Euclidean Distance (Back to the Math...complete with two loops)

Hi

Sorry to ask this as it appears to have been asked loads before! I am trying to construct a code with two loops to calculate the Euclidean Distance between elements of two matrices. I have code which uses repmat and avoids loops but now I need to show an element by element calculation.

I seem to be able to do this on a per element basis but I can't figure out how to 'automate' the loops. They need to be left in so that the Mathematics is much clearer.

% Loop Test Script
A=[2 3 4;1 6 5;4 3 1];
B=[5 1 2;3 2 6;8 0 7];
C=zeros(3);

for l=1:3                                   % rows
    for m=1:3                               % columns
        C(l,m)=sum(A(l,m)-B(l,m)).^2;       % Not working!!
    end
end
C

Here, element C(2,1) would be (1-5).^2 + (6-1).^2 + (5-2).^2. This is simple to implement in more advanced code.

Any help with keeping the loops and implementing this properly would be most welcome.

Sincerely

Joe

Answer 1

[Roger Stafford]

Apparently what you want is this:

 for l = 1:3
   for m = 1:3
     C(l,m) = sum((A(l,:)-B(m,:)).^2);
   end
 end

This would be squared euclidean distance. To get euclidean distance you need to take the square root of the 'C' values. Note that this is what the function 'pdist2' would give you.

[Joe Bannen]

Roger

Thanks for the response. I don't think this works either as this is yielding incorrect answers for C.

This is why loops drive me crazy!

Cheers

Joe

[Roger Stafford]

The code I gave you should agree with the example C(2,1) you described:

C(2,1) = (A(2,1)-B(1,1))^2+(A(2,2)-B(1,2))^2+(A(2,3)-B(1,3))^2
       =     (1-5).^2     +   (6-1).^2      +   (5-2).^2.

What should the full 3-by-3 C be in your example?

[Joe Bannen]

Roger

The answer should be:

ans =

     17.0000    6.0000   54.0000
     50.0000   21.0000   89.0000
      6.0000   27.0000   61.0000

In this case C(3,2) is (4-3).^2 + (3-2).^2 + (1-6).^2. Hence for this entry I take away and square on a term by term basis the 2nd Row in B from the 3rd Row in A.

I am trying to put into two loops the following standard Distance Matrix code:

function D=distm(A,B)
    % distm is the distance matrix
    % distm(A,B) is the matrix of the pairwise Euclidean distance
    % between the points in the rows of A and B
    [M,s]=size(A);
    [N,s]=size(B);
    D=repmat(sum(A.*A,2),1,N)-2*A*B'+repmat((sum(B.*B,2))',M,1);
    D=sqrt(D);

Thanks

Joe

[Roger Stafford]

The code I gave you gives the result you want:

C =
   17    6   54
   50   21   89
    6   27   61

However, if you want Euclidean distance, you must take the square root of each of these values.

[Joe Bannen]

Thanks

This works a treat for different matrices A and B.

It runs into a problem when A=B.

distm returns:

>> A=[2 3 4;1 6 5;4 3 1];
>> distm(A,A)

ans =

           0    3.3166    3.6056
      3.3166         0    5.8310
      3.6056    5.8310         0

But the Loop Test script yields:

C =

       0     3     1
       3     0     4
       1     4     0

Any thoughts?

Best wishes

Joe

Answer 2

[Chad Greene]

The euclidean distance is a strange term to apply to this particular problem. For each row,column pair you have a value in A and a value in B. For such a one-dimensional problem the euclidean distance is simply

C = abs(A-B)

You're taking the square of a sum of a difference in your calculation. That changes the units, whatever your units are, they become squared the way you're calculating C. Do your A and B matrices represent differences in separate dimensions? If so, perhaps this is what you want:

C = hypot(A,B)

which, if for some reason you need to calculate in a loop, could be done in a loop by

C(l,m) = hypot(A(l,m),B(l,m));

[Joe Bannen]

Chad

I am trying to show the explicit calculation and not use any inbuilt code that obscures the mathematics. The hypot code looks like a good compromise though.

This also seems to give incorrect values for the distance matrix which should be (in its squared format):

ans =

     17.0000    6.0000   54.0000
     50.0000   21.0000   89.0000
      6.0000   27.0000   61.0000

Cheers

Joe