Exploratory Data Analysis Methods in Functional MRI - Advance Image Processing in Magnetic Resonance Imaging - page 509

VT . The transformation matrix can be an orthogonal transforma-
tion, such that the rotated components are still linearly independent or oblique.
This operation is a search among the linear combinations of the PCs and can be
seen as a projection pursuit task. The interesting projections of the data can be
found manually or by using some graphical description of the data, by means of
information theoretic criteria or by using some a priori information about the
expected results. In Reference 32 the rotation matrix was applied to eigenimages
and was estimated from the data set after a first processing step using PCA. This
step was useful to obtain information about the images of interest, called factor
images . In Reference 31 the rotation was applied to PC found by the analysis of
fMRI data from a primate during the pharmacological stimulation of the dopam-
inergic nigrostriatal system. The first two PCs were associated with an effect due
to pharmacological stimulation but also contained abrupt changes, probably due
to movement. An oblique rotation was performed manually to separate the motion-
related component, obtaining a smooth drug-response profile and a component with
only the abrupt changes. The optimal rotation was also found by inspecting the
associated maps: expected brain regions that respond to drug stimulation for the
first component, and regions with rapid spatial changes in mean signal values for
movement.

or, dually, V T =

16.5 ICA

This [69] is one of the exploratory methods that has given rise to greater interest.
The reason lies in the assumption of statistical independence of the components,
and this guiding principle has proved to be useful in several applications ranging
from telecommunications to image feature extraction, from financial time series
analysis to artifacts separation in brain-imaging applications [70]. Similar to PCA,
the ICA model is generative and models the data as a linear combination of
components. The main difference with PCA is that the components are thought
to be statistically independent of each other instead of linearly independent. The
extraction of the independent components is based on an information theoretic
criterion and not on the maximization of variance explained by the orthogonal
components. As we have seen in Subsection 16.3.2, a problem with the application
of PCA to fMRI data is that the components related to brain activity of interest
may contribute very little to the overall variance. Moreover, a method based only
on second-order statistics may not reveal more complex patterns of activation;
these can be detected by methods such as ICA, which takes into account higher-
order statistics also, to assess the hypothesis of statistical independence [71]. ICA
was first used to solve the blind source separation (BSS) problem [72,73], in
which a set of sources is mixed linearly to form the observations. Both the sources
and the mixing process are unknown. If we denote the unknown sources by
a vector s

= { s 1 ( t ), … , s n ( t )}, the mixing matrix by A , and the data by x = { x 1 ( t ), … ,

x n ( t )}, the model can be written as

xAs = (16.29)