Wavelets

First I have to apologize for stopping the posts in the blog for some time. Life happened.
Anyway, back to the wavelets.
In a previous post, I explained why wavelets are superior to the windowed Fourier transform for my research project. The image below demonstrates the wealth of information I can get from the wavelet analysis. This particular image is an illustration from one-pixel wavelet analysis of solar images.

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Wavelets are a complex tool that needs multiple settings. The main principle behind wavelets is to take a relatively simple function, scale it to different sizes and pass through the signal with that function. The chosen function acts as a magnifying glass that ends up jigging a lovely little dance when it bumps into the pattern you set to discover.

In wavelet tools, such functions are called Mother wavelet, and any toolset that involves the wavelet analysis offers several different options for mother wavelet. Which one you will choose depends on the type of the analysis you’re trying to perform. In essence, it is good to have some idea of the shape of the pattern you’re trying to detect.
In my case, it is simple, all the past corpus of knowledge about Magneto-Hydrodynamics comes to my rescue. I know what the shape of the oscillations I’m trying to discover is.

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The shape of the Morlet wavelet, the one I used as my Mother wavelet. Chosen because it tapers down on both sides, increasing the chance to detect the dampened oscillations in the signal. 

In astrophysics and physics, we’re trying to determine not only that something is going on, but also to determine how the events are connected.
If you ever heard about the statistics and how the statistic is a fancy way of lying to you, well that comes from mixing correlation with causality.
First, means that two events are happening simultaneously, but second means that one event is causing the other. If two things happen simultaneously, that does not mean they are causing each other. (For example, the sun rises and then your alarm clock rings. They correlate, but your alarm clock does not ring because the sun rises. )
In astrophysics, there were many of such mixups, and I have to say that with a decline in education some persist even today. ( I’ll leave rant about this for some other post and go to the main point.)
Because those mixups held back physics and astrophysics for centuries, physicists developed the research approach where a hypothesis first has to pass mathematical check and then experimental check, only if it passes both it can be truly called theory. Thanks to that approach I know that I’m looking for waves of certain type and shape. I know what math says about the shape of possible waves and conditions that exist in the sun, so it is easier for me to pursue true and not some figments of my imaginations. I know that what I’m looking for has to match with the previous corpus of knowledge and explain the new thing I’m trying to explain. And I know mathematical expressions of their shape, so it is easy for me to pick the appropriate wavelet.

In my particular case, the first type of the oscillations I’ll look for are damped pressure waves, a sound wave outside of our hearing range.

Similar waves are used in seismology, successfully for years, so I have even that little help, I can see what wavelet they used for their research, and why they choose that particular wavelet. If their reasoning matches the conditions of my problem, I can apply the same wavelet.
And I did check, and it did match, we both are trying to detect the dampening pressure waves. So I went with their choice of Morlet wavelet.
The problem with this wavelet is that is continuous, meaning I’m doomed to 1D analysis, so I do plan to see is it possible to play with discreet wavelet since those can be used for multidimensional analysis.

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