Fractal analysis

{{Short description|Mathematical technique in data science}} {{Use American English|date = January 2019}} {{CS1 config|mode=cs1}} thumb | 220x124px | right | alt= 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,<ref name=":10">{{Cite journal|last1=Gerges|first1=Firas|last2=Geng|first2=Xiaolong|last3=Nassif|first3=Hani|last4=Boufadel|first4=Michel C.|title=Anisotropic Multifractal Scaling of Mount Lebanon Topography: Approximate Conditioning |date=2021|url=https://www.worldscientific.com/doi/abs/10.1142/S0218348X21501127|journal=Fractals|language=en|volume=29|issue=5|pages=2150112–2153322|doi=10.1142/S0218348X21501127|bibcode=2021Fract..2950112G |s2cid=234272453 |issn=0218-348X}}</ref> natural geometric objects, ecology and aquatic sciences,<ref name=":2">{{Cite book|title=Fractals and Multifractals in Ecology and Aquatic Science|last=Seuront|first=Laurent|date=2009-10-12|publisher=CRC Press|isbn=978-0-8493-2782-7|doi = 10.1201/9781420004243}}</ref> sound, market fluctuations,<ref name="time series2">{{cite book|title=Chaos and order in the capital markets: a new view of cycles, prices, and market volatility|last=Peters|first=Edgar|publisher=Wiley|year=1996|isbn=978-0-471-13938-6|location=New York}}</ref><ref name="mul20042">{{cite journal|last=Mulligan|first=R.|year=2004|title=Fractal analysis of highly volatile markets: an application to technology equities|journal=The Quarterly Review of Economics and Finance|volume=44|pages=155–179|doi=10.1016/S1062-9769(03)00028-0}}</ref><ref name="kam20142">{{cite journal|last=Kamenshchikov|first=S.|year=2014|title=Transport Catastrophe Analysis as an Alternative to a Monofractal Description: Theory and Application to Financial Crisis Time Series|journal=Journal of Chaos|volume=2014|pages=1–8|doi=10.1155/2014/346743|doi-access=free}}</ref> heart rates,<ref name="heart2">{{Cite journal|last1=Tan|first1=Can Ozan|last2=Cohen|first2=Michael A.|last3=Eckberg|first3=Dwain L.|last4=Taylor|first4=J. Andrew|year=2009|title=Fractal properties of human heart period variability: Physiological and methodological implications|journal=The Journal of Physiology|volume=587|issue=15|pages=3929–3941|doi=10.1113/jphysiol.2009.169219|pmc=2746620|pmid=19528254}}</ref> frequency domain in electroencephalography signals,<ref name="brain12">{{Cite journal|last1=Zappasodi|first1=Filippo|last2=Olejarczyk|first2=Elzbieta|last3=Marzetti|first3=Laura|last4=Assenza|first4=Giovanni|year=2014|title=Fractal Dimension of EEG Activity Senses Neuronal Impairment in Acute Stroke|journal=PLOS ONE|volume=9|issue=6|pages=3929–3941|bibcode=2014PLoSO...9j0199Z|doi=10.1371/journal.pone.0100199|pmc=4072666|pmid=24967904|doi-access=free}}</ref><ref name="brain22">{{Cite book|last1=Hisonothai|first1=M.|title=2008 30th Annual International Conference of the IEEE Engineering in Medicine and Biology Society|last2=Nakagawa|first2=M.|year=2008|isbn=978-1-4244-1814-5|volume=2008|pages=3880–3|doi=10.1109/IEMBS.2008.4650057|pmid=19163560|chapter=EEG signal classification method based on fractal features and neural network|s2cid=22136019}}</ref> digital images,<ref>Fractal Analysis of Digital Images [https://web.archive.org/web/20090131193936/http://rsbweb.nih.gov/ij/plugins/fraclac/FLHelp/Fractals.htm]</ref> molecular motion, and data science. Fractal analysis is now widely used in all areas of science.<ref>{{cite journal|title=Fractals: Complex Geometry, Patterns, and Scaling in Nature and Society|journal=Fractals: An Interdiscipinary Journal on the Complex Geometry of Nature|url=http://www.worldscinet.com/fractals/fractals.shtml|issn=1793-6543}}</ref> 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.<ref name="Mandelbrot19832">{{cite book|url=https://books.google.com/books?id=0R2LkE3N7-oC|title=The fractal geometry of nature|author=Benoît B. Mandelbrot|publisher=Macmillan|year=1983|isbn=978-0-7167-1186-5|access-date=1 February 2012}}</ref> 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. <ref name="Ali">{{cite book |last= Khalili Golmankhaneh|first= Alireza |date=2022 |title=Fractal Calculus and its Applications |url=https://worldscientific.com/worldscibooks/10.1142/12988#t=aboutBook|location=Singapore |publisher= World Scientific Pub Co Inc|page=328 |doi= 10.1142/12988 |isbn=978-981-126-110-7 |s2cid= 248575991 }}</ref>

