q-Gaussian distribution

This article is about the Tsallis q-Gaussian. For a different q-analog, see Gaussian q-distribution.

q-Gaussian
Probability density function
Parameters	$q < 3$ shape (real) $\beta > 0$ (real)
Support	$x \in (-\infty; +\infty)\!$ for $1\le q < 3$ $x \in \left[\pm {1 \over \sqrt{\beta(1-q)}}\right]$ 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	$0$
Variance	${ 1 \over {\beta (5-3q)}} \text{ for } q < {5 \over 3}$ $\infty \text{ for } {5 \over 3} \le q < 2$ $\text{Undefined for }2 \le q <3$
Skewness	$0 \text{ for } q < {3 \over 2}$
Ex. kurtosis	$6{q-1 \over 7-5q} \text{ for } q < {7 \over 5}$

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.

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. A generalized q-analog of the classical central limit theorem^[2] 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.^[3]

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 q-Gaussian has the probability density function ^[2]

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\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

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

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 .

Entropy

Just as the normal distribution is the maximum information entropy distribution for fixed values of the first moment $\operatorname{E}(X)$ and second moment $\operatorname{E}(X^2)$ (with the fixed zeroth moment $\operatorname{E}(X^0)=1$ 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. 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 $\beta \ne {1 \over {3-q}}$ , 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.

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

{\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.^[4] 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 $\beta = {1 \over {3-q}}$

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

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

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

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.^[5]

Finance

Financial return distributions in the New York Stock Exchange, NASDAQ and elsewhere are often interpreted as q-Gaussians.^[6]^[7]

Notes

↑ Tsallis, C. Nonadditive entropy and nonextensive statistical mechanics-an overview after 20 years. Braz. J. Phys. 2009, 39, 337–356
1 2 Umarov, Sabir; Tsallis, Constantino; Steinberg, Stanly (2008). "On a q-Central Limit Theorem Consistent with Nonextensive Statistical Mechanics" (PDF). Milan j. math. (Birkhauser Verlag) 76: 307–328. doi:10.1007/s00032-008-0087-y. Retrieved 2011-07-27.
↑ Hilhorst, H.J. (2010), "Note on a q-modified central limit theorem", Journal of Statistical Mechanics: Theory and Experiment 2010 (10): P10023, doi:10.1088/1742-5468/2010/10/P10023.
↑ 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)
↑ Douglas, P.; Bergamini, S.; Renzoni, F. (2006). "Tunable Tsallis Distributions in Dissipative Optical Lattices". Physical Review Letters 96 (11). doi:10.1103/PhysRevLett.96.110601.
↑ L. Borland, Option pricing formulas based on a non-Gaussian stock price model, Phys. Rev. Lett. 89, 098701 (2002)
↑ 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)

External links

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

Tsallis statistics

Constantino Tsallis Tsallis entropy Tsallis statistics Tsallis distribution q-Gaussian distribution q-exponential distribution q-Weibull distribution

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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