I'm trying to do research into application of neural networks to financial time series. One method that I often see being employed for denoising of one-dimensional time series data is some form of wavelet transformation. I've been trying to better understand this math concept, but I'm hitting a few walls.
One of the first things anyone says (in tutorials, etc.) about the wavelet transformation is that it produces information "localized in both time and frequency" – this seems impossible to me. Having (an admittedly quite basic) understanding of the Fourier Transformation, the natural relationship between frequency and time would imply a fundamental uncertainty when considering both at the same time. One cannot simultaneously know both the frequency of a wave and its position in time. Yet I hear that wavelets do just this.
How does this work? Some of the more esoteric mathematical notation is unfortunately a bit lost on me, so if anyone could offer a gentle introduction to wavelets, I'd appreciate it...
If we are talking about Fourier frequencies (sin- and cos- frequencies) yes then it is impossible. But all frequency concepts in our world are not Fourier frequencies.
For example an arbitrary well-enough-behaved periodic function has a Fourier series expansion which can include unlimited number of "overtones". It still in some sense has just one frequency. Which is 1 / period time. It would not have compact support in the Fourier domain but obviously it would be periodic and therefore have a "frequency" in any relevant sense of the word.
Wavelet and their scaling functions can have compact support in time domain - in that case their Fourier transform does not have compact support. But majority of the energy captured will be in a specific Fourier-frequency band. Say 90% or 99%.
But don't forget it is Fourier frequencies we are talking about! There is nothing "holy" about them which gives them some special right to trademark "frequency" concept. It is just a bunch of sines and cosines which happen to be practical because convolution becomes multiplication in this new space.
One way to understand why we get uncertainty principle in one domain is to consider multiplying with a box function. What happens in other domain is that function is convolved with a sinc function: $\frac{\sin(ax)}{ax}$. This function does not have compact support so neither will the result of the convolution.