[R] Latin Hypercube Sampling with a condition

[Rob Carnell]

Duarte Viana <viana.sptd <at> gmail.com> writes:

> 
> Hello all,
> 
> I am trying to do a Latin Hypercube Sampling (LHS) to a 5-parameter
> design matrix. I start as follows:
> 
> library(lhs)
> 
> p1<-randomLHS(1000, 5)
> 
> If I check the distribution of each parameter (column), they are
> perfectly uniformly distributed (as expected).For example,
> 
> hist(p1[,1])
> 
> Now the hard (maybe strange) question. I want the combination of the
> first three parameters to sum up to 1 (which obviously do not)
> 
> s<-p1[,1]+p1[,2]+p1[,3]
> 
> s==1
> 
> It occurred to me to divide each of these parameters with the sum
> (vector "s" above). However the uniform distribution is lost (example
> for parameter 1 - first column):
> 
> par1.transf<-p1[,1]/s
> 
> hist(par1.transf)
> 
> So, is there a way to maintain the random LHS (with uniformly
> distributed parameters) so that the refered condition is fulfilled?
> 
> Any suggestions would be much welcome.
> 
> Thanks,
> 
> Duarte
> 
> 

Duarte,

In my experience with Latin hypercube samples, most people draw the sample on 
a uniform hypercube and then transform the uniform cube to have new 
distributions on the margins.  The transformed distributions are not 
necessarily uniform.  It is possible to draw a Latin hypercube with correlated 
margins and I hope to add that to my package in the future.  I have also done 
transforms such that the transformed marginal distributions are correlated (as 
you have in your example).  I have not seen a correlated set of uniform 
marginal distributions such that the margins sum to one, however.  I'll make a 
quick example argument that explains the difficulty...

In two dimensions, you could draw this which is uniform and correlated.
x <- seq(0.05, 0.95, length=10)
y <- 1-x
all.equal(x+y, rep(1, length(x)))
hist(x)
hist(y)

But in three dimensions, it is hard to maintain uniformity because large 
samples on the first uniform margin overweight the small samples on the other 
margins.
x <- seq(0.05, 0.95, length=10)
y <- runif(length(x), 0, 1-x)
z <- 1-x-y
hist(x)
hist(y)
hist(z)

If you could explain why you want to maintain the uniformity on the margins, I 
might be able to suggest something different.

Rob