Normally distributed random numbers with fixed sum

Hi everybody,

I am looking for a way to create normally distributed numbers with a fixed sum.

I know the random fixed sum numer generator from FileExchange https://de.mathworks.com/matlabcentral/fileexchange/9700-random-vectors-with-fixed-sum.

However, I wasn't able to easily adjust it to my needs.

Basically, I want the same as randfixedsum with the addition that the function recevies values for mu and sigma as a vector of length n. I understand that using different standard deviations for n-different values will lead to unequal pairwise corrrelations between the generated values, but that's ok.

I understand it's maybe not a trivial problem. Any ideas?

Thanks

[Jeff Miller]

Just a comment that this isn't possible in general. With two X's, for example, you can't let them both have mu=0 yet have them produce a fixed sum of 1. Likewise, you can't let them have different sigmas if they produce a fixed sum--in that case they must have equal sigmas as well as a perfect negative correlation. With larger numbers of X's the constraints are more complicated, but I would guess that there are a lot more sets of impossible mus & sigmas than of possible ones.

Answer 1

[Richard Ott]

I tried this approach below that I translated from a post I found in an R forum

https://stats.stackexchange.com/questions/315109/draw-from-random-distributions-so-that-sum-of-values-is-fixed/536045#536045

It seems to be doing what I want, but I can't figure out how the covariance matrix was determined from the variances... Any ideas? I wanted to try corr2cov but don't have the financial toolbox.

mu = [10,20,30,40];

variances = [2,3,5,6];
% somehow the user uses these variances to built the following covariance
% matrix
sigma =[3.7838,-0.4865,-1.3514,-1.9459...
    ;-0.4865,7.9054,-3.0405,-4.3784...
    ;-1.3514,-3.0405,16.5541,-12.1622;...
    -1.9459,-4.3784,-12.1622,18.4865];

n = length(mu);
[u,s,v] = svd((sigma + sigma')/2);
s = round(s,2);
m = sum(sum(s > 0));

% Generate normals in the lower dimension
nsample = 1e3;
x = normrnd(0,1,m,nsample);

% Embed in the higher dimension and apply square root
% of sigma obtained from its SVD
x = [x; zeros(1,nsample)];
xmu = nan(length(mu),nsample);
for i = 1:nsample
    xmu(:,i) = x(:,i) + mu';
end
y = u * sqrt(s) * x;

for i = 1:nsample
    y(:,i) = y(:,i) + mu';
end

sum(y(:,1))

histogram(y(1,:))
hold on
histogram(y(2,:))
histogram(y(3,:))
histogram(y(4,:))

Answer 2

[Matt J]

Why not simply,

x(1:N-1)=randn(1,N-1);
x(N)=fixedValue-sum(x(1:N-1))

[DGM]

N = 10000;
fixedValue = 50;

x(1:N-1)=randn(1,N-1);
x(N)=fixedValue-sum(x(1:N-1));
histogram(x)

It's not so much a gaussian distribution with a fixed sum so much as it is a gaussian distribution with an another number in the same vector that offsets the sum.

[Matt J]

It's not so much a gaussian distribution with a fixed sum so much as it is a gaussian distribution with an another number in the same vector that offsets the sum.

I don't really see the distinction. It is clearly a Gaussian random vector and the sum is unvarying, as the test below shows

N = 10000;
fixedValue = 50;

for i=1:4
x(1:N-1)=randn(1,N-1);
x(N)=fixedValue-sum(x(1:N-1));
s=sum(x)
end

s = 50

s = 50

s = 50

s = 50