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Principal Component Analysis (PCA)[6] and Independent Component
Analysis (ICA)[7] are established exploratory methods for single subject analysis in Functional MR
imaging. The spatio-temporal decomposition of the data matrix (up to a given level ) in
both cases can be written:
|
(1) |
with different statistical properties according to the chosen method. Inferences for
location on the brain with components [1][2] and stimulus influence with
associated time course component (e.g. correlation with the paradigm) describe the
functional activation. For multi-subject analysis generalisations of these methods to 3-way arrays
are needed to get a decomposition of the data tensor of order 3 (space, time, and subject)
in a form:
|
(2) |
Testing the subject component would allow a population inference and can be viewed as a
spatio-temporal omnibus test. For the variance criterion (PCA) the PTA-k method [3, 4] offers a
decomposition like (2) and potentials to describe multi-subject fMRI data will be
illustrated. On the way to optimal rank one decomposition using ICA criterion, analyses of a
three-way data seen as a stacked two-way data offers a first step forward: the Single-ICA and
Multiple-ICA methods. With an analogy of a new interpretation of the algorithm used for PTA-k
method [3] we will also present a rank-one version of the Single-ICA. Other 3-way ICA
methods can be derived fixing the independence criterion in one dimension only, or on two
dimensions (space and time) with a similar algorithm. These latter will be
illustrated with a rank one version of the Multiple-ICA.
Next: Variance criterion
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Didier Leibovici
2001-09-06