Find groups with high cross correlation matrix in Matlab - matlab

Question

Given a lower triangular matrix (100x100) containg cross-correlation
values, where entry 'ij' is the correlation value between signal 'i'
and 'j' and so a high value means that these two signals belong to
the same class of objects, and knowing there are at most four distinct
classes in the data set, does someone know of a fast and effective way
to classify the data and assign all the signals to the 4 different
classes, rather than search and cross check all the entries against
each other? The following 7x7 matrix may help illustrate
the point:
1 0 0 0 0 0 0
.2 1 0 0 0 0 0
.8 .15 1 0 0 0 0
.9 .17 .8 1 0 0 0
.23 .8 .15 .14 1 0 0
.7 .13 .77 .83. .11 1 0
.1 .21 .19 .11 .17 .16 1
there are three classes in this example:
class 1: rows <1 3 4 6>,
class 2: rows <2 5>,
class 3: rows <7>

Answer 1

This is a good problem for hierarchical clustering. Using complete linkage clustering you will get compact clusters, all you have to do is determine the cutoff distance, at which two clusters should be considered different.
First, you need to convert the correlation matrix to a dissimilarity matrix. Since correlation is between 0 and 1, 1-correlation will work well - high correlations get a score close to 0, and low correlations get a score close to 1. Assume that the correlations are stored in an array corrMat
%# remove diagonal elements
corrMat = corrMat - eye(size(corrMat));
%# and convert to a vector (as pdist)
dissimilarity = 1 - corrMat(find(corrMat))';
%# decide on a cutoff
%# remember that 0.4 corresponds to corr of 0.6!
cutoff = 0.5;
%# perform complete linkage clustering
Z = linkage(dissimilarity,'complete');
%# group the data into clusters
%# (cutoff is at a correlation of 0.5)
groups = cluster(Z,'cutoff',cutoff,'criterion','distance')
groups =
2
3
2
2
3
2
1
To confirm that everything is great, you can visualize the dendrogram
dendrogram(Z,0,'colorthreshold',cutoff)

Answer 2

You can use the following method instead of creating the dissimilarity matrix.
Z = linkage(corrMat,'complete','correlation')
This allows Matlab to interpret your matrix as correlation distance and then, you can plot the dendrogram as follows:
dendrogram(Z);
One way to verify if your dendrogram is right or not is by checking its maximum height which should correspond to 1-min(corrMat). If the minimum value in corrMat is 0 then the maximum height of your tree should be 1. If the minimum value is -1 (negative correlation), the height should be 2.

Answer 3

Since it is given that there are going to be 4 groups, I'd start with a pretty simplistic two stage approach.
In the first stage you find the maximum correlation among any two elements, place those two elements in a group, then zero out their correlation in the matrix. Repeat, finding the next highest correlation among two elements and either adding those to an existing group or creating a new one until you have the correct number of groups.
Finally, check which elements aren't in a group, go to their column, and identify the highest correlation they have with any other group. If that element is in a group already, place them in that group as well, otherwise skip to the next element and come back to them later.
If there is interest or anything isn't clear I can add code later. Like I said, the approach is simplistic but if you don't need to verify the number of groups I think it should be effective.