Image Processing Reference
In-Depth Information
FIGURE 15.5 Schematic illustration of a general linear model of an fMRI experiment.
Each voxel's time course is modeled as the linear combination of condition-specific
predictors obtained by convolution of the ideal on-off response with a realistic model of
the hemodynamic response (see text).
or in matrix notation:
y
=
X
+
e.
(15.9)
L matrix
of the predictors (one row per observation, one column per model parameter),
Here, y is the T
×
1 column vector of the observations, X is the T
×
=
[
1
column vector of error terms. The matrix X is conventionally referred to as the
design matrix of the experiment. Most of the fMRI studies (with a single subject
or multiple subjects) that contain a baseline condition as well as several repetitions
of one or more experimental conditions may be easily expressed as a multiple
regression problem by defining an appropriate form for X (Figure 15.5). For
instance, for an experimental design with a baseline condition and five different
stimulation conditions, the design matrix X has five columns and one row for
each measurement time point. Each predictor is obtained by convolution of the
ideal box-car (on-off) response with a realistic model of the hemodynamic
β 1 ,
,
β l ,
,
β L ] T is the L
×
1 column vector of parameters, and e is the T
×
response (hemodynamic response function; see, e.g., Reference 18 and Reference 21 ).
Effects other than the expected task-related BOLD changes may also be modeled
in the design matrix. Normally, X includes a column consisting of 1's for all
measurements to account for the mean value of the voxel time course. Similarly,
X may include a column with linearly increasing values to account for linear
trends in the voxels' time courses.
Once the design matrix has been defined, the next step in the GLM analysis
consists of the estimation of the regression weights
such that the predicted
values y ' are as close as possible to the measured values y at each time point.
Let us denote with y ' the estimate of the time course Y for the regression values
'
y '
=
X
'
(15.10)
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