# Simulated amount datasets Several simulated datasets intended as reference examples for various conceptual and statistical models of compositions and amounts. data ```r data(SimulatedAmounts) ``` ## Format Data matrices with 60 cases and 3 or 5 variables. ## Source The datasets are simulated for this package and are under the GNU Public Library Licence Version 2 or newer. ## Details The statistical analysis of amounts and compositions is set to discussion. Four essentially different approaches are provided in this package around the classes "rplus", "aplus", "rcomp", "acomp". There is no absolutely "right" approach, since there is a conection between these approaches and the processes originating the data. We provide here simulated standard datasets and the corresponding simulation procedures following these several models to provide "good" analysis examples and to show how these models actually look like in data. The data sets are simulated according to correlated lognormal distributions (sa.lognormals, sa.lognormal5), winsorised correlated normal distributions (sa.tnormals, sa.tnormal5), Dirichlet distribution on the simplex (sa.dirichlet, sa.dirichlet5), uniform distribution on the simplex (sa.uniform, sa.uniform5), and a grouped dataset (sa.groups, sa.groups5) with three groups (given in sa.groups.area and sa.groups5.area) all distributed accordingly with a lognormal distribution with group-dependent means. We can imagine that amounts evolve in nature e.g. in part of the soil they are diluted and transported in a transport medium, usually water, which comes from independent source (the rain, for instance) and this new composition is normalized by taking a sample of standard size. For each of the datasets sa.X there is a corresponding sa.X `.dil` dataset which is build by simulating exactly that process on the corresponding sa.X dataset . The amounts in the sa.X `.dil` are given in ppm. This idea of a transport medium is a major argument for a compositional approach, because the total amount given by the sum of the parts is induced by the dilution given by the medium and thus non-informative for the original process investigated. If we imagine now these amounts flowing into a river and sedimenting, the different contributions are accumulated along the river and renormalized to a unit portion on taking samples again. For each of the dataset sa.X `.dil` there is a corresponding sa.X `.mix` dataset which is built from the corresponding sa.X dataset by simulating exactly that accumulation process. Mixing of different compositions is a major argument against the log based approaches (`aplus`, `acomp`) since mixing is a highly nonlinear operation in terms of log-ratios. ## Examples ```r data(SimulatedAmounts) plot.acomp(sa.lognormals) plot.acomp(sa.lognormals.dil) plot.acomp(sa.lognormals.mix) plot.acomp(sa.lognormals5) plot.acomp(sa.lognormals5.dil) plot.acomp(sa.lognormals5.mix) plot(acomp(sa.missings)) plot(acomp(sa.missings5)) #library(MASS) plot.rcomp(sa.tnormals) plot.rcomp(sa.tnormals.dil) plot.rcomp(sa.tnormals.mix) plot.rcomp(sa.tnormals5) plot.rcomp(sa.tnormals5.dil) plot.rcomp(sa.tnormals5.mix) plot.acomp(sa.groups,col=as.numeric(sa.groups.area),pch=20) plot.acomp(sa.groups.dil,col=as.numeric(sa.groups.area),pch=20) plot.acomp(sa.groups.mix,col=as.numeric(sa.groups.area),pch=20) plot.acomp(sa.groups5,col=as.numeric(sa.groups.area),pch=20) plot.acomp(sa.groups5.dil,col=as.numeric(sa.groups.area),pch=20) plot.acomp(sa.groups5.mix,col=as.numeric(sa.groups.area),pch=20) plot.acomp(sa.uniform) plot.acomp(sa.uniform.dil) plot.acomp(sa.uniform.mix) plot.acomp(sa.uniform5) plot.acomp(sa.uniform5.dil) plot.acomp(sa.uniform5.mix) plot.acomp(sa.dirichlet) plot.acomp(sa.dirichlet.dil) plot.acomp(sa.dirichlet.mix) plot.acomp(sa.dirichlet5) plot.acomp(sa.dirichlet5.dil) plot.acomp(sa.dirichlet5.mix) # The