Package MultiRNG. January 10, 2018

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1 Type Package Package MultiRNG January 10, 2018 Title Multivariate Pseuo-Ranom Number Generation Version 1.1 Date Author Hakan Demirtas, Rawan Allozi Maintainer Rawan Allozi Pseuo-ranom number generation for 11 multivariate istributions: Normal, t, Uniform, Bernoulli, Hypergeometric, Beta (Dirichlet), Multinomial, Dirichlet- Multinomial, Laplace, Wishart, an Inverte Wishart. License GPL-2 GPL-3 NeesCompilation no Repository CRAN Date/Publication :14:45 UTC R topics ocumente: MultiRNG-package raw.correlate.binary raw..variate.normal raw..variate.t raw..variate.uniform raw.irichlet raw.irichlet.multinomial raw.inv.wishart raw.multinomial raw.multivariate.hypergeometric raw.multivariate.laplace raw.wishart generate.point.in.sphere loc.min Inex 16 1

2 2 MultiRNG-package MultiRNG-package Multivariate Pseuo-Ranom Number Generation This package implements the algorithms escribe in Demirtas (2004) for pseuo-ranom number generation of 11 multivariate istributions. The following multivariate istributions are available: Normal, t, Uniform, Bernoulli, Hypergeometric, Beta (Dirichlet), Multinomial, Dirichlet- Multinomial, Laplace, Wishart, an Inverte Wishart. This package contains 11 main functions an 2 auxiliary functions. The methoology for each ranom-number generation proceure varies an each istribution has its own function. For multivariate normal, raw..variate.normal utilizes the Cholesky ecomposition an a vector of univariate normal raws an for multivariate t, raw..variate.t employs the Cholesky ecomposition an a vector of univariate normal an chi-square raws. raw..variate.uniform is base on cf of multivariate normal eviates (Falk, 1999) an raw.correlate.binary generates correlate binary variables using an algorithm evelope by Park, Park an Shin (1996) an makes use of the auxiliary function loc.min. raw.multivariate.hypergeometric utilizes sequential generation of succeeing conitionals which are univariate hypergeometric. Furthermore, raw.irichlet uses the ratios of gamma variates with a common scale parameter an raw.multinomial generates ata via sequential generation of marginals which are binomials. raw.irichlet.multinomial is a mixture istribution of a multinomial that is a realization of a ranom variable having a Dirichlet istribution. raw.multivariate.laplace is base on generation of a point s on the -imensional sphere an utilizes the auxiliary function generate.point.in.sphere. raw.wishart an raw.inv.wishart utlize Wishart variates that follow -variate normal istribution. Details Package: UnivRNG Type: Package Version: 1.1 Date: License: GPL-2 GPL-3 Author(s) Hakan Demirtas, Rawan Allozi Maintainer: Rawan Allozi <ralloz2@uic.eu> References Demirtas, H. (2004). Pseuo-ranom number generation in R for commonly use multivariate istributions. Journal of Moern Applie Statistical Methos, 3(2),

3 raw.correlate.binary 3 Falk, M. (1999). A simple approach to the generation of uniformly istribute ranom variables with prescribe correlations. Communications in Statistics, Simulation an Computation, 28(3), Park, C. G., Park, T., & Shin D. W. (1996). A simple metho for generating correlate binary variates. The American Statistician, 50(4), raw.correlate.binary Generation of Correlate Binary Data This function implements pseuo-ranom number generation for a multivariate Bernoulli istribution (correlate binary ata). raw.correlate.binary(,,prop.vec,corr.mat) prop.vec corr.mat Vector of means. Correlation matrix. A matrix of generate ata. References Park, C. G., Park, T., & Shin D. W. (1996). A simple metho for generating correlate binary variates. The American Statistician, 50(4), See Also loc.min 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) myata=raw.correlate.binary(=1e5,=3,prop.vec=propvec,corr.mat=cmat) apply(myata,2,mean)-propvec cor(myata)-cmat

4 4 raw..variate.normal raw..variate.normal Pseuo-Ranom Number Generation uner Multivariate Normal Distribution This function implements pseuo-ranom number generation for a multivariate normal istribution with pf f(x µ, Σ) = c exp ( 1 2 (x µ)t Σ 1 (x µ)) for < x < an c = (2π) /2 Σ 1/2, Σ is symmetric an positive efinite, where µ an Σ are the mean vector an the variance-covariance matrix, respectively. raw..variate.normal(,,mean.vec,cov.mat) mean.vec cov.mat Vector of means. Variance-covariance matrix. A matrix of generate ata. 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) myata=raw..variate.normal(=1e5,=3,mean.vec=meanvec,cov.mat=cmat) apply(myata,2,mean)-meanvec cor(myata)-cmat

