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I was reading through the documentation for the R wavelets package and doing a little experimentation and noticed that the modwt and mra functions have similar outputs, yet the coefficients are called different things. The modwt function produces wavelet and scaling coefficients (W and V), whereas the mra function produces detail and smooth coefficients (D and S).

library("wavelets")

mod = modwt(data.frame(X=c(1,2,3,4)), filter="haar", n.levels=3)
wtrma = mra(data.frame(X=c(1,2,3,4)), filter="haar", n.levels=3, method="modwt")

What I'm trying to figure out what the differences are between these two sets of coefficients and how they work together (assuming they do), especially since the mra function can take a method of "modwt".

I did see this explanation of scaling versus wavelet coefficients, however, I'm not sure if the example aligns with MODWT. I'm specifically trying to learn about multiresolution analysis as I understand it can fit with ensemble machine learning regression strategies and I saw some references to it specifically being used with MODWT, but I seem to be lacking the basics in terms of wavelet transforms. I think part of the confusion is that multiresolution analysis is a term used for wavelet transforms, however modwt is described in the R package as a type of wavelet transform and it's distinct from the mra function.

If anyone can suggest an online course, that would also be helpful.

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