paper/paper.md
In wagnerufs/qGaussian: The q-Gaussian Distribution
title: "qGaussian: An R package that deal with the Tsallis statistics"
authors:
- name: Wagner Santos de Lima
orcid: 0000-0001-8661-4828
affiliation: Universidade Federal de Sergipe
- name: Emerson Luis de Santa Helena
orcid: 0000-0002-9970-3544
affiliation: Universidade Federal de Sergipe
date: 04 April 2017
bibliography: paper.bib
Summary
Entropy is a fundamental concept in physics since it is in essence of second law of thermodynamics.
In the article 'Possible generalization of Boltzmann-Gibbs statistics' [@T88], was
postulated the Tsallis entropy $S_q[p(x)]=(1-\int_{-\infty}^{\infty}p^q(x)dx)/(q-1)$.
The $q$-Gaussian density function $p(x)$ arises from maximizing the $q$-entropy functional $S_q[p(x)]$ under constraints
[@tsallis] and [@H14] and it is important at the framework of the nonextensive statistical mechanics.
There is a broad literature where the nonextensive approach is used to model systems and/or explain many-body problems and
issues related to Chaos. In some cases, there exist strong evidences that theory works when include a nonextensive approach
and certified by a broad numerical decade from experimental measurements or numericals simulation, that let us obtain a $q$ value by an almost flawless curve fitting. In other cases, a lot of observational data are only suggestive of a nonextensive approach. A myriad of examples can be found in [@GT04] and [@tsallis].
The $q$-Gaussian probability density function, named here $q$PDF, with $q$-mean $\mu_q$ and
$q$-variance $\sigma_{q}$ can be written as:
\begin{equation}
p(x;\mu_q,\sigma_q)=\frac{1}{\sigma_q {\text B}\left(\frac{\alpha}{2},\frac{1}{2}\right)}
\sqrt{
\frac{|Z|}{u^{(1+1/Z)}}
}
\label{pdf}
\end{equation}
where $Z=(q-1)/(3-q)$,
\begin{equation}
\alpha=\left{
\begin{array}{ll}
1-1/Z& \textrm{if $q<1$}, \
1/Z & \textrm{if $1<q<3$}, \
\end{array}\right.
\end{equation}
$u(x)= 1+Z (x-\mu_q)^2/\sigma_{q}^{2}$, and $\text B(a,b)$ is the Beta function.
In the limit of $q\rightarrow 1$ a $q$PDF tends to a standard Gaussian distribution.
For $q < 1$, it is a compact support distribution, with $x \in [ \pm\sigma_q / \sqrt{-Z}]$. When $1 < q < 3$,
it is a heavy tail. In the last case, a power law asymptotic behaviour describes well this class of distribution.
Given $X=(x_1,x_2,...x_n)$ a random variable, the cumulative distribution function ($q$CDF), $F(x)=P[X < x]$
\begin{equation}\label{csum}
F(x) = \int_{-\infty}^{x}p(v)dv,
\end{equation}
could be represented through $\text I_{w}(a,b) = \text B_{w}(a,b)/\text B(a,b)$, the regularized incomplete Beta function,
where $\text B_{w}(a,b) = \int_{0}^{w} t^{a-1}(1-t)^{b-1}dt$, is the Incomplete Beta function.
\begin{equation}
\label{cdf}
F(x;\mu_q;\sigma_q) = \left{
\begin{array}{ll}
\frac{1}{2} \text I_{\beta}\left(\frac{\alpha}{2},\frac{1}{2}\right) & \textrm{if $x<\mu_q$}, \
1-\frac{1}{2} \text I_{\beta}\left(\frac{\alpha}{2},\frac{1}{2}\right) & \textrm{if $x> \mu_q$}, \
\end{array}\right.
\end{equation}
where
\begin{equation}
\beta=\left{
\begin{array}{ll}
u(x) & \textrm{if $q<1$}, \
1/u(x) & \textrm{if $1<q<3$}. \
\end{array}\right.
