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In statistics, a Tsallis distribution is a probability distribution derived from the maximization of the Tsallis entropy under appropriate constraints. There are several different families of Tsallis distributions, yet different sources may reference an individual family as "the Tsallis distribution". The q-Gaussian is a generalization of the Gaussian in the same way that Tsallis entropy is a generalization of standard Boltzmann–Gibbs entropy or Shannon entropy.^[1] Similarly, if the domain of the variable is constrained to be positive in the maximum entropy procedure, the q-exponential distribution is derived.

The Tsallis distributions have been applied to problems in the fields of statistical mechanics, geology, anatomy, astronomy, economics, finance, and machine learning. The distributions are often used for their heavy tails.

Note that Tsallis distributions are obtained as Box–Cox transformation^[2] over usual distributions, with deformation parameter $\lambda =1-q$ . This deformation transforms exponentials into q-exponentials.

Procedure

In a similar procedure to how the normal distribution can be derived using the standard Boltzmann–Gibbs entropy or Shannon entropy, the q-Gaussian can be derived from a maximization of the Tsallis entropy subject to the appropriate constraints.^[3]^[4]

Common Tsallis distributions

Notes

^ Tsallis, C. (2009) "Nonadditive entropy and nonextensive statistical mechanics-an overview after 20 years", Braz. J. Phys, 39, 337–356
^ Box, George E. P.; Cox, D. R. (1964). "An analysis of transformations". Journal of the Royal Statistical Society, Series B. 26 (2): 211–252. JSTOR 2984418. MR 0192611.
^ Umarov, Sabir; Tsallis, Constantino; Steinberg, Stanly (2008-12-01). "On a q-Central Limit Theorem Consistent with Nonextensive Statistical Mechanics". Milan Journal of Mathematics. 76 (1): 307–328. doi:10.1007/s00032-008-0087-y. ISSN 1424-9294. S2CID 55967725.
^ Prato, Domingo; Tsallis, Constantino (1999-08-01). "Nonextensive foundation of Lévy distributions". Physical Review E. 60 (2): 2398–2401. Bibcode:1999PhRvE..60.2398P. doi:10.1103/PhysRevE.60.2398. ISSN 1063-651X. PMID 11970038.

External links

Tsallis Statistics, Statistical Mechanics for Non-extensive Systems and Long-Range Interactions

[1] Tsallis, C. (2009) "Nonadditive entropy and nonextensive statistical mechanics-an overview after 20 years", Braz. J. Phys, 39, 337–356

[boxcox-2] Box, George E. P.; Cox, D. R. (1964). "An analysis of transformations". Journal of the Royal Statistical Society, Series B. 26 (2): 211–252. JSTOR 2984418. MR 0192611.

[3] Umarov, Sabir; Tsallis, Constantino; Steinberg, Stanly (2008-12-01). "On a q-Central Limit Theorem Consistent with Nonextensive Statistical Mechanics". Milan Journal of Mathematics. 76 (1): 307–328. doi:10.1007/s00032-008-0087-y. ISSN 1424-9294. S2CID 55967725.

[4] Prato, Domingo; Tsallis, Constantino (1999-08-01). "Nonextensive foundation of Lévy distributions". Physical Review E. 60 (2): 2398–2401. Bibcode:1999PhRvE..60.2398P. doi:10.1103/PhysRevE.60.2398. ISSN 1063-651X. PMID 11970038.

