Differences in ordinal ranking between absolute deviation and squared deviation

ordinal ranking.R

#Very simple fitness function.  Works sensibly only on 1 dimensional numeric vectors, but that is 
#ok for this example
fitness.hw<- function(tar.hw,meas.hw,type=c("abs","sqr")) {
	diff.hw<-tar.hw-meas.hw
	if (type=="abs") {
	rowSums(abs(diff.hw))
	}
	else if (type=="sqr") {
	rowSums(diff.hw^2)
	}
}
#Since we are looking at the large sample properties, the name "generation" is a misnomer,
#all the generations are random numbers from 0-255
generation<- array(0,c(400,11,500))
pop.random<-function() {t(replicate(dim(generation)[1], round(runif(11,0,255))))}
for (i in 1:500) {generation[,,i]<-pop.random()}
target<-utf8ToInt("Hello World")
target.mat<- t(matrix(target,nrow=11,ncol=dim(generation)[1]))

abs.fitness<- matrix(0,400,500)
sqr.fitness<- matrix(0,400,500)
diff.rank<- matrix(0,500,400)
for (i in 1:500) {
	abs.fitness[,i]<-fitness.hw(target.mat,generation[,,i],type="abs")
	sqr.fitness[,i]<-fitness.hw(target.mat,generation[,,i],type="sqr")
#This is the only (relatively) tricky part.  The first bit returns the index of sorted outcomes for the squared deviation
#method.  The second returns the index of sorted outcomes from the absolute difference method.  If ordering is preserved,
#the difference will be zero.  If ordering is preserved in expectation, the average difference will be zero.
	diff.rank[i,]<- sort(sqr.fitness[,i],index.return=TRUE)$ix-sort(abs.fitness[,i],index.return=TRUE)$ix
	}
#scatterplot and smoothed line of differences
scatter.smooth(x=1:length(colMeans(diff.rank)), y=colMeans(diff.rank),main="Smoothed Difference of Rank Ordering",ylab="Rank Difference",xlab="Absolute Value Rank Fitness",pch=20)
#indeed the mean of each order for 500 samples is VERY close to zero.
mean(colMeans(diff.rank))