Hello my name is Arnaud Delorme and this is part 1 of a series of video on
independent component analysis applied to electroencephalography time series so EEG
time series -- so this part is about what is ICA?
Independent component analysis is a signal processing method to separate
independent source linearly mixed in several sensors and so for example it
can be applied to many different types of signals -- it can be applied to for
instance data from microphones -- so if you have five microphones in a room
recording different sounds like for example here
and then you apply ICA to separate the sounds -- and based on their statistical
independence you can separate the sounds.
[ music playing...]. Second sound.
[other music playing...]. And then we have the voice:
"Independent component analysis is a new data tool that is sensitive to higher order
structure in multidimensional data" [other music with clapping...] and so all of these sounds,
all of these sounds were embedded in the in the original signal right here and based on
the statistical independence and the projection of these sounds on different
microphones, ICA was able to separate them. It's the same principle for EEG
where we have electrodes here instead of microphones and then here is
a representation of the cortex the skull and the skin -- so if we have an EEG source
here, a patch of local synchrony, that projects two different electrodes and
then we have another patch here that protects to the same electrodes,
it is similar to having different conversations projecting on different microphones.
So in this case as well we can apply ICA -- and that's called a cocktail party problem --
and so we showed earlier how the projection on different microphones can
be separated with ICA and then similarly the projection of
different brain sources on different scalp sensors can be separated with ICA.
So how does this work? Well ICA recovers a version of the original signal so we
have the data data is X -- that's the data we recorded -- on different channels --
and ICA is going to find for us W which is the mixing matrix and U which are
the original source activities - so the different
sounds in the case of source separation with sounds or different brain sources
and artifacts and in the case of electroencephalography brain data.
And while PCA simply decorelates the output and finds an orthogonal matrix
ICA attempts to make the output
statistically independent while placing that constraints on the matrix W so
we'll see that a little bit more detail later. But here is an example of two
sources -- so we have source A and source B -- and then we have a linear mixture of
these sources -- so this is one times A minus twice B -- this is one 1.57 times A
minus 0.33 times B -- so these are the signal from the microphones or
these are the signals from the scalp channels we recorded and we want to recover these sources.
So we apply ICA which you can find the estimated sources and the mixing matrix.
And it (ICA) can recover the sources -- the polarity is lost but we are going to look into that
in more detail in some of the lectures later. And so ICA is able
to recover the sources -- the original sources. So some historical remarks..
The first ICA paper was published 1996 by Herault and Jutten and then in 1995 we
had the first Infomax paper by Bell and Sejnowski. Amari and collaborators added
the natural gradient to the Infomax algorithm and we're going to see in
the next lecture and there were also other ICA algorithms developed such as
Jade by Jean Francois Cardoso in 1996. Applications of ICA to biomedical signals:
the first application to electroencephalography data was by Scott Makeig
and collaborators in 1996 and to fMRI data by McKeown and collaborator in 1998.
There are many different ICA algorithms. There are conferences on blinds source
separations and there are hundreds of ICA algorithms.
These are some of the most popular, for instance, you have separation of source based on
information theory -- that's Infomax -- based on Bayesian probability theory
like such as maximum likelihood estimation, negentropy maximization,
non linear PCA , statistical signal processing such as Jade, which saw before.
So which one is best, which one works the best? We actually have a paper
about applying all these algorithms to electroencephalography data and we're
going into that in some of the future lectures but on the theory -- at the theory level --
they'll also been papers published showing that on idealized data where the
source are truly independent and we have infinite time series, all of these
algorithms will be able to recover the sources -- so there was a first
paper in 1996 showing that Infomax and maximum likelihood estimation were
mathematically equivalent, then another one in 1999 by Lee and collaborators on
Infomax and negentropy and then another one on non-linear PCA and Informax in 1997.
So, they all work in theory and we'll see how that works in practice in some of the next lectures.
So this is to be continued in the next lecture and we'll talk about how ICA works?