Tsallis distribution

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 λ = 1 − q {\displaystyle \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

q-Gaussian

See q-Gaussian.

q-exponential distribution

See q-exponential distribution

q-Weibull distribution

See q-Weibull distribution

See also

Notes

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

Further reading

  • Juniper, J. (2007) "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

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