• Source: 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. 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 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.




Common Tsallis distributions






= q-Gaussian

=
See q-Gaussian.




= q-exponential distribution

=
See q-exponential distribution




= q-Weibull distribution

=
See q-Weibull distribution




See also


Constantino Tsallis
Tsallis statistics
Tsallis entropy




Notes






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

Kata Kunci Pencarian:

• Tsallis distribution
• Tsallis entropy
• Q-Gaussian distribution
• Normal distribution
• Q-exponential distribution
• Tsallis statistics
• Q-Weibull distribution
• Q distribution
• Constantino Tsallis
• List of statistics articles