Multivariate Pseudo-Random Number Generation

Description

This package implements the algorithms described in Demirtas (2004) for pseudo-random number generation of 11 multivariate distributions. The following multivariate distributions are available: Normal, t, Uniform, Bernoulli, Hypergeometric, Beta (Dirichlet), Multinomial, Dirichlet-Multinomial, Laplace, Wishart, and Inverted Wishart.

This package contains 11 main functions and 2 auxiliary functions. The methodology for each random-number generation procedure varies and each distribution has its own function. For multivariate normal, draw.d.variate.normal employs the Cholesky decomposition and a vector of univariate normal draws and for multivariate t, draw.d.variate.t employs the Cholesky decomposition and a vector of univariate normal and chi-squared draws. draw.d.variate.uniform is based on cdf of multivariate normal deviates (Falk, 1999) and draw.correlated.binary generates correlated binary variables using an algorithm developed by Park, Park and Shin (1996) and makes use of the auxiliary function loc.min. draw.multivariate.hypergeometric employs sequential generation of succeeding conditionals which are univariate hypergeometric. Furthermore, draw.dirichlet uses the ratios of gamma variates with a common scale parameter and draw.multinomial generates data via sequential generation of marginals which are binomials. draw.dirichlet.multinomial is a mixture distribution of a multinomial that is a realization of a random variable having a Dirichlet distribution. draw.multivariate.laplace is based on generation of a point s on the d-dimensional sphere and utilizes the auxiliary function generate.point.in.sphere. draw.wishart and draw.inv.wishart employs Wishart variates that follow d-variate normal distribution.

Details

Author(s)

Hakan Demirtas, Rawan Allozi, Ran Gao

Maintainer: Ran Gao <rgao8@uic.edu>

References

Demirtas, H. (2004). Pseudo-random number generation in R for commonly used multivariate distributions. Journal of Modern Applied Statistical Methods, 3(2), 485-497.

Falk, M. (1999). A simple approach to the generation of uniformly distributed random variables with prescribed correlations. Communications in Statistics, Simulation and Computation, 28(3), 785-791.

Park, C. G., Park, T., & Shin D. W. (1996). A simple method for generating correlated binary variates. The American Statistician, 50(4), 306-310.

Generation of Correlated Binary Data

Description

This function implements pseudo-random number generation for a multivariate Bernoulli distribution (correlated binary data).

Usage

draw.correlated.binary(no.row,d,prop.vec,corr.mat)

Arguments

Value

A no.row \times d matrix of generated data.

References

Park, C. G., Park, T., & Shin D. W. (1996). A simple method for generating correlated binary variates. The American Statistician, 50(4), 306-310.

See Also

loc.min

Examples

cmat<-matrix(c(1,0.2,0.3,0.2,1,0.2,0.3,0.2,1), nrow=3, ncol=3)
propvec=c(0.3,0.5,0.7)

mydata=draw.correlated.binary(no.row=1e5,d=3,prop.vec=propvec,corr.mat=cmat)
apply(mydata,2,mean)-propvec
cor(mydata)-cmat

Pseudo-Random Number Generation under Multivariate Normal Distribution

Description

This function implements pseudo-random number generation for a multivariate normal distribution with pdf

f(x|\mu,\Sigma)=c\exp{(-\frac{1}{2}(x-\mu)^{T}\Sigma^{-1}(x-\mu))}

for -\infty < x < \infty and c=(2\pi)^{-d/2}|\Sigma|^{-1/2}, \Sigma is symmetric and positive definite, where \mu and \Sigma are the mean vector and the variance-covariance matrix, respectively.

Usage

draw.d.variate.normal(no.row,d,mean.vec,cov.mat)

Arguments

Value

A no.row \times d matrix of generated data.

Examples

cmat<-matrix(c(1,0.2,0.3,0.2,1,0.2,0.3,0.2,1), nrow=3, ncol=3)
meanvec=c(0,3,7)
mydata=draw.d.variate.normal(no.row=1e5,d=3,mean.vec=meanvec,cov.mat=cmat)
apply(mydata,2,mean)-meanvec
cor(mydata)-cmat

Pseudo-Random Number Generation under Multivariate t Distribution

Description

This function implements pseudo-random number generation for a multivariate t distribution with pdf

f(x|\mu, \Sigma, \nu)=c\left(1+\frac{1}{\nu}(x-\mu)^{T}\Sigma^{-1}(x-\mu)\right)^{-(\nu+d)/2}

for -\infty < x < \infty and c=\frac{\Gamma((\nu+d)/2)}{\Gamma(\nu/2)(\nu\pi)^{d/2}}|\Sigma|^{-1/2}, \Sigma is symmetric and positive definite, \nu>0, where \mu, \Sigma, and \nu are the mean vector, the variance-covariance matrix, and the degrees of freedom, respectively.