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+In [[statistics]], a '''Tsallis distribution''' is a [[probability distribution]] derived from the maximization of the [[Tsallis entropy]] under appropriate constraints.  There are several different families of Tsallis distributions, yet different sources may reference an individual family as "the Tsallis distribution". The [[q-Gaussian]] is a generalization of the Gaussian in the same way that [[Tsallis entropy]] is a generalization of standard [[Entropy (statistical thermodynamics)|Boltzmann–Gibbs entropy]] or [[Entropy (information theory)|Shannon entropy]].<ref>Tsallis, C. (2009) "Nonadditive entropy and nonextensive statistical mechanics-an overview after 20 years", ''Braz. J. Phys'', 39, 337–356</ref> Similarly, if the domain of the variable is constrained to be positive in the [[Maximum entropy probability distribution|maximum entropy]] procedure, the [[q-exponential distribution]] is derived.
-{{expand|date=May 2009}}
-{{Probability distribution|
-  name       = Tsallis distribution |
-  type       = density |
-  pdf_image  = |
-  cdf_image  = |
-  parameters = <math>q</math> degree of non-extensivity ([[real number|real]]) |
-  support    = <math>x \in\mathbb{R}^{+*}</math> |
-  pdf        = |
-  cdf        = |
-  mean       = |
-  median     = |
-  mode       = |
-  variance   = |
-  skewness   = |
-  kurtosis   = |
-  entropy    = |
-  mgf        = |
-  char       = |
-}}
+The Tsallis distributions have been applied to problems in the fields of [[statistical mechanics]], [[geology]], [[anatomy]], [[astronomy]], [[economics]], [[finance]], and [[machine learning]]. The distributions are often used for their [[heavy tails]].
-The '''Tsallis distribution''' is a probability distribution arising from the optimisation of the [[Tsallis entropy]].
+Note that Tsallis distributions are obtained as [[Box–Cox transformation]]<ref name=boxcox>{{cite journal |last1=Box |first1=George E. P. |authorlink=George E. P. Box |last2=Cox |first2=D. R. |authorlink2=David Cox (statistician) |title=An analysis of transformations |journal=Journal of the Royal Statistical Society, Series B |volume=26 |issue=2 |pages=211–252 |year=1964 |mr=192611 |jstor=2984418 }}</ref> over usual distributions, with deformation parameter <math>\lambda=1-q</math>. This deformation transforms exponentials into q-exponentials.
-== Applications ==
-=== Physics ===
+==Procedure==
+In a similar procedure to how the [[normal distribution]] can be derived using the standard Boltzmann–Gibbs entropy or Shannon entropy, the [[Q-Gaussian distribution|q-Gaussian]] can be derived from a maximization of the [[Tsallis entropy]] subject to the appropriate constraints.<ref>{{Cite journal |last1=Umarov |first1=Sabir |last2=Tsallis |first2=Constantino |last3=Steinberg |first3=Stanly |date=2008-12-01 |title=On a q-Central Limit Theorem Consistent with Nonextensive Statistical Mechanics |url=https://doi.org/10.1007/s00032-008-0087-y |journal=Milan Journal of Mathematics |language=en |volume=76 |issue=1 |pages=307–328 |doi=10.1007/s00032-008-0087-y |s2cid=55967725 |issn=1424-9294}}</ref><ref>{{Cite journal |last1=Prato |first1=Domingo |last2=Tsallis |first2=Constantino |date=1999-08-01 |title=Nonextensive foundation of Lévy distributions |url=https://link.aps.org/doi/10.1103/PhysRevE.60.2398 |journal=Physical Review E |language=en |volume=60 |issue=2 |pages=2398–2401 |doi=10.1103/PhysRevE.60.2398 |pmid=11970038 |bibcode=1999PhRvE..60.2398P |issn=1063-651X}}</ref>
+==Common Tsallis distributions==
-=== Geography ===
+===q-Gaussian===
+See [[q-Gaussian]].
+===q-exponential distribution===
-=== Economics (econophysics) ===
+See [[q-exponential distribution]]
+===q-Weibull distribution===
-The Tsallis distribution has been used to describe the distribution of wealth (assets) between individuals <ref>Adrian A. Dragulescu [http://arxiv.org/abs/cond-mat/0307341 Applications of physics to economics and finance: Money, income, wealth, and the stock market] arXiv:cond-mat/0307341v2</ref>.
+See [[q-Weibull distribution]]
-== Notes ==
-{{reflist}}
 == See also ==
-=== Internal links ===
 * [[Constantino Tsallis]]
@@ Line 44: / Line 24: @@
 * [[Tsallis entropy]]
-=== External links ===
+== Notes ==
+{{reflist}}
+==Further reading==
+*Juniper, J. (2007) [http://www.fullemployment.net/publications/wp/2007/07-10.pdf "The Tsallis Distribution and Generalised Entropy: Prospects for Future Research into Decision-Making under Uncertainty"], Centre of Full Employment and Equity, The University of Newcastle, Australia
+*Shigeru Furuichi, Flavia-Corina Mitroi-Symeonidis, Eleutherius Symeonidis, On some properties of Tsallis hypoentropies and hypodivergences, Entropy, 16(10) (2014), 5377-5399; {{doi|10.3390/e16105377}}
+*Shigeru Furuichi, Flavia-Corina Mitroi, Mathematical inequalities for some divergences, Physica A 391 (2012), pp. 388-400, {{doi|10.1016/j.physa.2011.07.052}}; {{ISSN|0378-4371}}
+*Shigeru Furuichi, Nicușor Minculete, Flavia-Corina Mitroi, Some inequalities on generalized entropies, J. Inequal. Appl., 2012, 2012:226. {{doi|10.1186/1029-242X-2012-226}}
+== External links ==
 * [http://www.cscs.umich.edu/~crshalizi/notebooks/tsallis.html Tsallis Statistics, Statistical Mechanics for Non-extensive Systems and Long-Range Interactions]
+{{DEFAULTSORT:Tsallis Distribution}}
+{{Tsallis}}
 [[Category:Statistical mechanics]]
-[[Category:Continuous distributions]]
+[[Category:Types of probability distributions]]
+[[Category:Probability distributions with non-finite variance]]