I have made a hierarchical clustering and dived it into a distinct number of clusters. Now from each cluster I would like to pick one element representing the cluster best.
What would be a good method for doing this?
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I've figured out this is called a mediod and you get it by taking your original distance matrix ($D$). Then make a subset for the cluster ($c$) elements from this distance matrix which gives you a new (smaller) distance matrix. $$D^{c} \subset D$$ After that you can either take the sum of the rows or the columns, which in the end gives you the mediod (the element with the smallest sum, because it has the smallest distance to all other elements in the cluster). $$s_{j}^{c} = \sum_{i=1}^{n} D_{ij}^{c}\qquad\mbox{for }j=1,\ldots,n $$$$mediod^{c} = min(s^{c})$$You can have multiple mediods if they have the same sum.
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1$\begingroup$ Just a note: there's no reason you can't come up with a different kind of representative element. But the medoid is the vector-valued generalization of the median, so it's probably your best bet $\endgroup$ Feb 12, 2015 at 13:43