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September 2, 2010 17:09
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# multivariate normal example using Cholesky decomposition | |
require(Matrix) | |
#make this reproducible | |
set.seed(2) | |
#how many draws in our starting data set? | |
n <- 1e4 | |
# how many draws do we want from this distribution? | |
drawCount <- 1e4 | |
myData <- rnorm(n, 5, .6) | |
yourData <- myData + rnorm(n, 8, .25) | |
hisData <- myData + rnorm(n, 6, .4) | |
herData <- hisData + rnorm(n, 8, .35) | |
ourData <- data.frame(myData, yourData, hisData, herData) | |
# now we have raw correlations in the 70s, 80s, and 90s. Funny how that | |
# works out | |
cor(ourData) | |
ourData0 <- ourData | |
# shift the mean of ourData to zero | |
ourData0 <- as.data.frame(sweep(ourData,2,colMeans(ourData),"-")) | |
#Cholesky Decomposition of the covariance matrix | |
C <- chol(nearPD(cov(ourData0))$mat) | |
#create a matrix of random standard normals | |
Z <- matrix(rnorm(n * ncol(ourData)), ncol(ourData)) | |
#multiply the standard normals by the transpose of the Cholesky | |
X <- t(C) %*% Z | |
#look at the covariances | |
cov(as.matrix(t(X))) | |
cov(ourData0) | |
#woot! | |
myDraws <- data.frame(as.matrix(t(X))) | |
names(myDraws) <- names(ourData) | |
# this method handles the covariance (correlation and standard deviation) | |
# but we still need to shift the means of the samples. | |
# shift the mean of the draws over to match the starting data | |
myDraws <- as.data.frame(sweep(myDraws,2,colMeans(ourData),"+")) | |
#check the mean and sd | |
apply(myDraws, 2, mean) | |
apply(myDraws, 2, sd) | |
#let's look at the mean and standard dev of the starting data | |
apply(ourData, 2, mean) | |
apply(ourData, 2, sd) | |
# so myDraws contains the final draws | |
# let's check Kolmogorov-Smirnov between the starting data | |
# and the final draws | |
for (i in 1:ncol(ourData)){ | |
print(ks.test(myDraws[[i]], ourData[[i]])) | |
} | |
#look at the correlation matrices | |
cor(myDraws) | |
cor(ourData) | |
#it's fun to plot the variables and see if the PDFs line up | |
#It's a good sanity check. Using ggplot2 to plot | |
require(ggplot2) | |
# rearrange the data to be "tall" not "wide" | |
meltDraws <-melt(myDraws) | |
meltDraws$dataSource <- "simulated" | |
meltData <- melt(ourData) | |
meltData$dataSource <- "original" | |
plotData <- rbind(meltData, meltDraws) | |
qplot(value, colour=dataSource, data=plotData, geom="density")+ facet_wrap(~variable) | |
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