# [R] Latin Hypercube with condition sum = 1

Rainer M Krug r.m.krug at gmail.com
Tue Nov 25 15:54:45 CET 2008

```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
>
> ______________________________________________
> R-help at r-project.org mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help
> and provide commented, minimal, self-contained, reproducible code.
>

--
Rainer M. Krug, PhD (Conservation Ecology, SUN), MSc (Conservation
Biology, UCT), Dipl. Phys. (Germany)

Centre of Excellence for Invasion Biology
Faculty of Science
Natural Sciences Building
Private Bag X1
University of Stellenbosch
Matieland 7602
South Africa

```