== Underlying principles == Fractals generally have fractional dimensions, which serve as a measure of complexity that indicates the degree to which the objects fill the available space.<ref name="Mandelbrot19832" /><ref name=":0">{{Cite journal|last=Mandelbrot|first=B.|s2cid=15662830|date=1967-05-05|title=How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension|journal=Science|volume=156|issue=3775|pages=636–638|doi=10.1126/science.156.3775.636|pmid=17837158|issn=0036-8075|bibcode=1967Sci...156..636M|url=http://ena.lp.edu.ua:8080/handle/ntb/52473|access-date=2020-12-21|archive-date=2021-10-19|archive-url=https://web.archive.org/web/20211019193011/http://ena.lp.edu.ua:8080/handle/ntb/52473}}</ref> The fractal dimension measures the change in "size" of a fractal set with the changing observational scale, and is not limited by integer values.<ref name=":2" /> This is possible given that a smaller section of the fractal resembles the entirety, showing the same statistical properties at different scales.<ref name="Mandelbrot19832" /> 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).<ref name=":2" /> Whether the view of the fractal is expanding or contracting, the structure remains the same and appears equivalently complex.<ref name="Mandelbrot19832" /><ref name=":0" /> 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.<ref name=":7">{{Cite journal|last1=Goldberger|first1=Ary L|last2=Peng|first2=C.-K|last3=Lipsitz|first3=Lewis A|date=January 2002|title=What is physiologic complexity and how does it change with aging and disease?|journal=Neurobiology of Aging|volume=23|issue=1|pages=23–26|doi=10.1016/S0197-4580(01)00266-4|pmid=11755014|s2cid=17022186}}</ref>

''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.<ref name=":10" /><ref name="time series2" /><ref name="Mandelbrot19832" /> A common feature of all types of fractal analysis is the need for benchmark patterns against which to assess outputs.<ref name="benchmark2">{{Cite web|title=Digital Images in FracLac|publisher=ImageJ|access-date=2012-02-08|url=http://rsbweb.nih.gov/ij/plugins/fraclac/FLHelp/Images.htm|archive-url=https://web.archive.org/web/20111020190336/http://rsbweb.nih.gov/ij/plugins/fraclac/FLHelp/Images.htm|archive-date=2011-10-20}}</ref> 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.<ref name=":8">{{Cite journal|last1=MacIntosh|first1=Andrew J. J.|last2=Pelletier|first2=Laure|last3=Chiaradia|first3=Andre|last4=Kato|first4=Akiko|last5=Ropert-Coudert|first5=Yan|date=December 2013|title=Temporal fractals in seabird foraging behaviour: diving through the scales of time|journal=Scientific Reports|volume=3|issue=1|page=1884|doi=10.1038/srep01884|issn=2045-2322|pmc=3662970|pmid=23703258|bibcode=2013NatSR...3.1884M}}</ref>

== 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.<ref name=":3">{{Citation |last=Frontier |first=Serge |title=Developments in Numerical Ecology |chapter=Applications of Fractal Theory to Ecology |date=1987 |pages=335–378 |publisher=Springer Berlin Heidelberg |isbn=978-3-642-70882-4 |doi=10.1007/978-3-642-70880-0_9}}</ref><ref>{{Cite journal|last1=Scheuring|first1=István|last2=Riedi|first2=Rudolf H.|date=August 1994|title=Application of multifractals to the analysis of vegetation pattern|journal=Journal of Vegetation Science|volume=5|issue=4|pages=489–496|doi=10.2307/3235975|jstor=3235975|bibcode=1994JVegS...5..489S }}</ref><ref>{{Cite journal|last1=Seuront|first1=Laurent|last2=Lagadeuc|first2=Yvan|date=1998|title=Spatio-temporal structure of tidally mixed coastal waters: variability and heterogeneity|journal=Journal of Plankton Research|volume=20|issue=7|pages=1387–1401|doi=10.1093/plankt/20.7.1387|issn=0142-7873|doi-access=free}}</ref> 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.<ref name=":4">{{Cite journal|last1=Rutherford|first1=Kenneth M.D.|last2=Haskell|first2=Marie J.|last3=Glasbey|first3=Chris|last4=Jones|first4=R.Bryan|last5=Lawrence|first5=Alistair B.|date=September 2003|title=Detrended fluctuation analysis of behavioural responses to mild acute stressors in domestic hens|journal=Applied Animal Behaviour Science|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 ecosystems and to reveal some of the underlying structural mechanisms of nature.