Exploratory Data Analysis Methods in Functional MRI - Advance Image Processing in Magnetic Resonance Imaging - page 510

The problem is to find an unmixing matrix W , which can be thought of as
the inverse of the unknown matrix A (here we assume we can invert the system)
such that

sWx

= (16.30)

are an approximation of the sources s . In this model, each time series s i ( t ) is
assumed to be the set of observations of a random variable s i and the vectors s
and x are random vectors. The assumption of statistical independence means that
the knowledge about the value of a signal at a certain time cannot give us
information about the other sources' values. Hence, the joint probability density
of s factorizes, and we have

n

∏ 1

p () s = p i ( ) s (16.31)

i

i =

Note that this method does not require any a priori assumptions about the
shape or extent of the activations. Instead, it does require the underlying sources
be statistically independent. In the following sections we will see when this may
happen with fMRI data set. Another restriction is that the unmixing matrix is
square: This means that the number of unknown sources equals the number of
observed variables. Working with fMRI data, this restriction would lead to wrong
modeling, and the dimensionality reduction of the data has to be performed.

16.5.1 S PATIAL AND T EMPORAL ICA

As with PCA, ICA of fMRI data can be carried out in temporal or spatial domains
[33]. The data set can be decomposed into a set of spatial patterns of activations
associated with their own time courses, assuming statistical independence among
the spatial patterns or among the time courses. In spatial ICA the data matrix X
is p

n , where p is the number of time points and n is the voxels number. The
generative model of the data in (Equation 16.29) shows that the independent
components are a set of statistically independent images or spatial patterns of
activation, mixed linearly by the matrix A whose columns are the time courses
associated with each independent component. In temporal ICA, the data matrix is
n

×

p and its rows are the signal time courses in each voxel of the acquired volume.
The independent components are temporally independent time courses, and the
columns of the matrix A represent the spatial distribution of the temporal compo-
nents. In the first application of ICA to fMRI data, McKeown [34] suggests that
the spatial distribution of voxels whose activation is related to a task of interest
should be unrelated to the spatial distribution of artifacts that affect the signal,
such as physiological pulsations, subject-movement-related effects, and scanner
noise. Several other assumptions are made: the model (Equation 16.29) assumes

×