Tsallis distribution

q-Gaussian

Probability density function
Parameters	$q<3$ shape (real) $\beta >0$ scale (real)
Support	$x\in (-\infty ;+\infty )\!$ for $q\geq 1$ $x\in [\pm {1 \over {\sqrt {\beta (1-q)}}}]$ for $q<1$
PDF	${{\sqrt {\beta }} \over C_{q}}e_{q}^{-\beta x^{2}}$
Mean	$0{\text{ for }}q<2$, otherwise undefined
Median	$0$
Mode	$0$

The Tsallis distribution, also known as a q-Gaussian, is a probability distribution arising from the maximization of the Tsallis entropy under approproate constraints. 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. The standard gaussian distribution is recovered as $q$→$1$.

The q-gaussian has been applied to problems in the fields of statistical mechanics, geology, anatomy, astronomy, economics, finance, and machine learning. The distribution is often favored for its heavy tails in comparison to the gaussian for $1<q<3$.

The heavy tail regions are equivalent to the Student's t-distribution with a direct mapping between q and the degrees of freedom. A practioner using one of these distributions can therefore parameterize the same distribution in two different ways. The choice of the q-gaussian form may arise if the system is non-extensive, or if there is lack of a connection to small samples sizes.

Characterization

Probability density function

The q-gaussian has the probability density function
$f(x)={{\sqrt {\beta }} \over C_{q}}e_{q}(-\beta x^{2})$

where

$e_{q}(x)=[1+(1-q)x]^{1 \over 1-q}$

is the q-exponential and the normalization factor $C_{q}$ is given by

$C_{q}={{2{\sqrt {\pi }}\Gamma ({1 \over 1-q})} \over {(3-q){\sqrt {1-q}}\Gamma ({3-q \over 2(1-q)})}}$ for $-\infty <q<1$

$C_{q}={\sqrt {\pi }}$ for $q=1$

$C_{q}={{{\sqrt {\pi }}\Gamma ({3-q \over 2(q-1)})} \over {{\sqrt {q-1}}\Gamma ({1 \over q-1})}}$ for $1<q<3.$

Derivation

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.

Relationship to Student's T Distribution

. While it can be justified by an interesting alternative form of entropy, statistically it is a simple reparametrization of the Student's t-distribution introduced by W. Gosset in 1908 to describe small sample statistics. In Gosset's original presentation the degrees of freedom parameter $\nu$ was constrained to be a positive integer related to the sample size, but it is readily observed that Gosset's density function is valid for all real values of $\nu$. The reparametrization introduces the alternative parameter $q$ related to $\nu$ by

$q={\frac {(\nu +3)}{(\nu +1)}}$

with inverse

$\nu ={\frac {(3-q)}{(q-1)}}$

It is sometimes argued that the distribution is a generalization of the Student to negative and or non-integer degrees of freedom. However, the theory of the Student extends trivially to all real degrees of freedom, where the support of the distribution is now compact rather than infinite in the case of $\nu <0$. The formula for the T-density function is given in many standard texts and it is a simple matter to confirm the above formulae. The best starting point is Gosset's original work, discussed in the article on William Sealy Gosset.

Applications

Physics

It has been shown that the momentum distribution of cold atoms in dissipative optical lattices is a Tsallis distribution

Finance

Financial return distributions in the New York Stock Exchange, NASDAQ and elsewhere are often interpreted as q-Gaussians.

Economics (econophysics)

The Tsallis distribution has been used to describe the distribution of wealth (assets) between individuals .

See also

• Constantino Tsallis
• Tsallis statistics
• Tsallis entropy

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

External links

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