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Suppose that i have multidimensional dataset and performed some partitioning clustering on it. Is there any way to find out what objects in a particular cluster have in common (except the fact that clustering algorithm decided to put them together)?

I've read many times that clustering is not a well-posed problem in general and that one should not overinterpret its results but still people are trying to cluster multidimensional data and make some practical sense of the results. I just can't find any good source on how the interpretation is done in practice.

Any tips and resources will be highly appreciated.

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  • $\begingroup$ Many clustering algorithms are sensitive to the number of clusters defined, data order, or initial starting points, making interpreting them a bit arbitrary. At work, we sometimes have to formulate our own hypothesis on these clusters. (we are marketers, so perhaps the powerpoint is more important than the analysis itself.) $\endgroup$ – The Lyrist Jun 8 '18 at 18:22
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In my experience, with very high dimensional climate data, if I do something as easy as k-means clustering, I would probably first look at the silhouette values of the clusters. As you would probably know, a high silhouette value for the clusters would mean that they are well classified and distinctly different from each other, while a low silhouette value or even a negative one would mean the opposite.

In case these values are reasonable, you can then try to reduce the dimensionality of your data set by doing something like PCA (for starters or even something like t-SNE) and see if your clustering results are still that good. If they are so, then you have a sense that those components (PCs in case of PCA or the retained components in case of t-SNE) are probably dominating the feature space that is used for clustering.

In my very personal opinion, I would try to reduce(dimensionality) as much as I can and go about doing the clustering until I can visualize them in 3D or 2D. With t-SNE this can be done quite successfully for the right kind of data sets. Sometimes, if you are lucky, even comparing L2 distances between cluster centers and samples may yield something fruitful but that's hardly ever the case for high dimensional data.

I guess you're question is a little open ended and a discussion on such would be great. But, once again, if I understand it correctly, extracting the part of the feature space that dominates the " distance" metric for clustering is difficult to find algorithmically in my limited knowledge but some sense can be extracted if you try these exercises.

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You can write code to analyze this. But this is very much data specific.

In some cases like k-mrans, you already get cluster averages. So you can easily compare the averages.

If you don't have major correlations in your data, then histograms can be insightful. Make stacked histograms, colored per cluster, to see if there is a different distribution in each attribute. Etc. There are so many ways to do this, and they are not specific to clustering.

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