PCA or principal component analysis gave a huge contribution in fMRI studies since this type of image was introduced about 2 decades ago. In general, PCA is used for two main reasons:
- To extract brain regions from the fMRI
- To extract the regions that are activated within the same network.
We will go deeper into these later on after covering the basics of PCA.
*All references for PCA can be found
here.
What's behind PCA
Motivation
PCA is very popular for its dimensionality reduction and pattern recognition abilities for huge data. It can help us to express the data in a more meaningful way by highlighting the differences and similarities. Since it is hard to visualize huge data in graphical representation, doing PCA can help to reduce the less important dimensions with much loss of information.
Equation
In general, PCA can be viewed in a common simple matrix equation...
X = AS
X is what we have now, the fMRI huge data. The details of fMRI structure can be referred here (link to entry). From these huge data, we want to find groups of voxels that can be defined as a region or groups of regions that we can define as a network.
A is normally called as the mixing matrix, the matrix that can construct and deconstruct back the original image (in fMRI case). People usually use this to learn:
- when a region (part of the brain) is activated over time, or
- where a network is changing over time.
S, the matrix that is usually being analyzed in a static studies, is the separated sources matrix, where the data X, are grouped in different dimensions from the X's dimensions.
Characteristics...
... of the sources (spatial map)
Gaussian One of the key things in PCA is the original have to be Gaussian. And a lot real data is usually Gaussian.
Why do the data have to be Gaussian distributed to use PCA?
Explained by
Shlens, "The only zero-mean probability distribution that is fully described by the variance is the Gaussian distribution." As PCA is searching for the new dimensions that have the highest variability, we need variance to be part of the factor in characterizing the data.
Not sure if this is clear, if not, let me know. I will try to rephrase.
In practice, most of real data are usually Gaussian. But not all. So, if we use PCA on non-Gaussian data, we will get some weird results not the way we expect it to be.
Uncorrelated Another important characteristic that we need to understand is the sources have to be uncorrelated between each pair. We assume that each two sources (X and Y for example) that we will find through PCA have zero correlation-coefficient:
And having a zero correlation, this will lead to
zero covariance as can be seen in the equation below:
And more, we know that covariance is related to the joint expectation,
will lead to the independent joint expectation.
This is kind of too mathematical, I know. I'll try to do a separate entry on this topic if I remember next time.
- gaussian
- uncorrelated
- independent?
Components distribution
- orthogonal
Resources
http://stats.stackexchange.com/questions/2691/making-sense-of-principal-component-analysis-eigenvectors-eigenvalues/140579#140579
http://www.cs.otago.ac.nz/cosc453/student_tutorials/principal_components.pdf
http://stats.stackexchange.com/questions/32105/pca-of-non-gaussian-data
http://www.stat.cmu.edu/~cshalizi/uADA/13/reminders/uncorrelated-vs-independent.pdf
