Exploratory Data Analysis Methods in Functional MRI - Advance Image Processing in Magnetic Resonance Imaging - page 518

than the actual number may cause overlearning. This phenomenon is related to
the use of too many parameters in the model with respect to the amount of data
available. Data reduction is usually achieved by applying PCA to the observed
data and retaining the PCs that account for most of the variance in the data, i.e.,
more than 90% of the overall variance. The discarded components may be related
to noise only. As this may be not true in general, the PCA reduction process is
very critical. This operation allows simultaneous whitening of the observed data.
Given the data covariance matrix C x = E { xx } T , the matrix of its eigenvectors

U = ( u 1 , u 2 , u n ) and the diagonal matrix of the eigenvalues λλ , , … n , λ , the whit-

12

ening transform can be written as

WDU
S

=

− 12 / T

(16.48)

The data reduction operation can be performed by retaining m of the n
eigenvectors, usually the first ones that take into account most of the data variance.
The first m whitened PCs are given by

YDUX =

− mm 12 / T

(16.49)

where is formed by the first m columns of U , and the matrix D m is the
diagonal matrix with the first m eigenvalues of C x . This initial preprocessing
step is a very important and critical stage because using fewer dimensions than
the actual number or underestimating the model order may cause information
loss, whereas overestimating the model order may cause overlearning and
generate spurious components. If we try to extract more independent compo-
nents than the real number, we may find components with a single spike or
sparsle distributed. These solutions can be seen as extreme cases of non-
Gaussianity. The dimension reduction process requires caution, because though
reduction may be preferable from the point of view of computational demands,
information loss may result. As stated before, it is possible to use the number
of components depending on the percentage of the overall variance explained.
If the noise is supposed to contaminate all the observed mixtures in the same
way, it may be supposed that the eigenvalues will be affected by it. It will be
possible to detect a threshold above which all the eigenvalues will be statistically
equal: this threshold can be determined by visual inspection of a scree plot for
a change in the steepness of the plot. Other methods to determine the dimension
are information theoretic criteria such as AIC [65,86] and MDL [66] and Baye-
sian approaches [87,88]. In Reference 43 a clustering approach based on the k -
means algorithm was proposed for spatial ICA. The dimensionality reduction
consists of applying the clustering operation to the observed variables and using
the mean of each cluster as reduced data. This latter approach does not take
advantage of second-order statistics and has proved to be superior to PCA for
higher values of CNR, whereas for lower CNR, PCA was found to outperform
clustering. In Reference 40 ICA was applied to voxels belonging to the cortex.

U T