## Monday, 15 August 2011

### Visualizing discrete wavelet transforms: part II

Here is a process that takes the discrete wavelet transform (it happens to be the Daubechies 4 wavelet in this case rather than the Haar but the results are similar) of some fake data and plots the corresponding results. This is different from the maximum overlap discrete wavelet transform from the previous post.

The result looks like this.

(the z2 attribute is plotted as the colour using log scaling)

The bottom row is the clean signal (with scaling to make it show up), the 2nd row from the bottom is the noisy signal and the third row from the bottom is the result of the discrete wavelet transform. This is only included for completeness since it does not correspond in the original domain to the signal. For an interpretation of this I found the following in the code of the discrete wavelet transform (in file DiscreteWaveletTransformation.java)

In the case of the "normal" DWT the output value series has just one dimension, and these
coefficients are to be interpreted as follows: the first N/2 coefficients are the wavelet
coefficients of scale 1, the following N/4 coefficients of scale 2, the next N/8 of scale
4 etc. (dyadic subsampling of both the time and scale dimension). The last remaining
coefficient is the last scaling coefficient.

So this means the 4th row from the bottom corresponds to the coefficients of scale 1, the 5th row to scale 2 and so on. The coefficients have been replicated as many times as required to match the x scale.

Note that this differs from the view tradionally presented in the literature where the high frequencies are presented at the top. That's an exercise for another day.

The main difference between this and the previous MODWT example is the unpacking of the DWT result. For this I used a Groovy script. This takes 2 example sets as input and copies the correct parts of the first (the DWT result) into the second (the output that will eventually be de-pivoted).

The output example set that is fed into the Groovy script is created using a "Generate Data" operator since I found this to be the easiest way to generate the example set with the right number of rows and columns.

As before, the graphic shows that the algorithm has seen the presence of the low frequency signal at the expected location from x = 5000 and there is perhaps a hint that something has been spotted at x = 100.