MATLAB: Generate pair of random numbers with respect to a sum constraint

conditionconstraintgeneraterandom number generatorrngstatistics

Hello, all. So I am still gaining experience with Matlab and am currently trying to generate sequences of two numbers, x and y, such that their sum is <= 1. The naive way I immediately thought to do this would be by 1) generating x within the range [0, .5], and 2) keep generating y until it is < x, which would guarantee my condition, but introduce some significant statistical bias.

Is there some standard, statistically robust way to do this, guys? I would be very grateful for any of your input on this matter.

Best Answer

(Corrected) Assuming you restrict x and y to non-negative values, the set of x and y values for which x+y<=1 would be a triangular area in the xy plane. You can obtain an area-wise uniform distribution of x and y with the following.

For generating a single pair:

x = 1-sqrt(rand);
y = (1-x)*rand;

To get row vectors with n elements each:

x = 1-sqrt(rand(1,n));
y = (1-x).*rand(1,n);

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MATLAB: Generate random coordinates inside a convex polytope

This is actually easier than you think, IF you understand how to solve the necessary sub-problems. I even have code that does it, but I tend not to give it out, because people seem not to understand that toolbox. Interface is everything. Well, not everything, but important as hell. And since I've never had the ambition to create a complete new interface for those tools, they will languish on my list of things to do.

So, lets see if we can solve the problem. Is this a 2-d problem? 3-d? N-d? I'll show you how to do it in 2-d, because I can make pretty plots there that are easy to look at and verify we were successful. The extension to n-d is trivial.

First, create a convex tessellation.

xy = randn(10,2);
CH = convhulln(xy)
CH =
   1     8
   8    10
  10     6
   6     2
   2     7
   7     1

Those are edges, because I'm working in 2-d. n-d is the same idea.

plot(xy(:,1),xy(:,2),'bo');
hold on
plot([xy(CH(:,1),1),xy(CH(:,2),1)]',[xy(CH(:,1),2),xy(CH(:,2),2)]','r-')

Compute a point in the interior of the polygon. Since it is known to be a convex polygon (polyhedron in n-d), mean will suffice.

xycent = mean(xy,1)
xycent =
   0.026363      0.42644
nxy = size(xy,1)
ncent = nxy+1;
xy(ncent,:) = xycent;

now convert the polygon into a triangulation of that convex domain. Each edge will become a triangle, or (n-1)-simplex in n-d. (A k-simplex is defined by k+1 vertices. It may be full volume, or it may live in a k-manifold, embedded in a higher dimensional space.) These are now triangles, living in the 2-d data plane.

tri = [CH,repmat(ncent,ntri,1)]
tri =
   1     8    11
   8    10    11
  10     6    11
   6     2    11
   2     7    11
   7     1    11
figure
plot(xy(:,1),xy(:,2),'bo');
hold on
plot([xy(tri(:,1),1),xy(tri(:,2),1),xy(tri(:,3),1)]',[xy(tri(:,1),2),xy(tri(:,2),2),xy(tri(:,3),2)]','g-')

You can see one new point in that set, the one we added. It is a common vertex for every triangle.

Next, compute the volume of each simplex. In 2-d, it is just the area. This is mot easily done using determinants, with a loop. Yes, you might know that I hate determinants in general. But sometimes they can be useful, and I'm feeling too lazy.

Note that the loop below uses aR2016b feature. Earlier releases will need to use bsxfun.

V = zeros(1,ntri);
for ii = 1:ntri
  V(ii) = abs(det(xy(tri(ii,1:2),:) - xycent));
end
V
V =
       1.7526       1.2067      0.79587      0.94023       1.0889        2.993

So V is a list of areas (or volumes) for each simplex. Actually I needed to divide by factorial(ndim), where ndim is the dimension of the space we are living in to truly compute volume, but I need only something proportional to volume for the next computation. If that bothers you here, you could divide V by 2, but save the cpu cycles, because the next step is to normalize V to sum to 1.

V = V/sum(V)
V =
    0.19968      0.13747     0.090674      0.10712      0.12406      0.34099

All we need are the relative areas of each simplex.

We are almost done. Lets say we wish to generate M points in total.

M = 1000;

For each point, we first need to know which simplex it lives in. That will be proportional to the relative volume of the corresponding simplexes.

[~,~,simpind] = histcounts(rand(M,1),cumsum([0,V]));

So the bigger simplexes will have proportionally more points to fall in them.

FINALLY, generate the desired random points. Yes, I know this seems like a trip to get here, but most of it was me typing explanation and making pictures.

Again, the code that follows will require R2016b. Easy to fix though.

r1 = rand(M,1);
uv = xy(tri(simpind,1),:).*r1 + xy(tri(simpind,2),:).*(1-r1);
r2 = sqrt(rand(M,1));
uv = uv.*r2 + xy(tri(simpind,3),:).*(1-r2);

uv is the complete set of random points. Trust me? Yeah, I know, trust but verify.

figure
plot(xy(:,1),xy(:,2),'bo');
hold on
plot([xy(tri(:,1),1),xy(tri(:,2),1),xy(tri(:,3),1)]',[xy(tri(:,1),2),xy(tri(:,2),2),xy(tri(:,3),2)]','g-')
plot(uv(:,1),uv(:,2),'m.')

As you can see, the sampling is indeed uniform in that domain.

In 3 or more dimensions, just extend what I did as is appropriate. Ask if you have questions, but the extension really is easy. (Ok, there is one part that requires some thought by the programmer, and that is how to extend the very last sampling step into n-d. The trick is to understand why I used sqrt there.) Again, ask if you need the trick.

I suppose one day I should post code on the FEX that does this, but sampling from a convex n-dimensional polyhedron is not something for which I see people clamoring.

MATLAB: Create/choose a rondom point from an existed data matrix

I will give an answer in terms of a single triangle with vertices at P0, P1, and P2, (either in 2D or 3D.) Define

   r1 = rand;
   r2 = sqrt(rand); % <-- The square root operation is important here

Then the value P

   P = (1-r2)*P0+r2*(r1*P1+(1-r1)*P2);

will be a random point within the triangle P0P1P2, with a statistically uniform probability distribution therein.

I will leave the problem of doing this for the entire triangulation to you. It can easily be vectorized.

Best Answer

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MATLAB: Generate random coordinates inside a convex polytope

MATLAB: Create/choose a rondom point from an existed data matrix

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