Editing Fractal dimension (section)

== Estimating from real-world data ==
Many real-world phenomena exhibit limited or statistical fractal properties and fractal dimensions that have been estimated from [[Sample (statistics)|sampled]] data using computer-based [[fractal analysis]] techniques.  
Practically, measurements of fractal dimension are affected by various methodological issues and are sensitive to numerical or experimental noise and limitations in the amount of data. Nonetheless, the field is rapidly growing as estimated fractal dimensions for statistically self-similar phenomena may have many practical applications in various fields, including astronomy,<ref>{{Cite journal |last1=Caicedo-Ortiz |first1=H. E. |last2=Santiago-Cortes |first2=E. |last3=López-Bonilla |first3=J. |last4=Castañeda |first4=H. O. |year=2015 |title=Fractal dimension and turbulence in Giant HII Regions |journal=Journal of Physics: Conference Series |volume=582 |issue=1 |pages=1–5 |doi=10.1088/1742-6596/582/1/012049 |arxiv=1501.04911 |bibcode=2015JPhCS.582a2049C |doi-access=free}}</ref> acoustics,<ref name="pubmed.ncbi.nlm.nih.gov">{{cite journal |title=A Mathematical Approach to Correlating Objective Spectro-Temporal Features of Non-linguistic Sounds With Their Subjective Perceptions in Humans |year=2019 |pmid=31417350 |last1=Burns |first1=T. |last2=Rajan |first2=R. |journal=Frontiers in Neuroscience |volume=13 |page=794 |doi=10.3389/fnins.2019.00794 |pmc=6685481 |doi-access=free}}</ref><ref>{{Cite journal |last1=Maragos |first1=P. |last2=Potamianos |first2=A. |year=1999 |title=Fractal dimensions of speech sounds: Computation and application to automatic speech recognition |journal=The Journal of the Acoustical Society of America |volume=105 |issue=3 |pages=1925–1932 |bibcode=1999ASAJ..105.1925M |doi=10.1121/1.426738 |pmid=10089613}}</ref> architecture,<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=978-331900143-2 |pages=339–354 |url=https://www.researchgate.net/publication/271197753}}</ref><ref>{{Cite book |last=Bovill |first=Carl |title=Fractal geometry in architecture and design |date=1996 |publisher=Birkhäuser |isbn=978-0-8176-3795-8 |series=Design science collection |location=Boston}}</ref><ref>{{cite journal |last1=Vaughan |first1=Josephine |last2=Ostwald |first2=Michael j. |date=2014 |title=Measuring the significance of façade transparency in Australian regionalist architecture: A computational analysis of 10 designs by Glenn Murcutt |journal=Architectural Science Review |volume=57 |issue=4 |pages=249–259 |doi=10.1080/00038628.2014.940273 |hdl=1959.13/1293729|hdl-access=free }}</ref>   geology and earth sciences,<ref>{{Cite journal |last=Avşar |first=Elif |date=2020-09-01 |title=Contribution of fractal dimension theory into the uniaxial compressive strength prediction of a volcanic welded bimrock |url=https://doi.org/10.1007/s10064-020-01778-y |journal=Bulletin of Engineering Geology and the Environment |language=en |volume=79 |issue=7 |pages=3605–3619 |doi=10.1007/s10064-020-01778-y |bibcode=2020BuEGE..79.3605A |s2cid=214648440 |issn=1435-9537}}</ref> diagnostic imaging,<ref>{{Cite journal
| last1 = Landini | first1 = G.
| last2 = Murray | first2 = P. I.
| last3 = Misson | first3 = G. P.
