Wavelet multi-resolution analysis
I've been thinking a bit about multi-resolution analysis via the maximum overlap (or 'stationary') DWT. Now, the fact that you're projecting the data onto dilated bases and so several of your original data points are taking part in constructing the redundant set of coefficients clearly means that you're going to get some sort of correlations structure.
Ok, cool. But, it's still nice to do that because with the DWT you have a bunch of 'lining-up' problems and you can also be sensitive to the starting point and such. So, I get that my 'effective' number of data points in the analysis at that level is somewhere between the DWT number and the MODWT number* (and probably closer to the former), but it is strictly (right?) greater than the DWT number, so that's why I'm using it.
I had this idea, but I'm pretty sure it's wrong for some reason, but I can't figure out why. If someone could point it out, that would be great - because it's driving me nuts: If I have, say, in a MRA the details W1, W2, W3, and the smooth V3 using via some wavelet, some of the correlation in the coefficients will be due to the actual series. (Everyone likes to repeat how much it de-correlates FD series, but I also don't see many acf charts in the papers!) Then some of that correlation will be 'induced' by the transform. I feel like I should be able to whiten that series up a bit because I 'know' the induced correlation.
I feel like there has to be a mistake here for a few reasons.
1. I have't seen anyone mention it in a paper. I'm not that smart and this field isn't that new. People will publish anything no matter how small....
2. It feels like I'm getting something for nothing. Well, maybe not 'nothing' as I will be 'giving up' some of the points as I mentioned in (*), and not getting any 'extra' information. But, it seems somehow like I'd be coming out ahead a bit too much somewhere.
Re: Wavelet multi-resolution analysis
Ok, of course....I think it just hit me. Of course, right after I write it out. (That's probably what did it.)
Is this it: The goal of MRA is to de-correlate between time-scales, so you sometimes pay for that in correlation within scale. If I want to (basically) reverse that filter on that scale and de-correlate it, than that's dandy, but now it will be more correlated with the other scales?
Let me also say a bit more about what I'm doing. I'm trying to break down a few series, separate the time scales and then do some multivariate regression. I get some okay results and I've seen worse in the literature, but the fact that I've got this correlation structure, that I -know- the structure of, hanging around is annoying me.
Help! (And, many, many thanks!)