| Type: | Package |
| Title: | The q-Gaussian Distribution |
| Version: | 0.1.8 |
| Author: | Emerson Luis de Santa Helena <elsh@ufs.br> Wagner Santos de Lima <wagnersantos.ufs@hotmail.com> |
| Maintainer: | Wagner Santos de Lima <wagnersantos.ufs@hotmail.com> |
| Description: | Density, distribution function, quantile function and random generation for the q-gaussian distribution with parameters mu and sig. |
| License: | GPL-2 | GPL-3 [expanded from: GPL (≥ 2)] |
| Imports: | Rcpp (≥ 0.12.10), stats, robustbase, zipfR |
| LinkingTo: | Rcpp |
| LazyData: | true |
| RoxygenNote: | 6.0.1 |
| NeedsCompilation: | yes |
| Packaged: | 2018-07-12 18:31:56 UTC; elsh |
| Repository: | CRAN |
| Date/Publication: | 2018-07-12 18:50:06 UTC |
Chaotic, a random number generator of q-Gaussian random variables.
Description
Given a random number generator of q-Gaussian random variables for a range of q values, -8 < q < 3, based on deterministic map dynamics. To yield a 'q' value, a characteristic entropic index of the q-gaussian distributions.
Usage
Chaotic(n,q,v0,z0)
Arguments
n |
number of observations. If length(n) > 1, the length is taken to be the number required. |
q |
entropic index. |
v0 |
a random seed. |
z0 |
a random seed. |
Value
a number q < 3, and the standard error.
Author(s)
Emerson Luis de Santa Helena , Wagner Santos de Lima
References
Umeno, K., Sato, A., IEEE Transactions on Information Theory (Volume:59,Issue:5,May 2013).Chaotic Method for Generating q-Gaussian Random Variables.
See Also
Distributions for other standard distributions, including dt and dcauchy.
Distributions
Examples
t=Chaotic(100000,0,.1,.1)
hist(t,breaks=100)
The q-gaussian Distribution
Description
Density, distribution function, quantile function and random generation for the q-gaussian distribution with parameters mu and sig.
Usage
cqgauss(p, q = 0, mu = 0, sig = 1, lower.tail = TRUE)
Arguments
p |
vector of probabilities. |
q |
entropic index. |
mu |
a value for q-mean. |
sig |
a value for q-variance. |
lower.tail |
logical; if TRUE (default), probabilities are P[X <= x] otherwise, P[X > x]. |
Details
If q , mu and sig values are not specified, they assume the default values of 0, 0 and 1, respectively. Defining Z=(q-1)/(3-q), the q-gaussian distribution has density wrinten as
p(x) = (sig*Beta(alpha/2,1/2))^-1*(1+Z(x-mu)^2/sig^2)^-(1+1/Z)/2
where alpha = 1 - 1/Z when q < 1 and 1/Z when 1 < q < 3.
Value
dqgauss gives the density, pqgauss gives the distribution function, cqgauss gives the quantile function, and rqgauss generates random deviates.
Author(s)
Emerson Luis de Santa Helena , Wagner Santos de Lima
References
Thistleton, W.,Marsh, J. A.,Nelson, K.,Tsallis, C., (2007) IEEE Transactions on Information Theory, 53(12):4805
Tsallis, C., (2009) Introduction to Nonextensive Statistical Mechanics. Springer.
de Santa Helena, E. L., Nascimento, C. M., and Gerhardt, G. J., (2015) Alternative way to charecterize a q-gaussian distribution by a robust heavy tail measurement. Physica A, (435):44-50.
Manuscript submitted for publication (2016) qGaussian: Tools to Explore Applications of Tsallis Statistics
See Also
Distributions for other standard distributions, including dt and dcauchy.
