Infomax ICA demo with Seven Segmentica (EEGLAB plugin) on Youtube.
This is a tool to give a classical ICA demo for decomposing overlapped seven-segment number displays (0-9) with independent (and kurtotic) time series data with some white noise. Many of EEGLAB users have been using ICA in data preprocessing. But are you comfortable with explaining this demo for detail? I hope this demo provide a good proof of concept for beginners, an educational opportunity for learners, a quick validation of predictions for experts, and pure fun for everyone!
- A user specify ground-truth inputs. The default selections are "2" "4" "6", so let's use this selection as an example here.
- In the seven segment mapping, for example the first input "2" is represented by channel 1, 2, 7, 5, and 4, or [1 1 0 1 1 0 1]' in Matlab.
- Normalized pink noise (variance = 1) with user-specified kurtosis, say 8, and data length, say 10000, is generated and assigned to "2". This defines the ground-truth data matrix G2, which is [1 1 0 1 1 0 1]'*[pinknoise_length10000] in Matlab, creating 7x10000 variable. Note the critical fact in the current case ("2"), at this stage time series data for channel 1, 2, 7, 5, and 4 are perfectly correlated (identical), and the rest of the channels have just zeros.
- The pink noise is generated independently for each ground truth in the same way. Thus, G4 and G6 are generated for "4" and "6", respectively.
- In addition to the ground truth inputs G2, G4, G6, a white noise E with user-specified variance, say 0.3, are added to create the "mixed signal" M such that M = G2+G4+G6+E.
Note the correspondence to EEG recoreding--in this case, M corresponds to scalp recording (i.e., after mixing via volume conduction), and G2, G4, G6 correspond to latent independent 'effective sources' (i.e., before mixing). The location of each stroke in seven segment display corresponds to electrodes. But note also the critical difference--in standard EEG recording, you don't know ground truth. Therefore, even if you obtain ICA results, it is not immediately clear (and ultimately not clear either) if the decomposition is good or bad. However, in this demo because we have ground truth defined on the same seven segment number display, we can evaluate how good the decomposition quality is immediately and intuitively--if you can read the number, it means successful!
After ICA is performed, "min r.v." at the bottom of the first GUI page is provided. This is calculated by sum((D1-D2).^2)/sum(D2.^2) where D1 is 7x1 ground truth vector; for example, in the case of "2" it is [1 1 0 1 1 0 1]'. D2 is also 7x1 size but this is from obtained decomposition. 'Minimum r.v.' means that because ICA does not find correspondence between input ground truth and the output, the r.v. shown here is the smallest value across 10 comparisons against all the ground truths. Smaller the r.v., more similar the seven-segment representations are.
This measure is available only when you know ground truth. The correlation is calculated between the time series of the ground truth you used as input and IC activations.
This is also a measure available only when you know ground truth. PVAF is calculated as 100*100-var(Y1-Y2)/var(Y2) where Y1 is time series of ground truth and Y2 time series of IC activation. Note that this definition/calculation of PVAF is different from that is supported in envtopo(), in which PVAF = 100*100-mean(var(Z1-Z2))/mean(var(Z2)) where Z1 (scalp back-projection of selected ICs) and Z2 (scalp back-projection of all ICs) are both channel x time, and variance is calculated across channels, then averaged across time.
In this demo, note how ICA maximizes non-Gaussianity in the probability density function (PDF) plots in the top row of the second-page GUI. This PDF is calculated after normalizing the signal to Z-score. Because of central limit theorem (CLT), PDF of each channel of M becomes necessarily more Gaussian than each of 'effective sources' G.
Central`` ``Limit`` ``Theorem`` ``(CLT)
Mixing signals -> Approaching to Gaussianity (kurtosis == 3)
Unmixing signals -> Departure from Gaussianity (sub/super-Gaussian)
Therefore, as long as the mixing process exists, there is always possibility for ICA to work in. In fact, the mixing process is guaranteed in scalp-recorded EEG by volume conduction, which is a problem of how an electric field is formed between cortical surface and electrode surface.
Generally, ICA works better if 1) data are more non-Gaussian (bit in this demo, only super-Gaussianity, which is kurtosis>=3 is supported); 2) data are longer; 3) there is less noise.
Based on the above argument, it is clear that in order to achieve seven number decomposition, you need to set 1) very high kurtosis, 2) very long data, and/or 3) very little noise. Because this is simulation, you can set any arbitrary numbers here if you don't mind spending longer data for ICA. However, what is the minimum condition to decompose seven numbers? Consider Think about the following points.
- Try to find a set of limit parameters that barely performs successful decomposition--for example, for a given number of ground truth inputs, kurtosis 6 but not 5, data length 15000 but not 10000, or white noise variance 0.25 but not 0.30.
- Press "Mix them" and "Decompose the mixture"--does it change the result? How badly stochastic the results is?
- Change the choice of ground truth, without changing the total number of them. Does it affect the results? Does the number of strokes in the numbers matter? Is it easier to decompose "1" than "8"? Why?
- Change the number of ground truth inputs. How does it affect the result? Why?
- Note the order of the output numbers--is it always fixed? EEGLAB's runica() outputs results according to signal variance. Does it explain the output order?
- Look at the PDF plots. Why the original channel PDF plots show their peaks being non-zero, though they are Z-scored? Why are the peaks in the component PDF are aligned to be zero?
- How do ICA evaluations in spatial domain (r.v.) and time domain (r and PVAF) relate to each other? Is it possible that you get good result only one of the two?
- Set a very easy decomposition conditions with less inputs, long data, and less noise, but kurtosis == 3. What happens? If you don't know the answer, go back to this section.
- In EEGLAB equivalent dipole fitting using Fieldtrip, we see 'r.v.'. What is the ground truth used there?
Author: Makoto Miyakoshi, SCCN, INC, UCSD.