# Cluster analysis returns questionable results

A few weeks ago I ran a cluster analysis on some economic data and (unfortunately only) now I found out that some results do not make sense. For instance I analyzed the development of unemployment rates form end-2005 to mid-2011 for 32 countries. One cluster contained among others Germany (Figure 1) and Spain (Figure 2) and another cluster among others Estonia (Figure 3). I couldn't understand why Mathematica would put Germany and Spain together in one cluster, instead of Estonia and Spain. I just calculated the EuclideanDistance between Germany and Spain (17.3) as well as Spain and Estonia (9.1). These results underline my doubts on my cluster-analysis-results.

Here is the code with which I worked:

First I generate a random sample (n=1200) of the original data, to minimize the effect of the input-order in FindClusters (Note from Mathematica-Documentation: "The order of elements can have an effect on the clusters found")

data;  (*see below*)
Table[DeleteDuplicates[
Table[RandomSample[DeleteCases[data[[j]], {}]], {i, 1200}]], {j,Length[data]}];

Then I run FindCluster for the first time to determine the commonest length of each cluster.

ClustersUnSorted[list_]:=Table[Map[FindClusters,list[[i]]],{i, Length[list]}]

In my next step I say, that if the previous step found less than four clusters, I would like Mathematica instead to set the minimum number of clusters to 4.

ClusterLength[list_]:=Table[If[
Commonest[Map[Length,list[[i]]]][[1]]<4,
4, Commonest[Map[Length, list[[i]]]][[1]]],
{i, Length[list]}]

Then I run FindClusters again on the basis of the previously determined number of clusters.

ClustersUnSortedFixLength[list_,Flatten[ClusterLength_]]:=
Table[Map[FindClusters[#,ClusterLength[[i]]] &, list[[i]]], {i, Length[list]}]

The last two steps are there to get rid of duplicates as for each indicator I generated a random sample (see step 1).

Sort[Map[Sort, list[[j]][[i]]]], {i, Length[list[[j]]]}],
{j, Length[list]}]

CommonestCluster[list_] := Map[Commonest, list]

The last two steps can be easily comprehended by the following example (instead of RandomSample I could use Permutations[{1, 5, 3, 10, 100}] for a much more reliable result but since Permutations is limited to input with length less than 11 it is not applicable to my data.):

Table[RandomSample[{1, 5, 3, 10, 100}], {i, 1200}];
Map[FindClusters, %];
Table[Map[Sort, %[[i]]], {i, Length[%]}];
Map[Sort, %];
Commonest[%]

exemplary data (Note: the time series are not always of the same length.):

Country1GDP = Join[{{"Country1", "GDP"}}, RandomReal[{-1, 1}, {6}]];
Country2GDP = Join[{{"Country2", "GDP"}}, RandomReal[{-1, 1}, {6}]];
Country3GDP = Join[{{"Country3", "GDP"}}, RandomReal[{-1, 1}, {6}]];
Country4GDP = Join[{{"Country4", "GDP"}}, RandomReal[{-1, 1}, {5}]];
Country5GDP = Join[{{"Country5", "GDP"}}, RandomReal[{-1, 1}, {6}]];

Country1Imports =
Join[{{"Country1", "Imports"}}, RandomReal[{-1, 1}, {6}]];
Country2Imports =
Join[{{"Country2", "Imports"}}, RandomReal[{-1, 1}, {5}]];
Country3Imports =
Join[{{"Country3", "Imports"}}, RandomReal[{-1, 1}, {6}]];
Country4Imports =
Join[{{"Country4", "Imports"}}, RandomReal[{-1, 1}, {6}]];
Country5Imports =
Join[{{"Country5", "Imports"}}, RandomReal[{-1, 1}, {6}]];

CompleteGDP = {Country1GDP, Country2GDP, Country3GDP, Country4GDP,
Country5GDP};
CompleteImports = {Country1Imports, Country2Imports, Country3Imports,
Country4Imports, Country5Imports};

data = {CompleteGDP, CompleteImports};
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You know, you can supply multiple iterators to a single Table call... i.e., you can write Table[..., {i,...}, {j,...}] instead of Table[Table[...,{i,...}], {j,...}] – R. M. Mar 20 '12 at 0:50
@Mr.Wizard We must've edited at the same time :) It's ok, mine was more thorough =) Besides, the poster is probably not American, hence the use of s instead of z, which I think we should respect... – R. M. Mar 20 '12 at 0:51
Also, it would be helpful if you could give the same dataset you are using, namely what is data to start out with? (Sorry if I missed it) – tkott Mar 20 '12 at 1:34
I'll try to generate some exemplary data, as the original one is far too large. – John Mar 20 '12 at 8:43
@rm -rf: Actually, the equivalent to Table[Table[...,{i,...}],{j,...}] is Table[...,{j,...},{i,...}]. This constantly throws me off, as it does not simply allow to delete brackets, but one has to change the order of iterators as well. – Thomas Oct 1 '12 at 18:08

Clustering is a relatively unstable process. Points which exist near to cluster boundaries may have small Euclidean, or other, distances between them, but be on different sides of the local boundary. So, in and of themselves, point separation distance metrics may be misleading.

If clusters in the data overlap to any degree, a common case, then there is almost certain that some unavoidable mislabelling of cluster membership will occur in the region of overlap.

The exact specifics of how this mislabelling came about will depend on the distance metric used and the distribution and dimensionality of the original data.

Projecting the data down into a low dimensional space (1D,2d,3D) using PrincipleComponents[], or other similar dimensional reduction transform ... MDS etc, will allow you to plot the clusters and visualise some elements of their specific configurations. This may, or may not, be useful in deciding if the clustering you have is sensible.

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Something in my code (see above) was just basically wrong. I run another cluster analysis with the same configurations (random sample with n=1200 and Euclidean Square Distance) but with an new code. I'll post the new code later. However, I would be pleased for additional comments on the above code. :) – John Mar 24 '12 at 8:57