# How to generate a random data vector that follows a constraint

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Saifullah Khalid on 29 Sep 2019
Commented: Saifullah Khalid on 30 Sep 2019
I want to generate random data vector where that follows this constrainst. where and is a constant. I would appreciate any help.

Bruno Luong on 29 Sep 2019
Edited: Bruno Luong on 29 Sep 2019
This will generate uniform distribution (within to polytope)
N = 10;
Etotal = 0;
E = -log(rand(N+1,1));
E = E(1:N,:)./sum(E,1);

Bruno Luong on 29 Sep 2019
Here is a comparison of distribution with the two other methods proposed below to show the issue of non-uniformity if one doesn't pay attention N = 2;
Etotal = 1;
p = 3e3;
Ebias = rand(1,p) .* randfixedsum(N,p,Etotal,0,Etotal);
E = rand(1,p).^(1/N)*Etotal .* randfixedsum(N,p,1,0,1);
E2 = -log(rand(N+1,p)); E2 = E2(1:N,:)./sum(E2,1);
subplot(2,2,1)
plot(Ebias(1,:),Ebias(2,:),'.');
title('rand * randfixedsum')
axis equal
subplot(2,2,2)
plot(E(1,:),E(2,:),'.');
axis equal
title('sqrt(rand) * randfixedsum')
subplot(2,2,3)
plot(E2(1,:),E2(2,:),'.');
axis equal
title('exponential method')
Saifullah Khalid on 30 Sep 2019
Bruno Luong, thank very much for detailed answer and the insight.

Walter Roberson on 29 Sep 2019
Look in the File Exchange for Roger's randfixedsum(). Generate a vector with fixed sum . Multiply all of the elements by rand() to implement the <= part. (Though you might want to worry about the difficulty that rand() is never exactly 1, so if you generate a sum exactly equal to and multiply by rand() then the result can never exactly total Bruno Luong on 29 Sep 2019
IIUC you propose
E = rand()*Etotal * randfixedsum(N,1,1,0,1)
this will generate bad distribution (too many point close to 0 compare to uniform distribution), espetially for N >> 1.
Walter Roberson on 29 Sep 2019
I was thinking of
rand() * randfixedsum(N,1,Etotal,0,Etotal)
but your comment might still apply.
Bruno Luong on 29 Sep 2019
Both give the same non-uniform pdf
The "correct" one is
E = rand()^(1/N)*Etotal * randfixedsum(N,1,1,0,1);