Distributions
Examples
qv <- c(2.8,2.5,2,1.01,0,-5); nn <- 700
xrg <- sqrt((3-qv[6])/(1-qv[6]))
xr <- seq(-xrg,xrg,by=2*xrg/nn)
y0 <- dqgauss(xr,qv[6])
plot(xr,y0,ty='l',xlim=range(-4.5,4.5),ylab='p(x)',xlab='x')
for (i in 1:5){
if (qv[i]< 1) xrg <- sqrt((3-qv[i])/(1-qv[i]))
else xrg <- 4.5
vby <- 2*xrg/nn
xr <- seq(-xrg,xrg,by=2*xrg/nn)
y0 <- dqgauss(xr,qv[i])
points (xr,y0,ty='l',col=(i+1))
}
legend(2, 0.4, legend =c(expression(paste(q==-5)),expression(paste(q==0)),
expression(paste(q==1.01)),expression(paste(q==2)),expression(paste(q==2.5)),
expression(paste(q==2.8))),col = c(1,6,5,4,3,2), lty = c(1,1,1,1,1,1))
######
qv <- 0
rr <- rqgauss(2^16,qv)
nn <- 70
xrg <- sqrt((3-qv)/(1-qv))
vby <- 2*xrg/(nn)
xr <- seq(-xrg,xrg,by=vby)
hist (rr,breaks=xr,freq=FALSE,xlab="x",main='')
y <- dqgauss(xr)
lines(xr,y/sum(y*vby),cex=.5,col=2,lty=4)
The q-gaussian Distribution
Description
Density, distribution function, quantile function and random generation for the q-gaussian distribution with parameters mu and sig.
Usage
dqgauss(x, q = 0, mu = 0, sig = 1)
Arguments
x |
vector of quantiles. |
q |
entropic index. |
mu |
a value for q-mean. |
sig |
a value for q-variance. |
Details
If q , mu and sig values are not specified, they assume the default values of 0, 0 and 1, respectively. Defining Z=(q-1)/(3-q), the q-gaussian distribution has density wrinten as
p(x) = (sig*Beta(alpha/2,1/2))^-1*(1+Z(x-mu)^2/sig^2)^-(1+1/Z)/2
where alpha = 1 - 1/Z when q < 1 and 1/Z when 1 < q < 3.
Value
dqgauss gives the density, pqgauss gives the distribution function, cqgauss gives the quantile function, and rqgauss generates random deviates.
Author(s)
Emerson Luis de Santa Helena , Wagner Santos de Lima
References
Thistleton, W.,Marsh, J. A.,Nelson, K.,Tsallis, C., (2007) IEEE Transactions on Information Theory, 53(12):4805
Tsallis, C., (2009) Introduction to Nonextensive Statistical Mechanics. Springer.
de Santa Helena, E. L., Nascimento, C. M., and Gerhardt, G. J., (2015) Alternative way to charecterize a q-gaussian distribution by a robust heavy tail measurement. Physica A, (435):44-50.
Manuscript submitted for publication (2016) qGaussian: Tools to Explore Applications of Tsallis Statistics
See Also
Distributions for other standard distributions, including dt and dcauchy.
Distributions
Examples
qv <- c(2.8,2.5,2,1.01,0,-5); nn <- 700
xrg <- sqrt((3-qv[6])/(1-qv[6]))
xr <- seq(-xrg,xrg,by=2*xrg/nn)
y0 <- dqgauss(xr,qv[6])
plot(xr,y0,ty='l',xlim=range(-4.5,4.5),ylab='p(x)',xlab='x')
for (i in 1:5){
if (qv[i]< 1) xrg <- sqrt((3-qv[i])/(1-qv[i]))
else xrg <- 4.5
vby <- 2*xrg/nn
xr <- seq(-xrg,xrg,by=2*xrg/nn)
y0 <- dqgauss(xr,qv[i])
points (xr,y0,ty='l',col=(i+1))
}
legend(2, 0.4, legend =c(expression(paste(q==-5)),expression(paste(q==0)),
expression(paste(q==1.01)),expression(paste(q==2)),expression(paste(q==2.5)),
expression(paste(q==2.8))),col = c(1,6,5,4,3,2), lty = c(1,1,1,1,1,1))
######
qv <- 0
rr <- rqgauss(2^16,qv)
nn <- 70
xrg <- sqrt((3-qv)/(1-qv))
vby <- 2*xrg/(nn)
xr <- seq(-xrg,xrg,by=vby)
hist (rr,breaks=xr,freq=FALSE,xlab="x",main='')
y <- dqgauss(xr)
lines(xr,y/sum(y*vby),cex=.5,col=2,lty=4)
The q-gaussian Distribution
Description
Density, distribution function, quantile function and random generation for the q-gaussian distribution with parameters mu and sig.