<ref name=":0" /><ref>{{Cite journal|last1=Bradbury|first1=Rh|last2=Reichelt|first2=Re|date=1983|title=Fractal Dimension of a Coral Reef at Ecological Scales|journal=Marine Ecology Progress Series|volume=10|pages=169–171|doi=10.3354/meps010169|issn=0171-8630|bibcode=1983MEPS...10..169B|doi-access=free}}</ref><ref>{{Cite journal|last1=Hastings|first1=Harold M.|last2=Pekelney|first2=Richard|last3=Monticciolo|first3=Richard|last4=Vun Kannon|first4=David|last5=Del Monte|first5=Diane|date=January 1982|title=Time scales, persistence and patchiness|journal=Biosystems|volume=15|issue=4|pages=281–289|doi=10.1016/0303-2647(82)90043-0|pmid=7165795|bibcode=1982BiSys..15..281H |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.|s2cid=3140271|date=1997-04-04|title=A General Model for the Origin of Allometric Scaling Laws in Biology|journal=Science|volume=276|issue=5309|pages=122–126|doi=10.1126/science.276.5309.122|pmid=9082983}}</ref><ref>{{Cite journal|last1=West|first1=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|journal=Proceedings of the National Academy of Sciences|volume=106|issue=17|pages=7040–7045|doi=10.1073/pnas.0812294106|issn=0027-8424|pmc=2678466|pmid=19363160|bibcode=2009PNAS..106.7040W|doi-access=free}}</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|last1=Rieu|first1=Michel|author2-link=Garrison Sposito|last2=Sposito|first2=Garrison|date=1991|title=Fractal Fragmentation, Soil Porosity, and Soil Water Properties: II. Applications|journal=Soil Science Society of America Journal|volume=55|issue=5|page=1239|doi=10.2136/sssaj1991.03615995005500050007x|issn=0361-5995|bibcode=1991SSASJ..55.1239R}}</ref><ref>{{Cite journal|last1=Morse|first1=D. R.|last2=Lawton|first2=J. H.|last3=Dodson|first3=M. M.|last4=Williamson|first4=M. H.|date=April 1985|title=Fractal dimension of vegetation and the distribution of arthropod body lengths|journal=Nature|volume=314|issue=6013|pages=731–733|doi=10.1038/314731a0|issn=0028-0836|bibcode=1985Natur.314..731M|s2cid=4362382}}</ref><ref>{{Cite journal|last1=Li|first1=Xiaoyan|last2=Passow|first2=Uta|last3=Logan|first3=Bruce E|date=January 1998|title=Fractal dimensions of small (15–200 μm) particles in Eastern Pacific coastal waters|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|last1=Lovejoy|first1=S.|last2=Schertzer|first2=D.|date=May 2006|title=Multifractals, cloud radiances and rain|journal=Journal of Hydrology|volume=322|issue=1–4|pages=59–88|doi=10.1016/j.jhydrol.2005.02.042|bibcode=2006JHyd..322...59L}}</ref> Despite its extensive use, it still receives some criticism.<ref>{{Cite journal|last1=Halley|first1=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.|s2cid=6059069|date=2004-02-24|title=Uses and abuses of fractal methodology in ecology|journal=Ecology Letters|volume=7|issue=3|pages=254–271|doi=10.1111/j.1461-0248.2004.00568.x|bibcode=2004EcolL...7..254H |issn=1461-023X}}</ref><ref>{{Cite journal|last1=Bryce|first1=R. M.|last2=Sprague|first2=K. B.|date=December 2012|title=Revisiting detrended fluctuation analysis|journal=Scientific Reports|volume=2|issue=1|page=315|doi=10.1038/srep00315|pmid=22419991|pmc=3303145|issn=2045-2322|bibcode=2012NatSR...2..315B}}</ref>

=== Architecture, urban design and landscape design === In his publication ''The Fractal Geometry of Nature'',<ref>{{cite book |last1=Mandelbrot |first1=Benoit |title=The Fractal Geometry of Nature |date=1982 |publisher=Freeman |location=San Francisco|bibcode=1982fgn..book.....M }}</ref> Benoit Mandelbrot suggested fractal theory could be applied to architecture. In this context, Mandelbrot was talking about the self-similar feature of fractal objects, rather than fractal analysis. In 1996, Carl Bovill applied the box counting method of fractal analysis to Architecture.<ref>{{cite book |last1=Bovill |first1=Carl |title=Fractal Geometry in Architecture and Design |date=1996 |publisher=Birkhauser |location=Boston}}</ref> Bovill's work, using a manual version of box counting, has since been refined by others<ref>{{cite book |last1=Vaughan |first1=Josephine |last2=Ostwald |first2=Michael J. |title=Proceedings of the 15th Conference on Computer Aided Architectural Design Research in Asia (CAADRIA) |chapter=Refining a computational fractal method of analysis: Testing Bovill's architectural data |date=2010 |pages=29–38 |doi=10.52842/conf.caadria.2010.029|hdl=1959.13/927195 |isbn=978-988-19026-1-0 |hdl-access=free }}</ref> and computational approaches have been developed.<ref>{{cite book |last1=Lorenz |first1=Wolfgang |title=Proceedings of the 27th International Conference on Education and Research in Computer Aided Architectural Design in Europe (ECAADe) |chapter=Fractal Geometry of Architecture: Implementation of the Box-Counting Method in a CAD-Software |date=2009 |pages=697–704 |doi=10.52842/conf.ecaade.2009.697|isbn=978-0-9541183-8-9 }}</ref>

