I apologize if this is an overly-specific question. I've been trying to figure this out with no luck.
Some background: I'm working on a research project in computational neuroscience on functional connectivity between brain regions. To make a network graph, timeseries data from every brain region of interest (it can be BOLD signal for fMRI, EEG signals for electrophysiology, etc), is correlated against every other timeseries to create a correlation matrix. That symmetric matrix is then treated as the adjacency matrix for a graph that you can do a variety of different analysis on.
My question is this:
I have made two sets of graphs from the same set of timeseries. For one set, I have just correlated the raw, unfiltered timeseries (as is done in all the papers I've read). For the next set, as an experiment, I also made graphs after taking the norm of the Hilbert Transform of those timeseries (using the python scipy.signal.hilbert() and abs() functions). Those transformed series are correlated against each-other to make the next batch of graphs.
I had expected that the two sets of graphs should have similar characteristics, but that is not the result at all. When I analyze both sets of graphs, I get significantly different behaviors, which are often opposition (for example, the distribution of degrees in the Hilbert-transformed graphs is quite different than the degree distribution of their untransformed twins).
Can anyone help me come up with an intuitive sense of what's happening here? I've spent a lot of time reading up on Hilbert Transforms (removing negative frequencies, forcing an analytic signal etc), but I'm having a hard time understanding why this should effect graphs made from transformed signals, since I was under the impression that the Hilbert Transform should (theoretically) not result in loss of information.
The normed Hilbert Transform wasn't completely random: many papers I've read where researchers have done analysis of neural signals begin taking the norm of the Hilbert Transforming the data (although none have done that prior to making graphs).