# Cluster analysis with skewed distibutions

For my master's thesis I would like to use different clustering algorithms to cluster municipalities (as objects) in regard to their land-use characteristics (as variables).

Analyzing my data descriptively I noted that I have a lot of extremely left skewed distributions (for example a lot of municipalities have zero values for some land use options and other have very high values). This is also true after standardizing my data by area or population...

Can anyone give me some advice on how does this will affect my cluster analysis? I think it may be important regarding the choice of my distance measurement (for example the absence of values to be interpreted as a non-similaritiy).

I didn't found a lot of information in this case in the common literature.

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Welcome to the site. I fixed up your grammar and spelling. I also removed the signature since the site adds one automatically. – Peter Flom Jan 10 '13 at 10:56
@PeterFlom Hello Peter, thank you for the editing and the warm welcome :) – Joschi Jan 10 '13 at 13:32

You may want to spend more time on data preparation.

For example, one may argue that "area" inherently is a quadratic value (and "volume" is inherently cubic). And thus, in order to make attributes more comparable, a x_new = sqrt(x_old) transformation may be sensible for some attributes.

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Thats an fair argument. Still I don't see how its going to deal with the problem of the left screwed distributions. I think sqrt may make make the distribution even more left screwed since big values will get even bigger ...? – Joschi Jan 10 '13 at 14:15
sqrt will only make values smaller than 1 larger: sqrt(4)=2. So you need to get the normalization steps into the appropriate order (and hope that your input data wasn't already normalized incompatibly) – Anony-Mousse Jan 10 '13 at 14:20
oh...yes, that is true, of course I have to get the order right. But supossing that I still have left screwed distributions afterwards. Is this a problem for the analysis? – Joschi Jan 10 '13 at 14:37
Skewedness itself is not a problem for most clustering algorithms. But it may be indicative of e.g. inappropriate normalization. But all these (sqrt, z-score, minmax, any normalization!) are just heuristics. Or have you seen a proof that one normalization is more "correct" than another? – Anony-Mousse Jan 10 '13 at 14:58
no I have not seen such thing like a correct normalization method. The thing here is that my data itself has this distribution even without any kind of normalization because I have a lot of zero values. So I think it may be important to choose the distance measure properly since some interpret the non-existance of an attribute as similarity and others dont...but I'm not expirienced to adequatly incorporate this fact in the choise of the distance measure or the cluster algorithm... – Joschi Jan 10 '13 at 15:33