Usage

draw.d.variate.t(dof,no.row,d,mean.vec,cov.mat)

Arguments

Value

A no.row \times d matrix of generated data.

Examples

cmat<-matrix(c(1,0.2,0.3,0.2,1,0.2,0.3,0.2,1), nrow=3, ncol=3)
meanvec=c(0,3,7)
mydata=draw.d.variate.t(dof=5,no.row=1e5,d=3,mean.vec=meanvec,cov.mat=cmat)
apply(mydata,2,mean)-meanvec
cor(mydata)-cmat

Pseudo-Random Number Generation under Multivariate Uniform Distribution

Description

This function implements pseudo-random number generation for a multivariate uniform distribution with specified mean vector and covariance matrix.

Usage

draw.d.variate.uniform(no.row,d,cov.mat)

Arguments

Value

A no.row \times d matrix of generated data.

References

Falk, M. (1999). A simple approach to the generation of uniformly distributed random variables with prescribed correlations. Communications in Statistics, Simulation and Computation, 28(3), 785-791.

Examples

cmat<-matrix(c(1,0.2,0.3,0.2,1,0.2,0.3,0.2,1), nrow=3, ncol=3)
mydata=draw.d.variate.uniform(no.row=1e5,d=3,cov.mat=cmat)
apply(mydata,2,mean)-rep(0.5,3)
cor(mydata)-cmat

Pseudo-Random Number Generation under Multivariate Beta (Dirichlet) Distribution

Description

This function implements pseudo-random number generation for a multivariate beta (Dirichlet) distribution with pdf

f(x|\alpha_{1},...,\alpha_{d})=\frac{\Gamma(\sum_{j=1}^{d}\alpha_{j})}{\prod_{j=1}^{d}\Gamma(\alpha_{j})} \prod_{j=1}^{d}x_{j}^{\alpha_{j}-1}

for \alpha_{j}>0, x_{j}\geq 0, and \sum_{j=1}^{d}x_{j}=1, where \alpha_{1},...,\alpha_{d} are the shape parameters and \beta is a common scale paramter.

Usage

draw.dirichlet(no.row,d,alpha,beta)

Arguments

Value

A no.row \times d matrix of generated data.

Examples

alpha.vec=c(1,3,4,4)
mydata=draw.dirichlet(no.row=1e5,d=4,alpha=alpha.vec,beta=2)
apply(mydata,2,mean)-alpha.vec/sum(alpha.vec)

Pseudo-Random Number Generation under Dirichlet-Multinomial Distribution

Description

This function implements pseudo-random number generation for a Dirichlet-multinomial distribution. This is a mixture distribution that is multinomial with parameter \theta that is a realization of a random variable having a Dirichlet distribution with shape vector \alpha. N is the sample size and \beta is a common scale parameter.

Usage

draw.dirichlet.multinomial(no.row,d,alpha,beta,N)

Arguments

Value

A no.row \times d matrix of generated data.

See Also

draw.dirichlet, draw.multinomial

Examples

alpha.vec=c(1,3,4,4) ; N=3
mydata=draw.dirichlet.multinomial(no.row=1e5,d=4,alpha=alpha.vec,beta=2, N=3)
apply(mydata,2,mean)-N*alpha.vec/sum(alpha.vec)

Pseudo-Random Number Generation under Inverted Wishart Distribution

Description

This function implements pseudo-random number generation for an inverted Wishart distribution with pdf

f(x|\nu,\Sigma)=(2^{\nu d/2}\pi^{d(d-1)/4}\prod_{i=1}^{d}\Gamma((\nu+1-i)/2))^{-1}|\Sigma|^{\nu/2}|x|^{-(\nu+d+1)/2}\exp(-\frac{1}{2}tr(\Sigma x^{-1}))

x is positive definite, \nu \geq d, and \Sigma^{-1} is symmetric and positive definite, where \nu and \Sigma^{-1} are the degrees of freedom and the inverse scale matrix, respectively.

Usage

draw.inv.wishart(no.row,d,nu,inv.sigma)

Arguments

Value

A no.row \times d^2 matrix ofcontaining Wishart deviates in the form of rows. To obtain the Inverted-Wishart matrix, convert each row to a matrix where rows are filled first.

See Also

draw.wishart

Examples

mymat<-matrix(c(1,0.2,0.3,0.2,1,0.2,0.3,0.2,1), nrow=3, ncol=3)
draw.inv.wishart(no.row=1e5,d=3,nu=5,inv.sigma=mymat)

Pseudo-Random Number Generation under Multivariate Multinomial Distribution

Description

This function implements pseudo-random number generation for a multivariate multinomial distribution with pdf

f(x|\theta_{1},...,\theta_{d})=\frac{N!}{\prod x_{j}!}\prod_{j=1}^{d}\theta_{j}^{x_{j}}

for 0<\theta_{j}<1, x_{j}\geq 0, and \sum_{j=1}^{d}x_{j}=N, where \theta_{1},...,\theta_{d} are cell probabilities and N is the size.

