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I asked the same question at Cross Validated, here

I implemented a K-Medoid clustering algorithm recently; I have a number of points $x_1, ..., x_n$ which have various properties and a distance function $d$ that maps two points to some nonnegative number. The clustering works fine (with some number $k$ of cluster medoids where $k$ is a lot smaller than $n$), but I wanted to be able to change the clustering according to a property of my points, more specifically I'd like more clusters to appear where my points have a high value in some property. I tried scaling the distance matrix like this:

$newd (x_i, x_j) = p(x_i) * d(x_i, x_j)$
where $p(x)$ is the value of the property of $x$.

My expected result was to see more clusters around points with a high value of $p$, but instead, I get the exact same clusters. I'm probably missing something very basic here.

So, to sum up, my question: I'd like to see more clusters around points with a certain property. I assume I can achieve this by changing my distance function, but I don't quite understand how to do it?

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    $\begingroup$ If you want a cluster around x_i to be more likely, you need make the scaling factor smaller. So what about $d' = \frac{1}{p} d$ ? $\endgroup$ Oct 9, 2020 at 22:14

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That technique is commonly called weighted or observation-weighted k-means. The weighting changes how the cluster centers (medoids in your case) are calculated.

If you are using R, the WeightedCluster package has weighted K-Medoids implemented.

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