Modeling and Nonlinear Analysis in fMRI via Statistical Learning - Advance Image Processing in Magnetic Resonance Imaging - page 554

of biological time series and in particular fMRI. Here, we show how this nonlinear
mapping can be used to “restore” data that can then be subject to conventional
statistical analysis. The basic idea is to treat the input vectors as explanatory
variables and the output as the observed BOLD response. This allows us to
characterize the mapping between the explanatory and response variables in a way
that is analogous to linear mapping between the design matrix and response used
in conventional analyses with the GLM. Critically, the mapping obtained with
SVR can be arbitrarily complicated and nonlinear. The predicted responses can
represent any systematic relationship between the input variables and observed
signal, and they can, therefore, be regarded as having been “restored” or “de-
noised.” In this work, we start with inputs that encode where and when a brain
response was measured. Using SVR, one can then estimate the response for any
brain position at any time. In practice, this involves using a local clique of neigh-
boring voxels over the entire time series. The inputs are then augmented to include
regressors of the sort used in conventional analyses of fMRI time series. The
predicted response is then used as a data surrogate that enters classical analyses.

18.3 STATISTICAL LEARNING THEORY

Statistical learning plays a key role in the fields of statistics, data mining, artificial
intelligence, engineering, and other disciplines. SVM, introduced by Vapnik
(37,38) and studied by others (9,36), is a new and powerful learning methodology
that can deal with nonlinear classification (support vector classification [SVC])
and regression (SVR). It is systematic and principled, and it has begun to be
widely applied in the machine learning community. The idea is to learn a function

x

f between the input vector and the output scalar y from M examples. When the

output takes continuous real

values, it is SVR problem. The main feature of SVM is to map the input data to
a high-dimensional feature space through a nonlinear mapping

y takes binary values, the problem is SVC; when y

Φ . Then we learn

x

the function between the mapped data . Typically, this

function is nonlinear in the input data space but linear in the feature space.
Classification or regression is performed in this feature space. An intuitive dia-
gram of this mapping and its advantage in the SVC mode is shown in Figure 18.1 .
In order to model the continuous fMRI signal, we use SVR. Here we sketch
the ideas behind SVR; a more detailed description of SVR can be found in Smola
(36). Given

Φ () and the output y

… … x i

M input sample points xx

1 ,,,,,

x x , where ∈ℜ

z

, and M cor-

i M

responding scalar output values y , y , … , y , … , y , the aim is to find an approxi-

1 2 i M

mation or a regression function of the form

M

∑

yf x = () = α Kxx b ( , ) + (18.1)

i i

i = 1

to learn this input-output mapping from the set of training examples with high
generalizability. Here K is the kernel function, which is going to be explained later.