03 April 2017

Why Wavelets?

Next step in the analysis is detecting oscillations in the data set. First, I have to know what kind of signal I’m trying to discover and what I wish to know about it.
In essence, this post is an explanation of the thought process that led me to pick wavelets as my tool for analysis.
So, in my ideal case, I would love to know the strength, frequencies, location, timestamp and duration of the oscillation. Of course, later I plan to include information about the magnetic field as well, that will give me whole another layer of information about the events I’m trying to analyze. But that is something I have to add from the different dataset.
Next step was going through the toolset and figuring out which tool can give me what information. The most spread tool for dealing with oscillations is a Fourier transform.
The Fourier Transform can give me frequencies and strength. And if I cut my dataset into pixels across the time, I could get the location as well. But not the timestamp nor duration of the oscillation. And I need those to determine if there a correlation with the flare.
These two photos show an example of a change in pixel intensity in a solar image over the time( left) and one-dimensional Fourier transform(right). I can see strength and frequency. Location comes from the location of the pixel, but there is no way for me to tell when the different frequencies appeared over time.
There is also a trick people use with Fourier transform to give approximate of a time stamp. They cut their time series into small chunks and apply Fourier Transform on those chunks. This way they can approximate the time stamp. In a way, it is like you have a window that you slide over your data and peak at it. The problem is that you cannot make your window small enough to match your temporal resolution and at the same time detect frequencies of the longer period. So you have to have different sized windows for different frequencies you're trying to identify. See, this problem is promising to get complicated and computing intensive really fast.
And that's not all. Fourier transform has this particularity that imagines infinite signal. So some math tricks are necessary to prepare finite data set before Fourier transform. And there is a limit to those tricks. If you have more math ‘fix’ than your original data, then you get rubbish as a result of a transform. Actually, to be sure, it is best to have as little as possible of those math fixes, because that means, there is less of your time series you need to ignore due to potential ‘edge effects.' The math fix can give you false signal, that is called the edge effects in Fourier Transform, that’s why it is best to ignore what is going on on the edges of the analyzed time signal.
But mathematicians presented us with a new tool. Wavelets. This tool tells us all I need to know about the oscillations. Frequency, location, strength, duration and time stamp. And it does it neatly and fast.
So for me, it was a no-brainer to pick wavelets for my next step in the analysis.
However, such a sophisticated tool requires complex settings and loads of other thinking about how to pick those settings so that I get the information I need for my research.
I’ll talk about those in the next post.

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