Fractal analysis is one of the few quantitative analysis methods available to architects and designers to understand the visual complexity of buildings, urban areas and landscapes. Typical uses of fractal analysis of the built environment have been to understand the visual complexity of cities and skylines,<ref>{{cite journal |last1=Rodin |first1=Vladimir |last2=Rodin |first2=Elena |title=The fractal dimension of Tokyo's streets |journal=Fractals |date=2000 |volume=8 |issue=4 |pages=413–418 |doi=10.1142/S0218348X00000457}}</ref> the fractal dimensions of works of different architects <ref>{{cite book |last1=Ostwald |first1=Michael J. |last2=Vaughan |first2=Josephine |last3=Tucker |first3=Chris |title=Characteristic visual complexity: Fractal dimensions in the architecture of frank lloyd wright and le corbusier. In:Architecture and Mathematics from Antiquity to the Future: Volume II: The 1500s to the Future |date=2015 |publisher=Springer International Publishing |isbn=978-3-319-00143-2 |pages=339–354 |url=https://www.researchgate.net/publication/271197753}}</ref> and the landscape.<ref>{{cite journal |last1=Patuano |first1=A |last2=Tara |first2=A |title=Fractal geometry for landscape architecture: Review of methodologies and interpretations |journal=Journal of Digital Landscape Architecture |date=2020 |volume=5 |pages=72–80 |doi=10.14627/537690008 |isbn=<!-- --> }}</ref>

Combining the fractal analysis of ecology (see above) with fractal analysis of architecture, fractal dimensions have been used to explore the possible relationship between nature and architecture.<ref>{{cite journal |last1=Vaughan |first1=Josephine |last2=Ostwald |first2=Michael J. |title=Measuring the geometry of nature and architecture: comparing the visual properties of Frank Lloyd Wright's Fallingwater and its natural setting |journal=Open House International |date=2022 |volume=47 |issue=1 |pages=51–67 |doi=10.1108/OHI-01-2021-0011|bibcode=2022OHIng..47...51V }}</ref><ref>{{cite conference | last1 = Bourchtein | first1 = Andrei | last2 = Bourchtein | first2 = Ludmila | author2-link = Ludmila Bourchtein | last3 = Naoumova | first3 = Natalia | contribution = On fractal complexity of built and natural landscapes | doi = 10.1007/978-3-319-09129-7_33 | isbn = 978-3-319-09128-0 | pages = 437–452 | publisher = Springer | series = Lecture Notes in Computer Science | title = Computational Science and Its Applications – ICCSA 2014 – 14th International Conference, Guimarães, Portugal, June 30 – July 3, 2014, Proceedings, Part II | volume = 8580 | year = 2014}}</ref> Promising results suggest further research is needed in this area.

=== Animal behaviour === Patterns in animal behaviour exhibit fractal properties on spatial and temporal scales.<ref name=":8" /> 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=":2" /> Various animal movement signatures in their respective environments have been found to demonstrate spatially non-linear fractal patterns.<ref>{{Cite journal|last1=Catalan|first1=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|journal=Proceedings of the National Academy of Sciences|volume=100|issue=22|pages=12771–12775|doi=10.1073/pnas.2137243100|issn=0027-8424|pmc=240693|pmid=14566048|bibcode=2003PNAS..10012771B|doi-access=free}}</ref><ref>{{Cite journal|last1=Garcia|first1=F.|last2=Carrère|first2=P.|last3=Soussana|first3=J.F.|last4=Baumont|first4=R.|date=September 2005|title=Characterisation by fractal analysis of foraging paths of ewes grazing heterogeneous swards|journal=Applied Animal Behaviour Science|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, which has proven to be a more accurate description of animal movement for some species.