How to get only the diagonal of the matrix from these loops? - matlab

Question

I have the following code for a 8 dimensional empirical copula that creates a 8d matrix but I only need the diagonal of the matrix which is named EC in this code. Since this code is very slow, is there anyway that I can get "EC" without computing all of "ecop"?
function EC = ecopula8d(x)
[m n] = size(x);
y = sort(x);
for r=1:m
for q=1:m
for p=1:m
for o=1:m
for l=1:m
for k=1:m
for j=1:m
for i=1:m
ecop(i,j,k,l,o,p,q,r) = sum( (x(:,1)<=y(i,1)).*(x(:,2)<=y(j,2)).*(x(:,3)<=y(k,3)).*(x(:,4)<=y(l,4))...
.*(x(:,5)<=y(o,5)).*(x(:,6)<=y(p,6)).*(x(:,7)<=y(q,7)).*(x(:,8)<=y(r,8)) )/(m+1);
end
end
end
end
end
end
end
end
for i=1:m
EC(i,1)=ecop(i,i,i,i,i,i,i,i);
end

Answer 1

I haven't checked if your initial computation is correct (as in compared your implementation with the wikipedia article's formula), but your code should be equivalent to:
[m n] = size(x);
y = sort(x);
for i = 1:m
EC(i,1) = sum(all(bsxfun(#le, x, y(i,:)), 2), 1)/(m+1);
end

Answer 2

You can employ a completely vectorized solution with bsxfun -
EC = squeeze(sum(all(bsxfun(#le,x,permute(y,[3 2 1])),2),1))/(m+1)
The magic here happens with the use of permute enabling us to go full throttle on vectorization.
Here's a benchmarking test to compare this approach and the other partially vectorized approach with bsxfun on runtime efficiency -
x = rand(2000,2000);
y = sort(x);
m = size(x,1);
%// Warm up tic/toc.
for k = 1:100000
tic(); elapsed = toc();
end
disp('----------- With completely vectorized solution')
tic
EC1 = squeeze(sum(all(bsxfun(#le,x,permute(y,[3 2 1])),2),1))/(m+1);
toc, clear EC1
disp('----------- With partial vectorized solution')
tic
for i = 1:m
EC2(i,1) = sum(all(bsxfun(#le, x, y(i,:)), 2), 1)/(m+1);
end
toc
The runtimes thus obtained were -
----------- With completely vectorized solution
Elapsed time is 2.883594 seconds.
----------- With partial vectorized solution
Elapsed time is 4.752508 seconds.
One can pre-allocate for the other partially vectorized approach -
EC2 = zeros(m,1);
for i = 1:m
EC2(i,1) = sum(all(bsxfun(#le, x, y(i,:)), 2), 1)/(m+1);
end
The runtimes thus obtained weren't that different though -
----------- With completely vectorized solution
Elapsed time is 2.963835 seconds.
----------- With partial vectorized solution
Elapsed time is 4.620455 seconds.

Answer 3

Once of the approaches I would use is to convert N-D array into square 2-D(if possible) and then simply extract diagonal term as they should be equal in both cases:
EC=diag(reshape(ecop,size1,size2));
I would suggest to try use Python because numpy has really nice and efficient linear algebra package to deal with N-D arrays. Matlab is pretty slow in adding and updating its libraries.