Why normalization matters with K-Means

A question about K-means clustering in Clementine was posted here. I thought I knew the answer, but took the opportunity to prove it to myself.

I took the KDD-Cup 98 data and just looked at four fields: Age, NumChild, TARGET_D (the amount the recaptured lapsed donors gave) and LASTGIFT. I took only four to make the problem simpler, and chose variables that had relatively large differences in mean values (where normalization might matter). Also, another problem with the two monetary variables is that they are both skewed positively (severely so).

The following image shows the results of two clustering runs: the first with raw data, the second with normalized data using the Clementine K-Means algorithm. The normalization consisted of log transforms (for TARGET_D and LASTGIFT) and z-scores for all (the log transformed fields, AGE and NUMCHILD). I used the default of 5 clusters.

Here are the results in tabular form. Note that I’m reporting unnormalized values for the “normalized” clusters even though the actual clusters were formed by the normalized values. This is purely for comparative purposes.

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Note that:
1) the results are different, as measure by counts in each cluster
2) the unnormalized clusters are dominated by TARGET_D and LASTGIFT–one cluster contains the large values and the remaining have little variance.
3) AGE and NUMCHILD have some similar breakouts (40s with more children and 40s with fewer children for example).

So, the conclusion is (to answer the original question) K-Means in Clementine does not normalize the data. Since Euclidean distance is used, the clusters will be influenced strongly by the magnitudes of the variables, especially by outliers. Normalizing removes this bias. However, whether or not one desires this removal of bias depends on what one wants to find: sometimes if one would want a variable to influence the clusters more, one could manipulate the clusters precisely in this way, by increasing the relative magnitude of these fields.

One last issue that I didn’t explore here, is the effects of correlated variables (LASTGIFT and TARGET_D to some degree here). It seems to me that correlated variables will artificially bias the clusters toward natural groupings of those variables, though I have never proved the extent of this bias in a controlled way (maybe someone can point to a paper that shows this clearly).