I've read some posts about PCA applied on time series, but still a bit confused and I have the following questions(Suppose I am working with a time series of the return of 50 industries and I want to use a clustering algorithm to divide them into several group):
Say I have calculated the eigenvalue and eigenvector from the correlation matrix, and found that the first twenty eigenvalue account for 85% of the total, and I then use these twenty eigenvalue to approximated the original time series. I know if I choose all eigenvalues then I can get the identical original time series, but what information did I lose if I choose twenty of them specifically? What is the purpose to do so?
I found some post said we can always drop the first principle component(means we don't use it), why can we do that?
Can I interpret each eigenvalue as a trend in the market, for example the first principle component, can I derive whether the corresponding industry is in the same direction or different from the market trend based on the sign of its corresponding eigenvector, and if so, can I apply k-means to all industries by using the eigenvectors of the first few principle component to group them, is this make sense?
Welcome for any kind of hint or ideas, thanks.