<ref>{{Cite journal|last1=Humphries|first1=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|journal=Proceedings of the National Academy of Sciences|volume=109|issue=19|pages=7169–7174|doi=10.1073/pnas.1121201109|issn=0027-8424|pmc=3358854|pmid=22529349|bibcode=2012PNAS..109.7169H|doi-access=free}}</ref><ref>{{Cite journal|last1=Raposo|first1=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|journal=Journal of Physics A: Mathematical and Theoretical|volume=42|issue=43|article-number=434003|doi=10.1088/1751-8113/42/43/434003|issn=1751-8113|bibcode=2009JPhA...42Q4003R|s2cid=13887492 }}</ref><ref>{{Cite journal|last1=Viswanathan|first1=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=June 2001|title=Lévy flights search patterns of biological organisms|journal=Physica A: Statistical Mechanics and Its Applications|volume=295|issue=1–2|pages=85–88|doi=10.1016/S0378-4371(01)00057-7|bibcode=2001PhyA..295...85V}}</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=":7" /><ref name=":5">{{Cite journal|last=MacIntosh|first=Andrew James Jonathan|date=2014|title=The Fractal Primate|journal=Primate Research|volume=30|issue=1|pages=95–119|doi=10.2354/psj.30.011|issn=1880-2117|doi-access=free}}</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|last1=Burgunder|first1=Jade|last2=Petrželková|first2=Klára J.|last3=Modrý|first3=David|last4=Kato|first4=Akiko|last5=MacIntosh|first5=Andrew J.J.|date=August 2018|title=Fractal measures in activity patterns: Do gastrointestinal parasites affect the complexity of sheep behaviour?|journal=Applied Animal Behaviour Science|volume=205|pages=44–53|doi=10.1016/j.applanim.2018.05.014|s2cid=53475196 }}</ref><ref name=":9">{{Cite journal|last1=MacIntosh|first1=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|journal=Journal of the Royal Society Interface|volume=8|issue=63|pages=1497–1509|doi=10.1098/rsif.2011.0049|issn=1742-5689|pmc=3163426|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|last1=Cribb|first1=Nardi|last2=Seuront|first2=Laurent|date=September 2016|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|journal=Journal of Experimental Marine Biology and Ecology|volume=482|pages=118–127|doi=10.1016/j.jembe.2016.03.020|bibcode=2016JEMBE.482..118C |issn=0022-0981}}</ref> and parasitic infection in Japanese macaques<ref name=":9" /> 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|last1=Bradbury|first1=J. W.|last2=Vehrencamp|first2=S. L.|date=2014-05-01|title=Complexity and behavioral ecology|journal=Behavioral Ecology|volume=25|issue=3|pages=435–442|doi=10.1093/beheco/aru014|issn=1045-2249|doi-access=free}}</ref> 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.<ref name=":4" /><ref>{{Cite journal|last1=Alados|first1=C.L.|last2=Escos|first2=J.M.|last3=Emlen|first3=J.M.|s2cid=53184132|date=February 1996|title=Fractal structure of sequential behaviour patterns: an indicator of stress|journal=Animal Behaviour|volume=51|issue=2|pages=437–443|doi=10.1006/anbe.1996.0040|bibcode=1996AnBeh..51..437A }}</ref><ref>{{Cite journal|url=https://www.ingentaconnect.com/contentone/ufaw/aw/2004/00000013/a00101s1/art00014|title=Fractal analysis of animal behaviour as an indicator of animal welfare|last1=Rutherford|first1=K. M. D.|last2=Haskell|first2=M. J.|date=February 2004|journal=Animal Welfare|access-date=2019-03-27|last3=Glasbey|first3=C.|last4=Jones|first4=R. B.|last5=Lawrence|first5=A. B.|volume=13 |issue=1 |pages=99–103 |doi=10.1017/S0962728600014433 |s2cid=146350786 }}</ref>

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.<ref name=":5" /> 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:<ref>{{cite web|url=http://library.thinkquest.org/26242/full/ap/ap.html|title=Applications|access-date=2007-10-21|archive-url=https://web.archive.org/web/20071012223212/http://library.thinkquest.org/26242/full/ap/ap.html|archive-date=2007-10-12}}</ref> {{col-start}}{{col-break}} * Heart rate analysis<ref name="heart">{{Cite journal | last1 = Tan | first1 = Can Ozan | last2 = Cohen | first2 = Michael A. | last3 = Eckberg | first3 = Dwain L. | last4 = Taylor | first4 = J. Andrew | title = Fractal properties of human heart period variability: Physiological and methodological implications | doi = 10.1113/jphysiol.2009.169219 | journal = The Journal of Physiology | volume = 587 | issue = 15 | pages = 3929–3941 | year = 2009 | pmid = 19528254| pmc = 2746620}}</ref><ref>{{cite journal |last1=Hernández-Torres |first1=E. A. |last2=Zapata-Rodríguez |first2=U. G. |last3=Ruiz-Pinales |first3=J. M. |last4=Aguilar-Rivera |first4=J. A. |title=Fractal and Graphical Approach in Arrhythmia Pre-Diagnosis ECG Based |journal=IEEE Latin America Transactions |volume=19 |issue=12 |pages=2091–2100 |year=2021 |doi=10.1109/TLA.2021.9486438 |url=https://www.researchgate.net/publication/356890807_Fractal_and_Grapical_Approach_in_Arrhytmia_Pre-Diagnosis_ECG_Based |access-date=2025-11-04}}</ref> * Human gait, balance, and activity<ref>{{Cite journal|last1=Costa|first1=Isis da Silva|last2=Gamundí|first2=Antoni|last3=Miranda|first3=José G. Vivas|last4=França|first4=Lucas G. Souza|last5=Santana|first5=De|last6=Novaes|first6=Charles|last7=Montoya|first7=Pedro|date=2017|title=Altered Functional Performance in Patients with Fibromyalgia|journal=Frontiers in Human Neuroscience|language=en|volume=11|page=14|doi=10.3389/fnhum.2017.00014|pmid=28184193|pmc=5266716|issn=1662-5161|doi-access=free}}</ref><ref> {{cite journal |last1=França |first1=L. G. S. |last2=Montoya |first2=Pedro |last3=Miranda |first3=J. G. V.|date=2017 |title=On multifractals: a non-linear study of actigraphy data |journal=Physica A: Statistical Mechanics and Its Applications |volume=514 |pages=612–619 |arxiv=1702.03912|doi=10.1016/j.physa.2018.09.122 |s2cid=18259316 }}</ref> * Human anatomy<ref>{{Cite journal | last1 = Kędzia | first1 = A. | last2 = Derkowski | first2 = W. | title = Modern Methods of Neuroanatomical and Neurophysiological Research | doi = 10.1016/j.mex.2024.102881 | journal = MethodsX | volume = 13| issue = December | year = 2024 | article-number = 102881 | pmid = 39176151 | pmc = 11340600 }}</ref> * Diagnostic imaging<ref name="diagnostic imaging" /> * Cancer research<ref>{{Cite journal | last1 = Kam | first1 = Y. | last2 = Karperien | first2 = A. | last3 = Weidow | first3 = B. | last4 = Estrada | first4 = L. | last5 = Anderson | first5 = A. R. | last6 = Quaranta | first6 = V. | doi = 10.1186/1756-0500-2-130 | title = Nest expansion assay: A cancer systems biology approach to in vitro invasion measurements | journal = BMC Research Notes | volume = 2 | page = 130 | year = 2009 | pmid = 19594934| pmc =2716356 | doi-access = free }}</ref> * Fractal analysis of complex networks<ref>{{cite journal |last1=Xiao |first1=Xiongye |last2=Chen |first2=Hanlong |last3=Bogdan |first3=Paul |title=Deciphering the generating rules and functionalities of complex networks |journal=Scientific Reports |date=25 November 2021 |volume=11 |issue=1 |page=22964 |doi=10.1038/s41598-021-02203-4|pmid=34824290 |pmc=8616909 |bibcode=2021NatSR..1122964X }}</ref> * Classification of histopathology slides in medicine<ref name="medicine"> {{cite book |editor1-last = Losa |editor1-first= Gabriele A. |editor2-last= Nonnenmacher |editor2-first= Theo |title=Fractals in biology and medicine |url=https://books.google.com/books?id=t9l9GdAt95gC |access-date=1 February 2012 |year=2005 |publisher=Springer |isbn=978-3-7643-7172-2}}</ref> * Fractal landscape or Coastline complexity<ref name="Mandelbrot19832" /><ref name="coastline">{{Cite journal | last1 = Mandelbrot | first1 = B. | s2cid = 15662830 | title = How Long is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension | doi = 10.1126/science.156.3775.636 | journal = Science | volume = 156 | issue = 3775 | pages = 636–638 | year = 1967 | pmid = 17837158 | bibcode = 1967Sci...156..636M | url = http://ena.lp.edu.ua:8080/handle/ntb/52473 | access-date = 2020-12-21 | archive-date = 2021-10-19 | archive-url = https://web.archive.org/web/20211019193011/http://ena.lp.edu.ua:8080/handle/ntb/52473 }}</ref> * Electrical engineering<ref name="electrical engineering"> {{Cite journal | last1 = Li | first1 = H. | title = Fractal analysis of side channels for breakdown structures in XLPE cable insulation | doi = 10.1007/s10854-012-0988-y | journal = J Mater Sci: Mater Electron | volume = 24 | issue = 5 | pages = 1640–1643 | year = 2013 | s2cid = 136564926 }}</ref> * Enzyme/enzymology (Michaelis-Menten kinetics)<ref name="ReuveniGranek2008">{{cite journal|last1=Reuveni|first1=Shlomi|last2=Granek|first2=Rony|last3=Klafter|first3=Joseph|s2cid=16203048|title=Proteins: Coexistence