How do I analyze clustering post-PCA

Question

I had in mind to cluster stocks based on some risk indicators such as VaR, sharpe ratio or variance. In a first instance I was thinking to cluster those data points and analyze the results, because the idea was to segmentate stocks by risk levels (clusters) assuming that if all variables were related to risk I would be able to assess if cluster 1 corresponds to very low risk, 2 medium-low risk etc... and therefore asign stocks to different risk-acceptance users.

All those thoughts were before realizing that I need to run PCA and data normalization. Here are my questions:

1. Should I run PCA before the clustering due to risk indicators are high-correlated, or should I first cluster and finally perform PCA to visualize the clusters?
2. I performed PCA to my data before clustering and here is the plot I got: After seeing this my first thought was: ok, if I cluster the data now, how could I analyze the results and see if each cluster represents different risks levels? I do not have the risk indicators anymore instead I have PC1 and PC2. I am somehow convinced that if all data is related to risk, according to distance clustering, the clusters should contain risk related stocks but how to know which risk level have each cluster. This is critical because the main idea is that if the user wants low risk stocks I provide low risk stocks and no any other stock.

Answer 1

One important thing to note is that both PCA and k-means clustering rely on similar algorithms. PCA essentially finds the main axis of variance in high-dimensional data, while k-mean minimizes and maximizes within and between variance. Both of these algorithms are in some way "simplifying" the full variance into a simplified problem space.

So all that being said, I doubt the order is going to make much difference. I'd try both and see what happens. If your principal components capture a large percentage of the variance, then the clustering should be almost identical. It looks like you have essentially three clusters, mostly defined by PC1. If your principal components aren't capturing a large degree of variance, then you may see differences, and then it really depends on the context of your PCA. Sometimes PCA is theoretically viewed as "de-noising" your data, and allowing you to focus on the most important factors. This is particularly true when your variables are truly imperfect representations of latent variables. However, if your variables are not truly representative of some latent variables, then really PCA is just a visualization trick. Both numeric and theoretical concerns need to be taken into account in addressing this issue.