Introduction of various clustering techniques.

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prwaitbar off                % waitbar not needed here
delfigs                      % delete existing figures
randreset;                   % takes care of reproducability
prwarning off                % no warnings

Define a dataset

we use some standard routines to create 8 two-dimensional clusters. After creation the label information is removed.

m = 50;
a = prdataset(rand(m,2).*repmat([8,0.5],m,1))+repmat([1 -1],m,1);
a = [a; prdataset(rand(m,2).*repmat([8,0.5],m,1)+repmat([1.5 0.5],m,1))];
x = +gendats([m m],2,4).*repmat([1,1],2*m,1)+repmat([3 5],2*m,1);
a = [a; x];
y = +[gendatc(m); gendatc(m)+repmat([0 10],m,1)];
y = y.*repmat([0.2,0.35],2*m,1)+repmat([-1.5 1.5],2*m,1);
a = [a; y];
z = prdataset(+[gencirc(m,0.05)*7; gencirc(m,0.04)*2.5]);
z = z.*repmat([0.5,0.5],2*m,1)+repmat([16 3],2*m,1);
a = [a; z];
axis equal;
title('Original dataset');
a = prdataset(+a); % remove labels
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This dataset consist of:

partitional clustering by kmeans, find 8 clusters

[labs,b] = prkmeans(a,8,1,'rand');
subplot(2,2,1); scattern(b); axis equal
title('kmeans, iteration 1');
% The cluster labels are stored in labs. The dataset b contains the
% original objects, but labeled with the cluster labels after 1 iteration.
it = [2 2 100];
for n=1:3
  % Here the results after more iterations are computed. Each call is
  % initialised by the cluster labeling resulting from the previous call.
  [labs,b] = prkmeans(a,8,it(n),labs);
  % Note that for n=3 100 iterations are requested. The algorithm, however,
  % stops much earlier, when the result is stable.
  subplot(2,2,n+1); scattern(b); axis equal
  title(['kmeans, iteration ' num2str(it(n)+1)]);
title('kmeans, final result');

The figure illustrates that the kmeans algorithm creates spherical clusters with the same radius.

hierarchical clustering, dendrograms

The two most extreme versions are shown, single linkage and complete linkage. In this section the densrograms are computed.

d = sqrt(distm(a)); % The routines operate on the distance matrices

den = hclust(d,'single');
subplot(2,1,1); plotdg(den);
title('single linkage dendrogram');
set(gca,'xtick',[]); % removes the cluttered x-ticks

den = hclust(d,'complete');
subplot(2,1,2); plotdg(den);
title('complete linkage dendrogram');
set(gca,'xtick',[]); % removes the cluttered x-ticks
fontsize(14)         % default font size is not nice

Dendrograms show in the horizontal direction the objects in some order. In the vertical direction the cluster distance is shown when two clusters are merged using the single linkage or complete linkage definition.

The two dendrograms show the very different characteristics of the two procedures. By single linkage clusters grow gradually with more remotely located objects until they are merged with another cluster.

By complete linkage clusters of similar size (largest within distance) are created.

hierarchical clustering, 4 clusters

labs = hclust(d,'single',4);
subplot(2,1,1); scattern(prdataset(a,labs));
axis equal; title('single linkage,4 clusters');

labs = hclust(d,'complete',4);
subplot(2,1,2); scattern(prdataset(a,labs));
axis equal; title('complete linkage, 4 clusters');

Here the resulting clusterings show their characteristing: elongated clusters constituted by touching (single linkage) or spherically shaped clusters neglecting touching (complete linkage). Note that single linkage shows one cluster of a single object. This can be verified in the dendrogram.

hierarchical clustering, 8 clusters

r = [1 3 5 7 2 4 6 8]'; % trick to shuffle labels for better visualization.
labs = hclust(d,'single',8);
subplot(2,1,1); scattern(prdataset(a,r(labs)));
axis equal; title('single linkage, 8 clusters');

labs = hclust(d,'complete',8);
subplot(2,1,2); scattern(prdataset(a,r(labs)));
axis equal; title('complete linkage, 8 clusters');

Again: elongated clusters for single linkage and spherical clusters for complete linkage. Single linkage has three single object clusters.

mode seeking

This procedures searches the modes (local maxima) of the density. A nearest neighbor search is used to follow from each object the density gradient to a mode. The number of neighbors used for the search influences the number of clusters found: the more neighbors the less clusters.

labs = modeseek(d,20);
subplot(2,1,1); scattern(prdataset(a,labs));
axis equal; title(['mode seeking, 20 neighbors, ' num2str(max(labs)) ' clusters']);

labs = modeseek(d,10);
subplot(2,1,2); scattern(prdataset(a,labs));
axis equal; title(['mode seeking, 10 neighbors, ' num2str(max(labs)) ' clusters']);

For larger numbers of neighbors, clusters are combined as long as objects have neighbors in the other cluster having a higher density.