of Stability and Flexibility|journal=Physical Review Letters|volume=100|issue=20|article-number=208101|year=2008|issn=0031-9007|doi=10.1103/PhysRevLett.100.208101|pmid=18518581|bibcode=2008PhRvL.100t8101R}}</ref> * Generation of new music * Generation of various art forms *Search and rescue<ref name="search and rescue">{{cite book| contribution=An Algorithmic Approach to Generate After-disaster Test Fields for Search and Rescue Agents| author=Panteha Saeedi, and Soren A. Sorensen | title=Proceedings of the World Congress on Engineering 2009 | year=2009 |pages=93–98 | publisher=Newswood Limited | isbn=978-988-17-0125-1|url=http://www.iaeng.org/publication/WCE2009/WCE2009_pp93-98.pdf}}</ref> * Signal and image compression {{Col-break}}{{col-break}} * Urban growth<ref name="Chen2011">{{Cite journal | last1 = Chen | first1 = Yanguang <!-- editor is irrelevant here| editor1-last = Hernández Montoya | editor1-first = Alejandro Raúl -->| title = Modeling Fractal Structure of City-Size Distributions Using Correlation Functions | doi = 10.1371/journal.pone.0024791 | journal = PLOS ONE | volume = 6 | issue = 9 | article-number = e24791 | year = 2011 | pmid = 21949753 | pmc = 3176775|arxiv = 1104.4682 |bibcode = 2011PLoSO...624791C | doi-access = free}}</ref> *Neuroscience<ref name="neuroscience">{{Cite journal | last1 = Karperien | first1 = Audrey L. | last2 = Jelinek | first2 = Herbert F. | last3 = Buchan | first3 = Alastair M. | doi = 10.1142/S0218348X08003880 | title = Box-Counting Analysis of Microglia Form in Schizophrenia, Alzheimer's Disease and Affective Disorder | journal = Fractals | volume = 16 | issue = 2 | pages = 103–107 | year = 2008 }}</ref><ref> {{Cite journal|last1=França|first1=Lucas G. Souza|last2=Miranda|first2=José G. Vivas|last3=Leite|first3=Marco|last4=Sharma|first4=Niraj K.|last5=Walker|first5=Matthew C.|last6=Lemieux|first6=Louis|last7=Wang|first7=Yujiang|date=2018|title=Fractal and Multifractal Properties of Electrographic Recordings of Human Brain Activity: Toward Its Use as a Signal Feature for Machine Learning in Clinical Applications|journal=Frontiers in Physiology|language=en|volume=9|article-number=1767|doi=10.3389/fphys.2018.01767|pmid=30618789|pmc=6295567|issn=1664-042X|bibcode=2018arXiv180603889F|arxiv=1806.03889|doi-access=free}} </ref><ref name="cerebellum">{{Cite journal | last1 = Liu | first1 = Jing Z. | last2 = Zhang | first2 = Lu D. | last3 = Yue | first3 = Guang H. | doi = 10.1016/S0006-3495(03)74817-6 | title = Fractal Dimension in Human Cerebellum Measured by Magnetic Resonance Imaging | journal = Biophysical Journal | volume = 85 | issue = 6 | pages = 4041–4046 | year = 2003 | pmid = 14645092 | pmc = 1303704|bibcode = 2003BpJ....85.4041L }}</ref><ref>{{cite journal | last1 = Nikolić | first1 = D. | last2 = Moca | first2 = V.V. | last3 = Singer | first3 = W. | last4 = Mureşan | first4 = R.C. | year = 2008 | title = Properties of multivariate data investigated by fractal dimensionality | journal = Journal of Neuroscience Methods | volume = 172 | issue = 1| pages = 27–33 | doi=10.1016/j.jneumeth.2008.04.007| pmid = 18495248 | s2cid = 12268410 }}</ref> *Diagnostic imaging<ref name="diagnostic imaging">{{Cite journal | last1 = Karperien | first1 = Audrey | last2 = Jelinek | first2 = Herbert F. | last3 = Leandro | first3 = Jorge de Jesus Gomes | last4 = Soares | first4 = João V. B. | last5 = Cesar Jr | first5 = Roberto M. | last6 = Luckie | first6 = Alan | title = Automated detection of proliferative retinopathy in clinical practice | journal = Clinical Ophthalmology | volume = 2 | issue = 1 | pages = 109–122 | year = 2008 | pmid = 19668394 | pmc = 2698675 | doi = 10.2147/OPTH.S1579 | doi-access = free }}</ref> *Pathology<ref name="pathology">{{Cite journal | last1 = Smith | first1 = Robert F. | last2 = Mohr | first2 = David N. | last3 = Torres | first3 = Vicente E. | last4 = Offord | first4 = Kenneth P. | last5 = Melton III | first5 = L. Joseph | title = Renal insufficiency in community patients with mild asymptomatic microhematuria | journal = Mayo Clinic Proceedings | volume = 64 | issue = 4 | pages = 409–414 | year = 1989 | pmid = 2716356 | doi=10.1016/s0025-6196(12)65730-9 }}</ref><ref>{{cite journal|author=Al-Kadi O.S, Watson D.