Simplified illustration of blind-source separation algorithms
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See attached picture 'BMI500_lab11_q1.JPG'
I created a simple power method calculation of the eigenvalues as 'powerMethod.m'. Using that code, when you run the following:
powerMethod([1;1],C,1/100) you get the eigenvalues to be 3.9069 and 2.4614.
I didn't see that there was a power analysis code provided but it does the same thing.
See attached figure 'BMI_lab11_q2b.jpg'
I didn't change the code that much at all other than changing to a different example (example 2: SampleECG2) and test turning on or off the lowpass filter. Example 2 is different from the first example but the biggest change was from the lowpass filter. When the lowpass filter was removed, the time variant aspect of the signal was not filtered out and the final plots were not on a uniform/horizontal axis. When the lowpass filter is on, all plots are normalized/had the time variance removed and was much easier to see the harmonic signals.
It was a simple power method of analyzing the eigenvalues so I just changed the number of channels to 2. As expected, changing the number of channels changes it to a 2x2 matrix and 2x1 eigenvector. Additionally, the number of iterations was changed to 1000. The accuracy of the power method increased as the number of iterations increased, which was expected.
I only changed the sampling frequency (fs) from 500 to 1000 and the results were changed as we expected. The signal length was doubled from 1,500 to 3,000. Other than that, there really wasn't any unexpected changes. I also changed the g ('pow3', 'tanh', 'gauss', 'skew') and didn't see much differences between them except for the 'skew' where the signals were very different. Which seems to indicate that choosing the nonlinearity is fairly important.
Ran the code by halving the sampling frequency (fs) from 250 to 125 but did not see much of a difference if at all which may suggest that the code is robust to sampling frequency. I did run a second dataset (EEGdata2) but it was too much for my computer to handle so I stopped it early. However, the second dataset seemed to look more interesting as there were time variances within the dataset. JADE seems to be a BSS method or implementation.
I ran the last example for my ex05. As I understand, denoises the mixed maternal and fetal signals and then analyzes each based on the different components of the signal using different algorithms like JADE. Beyond this, I did not understand the code very well.
Chosen paper: A comparison of PCA, KPCA and ICA for dimensionality reduction in support vector machine
Summary: The paper discussed and showed the effectiveness of applying different component analyses namely, principal component analysis (PCA), kernel principal component analysis (KPCA) and independent component analysis (ICA) on a failry simple machine learning model (support vector machine (SVM)). While machine learning methods extract features, the importance of those features may be questionable and sometimes may be confounding noise. These component analysis methods allow the extraction of the most relevant features for the model to analyze and learn for their tasks. They have shown in their paper that KPCA performed the best which makes sense as in deep learning models, convolutional layers with kernels are used to extract features like edges. As I understand, the KPCA allows the extraction of specific principle components which can not be linearly separated. This increases the accuracy of SVM.
As for the pseudo code of the KPCA algorithm, it's pretty much the same as linear PCA (eigen analysis) but with the addition of the kernel that converts the data into a "feature space" to perform the linear PCA on.