5 raw..variate.t 5 raw..variate.t Pseuo-Ranom Number Generation uner Multivariate t Distribution This function implements pseuo-ranom number generation for a multivariate t istribution with pf ( f(x µ, Σ, ν) = c ) (ν+)/2 ν (x µ)t Σ 1 (x µ) for < x < an c = Γ((ν+)/2) Σ 1/2, Σ is symmetric an positive efinite, ν > 0, Γ(ν/2)(νπ) /2 where µ, Σ, an ν are the mean vector, the variance-covariance matrix, an the egrees of freeom, respectively. raw..variate.t(of,,,mean.vec,cov.mat) of mean.vec cov.mat Degrees of freeom. Vector of means. Variance-covariance matrix. A matrix of generate ata. 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) myata=raw..variate.t(of=5,=1e5,=3,mean.vec=meanvec,cov.mat=cmat) apply(myata,2,mean)-meanvec cor(myata)-cmat

6 6 raw..variate.uniform raw..variate.uniform Pseuo-Ranom Number Generation uner Multivariate Uniform Distribution This function implements pseuo-ranom number generation for a multivariate uniform istribution with specifie mean vector an covariance matrix. raw..variate.uniform(,,cov.mat) cov.mat Variance-covariance matrix. A matrix of generate ata. References Falk, M. (1999). A simple approach to the generation of uniformly istribute ranom variables with prescribe correlations. Communications in Statistics, Simulation an Computation, 28(3), cmat<-matrix(c(1,0.2,0.3,0.2,1,0.2,0.3,0.2,1), nrow=3, ncol=3) myata=raw..variate.uniform(=1e5,=3,cov.mat=cmat) apply(myata,2,mean)-rep(0.5,3) cor(myata)-cmat

7 raw.irichlet 7 raw.irichlet Pseuo-Ranom Number Generation uner Multivariate Beta (Dirichlet) Distribution This function implements pseuo-ranom number generation for a multivariate beta (Dirichlet) istribution with pf f(x α 1,..., α ) = Γ( j=1 α j) j=1 Γ(α j) j=1 x αj 1 j for α j > 0, x j 0, an j=1 x j = 1, where α 1,..., α are the shape parameters an β is a common scale paramter. raw.irichlet(,,alpha,beta) alpha beta Vector of shape parameters. Scale parameter common to variables. A matrix of generate ata. alpha.vec=c(1,3,4,4) myata=raw.irichlet(=1e5,=4,alpha=alpha.vec,beta=2) apply(myata,2,mean)-alpha.vec/sum(alpha.vec)

8 8 raw.irichlet.multinomial raw.irichlet.multinomial Pseuo-Ranom Number Generation uner Dirichlet-Multinomial Distribution This function implements pseuo-ranom number generation for a Dirichlet-multinomial istribution. This is a mixture istribution that is multinomial with parameter θ that is a realization of a ranom variable having a Dirichlet istribution with shape vector α. N is the sample size an β is a common scale parameter. raw.irichlet.multinomial(,,alpha,beta,n) alpha beta N Vector of shape parameters. Scale parameter common to variables. Sample size. A matrix of generate ata. See Also raw.irichlet, raw.multinomial alpha.vec=c(1,3,4,4) ; N=3 myata=raw.irichlet.multinomial(=1e5,=4,alpha=alpha.vec,beta=2, N=3) apply(myata,2,mean)-n*alpha.vec/sum(alpha.vec)

9 raw.inv.wishart 9 raw.inv.wishart Pseuo-Ranom Number Generation uner Inverte Wishart Distribution This function implements pseuo-ranom number generation for an inverte Wishart istribution with pf f(x ν, Σ) = (2 ν/2 π ( 1)/4 Γ((ν + 1 i)/2)) 1 Σ ν/2 x (ν++1)/2 exp( 1 2 tr(σx 1 )) i=1 x is positive efinite, ν, an Σ 1 is symmetric an positive efinite, where ν an Σ 1 are the egrees of freeom an the inverse scale matrix, respectively. raw.inv.wishart(,,nu,inv.sigma) nu inv.sigma Degrees of freeom. Inverse scale matrix. A 2 matrix ofcontaining Wishart eviates in the form of rows. To obtain the Inverte- Wishart matrix, convert each row to a matrix where rows are fille first. See Also raw.wishart mymat<-matrix(c(1,0.2,0.3,0.2,1,0.2,0.3,0.2,1), nrow=3, ncol=3) raw.inv.wishart(=1e5,=3,nu=5,inv.sigma=mymat)

