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# Variance criterion

For the variance criterion (PCA) the PTA-k method [3, 4] offers decompositions like (2). The optimisation scheme was derived from searching for singular values:
 (3)

where , , are the Hilbert spaces of finite dimensions (embedding of components in functional Hilbert spaces is also possible), the operation is the contraction of the the tensors considered, here equivalent to inner product in the tensor space. To search for the second singular values and Principal Tensor associated an orthogonality constraint is added [4].

Table 1: PTA-modes listing of the decomposition up to the third modes and associated solutions.
  ++++ PTA- 3 modes ++++ data= J12.gm16 576 100 12 just slice 16 ------Percent Rebuilt---- 23.34021% ------Percent Rebuilt from Selected ---- 17.01323% -no- --Sing Val-- --ssX-- --local Pct-- --Global Pct-- vs111 1 191.376 599772 6.1064 6.1064 576 vs111 100 12 3 85.062 70189 10.3088 1.2064 100 vs111 576 12 6 97.382 81266 11.6694 1.5812 100 vs111 576 12 7 87.138 81266 9.3434 1.2660 12 vs111 576 100 9 87.616 80815 9.4988 1.2799 vs222 11 133.299 440752 4.0314 2.9625 vs333 21 97.991 370027 2.5950 1.6010 12 vs333 576 100 29 77.825 42284 14.3237 1.0098 ++++ ++++ Shown are selected over 21 PT with var> 1% total 

An illustration of the output of the method using the PTAk package [4] follows. The table (1) gives the beginning of the decomposition (pruned). The first Principal Tensor shown on figure (2) is well correlated (the time component) to the paradigm of the experiment. Each component can be tested (with appropriate tests) separately to confirm BOLD activation at a population level, i.e. the correlation is significant showing some activation, related globally to the population (with no apparent differences due to sex), and high values1 locates activation on the left fronto-temporal of the brain (right on the image). The second Principal Tensor (figure 2) shows a motion artifact of one particular subject.

In fact the criterion of singular values (3) can be written in variance criterion form reminiscent of the optimisation for eigenvalues of a covariance" matrix:
 (4)

where here is a random matrix in from which observations (here voxels) were collected in . This random version is interesting in giving some differnets ways of implementing an algorithm and in describing the PTA-modes method in a statistical framework rather than in algebra. By analogy generalisation in term of a different optimisation index based on distribution of the component rather than only on the variance of can be derived.

Next: Negentropy criterion Up: tr01dl1 Previous: Introduction
Didier Leibovici 2001-09-06