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:

$$\sigma_1 = \max_{{\scriptstyle \begin{array}{l } \scriptstyle \left\Vert \ps... ...Vert \phi \right\Vert _G =1 \end{array}}}Y..(\psi \otimes \varphi \otimes \phi)$$

$$= Y..(B_1 \otimes T_1 \otimes S_1).$$ (3)

where $E$, $F$, $G$ 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-$3$modes listing of the decomposition up to the third $k$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.

Figure 1: PTA3-modes of the detrended time series and subject scaled of a 12 subjects fMRI data: activation

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:

$$\sigma_1^2 = \max_{{\scriptstyle \begin{array}{l } \scriptstyle \left\Vert \va... ...\left\Vert \phi \right\Vert _G =1 \end{array}}}E([Y'..(\varphi \otimes \phi)]2)$$

$$= \max_{{\scriptstyle \begin{array}{l } \scriptstyle \left\Vert \va... ...ray}}}E([Y'\otimes Y'])..([\varphi \otimes \phi]\otimes [\varphi \otimes \phi])$$

$$= \max_{{\scriptstyle \begin{array}{l } \scriptstyle \left\Vert \va... ... \end{array}}}E([Y'..\phi_1] \otimes [Y'..\phi_1]) .. (\varphi \otimes \varphi)$$

$$= \max_{{\scriptstyle \begin{array}{l } \scriptstyle \left\Vert \ph... ... \end{array}}}E([Y'..\varphi_1] \otimes [Y'..\varphi_1]) .. (\phi \otimes \phi)$$

$$= E(Y'\otimes Y')..(\varphi_1 \otimes \phi_1)$$ (4)

where here $Y'$ is a random matrix in $F\otimes G$ from which $N$ observations (here voxels) were collected in $Y$. This random version is interesting in giving some differnets ways of implementing an algorithm and in describing the PTA-$3$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.

Figure 2: PTA3-modes of the detrended time series and subject scaled (whole sequence masked brain )of a 12 subjects fMRI data: motion artefact