| title = Local connected fractal dimensions and lacunarity analyses of 60 degrees fluorescein angiograms
| journal = Investigative Ophthalmology & Visual Science
| volume = 36
| issue = 13
| pages = 2749–2755
| year = 1995
| pmid = 749909
}}</ref><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><ref>{{Cite journal |last1=Santiago-Cortés |first1=E.| last2=Martínez Ledezma |first2=J. L. |year=2016 |title=Fractal dimension in human retinas |url=https://jci.uniautonoma.edu.co/2016/2016-8.pdf |journal=Journal de Ciencia e Ingeniería |volume=8 |pages=59–65 |issn=2145-2628 |eissn=2539-066X}}</ref>
ecology,<ref>{{Cite journal |last1=Wildhaber |first1=Mark L. |last2=Lamberson |first2=Peter J. |last3=Galat |first3=David L. |date=2003-05-01 |title=A Comparison of Measures of Riverbed Form for Evaluating Distributions of Benthic Fishes |journal=North American Journal of Fisheries Management |volume=23 |issue=2 |pages=543–557 |doi=10.1577/1548-8675(2003)023<0543:acomor>2.0.co;2 |bibcode=2003NAJFM..23..543W |issn=1548-8675}}</ref> electrochemical processes,<ref>{{Cite journal |last1=Eftekhari |first1=A. |year=2004 |title=Fractal Dimension of Electrochemical Reactions |journal=Journal of the Electrochemical Society |volume=151 |issue=9 |pages=E291–E296 |doi=10.1149/1.1773583 |bibcode=2004JElS..151E.291E}}</ref>
image analysis,<ref>{{cite journal |author=Al-Kadi O. S., Watson D. |year=2008 |title=Texture Analysis of Aggressive and non-Aggressive Lung Tumor CE CT Images |url=http://sro.sussex.ac.uk/1919/1/tbme.pdf |url-status=dead |journal=IEEE Transactions on Biomedical Engineering |volume=55 |issue=7 |pages=1822–1830 |doi=10.1109/tbme.2008.919735 |pmid=18595800 |s2cid=14784161 |archive-url=https://web.archive.org/web/20140413124458/http://sro.sussex.ac.uk/1919/1/tbme.pdf |archive-date=2014-04-13 |access-date=2014-04-10}}</ref><ref>{{cite journal |author=Pierre Soille and Jean-F. Rivest |year=1996 |title=On the Validity of Fractal Dimension Measurements in Image Analysis |url=http://mdigest.jrc.ec.europa.eu/soille/soille-rivest96.pdf |url-status=dead |journal=Journal of Visual Communication and Image Representation |volume=7 |issue=3 |pages=217–229 |doi=10.1006/jvci.1996.0020 |issn=1047-3203 |archive-url=https://web.archive.org/web/20110720161245/http://mdigest.jrc.ec.europa.eu/soille/soille-rivest96.pdf |archive-date=2011-07-20}}</ref><ref>{{Cite journal |last1=Tolle |first1=C. R. |last2=McJunkin |first2=T. R. |last3=Gorsich |first3=D. J. |year=2003 |title=Suboptimal minimum cluster volume cover-based method for measuring fractal dimension |url=https://zenodo.org/record/1282294 |journal=IEEE Transactions on Pattern Analysis and Machine Intelligence |volume=25 |pages=32–41 |citeseerx=10.1.1.79.6978 |doi=10.1109/TPAMI.2003.1159944}}</ref><ref>{{Cite book |last1=Gorsich |first1=D. J. |last2=Tolle |first2=C. R. |last3=Karlsen |first3=R. E. |last4=Gerhart |first4=G. R. |editor-first1=Michael A. |editor-first2=Akram |editor-first3=Andrew F. |editor-last1=Unser |editor-last2=Aldroubi |editor-last3=Laine |year=1996|contribution=Wavelet and fractal analysis of ground-vehicle images |url=https://zenodo.org/record/1235586 |title=Wavelet Applications in Signal and Image Processing IV |series=Proceedings of the SPIE |volume=2825 |pages=109–119 |bibcode=1996SPIE.2825..109G |doi=10.1117/12.255224 |s2cid=121560110}}</ref> biology and medicine,<ref>{{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 = Smith | first1 = T. G. | last2 = Lange | first2 = G. D. | last3 = Marks | first3 = W. B. | doi = 10.1016/S0165-0270(96)00080-5 | title = Fractal methods and results in cellular morphology — dimensions, lacunarity and multifractals | journal = Journal of