Hello. My name is Arnaud Delorme, and this is a series on independent component
analysis applied to EEG time series, about the EEGLAB software.
And this is part 5 on source localizaton of ICA components.
So what is ICA localization of components?
So we have our electrodes and then we use ICA, we get our components, and for components
we get both time-series and scalp topography, as we've mentioned before.
With the scalp topography we can then do source localization.
And here I show two different types of source localization.
We can use dipoles, which are simple source points in the brain, and we can use,
also, distributed source reconstruction, such as Loreta.
So that's what's shown on the right.
And this is using the DIPFIT plugin of EEGLAB.
So if you imagine one neuron in the brain,
a pyramidal neuron, this modeled as a dipole, so it has a
plus and a minus polarity and then there is electric fields.
And if you have many of these, many of these neurons, the
electric field can be summed up and seen at the surface of the scalp.
So that's the schematic representation of what might
be happening with source-like dipoles.
And when we apply that to ICA, so this is an ICA scalp topography, and we can see
that this one should be possibly modeled using a dipole in this region and
this other one should be possibly modeled with a dipole of opposite
polarity in this region, and we can project these back inside the brain.
And this is of course a very rudimentary
method for localizing ICA components, but it's very useful because we can assess
the biological plausibility of these components.
So here ICA created a spatial filter for each temporarily independent brain source.
So to do this we need to compute residual variance.
So on the left here we have the actual data, on the right we have the
dipole projection, and what we're going to do to see if the ICA component can be
modeled using a single dipole, we're going to subtract the two maps and see
what's left, and basically that's called the residual variance.
If the fit is really good the residual variance is
very low. It's a number between 0 and 100%, or 0 and 1. And, if it's really bad it
means the residual variance is very high.
And we tried to see what happened when we use a lot of ICA components. When we
use ICA decompositions, are these components dipolar or not.
So here we did the relatively systematic study with 13
subjects, 71 electrodes, and we had a lot of data for this study.
We had 300,000 data points per subjects.
We tested, also, different ICA decomposition to see if one was performing better than
the others, and that's what I'm going to explain in these next slides.
And we localized these components and then we
computed the residual variance as I just showed you.
That's the publication here on the bottom.
These are the ICA algorithms we
tested, and a lot of them were in EEGLAB, some of them were in ICALAB, some
others we just downloaded on the internet.
We also tested other algorithms which are at the bottom which is, such as PCA, principal component analysis.
We tested also PROMAX, and whitening/sphering, which are not really blind source
separation algorithm, but are still linear decompositions.
These are examples of what we call good and bad components.
So you see at the top the scalp topography can be as low as 1% residual variance.
So it means the dipole, the single equivalent dipole,
models 99% of the scale topography for these components.
Then we have 1 with 2%, 4%, 8%, 16%, 32%, 64%, 84%.
And we can see that as we reach 84% the component scalp topography
look like a checkerboard, so it can't really be modeled using a single dipole.
So that's why there our variances is very high.
And this is the one of the main results of the paper.
So here we've plotted -- every single line, there's a different ICA algorithm --
and we plotted it on one axis
the dipole residual variance. And on the other axis
the percentage of ICA component cumulated.
So, for example, if we take this point here for the algorithm Amica, it means that at
10% residual variance, I have 50% of my components.
This is across all the 13 subjects.
And the other algorithms, they don't perform as well in
the sense that they don't have as many dipolar components.
Two curves also are interesting here.
The PCA curve, principal component analysis, you can see that it is on the bottom.
Except for the first and maybe the second PCA components, all the other
ones they look like checkerboard, so they can't really be
modeled using single sources.
We also have the other dotted line which is the raw EEG.
So when we sample the raw EEG
as at random latencies, we can see how it looks like.
And so, basically, it's not as dipolar as the ICA components.
It doesn't mean that because you're on top of the curve you're necessarily doing better.
There's some caveat, which we explained in the paper, about whitening, for example.
So I invite you to read the paper if you're interested in that.
What we're interested in seeing is how is this correlated with whether the
ICA algorithm performs well or not on the data?
And for this what we did is that we set thresholds for residual variance.
I'm going to show you a curve where we set the threshold at 5%,
but we also tried different thresholds 10%, 15%, etc.
And basically that allows us to compare
at one specific threshold the different ICA algorithms.
So to assess if the ICA algorithms are performing well, we use two measures.
That's the first one.
Here is the mutual information reduction.
And all this math I'm going to show you here,
and try to explain to you, they were derived by Jason Palmer in our paper.
