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Introduction

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 $r$) in both cases can be written:
\begin{displaymath}
X=\sum_i^r B_i \otimes T_i + \varepsilon
\end{displaymath} (1)

with different statistical properties according to the chosen method. Inferences for location on the brain with components $B_i$ [1][2] and stimulus influence with associated time course component $T_i$ (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:
\begin{displaymath}
Y=\sum_i^r B_i \otimes T_i \otimes S_i+ \varepsilon
\end{displaymath} (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 up previous
Next: Variance criterion Up: tr01dl1 Previous: tr01dl1
Didier Leibovici 2001-09-06