|title=Texture Analysis of Aggressive and non-Aggressive Lung Tumor CE CT Images|url=http://sro.sussex.ac.uk/1919/1/tbme.pdf|journal=IEEE Transactions on Biomedical Engineering|volume=55|issue=7|pages=1822–1830|year=2008|doi=10.1109/tbme.2008.919735|pmid=18595800|bibcode=2008ITBE...55.1822A |s2cid=14784161|access-date=2014-04-10|archive-url=https://web.archive.org/web/20140413124458/http://sro.sussex.ac.uk/1919/1/tbme.pdf|archive-date=2014-04-13}}</ref><ref>{{Cite journal | last1 = Landini | first1 = Gabriel | title = Fractals in microscopy | doi = 10.1111/j.1365-2818.2010.03454.x | journal = Journal of Microscopy | volume = 241 | issue = 1 | pages = 1–8 | year = 2011 | pmid = 21118245| s2cid = 40311727 }}</ref> *Geology<ref>{{Cite journal | last1 = Cheng | first1 = Qiuming | author-link = Qiuming Cheng| title = Multifractal Modeling and Lacunarity Analysis | journal = Mathematical Geology | volume = 29 | issue = 7 | pages = 919–932 | doi = 10.1023/A:1022355723781 | year = 1997 | bibcode = 1997MatG...29..919C | s2cid = 118918429 }}</ref> *Geography<ref name="Chen2011" /> *Archaeology<ref name = "archaeology">{{Cite journal | last1 = Burkle-Elizondo | first1 = Gerardo | last2 = Valdéz-Cepeda | first2 = Ricardo David | title = Fractal analysis of Mesoamerican pyramids | journal = Nonlinear Dynamics, Psychology, and Life Sciences | volume = 10 | issue = 1 | pages = 105–122 | year = 2006 | pmid = 16393505 }}</ref><ref>{{Cite journal | last1 = Brown | first1 = Clifford T. | last2 = Witschey | first2 = Walter R. T. | last3 = Liebovitch | first3 = Larry S. | title = The Broken Past: Fractals in Archaeology | doi = 10.1007/s10816-005-2396-6 | journal = Journal of Archaeological Method and Theory | volume = 12 | pages = 37–78 | year = 2005 | s2cid = 7481018 }}</ref> * Seismology<ref name="seismology">{{Cite journal | last1 = Vannucchi | first1 = Paola |author1-link=Paola Vannucchi| last2 = Leoni | first2 = Lorenzo | doi = 10.1016/j.epsl.2007.07.056 | title = Structural characterization of the Costa Rica décollement: Evidence for seismically-induced fluid pulsing | journal = Earth and Planetary Science Letters | volume = 262 | issue = 3–4 | pages = 413–428 | year = 2007 |bibcode = 2007E&PSL.262..413V | hdl = 2158/257208 | hdl-access = free }}</ref><ref> {{cite book |pages=128–140 |title=Critical phenomena in natural sciences: chaos, fractals, self-organization, and disorder: concepts and tools |author=Didier Sornette |year=2004 |publisher=Springer |isbn=978-3-540-40754-6}}<!--|access-date=2011-02-05--></ref> * Soil studies<ref name="soil">{{Cite journal | last1 = Hu | first1 = Shougeng | last2 = Cheng | first2 = Qiuming | last3 = Wang | first3 = Le | last4 = Xie | first4 = Shuyun | title = Multifractal characterization of urban residential land price in space and time | doi = 10.1016/j.apgeog.2011.10.016 | journal = Applied Geography | volume = 34 | pages = 161–170 | year = 2012 | bibcode = 2012AppGe..34..161H }}</ref> {{col-break}} * Computer and video game design, especially computer graphics for organic environments and as part of procedural generation * Fractography and fracture mechanics * Fractal antennas — Small size antennas using fractal shapes * Small angle scattering theory of fractally rough systems * Generation of patterns for camouflage, such as MARPAT * Digital sundial * Technical analysis of price series (see Elliott wave principle) {{col-end}} *Fractal calculus<ref name="Ali"/>

==See also== *Multifractal *Rescaled range *Analysis on fractals

==References== <references />

==Further reading== *[https://web.archive.org/web/20080920054002/http://rsb.info.nih.gov/ij/plugins/fraclac/FLHelp/Fractals.htm Fractals and Fractal Analysis] *[http://www.fch.vutbr.cz/lectures/imagesci/download_ejournal/01_O.Zmeskal.pdf Fractal analysis] *[http://www.trusoft.netmegs.com/ Benoit – Fractal Analysis Software] {{Webarchive|url=https://web.archive.org/web/20080517082415/http://www.trusoft.netmegs.com/ |date=2008-05-17 }} *[http://www.physionet.org/tutorials/fmnc/index.shtml Fractal Analysis Methods for Human Heartbeat and Gait Dynamics] {{Webarchive|url=https://web.archive.org/web/20160506052257/http://www.physionet.org/tutorials/fmnc/index.shtml |date=2016-05-06 }} {{Fractals}}

Category:Chaos theory Category:Dynamical systems Category:Dimension theory Category:Fractals