And basically you have the entropy, so the joint entropy of Y, which is the ICA
decomposition, is equal to the log of the determinant of the weight matrix, so
that's the ICA weight matrix, plus the entropy of X, which is the channel data.
If we looked at the mutual information of the ICA components, so we can decompose that further into
the univariate entropy of
every single component minus the joint entropy of all the components.
And if we replace the joint entropy of all the components
with the log of the determinant and plus the entropy of X,
the bottom equation is the one we get.
It becomes interesting when we look at this measure.
So this measure is the mutual information reduction.
We start from the channel space, we end up in the component space.
How much were we able to reduce information?
So we subtract the mutual information of X channels
minus the mutual information of the components.
And what we get is that we just use
the formula above -- which are here highlighted --
basically we get the mutual information of the channels
minus the mutual information of the components.
And what's very interesting here is that the part that's really hard to calculate,
which is the joint entropy of the channels.
So the parts that's really hard to calculate here cancels out.
And what's left is the log of the determinant of the
weight matrix, which is relatively easy to calculate, as well as, univariate
entropies, here on the bottom, which are also easy to calculate.
If you have the probability density distribution of the channels
and the components, you can easily calculate these measures.
So we now, we can calculate mutual information
reduction for every single ICA decomposition.
And we plotted that against the -- you remember I said we've said a threshold at 5% --
we plotted that against the number of ICA components which are dipolar at 5%.
And this is what the curve shows.
And basically the more independence the better the ICA decomposition,
the more biological the components are.
So this is, this was the main result of that paper.
The fact that this relationship is linear -- from our perspective -- is relatively by chance, because there is no real
connection between these two measures. One is about the statistics of the data,
mutual information reduction, and another one is a biophysical model.
So that's one measure.
And can we try another one?
So we tried another one, we tried the pairwise mutual information.
So this measure doesn't calculate all, doesn't take into
account all the higher-order joint entropy between components.
It's just sticking to second-order entropy.
As you can see mutual information between two components is the entropy
of the first one plus the entropy of the second one minus their joint entropy.
And as I mentioned in one of the previous lectures,
this joint entropy [between 2 components] is relatively easy to calculate.
This is a contingency table, here, for two variables.
We had stress and emotions, and I explained how you can use Shannon
formula by just summing over the probability in each cell times the log of
probability in each cell to get a measure for the joint entropy.
So this is perfectly doable.
When we get to three variables, four variables,
calculating the joint entropy becomes impossible.
Here we have 72.
We have 72 ICA decomposition.
So the mutual information reduction was taking care of that, but this is
an interesting relatively independent measure.
And, so, this is what we get, here.
So the remaining pairwise mutual information,
when we have lower values, we have more dipolar components.
And again, the Amica algorithms and other ICA algorithm we use in EEGLAB,
like Infomax, were on the left side of the curve, which means that their components
were more dipolar and they had less remaining pairwise mutual information.
The color here are added by
hand, they indicate family of algorithms.
So, again, this shows basically something very similar
when we're using the mutual information reduction.
So these are two independent ways to look at the data, which confirmed that ICA algorithm
which are more efficient tend to return more dipolar components.
So how do these components look like?
So these are six, seven classes of ICA components.
We have alpha rhythm, and on top we have the spectrum.
On the bottom we have the scalp
topography of these components.
We have frontal midline theta, we have a blink components, we have left
mu rhythm components, we have a right mu rhythm components, we have a lateral
eye movements, and we have an EMG component.
Do the scalp topographies look the same
across ICA decomposition or do they look very different.
And basically they look almost the same for the
different ICA decomposition, so that's what's shown here.
And I'm going to show you a quantified graphics.
On top, the image represents the scalp topography
correlation, and I just showed you the scalp topographies.
And the bottom represents the time activity correlation.
Are the time activities also correlated, or is it just the scalp topographies.
And we use basically Amica: we correlated everything with
Amica, because it was performing the best.
And on the bottom, here, we see the information reduction difference with
Amica, and we ordered the algorithm.
And we can see that most algorithms are very close to Amica.
And so here, on the bottom, we can see the algorithm which didn't work well.
So we have sphering.
Sphering is basically decorrelation of the data, and all of these algorithms didn't work well.
And the reason we think they didn't work well is because a lot of these algorithms
were developed with toy data, they were trying to separate one or two
components, and here we are using more than a thousand data points.
This is a much different problem.
There is 71 components to work with and also 300,000 data points, so maybe
the parameters for these algorithms was not optimized.
And so I want to thank you for your attention, and I'll see you on the next video.