Usage

draw.multinomial(no.row,d,theta,N)

Arguments

Value

A no.row \times d matrix of generated data.

Examples

theta.vec=c(0.3,0.3,0.25,0.15) ; N=4
mydata=draw.multinomial(no.row=1e5,d=4,theta=c(0.3,0.3,0.25,0.15),N=4)
apply(mydata,2,mean)-N*theta.vec

Pseudo-Random Number Generation under Multivariate Hypergeometric Distribution

Description

This function implements pseudo-random number generation for a multivariate hypergeometric distribution.

Usage

draw.multivariate.hypergeometric(no.row,d,mean.vec,k)

Arguments

Value

A no.row \times d matrix of generated data.

References

Demirtas, H. (2004). Pseudo-random number generation in R for commonly used multivariate distributions. Journal of Modern Applied Statistical Methods, 3(2), 485-497.

Examples

meanvec=c(10,10,12) ; myk=5
mydata=draw.multivariate.hypergeometric(no.row=1e5,d=3,mean.vec=meanvec,k=myk)
apply(mydata,2,mean)-myk*meanvec/sum(meanvec)

Pseudo-Random Number Generation under Multivariate Laplace Distribution

Description

This function implements pseudo-random number generation for a multivariate Laplace (double exponential) distribution with pdf

f(x|\mu,\Sigma,\gamma)=c\exp(-((x-\mu)^{T}\Sigma^{-1}(x-\mu))^{\gamma/2})

for -\infty < x < \infty and c=\frac{\gamma\Gamma(d/2)}{2\pi^{d/2}\Gamma(d/\gamma)}|\Sigma|^{-1/2}, \Sigma is symmetric and positive definite, where \mu, \Sigma, and \gamma are the mean vector, the variance-covariance matrix, and the shape parameter, respectively.

Usage

draw.multivariate.laplace(no.row,d,gamma,mu,Sigma)

Arguments

Value

A no.row \times d matrix of generated data.

References

Ernst, M. D. (1998). A multivariate generalized Laplace distribution. Computational Statistics, 13, 227-232.

See Also

generate.point.in.sphere

Examples

cmat<-matrix(c(1,0.2,0.3,0.2,1,0.2,0.3,0.2,1), nrow=3, ncol=3)
mu.vec=c(0,3,7)
mydata=draw.multivariate.laplace(no.row=1e5,d=3,gamma=2,mu=mu.vec,Sigma=cmat)

apply(mydata,2,mean)-mu.vec
cor(mydata)-cmat

Pseudo-Random Number Generation under Wishart Distribution

Description

This function implements pseudo-random number generation for a Wishart distribution with pdf

f(x|\nu,\Sigma)=(2^{\nu d/2}\pi^{d(d-1)/4}\prod_{i=1}^{d}\Gamma((\nu+1-i)/2))^{-1}|\Sigma|^{-\nu/2}|x|^{(\nu-d-1)/2}\exp(-\frac{1}{2}tr(\Sigma^{-1}x))

x is positive definite, \nu \geq d, and \Sigma is symmetric and positive definite, where \nu and \Sigma are positive definite and the scale matrix, respectively.

Usage

draw.wishart(no.row,d,nu,sigma)

Arguments

Value

A no.row \times d^2 matrix of Wishart deviates in the form of rows.To obtain the Wishart matrix, convert each row to a matrix where rows are filled first.

See Also

draw.d.variate.normal

Examples

mymat<-matrix(c(1,0.2,0.3,0.2,1,0.2,0.3,0.2,1), nrow=3, ncol=3)
draw.wishart(no.row=1e5,d=3,nu=5,sigma=mymat)

Point Generation for a Sphere

Description

This function generates s points on a d-dimensional sphere.

Usage

generate.point.in.sphere(no.row,d)

Arguments

Value

A no.row \times d matrix of coordinates of points in sphere.

References

Marsaglia, G. (1972). Choosing a point from the surface of a sphere. Annals of Mathematical Statistics, 43, 645-646.

Examples

generate.point.in.sphere(no.row=1e5,d=3)

Minimum Location Finder

Description

This function identifies the location of the minimum value in a square matrix.

Usage

loc.min(my.mat,d)

Arguments

Value

A vector containing the row and column number of the minimum value.

Examples

cmat<-matrix(c(1,0.2,0.3,0.2,1,0.2,0.3,0.2,1), nrow=3, ncol=3)
loc.min(my.mat=cmat, d=3)