\end{equation}
In third, the quantile function ($q$QF) is obtained as an inverse function of $y=F(x)$,
$w=\text I^{-1}_{2y}(\frac{\alpha}{2},\frac{1}{2})$:
\begin{equation}\label{qdf}
C(y;\mu_q,\sigma_q) = \left{
\begin{array}{ll}
\mu_q-\sigma_q^2 \sqrt{\left( \gamma -1\right ) /Z} & \textrm{if $y<1/2$}, \
\mu_q + \sigma_q^2 \sqrt{\left( \gamma -1\right ) /Z} & \textrm{if $1-y<1/2$}, \
\end{array}\right.
\end{equation}
where
\begin{equation}
\gamma=\left{
\begin{array}{ll}
\text I^{-1}{2y}\left(\frac{\alpha}{2},\frac{1}{2}\right) & \textrm{if $q<1$}, \
1/\text I^{-1}{2y}\left(\frac{\alpha}{2},\frac{1}{2}\right) & \textrm{if $1<q<3$}. \
\end{array}\right.
\end{equation}
The random numbers generator from a $q$PDF can be implemented in different ways. The straightforward method use the
quantile function to creates random sample elements $x_i=C(y_i;\mu_q,\sigma_q)$, where $y_i \in [0,1]$ are obtained from
a uniform random number. In the second way, we use the Box-Muller algorithm as presented in [@thistleton] with
the Mersenne-Twister algorithm as a uniform random number generator to create a $q$-Gaussian random variable
$X\equiv N_q(\mu_q,\sigma_q) \equiv \mu_q+\sigma_q N_q(0,1)$ where $N_q(0,1)$ is called standard $q$-Gaussian.
The classical kurtosis methods, when applied to a data set, are very sensitive to outlying values, however, it is
possible to diminish this problem at a measurement of the tail heaviness by using robust statistic concepts. To doing that,
given a sorted sample ${x_1< \dots < \tilde{x}< \dots < x_n}$ from a univariate distribution with median $\tilde x$,
[@BHS06] established the medcouple to evaluates the tail weight when applied to ${x_{1} < \dots < \tilde{x}}$ and
${\tilde{x}< \dots < x_n}$. This procedure can be applied to characterize the $q$-Gaussian with heavy tail and
compact support. To this end, [@santahelena]
aiming to identify a $q$-Gaussian distribution at empirical data, proposed a relationship between medcouple and $q$ value obtained by curve fitting.
The main goal of the package \pkg{qGaussian} it is lets us to compute $q$PDF (\ref{pdf}), $q$CDF (\ref{cdf})
and $q$QF (\ref{qdf}) as same time generates random numbers from a $q$-Gaussian distribution parametrised by $q$ value.
To compute the Beta function and its inverse it is necessary the
\pkg{zipfR} package and \pkg{robustbase} is the package of robust statistic to implement a tail weight measurement.
\begin{table}[!h]
\begin{center}
\begin{tabular}{cc}
\toprule
Quantity & R's commands \
\midrule
$ p(x) $ & \code{ dqgauss(x,q,mu,sig) } \
$F(x) $ & \code{pqgauss(x,q,mu,sig,lower.tail=T)} \
$C(y)$ & \code{cqgauss(y,q,mu,sig,lower.tail=T) } \
$x_{i}$ & \code{rqgauss(n,q,mu,sig,meth="Box-Muller")} \
$q $ & \code{qbymc(X)} \
\bottomrule
\end{tabular}
\end{center}
%\vskip -.5 cm
\caption{Sintaxe of R's commands for each output quantities}
\label{tab}
\end{table}
In table \ref{tab}, the input argument \code{x} represent a vector of quantiles for instructions \code{dqgauss(x,..)} and
\code{pqgauss(x,..)} while \code{y} represent a vector of probabilities and \code{n} the length sample, for the random
number generator. The parameters \code{q}, \code{mu} and \code{sig} are the entropic index,
$q$-mean and $q$-variance, respectively,
assuming the default values $(0,0,1)$. The medcouple is used into the \code{qbymc(X)} code to estimate $q$
value and standard error, receiving a random variable \code{X}, from the class "vector", as input.
