Image Processing Reference
In-Depth Information
context means each observation, by means of the appropriate score on the PC of
interest. Usually, these are bidimensional plots with the first two PCs as orthog-
onal axes, but if other components are relevant, more plots may be needed. The
relevance of PCs is usually quantified by the percentage of variance explained
by the components, determined using the eigenvalues spectrum. A plot of
∑
∑
k
s
2
i
i
=
1
(16.27)
r
s
2
i
i
=
1
shows the percentage of total variance explained by the first
k
principal components.
Usually, the first PCs account for most of the variance, and the noise level can be
evaluated by examining the flattening of the plot. However, it is important to keep
in mind that a classification based on variance may be of no interest. In fact,
although some components may be related to interesting physiological phenom-
ena, others may have originated from movement effects or other physiological
sources [64]. Moreover, movement-related components can cause large signal
changes in the data set, contributing heavily to the overall variance, whereas
activation-related signal changes may explain a smaller percentage of the overall
variance. Usually, out-of-brain voxels are masked in order not to influence the PC
transform and hide some interesting activations. In order to enhance the extraction
of an activation that may be located in a specific region, it is possible to select
an ROI. In Reference 29, a finger-tapping study is reported: a PCA of all brain
voxels resulted in a second PC highly correlated with the task. The percentage of
the variance explained by the component was expressed as where
s
2
is the
eigenvalue associated with the second PC, and
s
1
is the eigenvalue of the PC
explaining the larger variance fraction and was found to be 0.01. The selection
of a smaller ROI centered around the activation yielded a fractional value of
0.038. In order to find the relevant number of PCs, information theoretic criteria
such as Akaike's information criteria (AIC) [65], minimum description length
(MDL) [66], and Bayesian model selection [67] can be used. These methods can
also be used in combination with PCA before applying ICA to fMRI data in order
to perform dimensionality reduction. However, the PC found could not be rep-
resentative of real physical quantities [32]: because the decomposition is based
on variance partitioning, this general assumption cannot be used to interpret the
final results. Some attempts can be made to identify the sources of interest, such
as applying rotations to the components found by the PCA approach [68]. The
eigenvectors, or equivalently, the eigenimages can be regarded as a new coordinate
system for the data set. Along the directions of the first eigenvector (or the first
eigenimage) the measurements show larger variability; the second direction is
the orthogonal direction, along which the remaining variability is maximally
explained, and so on. The rotation operation can be mathematically described by
a transformation matrix
T
, so that the rotated components are
(/),
s
21
U
T
=
T
(16.28)
Search WWH ::
Custom Search