# [R] A vector of normal distributed values with a sum-to-zero constraint

Greg Snow 538280 at gmail.com
Tue Apr 1 20:55:50 CEST 2014

```Here is one approach to generating a set (or in this case multiple
sets) of normals that sum to 0 (with a little round off error) and
works for an odd number of points:

v <- matrix(-1/8, 9, 9)
diag(v) <- 1
eigen(v)
x <- mvrnorm(100,mu=rep(0,9), Sigma=v, empirical=TRUE)
rowSums(x)
range(.Last.value)
hist(x)
sd(x)
mean(x)
apply(x,2,sd)

the key is to find the value of the off diagonals in the covariance
matrix that gives you exactly one eigenvalue that is equal to 0 (or
close enough with rounding) and all the others are positive.  There is
probably a mathematical formula that gives the exact value to use, but
I found one that works with a little trial and error (it will change
for different sample sizes).

On Tue, Apr 1, 2014 at 6:56 AM, Marc Marí Dell'Olmo
<marceivissa at gmail.com> wrote:
> Dear all,
>
> Anyone knows how to generate a vector of Normal distributed values
> (for example N(0,0.5)), but with a sum-to-zero constraint??
>
> The sum would be exactly zero, without decimals.
>
>
>> l <- 1000000
>> aux <- rnorm(l,0,0.5)
>> s <- sum(aux)/l
>> aux2 <- aux-s
>> sum(aux2)
>  -0.000000000006131392
>>
>> aux<- -sum(aux[2:l])
>> sum(aux)
>  -0.00000000000003530422
>
>
> but the sum is not exactly zero and not all parameters are N(0,0.5)
> distributed...
>
> Perhaps is obvious but I can't find the way to do it..
>
> Thank you very much!
>
> Marc
>
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