Help for package qGaussian

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.

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

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

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

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.

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

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)

Chaotic, a random number generator of q-Gaussian random variables.

Description

Usage

Arguments

Value

Author(s)

References

See Also

Examples

The q-gaussian Distribution

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

The q-gaussian Distribution

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

The q-gaussian Distribution

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

qbymc, a q value estimator founded upon medcouple.

Description

Usage

Arguments

Value

Author(s)

References

See Also

Examples

The q-gaussian Distribution

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples