentone/ufaw/aw/2004/00000013/a00101s1/art00014|title=Fractal analysis of animal behaviour as an indicator of animal welfare|last=Rutherford|first=K. M. D.|last2=Haskell|first2=M. J.|date=2004-2|website=www.ingentaconnect.com|language=en|access-date=2019-03-27|last3=Glasbey|first3=C.|last4=Jones|first4=R. B.|last5=Lawrence|first5=A. B.}}</ref>

This approach is more objective than classical behaviour measurements, such as [[Frequency|frequency-based]] observations that are limited by the counts of behaviours, but is able to delve into the underlying reason for the behaviour.<ref name=":5" /> Another important advantage of fractal analysis is the ability to monitor the health of [[Wildlife|wild]] and free-ranging animal populations in their natural habitats without invasive measurements.

== Applications include: ==
Applications of fractal analysis include:<ref>{{cite web|url=http://library.thinkquest.org/26242/full/ap/ap.html|title=Applications|accessdate=2007-10-21|deadurl=yes|archiveurl=https://web.archive.org/web/20071012223212/http://library.thinkquest.org/26242/full/ap/ap.html|archivedate=2007-10-12|df=}}</ref>
Applications of fractal analysis include:<ref>{{cite web|url=http://library.thinkquest.org/26242/full/ap/ap.html|title=Applications|accessdate=2007-10-21|deadurl=yes|archiveurl=https://web.archive.org/web/20071012223212/http://library.thinkquest.org/26242/full/ap/ap.html|archivedate=2007-10-12|df=}}</ref>
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Revision as of 15:38, 10 April 2019

Fractal analysis is assessing fractal characteristics of data; used in data science. 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 networks. 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.[11][12] The fractal dimension measures the change in "size" of a fractal set with the changing observational scale, and is not limited by integer values.[13] 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).[13] Whether the view of the fractal is expanding or contracting, the structure remains the same and appears equivalently complex.[11][12] 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.[15][11] A common feature of all types of fractal analysis is the need for benchmark patterns against which to assess outputs.[16] 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.[17] It is suggested to use more than one approach in order to compare results and increase the robustness of ones 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.[18][19][20] 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.[21] 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.[22][23][24] 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.[25][26] 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.[27][28][29][30] Despite its extensive use, it still receives some criticism.[31][32]

Animal behaviour

Patterns in animal behaviour exhibit fractal properties on spatial and temporal scales.[33] Fractal analysis helps in understanding the behaviour of animals and how they interact with their environments on multiple scales in space and time.[34] Various animal movement signatures in their respective environments have been found to demonstrate spatially non-linear fractal patterns.[35][36] 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.[37][38][39]

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.[40][41] 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.[42][43] For example, it has been used to assess welfare of domestic hens,[21] stress in bottlenose dolphins in response to human disturbance,[44] and parasitic infection in Japanese macaques[45] and sheep.[42] The research is furthering the field of behavioural ecology by simplifying and quantifying very complex relationships.[46] 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.[21][47][48]

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.[41] 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:[49]

See also

References

  1. ^ Seuront, Laurent (2009-10-12). Fractals and Multifractals in Ecology and Aquatic Science. CRC Press. ISBN 9780849327827.
  2. ^ 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.{{cite journal}}: CS1 maint: unflagged free DOI (link)
  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.{{cite journal}}: CS1 maint: unflagged free DOI (link)
  7. ^ Hisonothai, M.; Nakagawa, M. (2008). EEG signal classification method based on fractal features and neural network. Vol. 2008. pp. 3880–3. doi:10.1109/IEMBS.2008.4650057. ISBN 978-1-4244-1814-5. PMID 19163560. {{cite book}}: |journal= ignored (help)
  8. ^ Fractal Analysis of Digital Images http://rsbweb.nih.gov/ij/plugins/fraclac/FLHelp/Fractals.htm
  9. ^ "Fractals: Complex Geometry, Patterns, and Scaling in Nature and Society". ISSN 1793-6543. {{cite journal}}: Cite journal requires |journal= (help)
  10. ^ Benoît B. Mandelbrot (1983). The fractal geometry of nature. Macmillan. ISBN 978-0-7167-1186-5. Retrieved 1 February 2012.
  11. ^ a b c d Benoît B. Mandelbrot (1983). The fractal geometry of nature. Macmillan. ISBN 978-0-7167-1186-5. Retrieved 1 February 2012.
  12. ^ a b 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.
  13. ^ a b Seuront, Laurent (2009-10-12). Fractals and Multifractals in Ecology and Aquatic Science. CRC Press. ISBN 9780849327827.
  14. ^ Goldberger, Ary L; Peng, C.-K; Lipsitz, Lewis A (2002-1). "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. {{cite journal}}: Check date values in: |date= (help)
  15. ^ 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.
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Further reading

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