data was simulated with the following commands: #library(MASS) dilution <- function(x) {clo(cbind(x,exp(rnorm(nrow(x),5,1))))[,1:ncol(x)]*1E6} seqmix <- function(x) {clo(apply(x,2,cumsum))*1E6} vars <- c("Cu","Zn","Pb") vars5 <- c("Cu","Zn","Pb","Cd","Co") sa.lognormals <- structure(exp(matrix(rnorm(3*60),ncol=3) %*% chol(matrix(c(1,0.8,-0.2,0.8,1, -0.2,-0.2,-0.2,1),ncol=3))+ matrix(rep(c(1:3),each=60),ncol=3)), dimnames=list(NULL,vars)) plot.acomp(sa.lognormals) pairs(sa.lognormals) sa.lognormals.dil <- dilution(sa.lognormals) plot.acomp(sa.lognormals.dil) pairs(sa.lognormals.dil) sa.lognormals.mix <- seqmix(sa.lognormals.dil) plot.acomp(sa.lognormals.mix) pairs(sa.lognormals.mix) sa.lognormals5 <- structure(exp(matrix(rnorm(5*60),ncol=5) %*% chol(matrix(c(1,0.8,-0.2,0,0, 0.8,1,-0.2,0,0, -0.2,-0.2,1,0,0, 0,0,0,5,4.9, 0,0,0,4.9,5),ncol=5))+ matrix(rep(c(1:3,-2,-2),each=60),ncol=5)), dimnames=list(NULL,vars5)) plot.acomp(sa.lognormals5) pairs(sa.lognormals5) sa.lognormals5.dil <- dilution(sa.lognormals5) plot.acomp(sa.lognormals5.dil) pairs(sa.lognormals5.dil) sa.lognormals5.mix <- seqmix(sa.lognormals5.dil) plot.acomp(sa.lognormals5.mix) pairs(sa.lognormals5.mix) sa.groups.area <- factor(rep(c("Upper","Middle","Lower"),each=20)) sa.groups <- structure(exp(matrix(rnorm(3*20*3),ncol=3) %*% chol(0.5*matrix(c(1,0.8,-0.2,0.8,1, -0.2,-0.2,-0.2,1),ncol=3))+ matrix(rep(c(1,2,2.5,2,2.9,5,4,2,5), each=20),ncol=3)), dimnames=list(NULL,c("clay","sand","gravel"))) plot.acomp(sa.groups,col=as.numeric(sa.groups.area),pch=20) pairs(sa.lognormals,col=as.numeric(sa.groups.area),pch=20) sa.groups.dil <- dilution(sa.groups) plot.acomp(sa.groups.dil,col=as.numeric(sa.groups.area),pch=20) pairs(sa.groups.dil,col=as.numeric(sa.groups.area),pch=20) sa.groups.mix <- seqmix(sa.groups.dil) plot.acomp(sa.groups.mix,col=as.numeric(sa.groups.area),pch=20) pairs(sa.groups.mix,col=as.numeric(sa.groups.area),pch=20) sa.groups5.area <- factor(rep(c("Upper","Middle","Lower"),each=20)) sa.groups5 <- structure(exp(matrix(rnorm(5*20*3),ncol=5) %*% chol(matrix(c(1,0.8,-0.2,0,0, 0.8,1,-0.2,0,0, -0.2,-0.2,1,0,0, 0,0,0,5,4.9, 0,0,0,4.9,5),ncol=5))+ matrix(rep(c(1,2,2.5, 2,2.9,5, 4,2.5,0, -2,-1,-1, -1,-2,-3), each=20),ncol=5)), dimnames=list(NULL, vars5)) plot.acomp(sa.groups5,col=as.numeric(sa.groups5.area),pch=20) pairs(sa.groups5,col=as.numeric(sa.groups5.area),pch=20) sa.groups5.dil <- dilution(sa.groups5) plot.acomp(sa.groups5.dil,col=as.numeric(sa.groups5.area),pch=20) pairs(sa.groups5.dil,col=as.numeric(sa.groups5.area),pch=20) sa.groups5.mix <- seqmix(sa.groups5.dil) plot.acomp(sa.groups5.mix,col=as.numeric(sa.groups5.area),pch=20) pairs(sa.groups5.mix,col=as.numeric(sa.groups5.area),pch=20) sa.tnormals <- structure(pmax(matrix(rnorm(3*60),ncol=3) %*% chol(matrix(c(1,0.8,-0.2,0.8,1, -0.2,-0.2,-0.2,1),ncol=3))+ matrix(rep(c(0:2),each=60),ncol=3),0), dimnames=list(NULL,c("clay","sand","gravel"))) plot.rcomp(sa.tnormals) pairs(sa.tnormals) sa.tnormals.dil <- dilution(sa.tnormals) plot.acomp(sa.tnormals.dil) pairs(sa.tnormals.dil) sa.tnormals.mix <- seqmix(sa.tnormals.dil) plot.acomp(sa.tnormals.mix) pairs(sa.tnormals.mix) sa.tnormals5 <- structure(pmax(matrix(rnorm(5*60),ncol=5) %*% chol(matrix(c(1,0.8,-0.2,0,0, 0.8,1,-0.2,0,0, -0.2,-0.2,1,0,0, 0,0,0,0.05,0.049, 0,0,0,0.049,0.05),ncol=5))+ matrix(rep(c(0:2,0.1,0.1),each=60),ncol=5),0), dimnames=list(NULL, vars5)) plot.rcomp(sa.tnormals5) pairs(sa.tnormals5) sa.tnormals5.dil <- dilution(sa.tnormals5) plot.acomp(sa.tnormals5.dil) pairs(sa.tnormals5.dil) sa.tnormals5.mix <- seqmix(sa.tnormals5.dil) plot.acomp(sa.tnormals5.mix) pairs(sa.tnormals5.mix) sa.dirichlet <- sapply(c(clay=0.2,sand=2,gravel=3),rgamma,n=60) colnames(sa.dirichlet) <- vars plot.acomp(sa.dirichlet) pairs(sa.dirichlet) sa.dirichlet.dil <- dilution(sa.dirichlet) plot.acomp(sa.dirichlet.dil) pairs(sa.dirichlet.dil) sa.dirichlet.mix <- seqmix(sa.dirichlet.dil) plot.acomp(sa.dirichlet.mix) pairs(sa.dirichlet.mix) sa.dirichlet5 <- sapply(c(clay=0.2,sand=2,gravel=3,humus=0.1,plant=0.1),rgamma,n=60) colnames(sa.dirichlet5) <- vars5 plot.acomp(sa.dirichlet5) pairs(sa.dirichlet5) sa.dirichlet5.dil <- dilution(sa.dirichlet5) plot.acomp(sa.dirichlet5.dil) pairs(sa.dirichlet5.dil) sa.dirichlet5.mix <- seqmix(sa.dirichlet5.dil) plot.acomp(sa.dirichlet5.mix) pairs(sa.dirichlet5.mix) sa.uniform <- sapply(c(clay=1,sand=1,gravel=1),rgamma,n=60) colnames(sa.uniform) <- vars plot.acomp(sa.uniform) pairs(sa.uniform) sa.uniform.dil <- dilution(sa.uniform) plot.acomp(sa.uniform.dil) pairs(sa.uniform.dil) sa.uniform.mix <- seqmix(sa.uniform.dil) plot.acomp(sa.uniform.mix) pairs(sa.uniform.mix) sa.uniform5 <- sapply(c(clay=1,sand=1,gravel=1,humus=1,plant=1),rgamma,n=60) colnames(sa.uniform5) <- vars5 plot.acomp(sa.uniform5) pairs(sa.uniform5) sa.uniform5.dil <- dilution(sa.uniform5) plot.acomp(sa.uniform5.dil) pairs(sa.uniform5.dil) sa.uniform5.mix <- seqmix(sa.uniform5.dil) plot.acomp(sa.uniform5.mix) pairs(sa.uniform5.mix) tmp<-set.seed(1400) A <- matrix(c(0.1,0.2,0.3,0.1),nrow=2) Mvar <- 0.1*ilrvar2clr(A %*% t(A)) Mcenter <- acomp(c(1,2,1)) typicalData <- rnorm.acomp(100,Mcenter,Mvar) # main population colnames(typicalData)<-c("A","B","C") # A dataset without outliers sa.outliers1 <- acomp(rnorm.acomp(100,Mcenter,Mvar)) # A dataset with 10% data with a large error in the first component sa.outliers2 <- acomp(rbind(typicalData+rbinom(100,1,p=0.1)*rnorm(100)*acomp(c(4,1,1)))) # A dataset with a single outlier sa.outliers3 <- acomp(rbind(typicalData,acomp(c(0.5,1.5,2)))) colnames(sa.outliers3)<-colnames(typicalData) tmp<-set.seed(30) rcauchy.acomp <- function (n, mean, var){ D <- gsi.getD(mean)-1 perturbe(ilrInv(matrix(rnorm(n*D)/rep(rnorm(n),D), ncol = D) %*% chol(clrvar2ilr(var))), mean) } # A dataset with a Cauchy type distribution sa.outliers4 <- acomp(rcauchy.acomp(100,acomp(c(1,2,1)),Mvar/4)) colnames(sa.outliers4)<-colnames(typicalData) # A dataset with like sa.outlier2 but a differently strong distortions sa.outliers5 <- acomp(rbind(unclass(typicalData)+outer(rbinom(100,1,p=0.1)*runif(100),c(0.1,1,2)))) # A dataset with a second population sa.outliers6 <- acomp(rbind(typicalData,rnorm.acomp(20,acomp(c(4,4,1)),Mvar))) # Missings sa.missings <- simulateMissings(sa.lognormals,dl=0.05,MAR=0.05,MNAR=0.05,SZ=0.05) sa.missings[5,2]<-BDLvalue sa.missings5 <- simulateMissings(sa.lognormals5,dl=0.05,MAR=0.05,MNAR=0.05,SZ=0.05) sa.missings5[5,2]<-BDLvalue objects(pattern="sa.*") ``` ## References Aitchison, J. (1986) **The Statistical Analysis of Compositional Data** Monographs on Statistics and Applied Probability. Chapman & Hall Ltd., London (UK). 416p. Rehder, S. and U. Zier (2001) Letter to the Editor: Comment on ''Logratio Analysis and Compositional Distance'' by J. Aitchison, C. Barcel\'o -Vidal, J.A. Mart\'in-Fern\'andez and V. Pawlowsky-Glahn, **Mathematical Geology**, 33 (7), 845-848. Zier, U. and S. Rehder (2002) Some comments on log-ratio transformation and compositional distance, **Terra Nostra**, Schriften der Alfred Wegener-Stiftung, 03/2003 ## Author(s) K.Gerald v.d. Boogaart [http://www.stat.boogaart.de](http://www.stat.boogaart.de)

SimulatedAmounts dataset