Usage
pqgauss(x, q = 0, mu = 0, sig = 1, lower.tail = TRUE)
Arguments
x |
vector of quantiles. |
q |
entropic index. |
mu |
a value for q-mean. |
sig |
a value for q-variance. |
lower.tail |
logical; if TRUE (default), probabilities are P[X <= x] otherwise, P[X > x]. |
Details
If q , mu and sig values are not specified, they assume the default values of 0, 0 and 1, respectively. Defining Z=(q-1)/(3-q), the q-gaussian distribution has density wrinten as
p(x) = (sig*Beta(alpha/2,1/2))^-1*(1+Z(x-mu)^2/sig^2)^-(1+1/Z)/2
where alpha = 1 - 1/Z when q < 1 and 1/Z when 1 < q < 3.
Value
dqgauss gives the density, pqgauss gives the distribution function, cqgauss gives the quantile function, and rqgauss generates random deviates.
Author(s)
Emerson Luis de Santa Helena , Wagner Santos de Lima
References
Thistleton, W.,Marsh, J. A.,Nelson, K.,Tsallis, C., (2007) IEEE Transactions on Information Theory, 53(12):4805
Tsallis, C., (2009) Introduction to Nonextensive Statistical Mechanics. Springer.
de Santa Helena, E. L., Nascimento, C. M., and Gerhardt, G. J., (2015) Alternative way to charecterize a q-gaussian distribution by a robust heavy tail measurement. Physica A, (435):44-50.
Manuscript submitted for publication (2016) qGaussian: Tools to Explore Applications of Tsallis Statistics
See Also
Distributions for other standard distributions, including dt and dcauchy.
Distributions
Examples
qv <- c(2.8,2.5,2,1.01,0,-5); nn <- 700
xrg <- sqrt((3-qv[6])/(1-qv[6]))
xr <- seq(-xrg,xrg,by=2*xrg/nn)
y0 <- dqgauss(xr,qv[6])
plot(xr,y0,ty='l',xlim=range(-4.5,4.5),ylab='p(x)',xlab='x')
for (i in 1:5){
if (qv[i]< 1) xrg <- sqrt((3-qv[i])/(1-qv[i]))
else xrg <- 4.5
vby <- 2*xrg/nn
xr <- seq(-xrg,xrg,by=2*xrg/nn)
y0 <- dqgauss(xr,qv[i])
points (xr,y0,ty='l',col=(i+1))
}
legend(2, 0.4, legend =c(expression(paste(q==-5)),expression(paste(q==0)),
expression(paste(q==1.01)),expression(paste(q==2)),expression(paste(q==2.5)),
expression(paste(q==2.8))),col = c(1,6,5,4,3,2), lty = c(1,1,1,1,1,1))
######
qv <- 0
rr <- rqgauss(2^16,qv)
nn <- 70
xrg <- sqrt((3-qv)/(1-qv))
vby <- 2*xrg/(nn)
xr <- seq(-xrg,xrg,by=vby)
hist (rr,breaks=xr,freq=FALSE,xlab="x",main='')
y <- dqgauss(xr)
lines(xr,y/sum(y*vby),cex=.5,col=2,lty=4)
qbymc, a q value estimator founded upon medcouple.
Description
Given a random data set, the 'qbymc' uses the medcouple, a robust measure of tail weights, to yield a 'q' value, a characteristic entropic index of the q-gaussian distributions.
Usage
qbymc(x)
Arguments
x |
numeric vector |
Value
a number q < 3, and the standard error.