10 10 raw.multinomial raw.multinomial Pseuo-Ranom Number Generation uner Multivariate Multinomial Distribution This function implements pseuo-ranom number generation for a multivariate multinomial istribution with pf f(x θ 1,..., θ ) = N! xj! j=1 θ xj j for 0 < θ j < 1, x j 0, an j=1 x j = N, where θ 1,..., θ are cell probabilities an N is the size. raw.multinomial(,,theta,n) theta Vector of cell probabilities. N Sample Size. Must be at least 2. A matrix of generate ata. theta.vec=c(0.3,0.3,0.25,0.15) ; N=4 myata=raw.multinomial(=1e5,=4,theta=c(0.3,0.3,0.25,0.15),n=4) apply(myata,2,mean)-n*theta.vec

11 raw.multivariate.hypergeometric 11 raw.multivariate.hypergeometric Pseuo-Ranom Number Generation uner Multivariate Hypergeometric Distribution This function implements pseuo-ranom number generation for a multivariate hypergeometric istribution. raw.multivariate.hypergeometric(,,mean.vec,k) mean.vec k Number of items in each category. Number of items to be sample. Must be a positive integer. A matrix of generate ata. References Demirtas, H. (2004). Pseuo-ranom number generation in R for commonly use multivariate istributions. Journal of Moern Applie Statistical Methos, 3(2), meanvec=c(10,10,12) ; myk=5 myata=raw.multivariate.hypergeometric(=1e5,=3,mean.vec=meanvec,k=myk) apply(myata,2,mean)-myk*meanvec/sum(meanvec)

12 12 raw.multivariate.laplace raw.multivariate.laplace Pseuo-Ranom Number Generation uner Multivariate Laplace Distribution This function implements pseuo-ranom number generation for a multivariate Laplace (ouble exponential) istribution with pf f(x µ, Σ, γ) = c exp( ((x µ) T Σ 1 (x µ)) γ/2 ) for < x < an c = γγ(/2) 2π /2 Γ(/γ) Σ 1/2, Σ is symmetric an positive efinite, where µ, Σ, an γ are the mean vector, the variance-covariance matrix, an the shape parameter, respectively. raw.multivariate.laplace(,,gamma,mu,sigma) gamma mu Sigma Shape parameter. Vector of means. Variance-covariance matrix. A matrix of generate ata. References Ernst, M. D. (1998). A multivariate generalize Laplace istribution. Computational Statistics, 13, See Also generate.point.in.sphere 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) myata=raw.multivariate.laplace(=1e5,=3,gamma=2,mu=mu.vec,sigma=cmat) apply(myata,2,mean)-mu.vec cor(myata)-cmat

13 raw.wishart 13 raw.wishart Pseuo-Ranom Number Generation uner Wishart Distribution This function implements pseuo-ranom number generation for a Wishart istribution with pf f(x ν, Σ) = (2 ν/2 π ( 1)/4 Γ((ν + 1 i)/2)) 1 Σ ν/2 x (ν 1)/2 exp( 1 2 tr(σ 1 x)) i=1 x is positive efinite, ν, an Σ is symmetric an positive efinite, where ν an Σ are positive efinite an the scale matrix, respectively. raw.wishart(,,nu,sigma) nu sigma Degrees of freeom. Scale matrix. A 2 matrix of Wishart eviates in the form of rows.to obtain the Wishart matrix, convert each row to a matrix where rows are fille first. See Also raw..variate.normal mymat<-matrix(c(1,0.2,0.3,0.2,1,0.2,0.3,0.2,1), nrow=3, ncol=3) raw.wishart(=1e5,=3,nu=5,sigma=mymat)

14 14 loc.min generate.point.in.sphere Point Generation for a Sphere This function generates s points on a -imensional sphere. generate.point.in.sphere(,) A matrix of coorinates of points in sphere. References Marsaglia, G. (1972). Choosing a point from the surface of a sphere. Annals of Mathematical Statistics, 43, generate.point.in.sphere(=1e5,=3) loc.min Minimum Location Finer This function ientifies the location of the minimum value in a square matrix. loc.min(my.mat,) my.mat A square matrix. Dimensions of the matrix.

15 loc.min 15 A vector containing the row an column number of the minimum value. 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, =3)

16 Inex raw.correlate.binary, 3 raw..variate.normal, 4, 13 raw..variate.t, 5 raw..variate.uniform, 6 raw.irichlet, 7, 8 raw.irichlet.multinomial, 8 raw.inv.wishart, 9 raw.multinomial, 8, 10 raw.multivariate.hypergeometric, 11 raw.multivariate.laplace, 12 raw.wishart, 9, 13 generate.point.in.sphere, 12, 14 loc.min, 3, 14 MultiRNG (MultiRNG-package), 2 MultiRNG-package, 2 16