q-Gaussian distribution

The q-Gaussian is a probability distribution arising from the maximization of the Tsallis entropy under appropriate constraints. It is one example of a 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. The normal 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.Template:Cn The distribution is often favored for its heavy tails in comparison to the Gaussian for 1 < q < 3. For $q<1$ the q-Gaussian distribution is the PDF of a bounded random variable. This makes in biology and other domains the q-Gaussian distribution more suitable than Gaussian distribution to model the effect of external stochasticity. A generalized q-analog of the classical central limit theorem was proposed in 2008, in which the independence constraint for the i.i.d. variables is relaxed to an extent defined by the q parameter, with independence being recovered as q → 1. However, a proof of such a theorem is still lacking.

In the heavy tail regions, the distribution is equivalent to the Student's t-distribution with a direct mapping between q and the degrees of freedom. A practitioner 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 standard q-Gaussian has the probability density function

$f(x)=\frac{\sqrt{\beta }}{C_q}{e}_q(-\beta x^2)$

where

$e_q(x)=[1+(1-q)x{]}_{+}^{\frac{1}{1-q}}$

is the q-exponential and the normalization factor $C_q$ is given by

$C_q=\frac{2\sqrt{\pi }\mathrm{\Gamma }\left(\frac{1}{1-q}\right)}{(3-q)\sqrt{1-q}\mathrm{\Gamma }\left(\frac{3-q}{2(1-q)}\right)}\text{ for }-\mathrm{\infty }<q<1$

$C_q=\sqrt{\pi }\text{ for }q=1$

$C_q=\frac{\sqrt{\pi }\mathrm{\Gamma }\left(\frac{3-q}{2(q-1)}\right)}{\sqrt{q-1}\mathrm{\Gamma }\left(\frac{1}{q-1}\right)}\text{ for }1<q<3.$

Note that for $q<1$ the q-Gaussian distribution is the PDF of a bounded random variable.

Cumulative density function

For $1<q<3$ cumulative density function is

$F(x)=\frac{1}{2}+\frac{\sqrt{q-1}\mathrm{\Gamma }\left(\frac{1}{q-1}\right)x\sqrt{\beta }{}_2{F}_1(\frac{1}{2},\frac{1}{q-1};\frac{3}{2};-(q-1)\beta x^2)}{\sqrt{\pi }\mathrm{\Gamma }\left(\frac{3-q}{2(q-1)}\right)},$

where ${}_2{F}_1(a,b;c;z)$ is the hypergeometric function. As the hypergeometric function is defined for Template:Math but x is unbounded, Pfaff transformation could be used.

For $q<1$, $$F(x)=\{\begin{array}{cc}0& x<-\frac{1}{\sqrt{\beta (1-q)}},\\ \frac{1}{2}+\frac{\sqrt{1-q}\mathrm{\Gamma }\left(\frac{5-3q}{2(1-q)}\right)x\sqrt{\beta }{}_2{F}_1(\frac{1}{2},\frac{1}{q-1};\frac{3}{2};-(q-1)\beta x^2)}{\sqrt{\pi }\mathrm{\Gamma }\left(\frac{2-q}{1-q}\right)}& -\frac{1}{\sqrt{\beta (1-q)}}<x<\frac{1}{\sqrt{\beta (1-q)}},\\ 1& x>\frac{1}{\sqrt{\beta (1-q)}}.\end{array}$$

Entropy

Just as the normal distribution is the maximum information entropy distribution for fixed values of the first moment $\mathrm{E}(X)$ and second moment $\mathrm{E}(X^2)$ (with the fixed zeroth moment $\mathrm{E}(X^0)=1$ corresponding to the normalization condition), the q-Gaussian distribution is the maximum Tsallis entropy distribution for fixed values of these three moments.

Related distributions

Student's t-distribution

While it can be justified by an interesting alternative form of entropy, statistically it is a scaled 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 ν 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 ν.Template:Citation needed The scaled reparametrization introduces the alternative parameters q and β which are related to ν.

Given a Student's t-distribution with ν degrees of freedom, the equivalent q-Gaussian has

$q=\frac{\nu +3}{\nu +1}\text{ with }\beta =\frac{1}{3-q}$

with inverse

$\nu =\frac{3-q}{q-1},\text{ but only if }\beta =\frac{1}{3-q}.$

Whenever $\beta \ne \frac{1}{3-q}$, the function is simply a scaled version of Student's t-distribution.

It is sometimes argued that the distribution is a generalization of Student's t-distribution to negative and or non-integer degrees of freedom. However, the theory of Student's t-distribution extends trivially to all real degrees of freedom, where the support of the distribution is now compact rather than infinite in the case of ν < 0.Template:Citation needed

Three-parameter version

As with many distributions centered on zero, the q-Gaussian can be trivially extended to include a location parameter μ. The density then becomes defined by

$\frac{\sqrt{\beta }}{C_q}{e}_q(-\beta (x-\mu {)}^2).$

Generating random deviates

The Box–Muller transform has been generalized to allow random sampling from q-Gaussians. The standard Box–Muller technique generates pairs of independent normally distributed variables from equations of the following form.

$Z_1=\sqrt{-2\ln (U_1)}\cos (2\pi U_2)$

$Z_2=\sqrt{-2\ln (U_1)}\sin (2\pi U_2)$

The generalized Box–Muller technique can generates pairs of q-Gaussian deviates that are not independent. In practice, only a single deviate will be generated from a pair of uniformly distributed variables. The following formula will generate deviates from a q-Gaussian with specified parameter q and $\beta =\frac{1}{3-q}$

$Z=\sqrt{-2{\text{ ln}}_{q^{\prime }}(U_1)}\text{ cos}(2\pi U_2)$

where ${\text{ ln}}_q$ is the q-logarithm and $q^{\prime }=\frac{1+q}{3-q}$

These deviates can be transformed to generate deviates from an arbitrary q-Gaussian by

$Z^{\prime }=\mu +\frac{Z}{\sqrt{\beta (3-q)}}$

Applications

Physics

It has been shown that the momentum distribution of cold atoms in dissipative optical lattices is a q-Gaussian.

The q-Gaussian distribution is also obtained as the asymptotic probability density function of the position of the unidimensional motion of a mass subject to two forces: a deterministic force of the type $F_1(x)=-2x/(1-x^2)$ (determining an infinite potential well) and a stochastic white noise force $F_2(t)=\sqrt{2(1-q)}\xi (t)$, where $\xi (t)$ is a white noise. Note that in the overdamped/small mass approximation the above-mentioned convergence fails for $q<0$, as recently shown.

Finance

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

See also

• Constantino Tsallis
• Tsallis statistics
• Tsallis entropy
• Tsallis distribution
• q-exponential distribution
• Q-Gaussian process

Further reading

• Juniper, J. (2007) Template:Cite web, 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