# SimulatedAmounts: Simulated amount datasets In compositions: Compositional Data Analysis

## Description

Several simulated datasets intended as reference examples for various conceptual and statistical models of compositions and amounts.

## Usage

 `1` ```data(SimulatedAmounts) ```

## Format

Data matrices with 60 cases and 3 or 5 variables.

## 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.

## Author(s)

K.Gerald v.d. Boogaart http://www.stat.boogaart.de

## Source

The datasets are simulated for this package and are under the GNU Public Library Licence Version 2 or newer.

## References

http://www.stat.boogaart.de/compositions/data

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

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279``` ```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.*") ```

compositions documentation built on May 30, 2017, 3:25 a.m.