[SOUND] So they often use multivariate decomposition
methods to study functional connectivity.
These methods provide a decomposition of the data into separate components.
And they can be used to find coherent brain networks and
provide information on how different brain regions interact with one another.
The most common decomposition methods are principal components analysis, or PCA, and
independent components analysis, or ICA.
So throughout, we're going to organize the fMRI data in an M x N matrix X.
The row dimension is the number of time points and
the column dimension the number of voxels.
So we're just going to put all the data together in a time by voxel matrix.
So here in contrast to say the GLM mode where we analyze
each voxel at a time, here we're going to analyze all the voxels simultaneously.
So principle components analysis Is a mutivariate procedure
concerned with explaining the variance,
co-variance structure of a high dimensional random vector.
So in PCA, a set of correlated variables are transformed into a set of uncorrelated
variables ordered by the amount of variability in the data that they explain.
So in fMRI principal component analysis involves finding the spatial modes,
or eigenimages, in the data.
These are the patterns that account for
most of the various-covariance structure in the data.
And they're ranked in order of the amount of variation that they explain.
The eigenimages images can be contained using singular value decomposition or
SVD, which decomposes the data into two sets of orthogonal vectors that
correspond to patterns in space and time.
So, the single value of decomposition is an operation that decomposes the matrix X,
into three other matrices.
So, we write X is equal to U times S times V transpose,
where V transpose V is equal to the identity matrix and
U transpose U is also equal to the identity matrix.
And S is a diagonal matrix whose elements are called singular values.
So, pictorially, we can write this as follows.
We can take our matrix, X, which again was time by voxels,
and decompose it into three matrices, U, S, and V.
Pictorially we can represent the singular value decomposition as follows.
Here we take the matrix X which is again time by voxels, and
we can separate it into U, S, and V Transpose.
So here what I'm going to claim is that the columns of V or
the rows of V transpose are the Eigenimages, and
the columns of U represent the corresponding time courses.
So these are the time courses that correspond to the respective Eigenimages.
So, we can write this as X is equal to U times S V transpose, but because of
the diagonal nature of S we can also decompose it into each column of U and V.
We can write S1 times U1 V1 transpose, etc for each of the subsequent columns.
Here we see a real data example.
Here we have x is the first image here.
And this can be decomposed into a number of sub-matrices as
indicated on the previous slide.
The first sub-matrix consists of S1 which is a scaler
times U1 which is a time course corresponding to the first Eigenimage.
And v1 transpose which is the first Eigenimage.
Then we have the second sub-matrix which is S2 times U2.
Which is the time course corresponding to the second Eigenimage V2
transpose which is the second Eigenimage etc., etc.
Now, each of these Vs Have a length of the number of voxels, and they can be
sort of reconstructed into images corresponding to the spacial modes here.
So here we see the first Eigenimage which is the V1 transpose here,
and we have U1 transpose which is the corresponding time course.
Similarly, we get the second Eigenimage, and
it's corresponding time of course, etc., etc.
So if we do this, we can get several different
temporal components corresponding to each of the columns of U,
and we get the corresponding Eigenimages below.
And here's an example of a PCA analysis.
Here we see the first four Eigenimage on the bottom, so
in the bottom panel we see four rows, one for each Eigenimage.
And on the top panel we see the corresponding time courses.
And to the right of these time courses, you see percentages.
Those percentages are the percent of variation, explained by each
of the components, and they're related to the values of S in the singular matrix.
>> So independent component analysis, or ICA, is a family of techniques used to
extract independent signals from some source signal.
ICA provides a method to blindly separate the data into spatially independent
components.
Here, the key assumption is that the data set consists of p
spatially independent components which are linearly mixed, but spatially fixed.
The ICA model differs a little bit from what we used in PCA.
Here, the matrix x is decomposed into two matrices, A and S.
Here A is referred to as the mixing matrix and S as the source matrix.
So our goal is ultimately to use this information to find
an un-mixing matrix W such that Y is equal to WX,
provides a good approximation to S which are these independent sources.
If the mixing matrix is known, the problem is straight forward and almost trivial.
However, ICA tries to solve this problem without knowing the mixing parameters.
So instead, what it does is it exploits some key assumptions.
First it assumes there is linear mixing of sources,
then it assumes that the components SI are statistically independent of one another.
And it assumes that the components are non-Gaussian, or most 1.
>> It can be Gaussian.
When applying ICA for fMRIs,
assume that fMRI data can be modeled by identifying sets of voxels whose activity
vary both over time and are different from activity in other sets.
We try to decompose the data into spatially independent component maps
with a set of corresponding time courses.
Here's the kind of cartoon image here we have x, which again is time by voxels,
that's our data.
And we have two matrixes a and s.
So again, S represents the spatially independent components, one for each row.
The columns of A represent the time courses corresponding to the spatially
independent components.
So the first column corresponds to the first row of S.
And what we want to do here is use an ICA algorithm to find both A and S.
Here's an example fitting ICA.
And here's two different components.
So this corresponds to two different spatial components
from the ICA decomposition.
And first we see a task related component.
And in a second, we see a more noise component.
And see here, you can see that this is a noise component by a lot of activation
around the edges of the brain, which is probably due to emotion-related artifacts.
Here's an example of eight of the most common and
consistently identified resting state networks, which are identified by ICA.
And so this week, we looked at the previous lecture on resting state fMRI.
And we showed these results.
And here, we can now come back to this and say that this was obtained using ICA.
So what's the difference between PCA and ICA?
Well PCA assumes an orthonormality constraint.
In contrast ICA assumes statistical independence among a collection of spatial
patterns.
So independence is a stronger requirement than orthonormality.
However in ICA, the spatially independent components are not ranked in order of
importance such as they are when performing PCA.
So it behooves you to go through all the components after the fact and find out
which ones are important and which ones are just related to noise and whatnot.
Okay, so that's the end of this module.
Here we've introduced principle component analysis and
independent component analysis.
So these are two ways of taking the full time by voxel data and
finding interesting patterns of activation in it.
And so, these are commonly used in functional connectivity analysis.
Okay, in the next module, we'll talk a little bit about dynamic connectivity.
See you then, bye.
[SOUND]