# Assign weights to variables in cluster analysis

I want to assign different weights to the variables in my cluster analysis, but my program (Stata) doesn't seem to have an option for this, so I need to do it manually.

Imagine 4 variables A, B, C, D. The weights for those variables should be

w(A)=50%
w(B)=25%
w(C)=10%
w(D)=15%


I am wondering whether one of the following two approaches would actually do the trick:

1. First I standardize all variables (e.g. by their range). Then I multiply each standardized variable with their weight. Then do the cluster analysis.
2. I multiply all variables with their weight and standardize them afterwards. Then do the cluster analysis.

Or are both ideas complete nonsense?

[EDIT] The clustering algorithms (I try 3 different) I wish to use are k-means, weighted-average linkage and average-linkage. I plan to use weighted-average linkage to determine a good number of clusters which I plug into k-means afterwards.

• Both ways are generally not correct. Multiplying values of variables is not equivalent to weighting variable importance for clustering. If the program doesn't have weighting option you could do it sometimes with data as you wish - but this depend on the exact nature of your clustering. So, describe (in your question) details of your clustering: what algorithm and method you are going to use. – ttnphns Nov 27 '13 at 14:49
• Note that the easiest and universal way to weight variables (and the weights are integers or can be made integers) would be simply to propagate the variables times those weights. In your example, you can take 50 As, 25 Bs, 10 Cs, 15 Ds in your clustering. – ttnphns Nov 27 '13 at 17:16
• Or, the alternative: If you use clustering based on euclidean measure or you use k-means, multiply each variable by the sq. root of its weight. This multipication should be, of course, done after any pre-processing (such as standardization) you might want to do before clustering. – ttnphns Nov 27 '13 at 17:23

One way to assign a weight to a variable is by changing its scale. The trick works for the clustering algorithms you mention, viz. k-means, weighted-average linkage and average-linkage.

Kaufman, Leonard, and Peter J. Rousseeuw. "Finding groups in data: An introduction to cluster analysis." (2005) - page 11:

The choice of measurement units gives rise to relative weights of the variables. Expressing a variable in smaller units will lead to a larger range for that variable, which will then have a large effect on the resulting structure. On the other hand, by standardizing one attempts to give all variables an equal weight, in the hope of achieving objectivity. As such, it may be used by a practitioner who possesses no prior knowledge. However, it may well be that some variables are intrinsically more important than others in a particular application, and then the assignment of weights should be based on subject-matter knowledge (see, e.g., Abrahamowicz, 1985).

On the other hand, there have been attempts to devise clustering techniques that are independent of the scale of the variables (Friedman and Rubin, 1967). The proposal of Hardy and Rasson (1982) is to search for a partition that minimizes the total volume of the convex hulls of the clusters. In principle such a method is invariant with respect to linear transformations of the data, but unfortunately no algorithm exists for its implementation (except for an approximation that is restricted to two dimensions). Therefore, the dilemma of standardization appears unavoidable at present and the programs described in this book leave the choice up to the user

Abrahamowicz, M. (1985), The use of non-numerical a pnon information for measuring dissimilarities, paper presented at the Fourth European Meeting of the Psychometric Society and the Classification Societies, 2-5 July, Cambridge (UK).

Friedman, H. P., and Rubin, J. (1967), On some invariant criteria for grouping data. J . Amer. Statist. ASSOC6.,2 , 1159-1178.

Hardy, A., and Rasson, J. P. (1982), Une nouvelle approche des problemes de classification automatique, Statist. Anal. Donnies, 7, 41-56.

• Your first reference is mangled somehow: Leonard Kaufman and Peter J. Rousseeuw are the authors of the book you link to. – Nick Cox Nov 27 '13 at 16:25
• Oh thanks for pointing this out... I got screwed by Lavoisier, which made a mistake on their page "Auteurs : SEWELL Grandville, ROUSSEEUW Peter J.", which in turn screwed Gscholar which I was using for getting the reference. – Franck Dernoncourt Nov 27 '13 at 16:28
• Thanks @FranckDernoncourt! If the scale (and thus the range) of the variable determines its weight, wouldn't approach 1.) in my initial question be a somehow correct solution? – SPi Nov 27 '13 at 16:59
• Yes approach 1 is the right one, and correspond to what Kaufman, Leonard, and Peter J. Rousseeuw are saying in the paragraphs I quoted in the answer. Approach 2 would be useless as the standardization removes the weights :) – Franck Dernoncourt Nov 27 '13 at 17:01