add error to a transformed vector of dependent values

Question

I would like to simulate (add error/rnorm) a transformed vector of dependent values, but I don't know how to keep the transformation's properties while doing so. I made a toy example to demonstrate my problem.

I have a vector of interval observations (obs) which I transform for modelling:

set.seed(123)
sd=0.1
obs=rnorm(10,10,3)  # each value is an age class

#transform observations (in reality something more complex based on cumulative logits)
obs=obs/sum(obs)  

These go into a model that estimates the standard deviation based on the transformation:

# model
predict=function(x){
    pred=c(1:10)^x
    pred=pred/sum(pred)
    return(pred)
}

model= function(x){
    nll=-sum(mapply(dnorm,predict(x),obs,sd))  #sd is estimated in reality
    return(nll)
}

mypar=optim(0,model,lower=0,upper=2,method='Brent')$par

# from my model I get predictions
out=predict(mypar)

# I would now like to simulate observations like this :
# (in reality I do this for predicted future values)
simu=mapply(rnorm,1,out,sd)
sum(simu)
 [1] 1.208622

But if I do this than my simulations of course don't follow the transformation rule anymore... In this toy case, the sum of simu should still be one.

I could do an inverse transformation of the predicted values and simulate that, but than my sd is not "appropriate" anymore.

How do I deal with this? Do I need to transform my sd somehow while doing the above (and if so how)? Or is there an other method?

Answer 1

This question cannot be deleted because it has a bounty, though I would like to do so because the problem arose from a tiny mistake in my inverse transformation function. Now it works fine; I can add error and the sum of observations is one.