Calculating the sum of squared within cluster distances

I would like to calculate the "the sum of pair-wise distance between all maps of a given cluster" Murray et al., 2008, to further calculate the Krzanowski-Lai criterion in Matlab. In Murray et al. (2008) the following equation is given for the sum of (squared) distances:

u is a specific the measured values of one time point (one observation, in my case the measured potentials, since I am working with EEG data).

v is a specific the measured values of another time point then in u.

I found a code, which already implemented the Krzanowski-Lai criterion for my specific data in calc_fitmeas provided by atpoulsen. Thereby, the squared sum of within distances are calculated as following:

Nk = size(cluster,2);
        clstrsq = dot(cluster,cluster,1);
        %sum of pair-wise distance between all maps of cluster k
        Dk = sum(sum(bsxfun(@plus,clstrsq',clstrsq)-2*(cluster'*cluster)));

`Cluster` is a vector containing the data for a specific cluster. Columns corresponds to observations, rows to variables. For example, if I have 100 observations and for each observation 10 variables:

cluster = randn(10,100)

I have got no good training in linear algebra and math, thus I have got some questions about the implementation.

First, what kind of distance is calculated here.

Second, I do not understand how this code relates to the formula given above. Can some confirm, that this code corresponds to the formula and explain me the deviration?

[Basil]

I found out, that it is just the 2nd binomial formula.

Answer 1

[Matt J]

The method you've posted is highly inefficient. You should just do,

Dk=2*width(cluster)*(norm(cluster-mean(cluster,2),'fro')^2 )

[Matt J]

Timing test:

cluster=rand(10,6000);

tic
Dk0 = baselineMethod(cluster);
toc

Elapsed time is 0.364334 seconds.

tic;
Dk=2*width(cluster)*(norm(cluster-mean(cluster,2),'fro')^2 );
toc

Elapsed time is 0.005465 seconds.

relativeError=abs(Dk0-Dk)/Dk0

relativeError = 3.7123e-16

function Dk=baselineMethod(cluster)

        clstrsq = dot(cluster,cluster,1);
        %sum of pair-wise distance between all maps of cluster k
        Dk = sum(sum(bsxfun(@plus,clstrsq',clstrsq)-2*(cluster'*cluster)));

end

[Basil]

Thanks a lot for your efficient solution!