Q-Gaussian distribution

This article is about the Tsallis q-Gaussian. For a different q-analog, see Gaussian q-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.^[1] The normal distribution is recovered as q → 1.

q-Gaussian

Probability density function
Parameters	${\displaystyle q<3}$ shape (real) ${\displaystyle \beta >0}$ (real)
Support	${\displaystyle x\in (-\infty$ ;+\infty )\!} for ${\displaystyle 1\leq q<3}$ ${\displaystyle x\in \left[\pm {1 \over {\sqrt {\beta (1-q)}}}\right]}$ for ${\displaystyle q<1}$
PDF	${\displaystyle {{\sqrt {\beta }} \over C_{q}}e_{q}({-\beta x^{2}})}$
CDF	see text
Mean	${\displaystyle 0{\text{ for }}q<2}$, otherwise undefined
Median	${\displaystyle 0}$
Mode	${\displaystyle 0}$
Variance	${\displaystyle {1 \over {\beta (5-3q)}}{\text{ for }}q<{5 \over 3}}$ ${\displaystyle \infty {\text{ for }}{5 \over 3}\leq q<2}$ ${\displaystyle {\text{Undefined for }}2\leq q<3}$
Skewness	${\displaystyle 0{\text{ for }}q<{3 \over 2}}$
Excess kurtosis	${\displaystyle 6{q-1 \over 7-5q}{\text{ for }}q<{7 \over 5}}$

The q-Gaussian has been applied to problems in the fields of statistical mechanics, geology, anatomy, astronomy, economics, finance, and machine learning.^{[citation needed]} The distribution is often favored for its heavy tails in comparison to the Gaussian for 1 < q < 3. For ${\displaystyle q<1}$ the q-Gaussian distribution is the PDF of a bounded random variable. This makes in biology and other domains^[2] 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^[3] 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.^[4]

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 ^[3]

${\displaystyle f(x)={{\sqrt {\beta }} \over C_{q}}e_{q}(-\beta x^{2})}$

where

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

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

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

${\displaystyle C_{q}={\sqrt {\pi }}{\text{ for }}q=1\,}$

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

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

Cumulative density function

For ${\displaystyle 1<q<3}$ cumulative density function is ^[5]

${\displaystyle F(x)={\frac {1}{2}}+{\frac {{\sqrt {q-1}}\,\Gamma \left({1 \over q-1}\right)x{\sqrt {\beta }}\,{}_{2}F_{1}\left({\tfrac {1}{2}},{\tfrac {1}{q-1}};{\tfrac {3}{2}};-(q-1)\beta x^{2}\right)}{{\sqrt {\pi }}\,\Gamma \left({3-q \over 2(q-1)}\right)}},}$

where ${\displaystyle {}_{2}F_{1}(a,b;c;z)}$ is the hypergeometric function. As the hypergeometric function is defined for |z| < 1 but x is unbounded, Pfaff transformation could be used.

For ${\displaystyle q<1}$, $${\displaystyle F(x)={\begin{cases}0&x<-{\frac {1}{\sqrt {\beta (1-q)}}},\\{\frac {1}{2}}+{\frac {{\sqrt {1-q}}\,\Gamma \left({5-3q \over 2(1-q)}\right)x{\sqrt {\beta }}\,{}_{2}F_{1}\left({\tfrac {1}{2}},{\tfrac {1}{q-1}};{\tfrac {3}{2}};-(q-1)\beta x^{2}\right)}{{\sqrt {\pi }}\,\Gamma \left({2-q \over 1-q}\right)}}&-{\frac {1}{\sqrt {\beta (1-q)}}}<x<{\frac {1}{\sqrt {\beta (1-q)}}},\\1&x>{\frac {1}{\sqrt {\beta (1-q)}}}.\end{cases}}}$$

Entropy

Just as the normal distribution is the maximum information entropy distribution for fixed values of the first moment ${\displaystyle \operatorname {E} (X)}$ and second moment ${\displaystyle \operatorname {E} (X^{2})}$ (with the fixed zeroth moment ${\displaystyle \operatorname {E} (X^{0})=1}$ 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.^{[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

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

with inverse

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

Whenever ${\displaystyle \beta \neq {1 \over {3-q}}}$, 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.^{[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

${\displaystyle {{\sqrt {\beta }} \over C_{q}}e_{q}({-\beta (x-\mu )^{2}}).}$

Generating random deviates

The Box–Muller transform has been generalized to allow random sampling from q-Gaussians.^[6] The standard Box–Muller technique generates pairs of independent normally distributed variables from equations of the following form.

