Reweighting features in PAM/K-means clustering As stated in Hennig et al. 2016 Handbook of cluster analysis:

If for subject matter reasons some variables are more important than
  others regardless of the within-variable variation, one could reweight
  them by multiplying them with constants reflecting the relative
  importance after having standardized their data-driven impact.

I feel that this is related to my data which I want to cluster using K-medoids algorithm, but I don't know the exact relationship between variables, i.e. I know that $var2$ should be more important than $var1$, but it is unknown if $var2$ is twice much important as $var1$ or maybe threefold, or even fourfold. Is there any established method or a measure to assess what should be the value of this weight other than an eye-test?
 A: You can't statistically measure which variable is more important on an unsupervised setting.
What is more important: shoe size, or income? Probably income, but statistics cannot "prove" this, or quantify the weights. Maybe you are a shoe salesman for oversized shoes, and want to pick that district where most people have oversized feet. It's user and problem dependent, so it will be subjective and require experience.
A: @jakes, I disagree with @anony-mousse on the statement, "You can't statistically measure which variable is more important on an unsupervised setting." This is wrong! What do you think Principal Component Analysis (PCA) does. PCA is an unsupervised dimensionality reduction algorithm. It works by extracting the variables that contain the maximum variance in them by doing an orthogonal transformation of the original variables. Besides, there exist scores of unsupervised algorithms that can be used for detecting and proving relationships between variables in an unsupervised setting. 
Again, "What is more important: shoe size, or income? Probably income, but statistics cannot "prove" this, or quantify the weights." Again, you are wrong. Statistics can prove this relationship provided how one has coded the variables and then apply an appropriate algorithm.  
