Exploratory Data Analysis Methods in Functional MRI - Advance Image Processing in Magnetic Resonance Imaging - page 504

repetition task, confirming the hypothesis of correlations among the regions
individuated by the eigenimage and the verbal fluency task. Another eigenimage,
with positive loadings in anterior cingulate, showed a monotonic decreasing
temporal pattern and was thought to be related to some attentional or perceptual
change. It is important to stress that this component could not be easily detected
by a model-driven approach, which could have been used for detecting task-
related changes, because it may not be expected in advance.

16.4.1 S PATIAL AND T EMPORAL PCA

As stated earlier, the duality inherent in fMRI data sets can be found in the multi-
variate approaches: fMRI data are arranged in a matrix where each image or volume
of the sequence is transformed in a row or in a column, depending on whether they
are considered as a time sequence of images or volumes or as a spatial distribution
of time courses. In multivariate methods, the data matrix X can be seen as a
collection of measurements of some observed variables x i . These can be the time
points or the voxels values. PCA aims to find a smaller set of variables, as a
linear combination of the original ones, that accounts for the variance in the data
set. The first principal component (PC) is the one that has the greatest variance
of all the possible linear combinations of the original variables, with the constraint
that the combination weights form a unit vector. The second PC has the largest
variance of all the possible linear combinations under the constraint of being
orthogonal to the previous one. This procedure can be repeated up to the maxi-
mum number of variables r min( n , p ), where r is the data matrix rank. In data
reduction methods, a fewer number of variables that explain the largest portion
of variance in the data set can be chosen. This reduction operation can be
performed also in the field of fMRI data analysis even if the local variability or
low-intensity signal changes may be not represented [29]. In fact, low-percentage
signal variations may be discarded in the reduction process because their influence
on the overall signal variance is small. In the same way, the influence of small
activation regions may result in percentages that are small with respect to other
sources of variance, both physiological or artifactual, distributed across brain
voxels. These issues will be addressed further later. If we consider the voxels as
variables, the first principal component can be written as

r

∑

y = u x

i

(16.17)

1 1 i

k = 1

where is a unity vector of the weights, and x i 's are the original
variables. The weights, or loadings, of vector u 1 can be found as the eigenvector
of the covariance matrix of the data

u 1 = {, uu , … , u n }

11 21 1

Cx = E {( − µ )( x − µ ) }

T

(16.18)

x x x