Examples can be found in https://arxiv.org/abs/1703.06172
References
wagnerufs/qGaussian documentation built on May 3, 2019, 1:50 p.m.
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title: "qGaussian: An R package that deal with the Tsallis statistics"
authors: - name: Wagner Santos de Lima orcid: 0000-0001-8661-4828 affiliation: Universidade Federal de Sergipe - name: Emerson Luis de Santa Helena orcid: 0000-0002-9970-3544 affiliation: Universidade Federal de Sergipe
date: 04 April 2017
bibliography: paper.bib
Summary
Entropy is a fundamental concept in physics since it is in essence of second law of thermodynamics. In the article 'Possible generalization of Boltzmann-Gibbs statistics' [@T88], was postulated the Tsallis entropy $S_q[p(x)]=(1-\int_{-\infty}^{\infty}p^q(x)dx)/(q-1)$. The $q$-Gaussian density function $p(x)$ arises from maximizing the $q$-entropy functional $S_q[p(x)]$ under constraints [@tsallis] and [@H14] and it is important at the framework of the nonextensive statistical mechanics.
There is a broad literature where the nonextensive approach is used to model systems and/or explain many-body problems and issues related to Chaos. In some cases, there exist strong evidences that theory works when include a nonextensive approach and certified by a broad numerical decade from experimental measurements or numericals simulation, that let us obtain a $q$ value by an almost flawless curve fitting. In other cases, a lot of observational data are only suggestive of a nonextensive approach. A myriad of examples can be found in [@GT04] and [@tsallis].
The $q$-Gaussian probability density function, named here $q$PDF, with $q$-mean $\mu_q$ and $q$-variance $\sigma_{q}$ can be written as:
\begin{equation} p(x;\mu_q,\sigma_q)=\frac{1}{\sigma_q {\text B}\left(\frac{\alpha}{2},\frac{1}{2}\right)} \sqrt{ \frac{|Z|}{u^{(1+1/Z)}} } \label{pdf} \end{equation} where $Z=(q-1)/(3-q)$, \begin{equation} \alpha=\left{ \begin{array}{ll} 1-1/Z& \textrm{if $q<1$}, \ 1/Z & \textrm{if $1<q<3$}, \ \end{array}\right. \end{equation} $u(x)= 1+Z (x-\mu_q)^2/\sigma_{q}^{2}$, and $\text B(a,b)$ is the Beta function.
In the limit of $q\rightarrow 1$ a $q$PDF tends to a standard Gaussian distribution. For $q < 1$, it is a compact support distribution, with $x \in [ \pm\sigma_q / \sqrt{-Z}]$. When $1 < q < 3$, it is a heavy tail. In the last case, a power law asymptotic behaviour describes well this class of distribution.
Given $X=(x_1,x_2,...x_n)$ a random variable, the cumulative distribution function ($q$CDF), $F(x)=P[X < x]$
\begin{equation}\label{csum} F(x) = \int_{-\infty}^{x}p(v)dv, \end{equation}
could be represented through $\text I_{w}(a,b) = \text B_{w}(a,b)/\text B(a,b)$, the regularized incomplete Beta function, where $\text B_{w}(a,b) = \int_{0}^{w} t^{a-1}(1-t)^{b-1}dt$, is the Incomplete Beta function.