Neuroscience Methods | volume = 69 | issue = 2 | pages = 123–136 | year = 1996 | pmid = 8946315 | s2cid = 20175299 | url = https://zenodo.org/record/1259855 }}</ref><ref>{{Cite journal | last1 = Li | first1 = J. | last2 = Du | first2 = Q. | last3 = Sun | first3 = C. | doi = 10.1016/j.patcog.2009.03.001 | title = An improved box-counting method for image fractal dimension estimation | journal = Pattern Recognition | volume = 42 | issue = 11 | pages = 2460–9 | year = 2009 | bibcode = 2009PatRe..42.2460L }}</ref> neuroscience,<ref name="10.12688/f1000research.6590.1">{{cite journal |title=Combining complexity measures of EEG data: multiplying measures reveal previously hidden information |year=2015 |pmc=4648221 |last1=Burns |first1=T. |last2=Rajan |first2=R. |journal=F1000Research |volume=4 |page=137 |doi=10.12688/f1000research.6590.1 |pmid=26594331 |doi-access=free }}</ref><ref name="doi10.1007/s11682-008-9057-9" /> [[Fractal dimension on networks|network analysis]], physiology,<ref name="doi10.1364/boe.1.000268"/> physics,<ref>{{Cite journal | last1 = Dubuc | first1 = B. | last2 = Quiniou | first2 = J. | last3 = Roques-Carmes | first3 = C. | last4 = Tricot | first4 = C. | last5 = Zucker | first5 = S. | title = Evaluating the fractal dimension of profiles | doi = 10.1103/PhysRevA.39.1500 | journal = Physical Review A | volume = 39 | issue = 3 | pages = 1500–1512 | year = 1989 | pmid = 9901387 |bibcode = 1989PhRvA..39.1500D }}</ref><ref>{{Cite journal | last1 = Roberts | first1 = A. | last2 = Cronin | first2 = A. | doi = 10.1016/S0378-4371(96)00165-3 | title = Unbiased estimation of multi-fractal dimensions of finite data sets | journal = Physica A: Statistical Mechanics and Its Applications | volume = 233 | issue = 3–4 | pages = 867–878 | year = 1996 |bibcode = 1996PhyA..233..867R | arxiv = chao-dyn/9601019 | s2cid = 14388392 }}</ref> and [[Riemann hypothesis|Riemann zeta zeros]].<ref name="Shanker">{{Cite journal | last1 = Shanker | first1 = O. | title = Random matrices, generalized zeta functions and self-similarity of zero distributions | doi = 10.1088/0305-4470/39/45/008 | journal = Journal of Physics A: Mathematical and General | volume = 39 | issue = 45 | pages = 13983–13997 | year = 2006 |bibcode = 2006JPhA...3913983S }}</ref> Fractal dimension estimates have also been shown to correlate with [[Lempel–Ziv complexity]] in real-world data sets from psychoacoustics and neuroscience.<ref name="10.12688/f1000research.6590.1"/><ref name="pubmed.ncbi.nlm.nih.gov"/>

An alternative to a direct measurement is considering a mathematical model that resembles formation of a real-world fractal object. In this case, a validation can also be done by comparing other than fractal properties implied by the model, with measured data. In [[colloid]]al physics, systems composed of particles with various fractal dimensions arise. To describe these systems, it is convenient  to speak about a [[Probability distribution|distribution]] of fractal dimensions and, eventually, a time evolution of the latter: a process that is driven by a complex interplay between [[Particle aggregation|aggregation]] and [[Coalescence (chemistry)|coalescence]].<ref>{{Cite journal | doi = 10.1002/mats.201300140 | title = Population Balance Modeling of Aggregation and Coalescence in Colloidal Systems | journal = Macromolecular Theory and Simulations | volume = 23 | issue = 3 | pages = 170–181 | year = 2014 | last1 = Kryven | first1 = I. | last2 = Lazzari | first2 = S. | last3 = Storti | first3 = G. | url = http://dare.uva.nl/personal/pure/en/publications/population-balance-modeling-of-aggregation-and-coalescence-in-colloidal-systems(05340e78-b40a-4bd0-ac6a-7f044fff1617).html}}</ref>