[R] Latin Hypercube with condition sum = 1

[Rainer M Krug]

On Tue, Nov 25, 2008 at 4:16 PM, Rob Carnell <carnellr at battelle.org> wrote:
> Rainer M Krug <r.m.krug <at> gmail.com> writes:
>
>>
>> Hi
>>
>> I want to du a sensitivity analysis using Latin Hypercubes. But my
>> parameters have to fulfill two conditions:
>>
>> 1) ranging from 0 to 1
>> 2) have to sum up to 1
>>
>> So far I am using the lhs package and am doing the following:
>>
>> library(lhs)
>> ws <- improvedLHS(1000, 7)
>> wsSums <- rowSums(ws)
>> wss <- ws / wsSums
>>
>> but I think I can't do that, as after the normalization
>>
>> > min(wss)
>> [1] 0.0001113015
>> > max(wss)
>> [1] 0.5095729
>>
>> Therefore my question: how can I create a Latin Hypercube whicgh
>> fulfills the conditions 1) and 2)?
>>
>> Thanks a lot
>>
>> Rainer
>>
>
> Rainer,
>
> Your original solution meets your two conditions.  The problem for you (I
> think) is that you'd like the result to have values near zero and near one.
>
> I have an imperfect solution to your problem using a Dirichlet distribution.
> The Dirichlet seems to keep the range of the values larger once they are
> normalized.  The result is not uniformly distributed on (0,1) anymore, but
> instead is Dirichlet distributed with the parameters alpha.  The Latin
> properties are maintained.

Thanks a lot Rob.
I'll look into the solution which sounds good.

Thanks

Rainer
>
> require(lhs)
>
> qdirichlet <- function(X, alpha)
> {
>  # qdirichlet is not an exact quantile function since the quantile of a
>  #  multivariate distribtion is not unique
>  # qdirichlet is also not the quantiles of the marginal distributions since
>  #  those quantiles do not sum to one
>  # qdirichlet is the quantile of the underlying gamma functions, normalized
>  # This has been tested to show that qdirichlet approximates the dirichlet
>  #  distribution well and creates the correct marginal means and variances
>  #  when using a latin hypercube sample
>  lena <- length(alpha)
>  stopifnot(is.matrix(X))
>  sims <- dim(X)[1]
>  stopifnot(dim(X)[2] == lena)
>  if(any(is.na(alpha)) || any(is.na(X)))
>    stop("NA values not allowed in qdirichlet")
>
>  Y <- matrix(0, nrow=sims, ncol=lena)
>  ind <- which(alpha != 0)
>  for(i in ind)
>  {
>    Y[,i] <- qgamma(X[,i], alpha[i], 1)
>  }
>  Y <- Y / rowSums(Y)
>  return(Y)
> }
>
> X <- randomLHS(1000, 7)
> Y <- qdirichlet(X, rep(1,7))
> stopifnot(all(abs(rowSums(Y)-1) < 1E-12))
> range(Y)
>
> ws <- randomLHS(1000, 7)
> wsSums <- rowSums(ws)
> wss <- ws / wsSums
> stopifnot(all(abs(rowSums(wss)-1) < 1E-12))
> range(wss)
>
> I hope this helps!
> Rob
>
> Rob Carnell
> Battelle
> Principal Research Scientist
>
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-- 
Rainer M. Krug, PhD (Conservation Ecology, SUN), MSc (Conservation
Biology, UCT), Dipl. Phys. (Germany)

Centre of Excellence for Invasion Biology
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