\begin{equation} \label{cdf} F(x;\mu_q;\sigma_q) = \left{ \begin{array}{ll} \frac{1}{2} \text I_{\beta}\left(\frac{\alpha}{2},\frac{1}{2}\right) & \textrm{if $x<\mu_q$}, \ 1-\frac{1}{2} \text I_{\beta}\left(\frac{\alpha}{2},\frac{1}{2}\right) & \textrm{if $x> \mu_q$}, \ \end{array}\right. \end{equation}
where
\begin{equation} \beta=\left{ \begin{array}{ll} u(x) & \textrm{if $q<1$}, \ 1/u(x) & \textrm{if $1<q<3$}. \ \end{array}\right. \end{equation}
In third, the quantile function ($q$QF) is obtained as an inverse function of $y=F(x)$, $w=\text I^{-1}_{2y}(\frac{\alpha}{2},\frac{1}{2})$:
\begin{equation}\label{qdf} C(y;\mu_q,\sigma_q) = \left{ \begin{array}{ll} \mu_q-\sigma_q^2 \sqrt{\left( \gamma -1\right ) /Z} & \textrm{if $y<1/2$}, \ \mu_q + \sigma_q^2 \sqrt{\left( \gamma -1\right ) /Z} & \textrm{if $1-y<1/2$}, \ \end{array}\right. \end{equation} where
\begin{equation} \gamma=\left{ \begin{array}{ll} \text I^{-1}{2y}\left(\frac{\alpha}{2},\frac{1}{2}\right) & \textrm{if $q<1$}, \ 1/\text I^{-1}{2y}\left(\frac{\alpha}{2},\frac{1}{2}\right) & \textrm{if $1<q<3$}. \ \end{array}\right. \end{equation}
The random numbers generator from a $q$PDF can be implemented in different ways. The straightforward method use the quantile function to creates random sample elements $x_i=C(y_i;\mu_q,\sigma_q)$, where $y_i \in [0,1]$ are obtained from a uniform random number. In the second way, we use the Box-Muller algorithm as presented in [@thistleton] with the Mersenne-Twister algorithm as a uniform random number generator to create a $q$-Gaussian random variable $X\equiv N_q(\mu_q,\sigma_q) \equiv \mu_q+\sigma_q N_q(0,1)$ where $N_q(0,1)$ is called standard $q$-Gaussian.
The classical kurtosis methods, when applied to a data set, are very sensitive to outlying values, however, it is possible to diminish this problem at a measurement of the tail heaviness by using robust statistic concepts. To doing that, given a sorted sample ${x_1< \dots < \tilde{x}< \dots < x_n}$ from a univariate distribution with median $\tilde x$, [@BHS06] established the medcouple to evaluates the tail weight when applied to ${x_{1} < \dots < \tilde{x}}$ and ${\tilde{x}< \dots < x_n}$. This procedure can be applied to characterize the $q$-Gaussian with heavy tail and compact support. To this end, [@santahelena] aiming to identify a $q$-Gaussian distribution at empirical data, proposed a relationship between medcouple and $q$ value obtained by curve fitting.
The main goal of the package \pkg{qGaussian} it is lets us to compute $q$PDF (\ref{pdf}), $q$CDF (\ref{cdf}) and $q$QF (\ref{qdf}) as same time generates random numbers from a $q$-Gaussian distribution parametrised by $q$ value. To compute the Beta function and its inverse it is necessary the \pkg{zipfR} package and \pkg{robustbase} is the package of robust statistic to implement a tail weight measurement.
\begin{table}[!h] \begin{center} \begin{tabular}{cc} \toprule Quantity & R's commands \ \midrule $ p(x) $ & \code{ dqgauss(x,q,mu,sig) } \ $F(x) $ & \code{pqgauss(x,q,mu,sig,lower.tail=T)} \ $C(y)$ & \code{cqgauss(y,q,mu,sig,lower.tail=T) } \ $x_{i}$ & \code{rqgauss(n,q,mu,sig,meth="Box-Muller")} \ $q $ & \code{qbymc(X)} \ \bottomrule \end{tabular} \end{center} %\vskip -.5 cm \caption{Sintaxe of R's commands for each output quantities} \label{tab} \end{table}
In table \ref{tab}, the input argument \code{x} represent a vector of quantiles for instructions \code{dqgauss(x,..)} and \code{pqgauss(x,..)} while \code{y} represent a vector of probabilities and \code{n} the length sample, for the random number generator. The parameters \code{q}, \code{mu} and \code{sig} are the entropic index, $q$-mean and $q$-variance, respectively, assuming the default values $(0,0,1)$. The medcouple is used into the \code{qbymc(X)} code to estimate $q$ value and standard error, receiving a random variable \code{X}, from the class "vector", as input. Examples can be found in https://arxiv.org/abs/1703.06172
References
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