I think I finally have the answer!

The short version is that only the product of eta and sigma affects analysis using Morlet wavelets. It makes absolutely no difference whether an analysis uses sigma = 0.5 and eta = 10.0 or if it uses sigma = 1.0 and eta = 5.0.

Formally, though, the wavelet transforms can take different values in the two cases, because the most commonly used normalization depends on eta and sigma separately, and not just their product. But since the same normalization will be used on two different measurements (different samples, different locations on a sample, or a sample at different times), or on a sample and a theoretical model, it has no practical impact. The scale on a contour plot of the wavelet transform will change, but that scale isn’t even included in some publications, because it’s essentially arbitrary. That’s especially true since there are different normalization schemes out there: https://cp.copernicus.org/preprints/cp-2019-105/cp-2019-105-supplement.pdf.

I arrived at these conclusions by examining the mathematics. Can anyone who has actually worked with Morlet wavelets confirm that’s the behavior? I could imagine that some piece of software might implement wavelets in such a way that different combinations of sigma and eta that have the same product might nevertheless yield very slightly different results just because of computational effects (e.g. differences in interpolation and such).

Best,

Scott Calvin
Lehman College of the City University of New York

On Aug 6, 2022, at 7:29 AM, Xinyu Luo <xinyuluo@zju.edu.cn> wrote:

Hello Scott,

  Thanks! I actually have the same question. I'm a graduate student. I learned the wavelet transform from the website of ESRF (and I learned XAFS from your book!). I can share my opinions on this question.

  There is a mother wavelet called cauchy [Muñoz, M.et al, Am. Mineral. 2003, 88 (4), 694], which seems to be pretty nice because only one parameter used to manipulate the resolution---n. And that's where I started. After reading more literatures and I was taught that wavelet analysis can be more powerful if the mother function looks more like to the path of interested. So probably, the two parameters of Morlet can be used to ,mimic the path of interest generated by FEFF. I wrote an email before to ask about how to realize the steps described in a literature[Funke, H. et al, J. Synchrotron Radiat. 2007, 14 (5), 426]. You can see that they customized a mother function on the base of FEFF to surpass the resolution of Morlet. If I understand it correctly, the uncertainty limit, though which is a law in physics, can't be compared on different Mother functions. And go back to your question, perhaps you can an optimized choice of sigma and eta with a fixed (sigma)(eta). BUT I DON'T KNOW HOW TO GET THAT. It's clumsy to try a bunch of value and I hope someone can teach me on how to choose a starting value of sigma, or eta.


Best

Xinyu



-----Original Messages-----
From:"Scott Calvin" <dr.scott.calvin@gmail.com>
Sent Time:2022-08-06 01:41:17 (Saturday)
To: ifeffit@millenia.cars.aps.anl.gov
Cc:
Subject: [Ifeffit] Parameters for parent function in wavelet transform analysis

Hi all,

I’ve never actually tried wavelet transform analysis before, so I’m trying to understand it better, but there’s one aspect that’s puzzling me:

The Morlet wavelet has two parameters, eta and sigma. Eta controls how fast it wiggles, and sigma how quickly the function dies out. (Essentially, it’s like the basis function for a Fourier transform multiplied by a Gaussian envelope.)

In wavelet analysis we pick a parent function, which is often a Morlet wavelet with particular values of eta and sigma that we have chosen. 

We then create a set of child functions from the parent function by shifting and dilating the function in k-space.

Each child function is then used as a basis function for a transform of chi(k) calculated by integrating over a range of k-values, much like what is done to calculate a Fourier transform. Since the Morlet wavelet is localized in k-space, and the child functions are shifted to focus on different regions of k-space, our wavelet transform produces plots which are a function of k. But since dilating the parent function by different amounts yields child functions with different frequencies, the plots are also a function of R. Therefore the result is a two-dimensional (k and R) contour plot.

So far, so good. 

But what I’m wondering about is the effect of the initial choice of eta and sigma for the parent function.

The dimensionless product of eta and sigma has a clear effect. If (eta)(sigma) is small, the parent function will not have very many oscillations of significant size; it it’s large, it will have a lot of oscillations. So if we want good resolution in R (and thus relatively poor resolution in k), we want a big value of (eta)(sigma). In the limit as (eta)(sigma) becomes arbitrarily large, we recover the Fourier transform. If we want good resolution in k, we use a small value for (eta)(sigma). In the limit as (eta)(sigma) becomes arbitrarily small we recover chi(k).

That still makes sense to me!

But what difference do sigma and eta make individually? In other words, how is sigma = 0.5 and eta = 10.0 different from sigma = 1.0 and eta = 5.0? Since we end up creating child functions that dilate the parent function anyway, I can’t see that it should have any effect at all. 

And if those two parameters don’t make an independent effect, why do we pretend there are two independent-looking parameters? 

I’ve seen at least one early paper that suggests choosing eta so that it’s close to the path length you’re most interested in probing, but I can’t see how that actually makes any difference, given the dilation.

I expect there’s a good chance I’m missing something obvious, or have a fundamental misunderstanding of part of the process, and look forward to learning more!

Best,

Scott Calvin
Lehman College of the City University of New York

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