Modeling and Nonlinear Analysis in fMRI via Statistical Learning - Advance Image Processing in Magnetic Resonance Imaging - page 557

subset of the training data. This type of representation is especially useful for
high-dimensional input spaces.

18.4 FMRI DATA ANALYSIS AND MODELING

THROUGH SVR

SVR has recently been applied to system identification, nonlinear system predic-
tion, and face detection with good results (18,26,31). Comparisons of SVR with
several existing regression techniques, including polynomial approximation,
RBFs, and neural networks have been carried out (31). Initial attempts that directly
use SVM have also been achieved for modeling hemodynamic response (5) and
for comparing and classifying the patterns of fMRI activations (10,17,24). How-
ever, the application of SVR in the context of fMRI analysis has not yet been
exploited, which is now introduced and developed in this work (39,40).

18.4.1 D R

ATA EPRESENTATION

We formulate fMRI data as spatially windowed continuous 4-dimentional (4-D)
functions. That is, the fMRI data is divided into many small windows, such as a
3 3 region within which the entire time series is included. Each input (the

training data) within a window is a 4-D vector equal to the row, column, slice,
and time indices of a voxel. The output is the corresponding intensity. We approx-
imate and recover all training data within the respective windows using SVR.
The detailed formulation follows.
Let

× 3 ×

y ( u , v , w , t ) be the fMRI signal of voxel [ u , v , w ]

T

at a given time point

t are the respective row, column, and slice coordinates of the

data. If the 4-D fMRI data size is

, where u , v , and w

S × S v × S w × S t , where S t is the total number

u

x

of time points, the corresponding input vector is represented as

xuvwt

= [,, ,] uSv

T

, ∈ [, 1 ], ∈ [, 1 SwSt ], ∈ [, 1 ], ∈ [, ] 1 S . (18.7)

u v w t

Within each spatiotemporal window of size , we have
M input samples , where , and the respective scalar output
. SVR is used to restore the training examples within the window.
Local intrinsic spatiotemporal correlations are accounted for during the regression
by controlling function smoothness and training error through the parameter C
(Equation 18.4). In order to compensate for the spatial correlation between neigh-
boring windows, spatially overlapped windows are used (in all three dimensions)
so that the recovered intensities over the overlapped voxels are averaged from the
corresponding windows.

MMMSM
u

×× ×=

v w t

… … x i ∈ℜ

xx

1 ,,,,,

x x

4

i M

yy

1 ,,,,,

…… y y

i M

18.4.2 T EMPORAL M ODELING

Without loss of generality, we assume an on-off boxcar function as our model
variable corresponding to a simple block-design paradigm, which contains p zeros
or ones during each off or on period and c repetitions or cycles of these two