How can I create a matrix where every combination of numbers between 0 and 1 (step x) is used and every row sums to one?

Hello everybody,

I'm looking for a way to create a matrix where every row sums to one and every combination of numbers between zero and one is used given a specified step parameter.

Example: 3 Columns; Step 0.5:

Row 1:[0 0.5 0.5] Row 2:[0.5 0 0.5] Row 3:[0.5 0.5 0]

I'm using the following code (for 6 columns) which is too slow given a small step parameter:

i=1;

for a = 0:step:0.5
  for b = 0:step:0.5
      for c = 0:step:0.5
          for d = 0:step:0.5
              for e = 0:step:0.5
                  for f = 0:step:0.5
                      weights(i,:)=[a b c d e f];
                      i = i+1;
                  end
              end
          end
      end
  end
 end

clear a b c d e f i 

ind = sum(weights,2) == 1;
weights = weights(ind,:);

Thank you for your help!

Answer 1

[Andrei Bobrov]

for step stp = .5

stp = .5;
ii = cell(1,6);
[ii{:}] = ndgrid(0:stp:1);
out = cell2mat(cellfun(@(x)x(:),ii(:,end:-1:1),'un',0));
out = out(sum(out,2) == 1,:);

OR

x = 0:stp:1;
out = x(fliplr(fullfact(numel(x)*ones(1,6))));
out = out(sum(out,2) == 1,:);

[Titus Edelhofer]

Nice!

Titus

Answer 2

[Teja Muppirala]

I like andrei's method, but it will will run out of memory quickly when step is small!

Here is a better implementation of the loop method. This runs very quickly and uses very little memory.

i=1;
weights = [];
step = .5/6;
t = 0.5 + 0.0000000001;

for a = 0:step:t
  for b = 0:step:t-a
      for c = 0:step:t-a-b
          for d = 0:step:t-a-b-c
              for e = 0:step:t-a-b-c-d
                  if a+b+c+d+e < t
                      weights(:,i)=[a; b; c; d; e; 0.5-a-b-c-d-e];
                      i = i+1;
                  end
              end
          end
      end
  end
end

weights = weights';

Notes:

1. Due to floating point arithmetic, sometimes the sum won't come out to exactly == 0.5, that's why you need some sort of tolerance. (For example, 0.1 + 0.2 ~= 0.3).

2. If you are going to grow matrices without preallocationg, grow them in the last dimension

tic
weights = [];
for n = 1:50000
weights(:,n) = [1; 1; 1; 1; 1; 1];
end
toc %Faster

tic
weights = [];
for n = 1:50000
weights(n,:) = [1 1 1 1 1 1];
end % Slower
toc % Slower

Answer 3

[Christian Maschner]

That's exactly what I was looking for! Thanks a lot!

Answer 4

[Christian Maschner]

It works for steps <0.02! That's very helpful! Thank you Teja!

But the rows do not sum to one.

[Teja Muppirala]

Oops. Yeah, I made them sum to 0.5. To make them sum up to 1, just change two places:

t = 0.5 + 0.0000000001;

to

t = 1.0 + 0.0000000001;

and

0.5-a-b-c-d-e

to

1-a-b-c-d-e