Author(s)
Emerson Luis de Santa Helena , Wagner Santos de Lima
References
de Santa Helena, E. L., Nascimento, C. M., and Gerhardt, G. J., (2015) Alternative way to charecterize a q-gaussian distribution by a robust heavy tail measurement. Physica A, (435):44-50.
See Also
Robustbase for medcouple.
mc
Examples
set.seed(0002)
rr <- rqgauss(1000,1.333)
qbymc(rr)
The q-gaussian Distribution
Description
Density, distribution function, quantile function and random generation for the q-gaussian distribution with parameters mu and sig.
Usage
rqgauss(n, q = 0, mu = 0, sig = 1, meth = "Box-Muller")
Arguments
n |
number of observations. If length(n) > 1, the length is taken to be the number required. |
q |
entropic index. |
mu |
a value for q-mean. |
sig |
a value for q-variance. |
meth |
method used at random generator |
Details
If q , mu and sig values are not specified, they assume the default values of 0, 0 and 1, respectively. Defining Z=(q-1)/(3-q), the q-gaussian distribution has density wrinten as
p(x) = (sig*Beta(alpha/2,1/2))^-1*(1+Z(x-mu)^2/sig^2)^-(1+1/Z)/2
where alpha = 1 - 1/Z when q < 1 and 1/Z when 1 < q < 3.
For different methods use: meth = "Chaotic" , meth = "Quantile" and meth = "Box-Muller"
Value
dqgauss gives the density, pqgauss gives the distribution function, cqgauss gives the quantile function, and rqgauss generates random deviates.
Author(s)
Emerson Luis de Santa Helena , Wagner Santos de Lima
References
Umeno, K., Sato, A., IEEE Transactions on Information Theory (Volume:59,Issue:5,May 2013).Chaotic Method for Generating q-Gaussian Random Variables.
Thistleton, W.,Marsh, J. A.,Nelson, K.,Tsallis, C., (2007) IEEE Transactions on Information Theory, 53(12):4805
Tsallis, C., (2009) Introduction to Nonextensive Statistical Mechanics. Springer.
de Santa Helena, E. L., Nascimento, C. M., and Gerhardt, G. J., (2015) Alternative way to characterize a q-gaussian distribution by a robust heavy tail measurement. Physica A, (435):44-50.
de Lima, Wagner S., de Santa Helena, E. L., qGaussian: Tools to Explore Applications of Tsallis Statistics. arXiv:1703.06172
See Also
Distributions for other standard distributions, including dt and dcauchy.
Distributions
Examples
qv <- c(2.8,2.5,2,1.01,0,-5); nn <- 700
xrg <- sqrt((3-qv[6])/(1-qv[6]))
xr <- seq(-xrg,xrg,by=2*xrg/nn)
y0 <- dqgauss(xr,qv[6])
plot(xr,y0,ty='l',xlim=range(-4.5,4.5),ylab='p(x)',xlab='x')
for (i in 1:5){
if (qv[i]< 1) xrg <- sqrt((3-qv[i])/(1-qv[i]))
else xrg <- 4.5
vby <- 2*xrg/nn
xr <- seq(-xrg,xrg,by=2*xrg/nn)
y0 <- dqgauss(xr,qv[i])
points (xr,y0,ty='l',col=(i+1))
}
legend(2, 0.4, legend =c(expression(paste(q==-5)),expression(paste(q==0)),
expression(paste(q==1.01)),expression(paste(q==2)),expression(paste(q==2.5)),
expression(paste(q==2.8))),col = c(1,6,5,4,3,2), lty = c(1,1,1,1,1,1))
######
qv <- 0
rr <- rqgauss(2^16,qv)
nn <- 70
xrg <- sqrt((3-qv)/(1-qv))
vby <- 2*xrg/(nn)
xr <- seq(-xrg,xrg,by=vby)
hist (rr,breaks=xr,freq=FALSE,xlab="x",main='')
y <- dqgauss(xr)
lines(xr,y/sum(y*vby),cex=.5,col=2,lty=4)