I am trying to understand the use of PCA in a recent journal article titled "Mapping brain activity at scale with cluster computing" Freeman et al., 2014. They use PCA on time series data, and use the PCA weights to create a map of the brain.

The data is trial-average imaging data, stored as a matrix (called Y_hat in the paper) with n voxels (or imaging locations in the brain) x t_hat time points (the length of a single stimulation to the brain).

They use the SVD resulting in Y_hat = USV^T (^T indicating transpose of matrix V).

The authors state "The principal components (the columns of V) are vectors of length t_hat, and the scores (the columns of U) are vectors of length n (number of voxels), describing the projection of each voxel on the direction given by the corresponding component, forming projections on the volume, i.e. whole-brain maps"

So the PCs are vectors of length t_hat. How can I interpret that the "first principal component explains the most variance" as is commonly expressed in tutorials of PCA. We started with a matrix of many highly correlated time-series - how does a single PC time series explain variance in the original matrix? I understand the whole "rotation of a gaussian cloud of points to the most-varied axis" thing, but am unsure how this relates to time-series. What do the authors mean by direction when they state: "the scores (the columns of U) are vectors of length n (number of voxels), describing the projection of each voxel on the direction given by the corresponding component"? How can a principal component time course have a direction? "To see an example of an example of the resulting times series from linear combinations of principle components 1 and 2 and the associated brain map, go to the following link and mouse over on the the dots in the XY plot: http://research.janelia.org/zebrafish/pca.html

My second question is related to the (state-space) trajectories they create using the principal component scores: ttp://research.janelia.org/zebrafish/trajectories.html (link intentionally missing leading h due to only being allowed 2 links)

These are created by taking first 2 scores (in the case of the "optomotor" example I have outlined above) and project the individual trials (used to create the trial-averaged matrix described above) into the principal subspace by the equation:

J = (U^T)Y (^T indicating transpose of matrix U)

As you can see by the linked movies, each trace in state space represents the activity of the brain as a whole.

Can someone provide the intuition for what each "frame" of the state space movie means with regards to the link (pca.ht) provided above that associates the XY plot of the scores of the first 2 PCs. What does it means at a given "frame" for 1 trial of the experiment to be in 1 position in the XY state-space and another trial to be in another position? How do the XY plot positions in the movies relate to the principle component traces in the "pca.ht" link mentioned in the first part of my question?

Thanks in advance!