${\displaystyle Z_{1}={\sqrt {-2\ln(U_{1})}}\cos(2\pi U_{2})}$

${\displaystyle Z_{2}={\sqrt {-2\ln(U_{1})}}\sin(2\pi U_{2})}$

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 ${\displaystyle \beta ={1 \over {3-q}}}$

${\displaystyle Z={\sqrt {-2{\text{ ln}}_{q'}(U_{1})}}{\text{ cos}}(2\pi U_{2})}$

where ${\displaystyle {\text{ ln}}_{q}}$ is the q-logarithm and ${\displaystyle q'={{1+q} \over {3-q}}}$

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

${\displaystyle Z'=\mu +{Z \over {\sqrt {\beta (3-q)}}}}$

Applications

Physics

It has been shown that the momentum distribution of cold atoms in dissipative optical lattices is a q-Gaussian.^[7]

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 ${\textstyle F_{1}(x)=-2x/(1-x^{2})}$ (determining an infinite potential well) and a stochastic white noise force ${\textstyle F_{2}(t)={\sqrt {2(1-q)}}\xi (t)}$, where ${\displaystyle \xi (t)}$ is a white noise. Note that in the overdamped/small mass approximation the above-mentioned convergence fails for ${\displaystyle q<0}$, as recently shown.^[8]

Finance

Financial return distributions in the New York Stock Exchange, NASDAQ and elsewhere have been interpreted as q-Gaussians.^[9][10]

See also

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

Notes

1. Tsallis, C. Nonadditive entropy and nonextensive statistical mechanics-an overview after 20 years. Braz. J. Phys. 2009, 39, 337–356
2. d'Onofrio A. (ed.) Bounded Noises in Physics, Biology, and Engineering. Birkhauser (2013)
3. Umarov, Sabir; Tsallis, Constantino; Steinberg, Stanly (2008). "On a q-Central Limit Theorem Consistent with Nonextensive Statistical Mechanics" (PDF). Milan J. Math. 76. Birkhauser Verlag: 307–328. doi:10.1007/s00032-008-0087-y. S2CID 55967725. Retrieved 2011-07-27.
4. Hilhorst, H.J. (2010), "Note on a q-modified central limit theorem", Journal of Statistical Mechanics: Theory and Experiment, 2010 (10): 10023, arXiv:1008.4259, Bibcode:2010JSMTE..10..023H, doi:10.1088/1742-5468/2010/10/P10023, S2CID 119316670.
5. https://reference.wolframcloud.com/language/ref/TsallisQGaussianDistribution.html ^{[bare URL]}
6. W. Thistleton, J.A. Marsh, K. Nelson and C. Tsallis, Generalized Box–Muller method for generating q-Gaussian random deviates, IEEE Transactions on Information Theory 53, 4805 (2007)
7. Douglas, P.; Bergamini, S.; Renzoni, F. (2006). "Tunable Tsallis Distributions in Dissipative Optical Lattices" (PDF). Physical Review Letters. 96 (11): 110601. Bibcode:2006PhRvL..96k0601D. doi:10.1103/PhysRevLett.96.110601. PMID 16605807.
8. Domingo, Dario; d’Onofrio, Alberto; Flandoli, Franco (2017). "Boundedness vs unboundedness of a noise linked to Tsallis q-statistics: The role of the overdamped approximation". Journal of Mathematical Physics. 58 (3). AIP Publishing: 033301. arXiv:1709.08260. Bibcode:2017JMP....58c3301D. doi:10.1063/1.4977081. ISSN 0022-2488. S2CID 84178785.
9. Borland, Lisa (2002-08-07). "Option Pricing Formulas Based on a Non-Gaussian Stock Price Model". Physical Review Letters. 89 (9). American Physical Society (APS): 098701. arXiv:cond-mat/0204331. Bibcode:2002PhRvL..89i8701B. doi:10.1103/physrevlett.89.098701. ISSN 0031-9007. PMID 12190447. S2CID 5740827.
10. L. Borland, The pricing of stock options, in Nonextensive Entropy – Interdisciplinary Applications, eds. M. Gell-Mann and C. Tsallis (Oxford University Press, New York, 2004)

Further reading

• Juniper, J. (2007) "The Tsallis Distribution and Generalised Entropy: Prospects for Future Research into Decision-Making under Uncertainty" (PDF). Archived from the original (PDF) on 2011-07-06. Retrieved 2011-06-24., 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