The problem is a relatively easy one, as long as the total sum does not exceed 1. It gets nasty above that.
I'll look at the problem in TWO dimensions, that is, imagine I want to find random numbers that lie within the unit square [0,1]X[0,1], such that the sum, x1 + x2 does not exceed 1.
The set of viable solutions lies within a triangle, the triangle with vertices [0, 0], [0,1], and [1,0]. So an isosceles triangle, with legs of length 1.
This is a different problem than what randfixedsum solves, in a subtle way. It solves the problem where the sum is constrained to be exactly 1. So you cannot use randfixedsum directly. (Though you CAN use it, with a subtle tweak, as long as the maximum on the sum does not exceed 1. That upper limit is important.)
Your admissable set is shown in the figure:
fill([0 0 1 0],[0 1 0 0],'r')
We want to see that the 2-d case is just a simple variation of the n-d case. In n-dimensions, 3 for example, the admissable set becomes an isosceles tetrahedron, so a 3-simplex with legs connecting the four points [0,0,0], [0,0,1], [0,1,0], and [1,0,0]. The number of dimensions is just the number of total elements in your vectors. So, if your vector is of length 100, then the points live in a 100 dimensional space. And then the admissable set will live in a 100-simplex, composed of 101 vertices. Hard to visualize, but a basic extension of the red triangle.
So that is your goal. Again, randfixedsum does not solve that specific problem. But it is close, in a sense.
How can we solve your problem? The idea is we can start with a set that sums to 1. Then take a nonlinear combination of those points with the point at the origin. And since the origin is at zero, it is simple. In two-dimensions, this reduces to:
ndim = 2;
nsample = 10000;
S = randfixedsum(ndim,nsample,1,0,1);
p = rand(1,nsample).^(1/ndim);
S = S.*p;
plot(S(1,:),S(2,:),'.')
Note that this will fail miserably if the maximum sum is greater than 1.
Be careful now. Because the first thing you might do is look at the distribution of the sum of those numbers. Even in the 2-dimensional case, the expected median value for the sum will be
But in 100 dimensions, the median of the sums will be quite close to 1.
ndim = 100;
nsample = 10000;
S = randfixedsum(ndim,nsample,1,0,1);
p = rand(1,nsample).^(1/ndim);
S = S.*p;
median(sum(S,1))
ans =
0.99305
These points are still uniformly distributed in theory in the domain in question. But you need to understand that the probability of finding a point inside the isosceles 100-hypersimplex with leg length 0.5 is VERY small. That would be:
In fact, what is the the probability of finding a point in that uniformly distributed simplex, such that NONE of the variables x1,...,x100 exceed 0.99?
That happens only about 37% of the time.
So don't compute the sum of the samples, and be surprised or disappointed. It will get even worse in a much higher number of dimensions.
Oh, and again, things go to hell if that sum exceeds 1, because then the domain that contains the points is not a single hyper simplex, but much more complex.
