Exploratory Data Analysis Methods in Functional MRI - Advance Image Processing in Magnetic Resonance Imaging - page 507

with s i being the eigenvalues. PC scores for the time points, which represent the
contribution of eigenimages to each time point, are given by X T U . These scores
can be interpreted as the original data in the space of u i . From Equation 16.26 it
is possible to see that these are the loadings, apart from a scaling factor given by
the eigenvalues, of the eigenvectors v i . The same can be done for the PC scores
for the voxels that are given by XV . These can be interpreted as the contribution
of each eigenvector in each voxel. It is easy to show that the loadings of the
eigenvectors u i equal the scores, apart from a scaling factor, of the eigenimages.
Although these approaches, temporal and spatial covariance based, seem equiv-
alent, they differ largely in fMRI data analysis applications in the maximum
number of computations needed to get the eigenimages starting from the matrix
XX T . In fact, in fMRI data, the number of voxels is far larger than the number of
scans. However, when applied in the temporal domain, i.e., looking at the time
series in each voxel as observations, the analysis is performed on a subset of all
brain voxels. In Reference 62 this method was applied to a subset of voxels that
showed significant differences across the alternating conditions. This allowed a
reduction in the computational costs, justified by the fact that voxels that do not
contribute to the measured variance cannot contribute to the covariance. In
Reference 28, PCA by means of SVD was applied to an fMRI data set. In this
work, the analysis was applied to activated voxels found by the regression method.
PCA was applied considering the time points as variables and the voxels as
observations. This resulted in eigenvectors whose loadings were the time evolu-
tion of the component, and a set of scores for each PC in each voxel.

16.4.1.1 Preprocessing of fMRI Data before PCA

Besides the usual preprocessing step mentioned in Section 16.2, several other
preprocessing steps can be performed: we have already mentioned that it is
possible to center the time series or the images, or both. Another preprocessing
step that can be performed is the normalization of the time series, i.e., subtracting
the mean and dividing by the standard deviation, in order to reduce the effect in
the overall variance of spatially varying random signal fluctuations. Linear regres-
sion can be used to remove low-frequency fluctuations or signal drifts as in
Reference 31, whereas in Reference 28, in order to reduce the effect of the noise
present in the data, PCA was applied to a set of fitted time series using six
sinusoidal regression parameters, a drift term, and a constant term.

16.4.2 I NTERPRETATION OF THE PCA D ECOMPOSITION

Once a decomposition is found, the experimenter has to decide which components
are interesting: this can be done using information about the experimental design,
for example, the correlations between the paradigm and the time course of a PC
or using a hypothesis about the activation regions. Results from PCA can be
reported as loadings of the eigenimages on anatomical images or as scores on
the PC of interest, depending upon the approach chosen. Another method makes
use of PC plots: these plots are used to identify each individual, which in this