I have a couple of quick questions about PCA:
- Does the PCA assume that the dataset is Gaussian?
- What happens when I apply a PCA to inherently non-linear data?
Given a dataset, the process is to first mean-normalize, set the variance to 1, take an SVD, reduce rank, and finally map the dataset into the new reduced-rank space. In the new space, each dimension corresponds to a "direction" of maximal variance.
- But is the correlation of that dataset in the new space always zero, or is that only true for data that is inherently Gaussian?
Suppose I have two datasets, "A" and "B", where "A" corresponds to randomly sampled points taken from a Gaussian, while "B" corresponds to points randomly sampled from another distribution (say Poisson).
- How does the PCA(A) compare to the PCA(B)?
- By looking at the points in the new space, how would I determine that the PCA(A) corresponds to points sampled from a Gaussian, while PCA(B) corresponds to points sampled from a Poisson?
- Is the correlation of the points in "A" 0?
- Is the correlation of points in "B" also 0?
- More importantly, am I asking the "right" question?
- Should I look at the correlation, or is there another metric that I should consider?