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3.2 Data pre-processing for tensor-PICA

As in the 2-D case, the data will be voxel-wise de-trended (using Gaussian-weighted least squares straight line fitting; [Marchini and Ripley, 2000]) and de-meaned separately for each data set $k$ before the tensor-PICA decomposition. In order to compare voxel locations between subjects/sessions, the individual data sets need to be co-registered into a common space, typically defined by a high-resolution template image. We do not, however, necessarily need to re-sample to the higher resolution and can keep the data at the lower EPI resolution in order to reduce computational load. After transformation into a common space, the data is temporally normalised by the estimated voxel-wise noise covariances $\mbox{\protect\boldmath$V$}_k^{-1/2}={\rm diag}(\sigma_{1,k},\dots,\sigma_{J,k})$ using the iterative approximation of the noise covariance matrix from a standard 2-PICA decomposition. This will normalise the voxel-wise variance both within a set of voxels from a single subject/session and between subjects/sessions. The voxel-wise noise variances need to be estimated from the residuals of an initial PPCA decomposition. This, however, cannot simply be done by calculating the individual data covariance matrices $\mbox{\protect\boldmath$R$}_{..k}\propto
\mbox{\protect\boldmath$X$}_{..k}\mbox{\protect\boldmath$X$}_{..k}^{\mbox{\scriptsize\textit{\sffamily {t}}}}$.

Within the tensor-PICA framework, the temporal modes (contained in $\mbox{\protect\boldmath$A$}$) are assumed to describe the temporal characteristics of a given process $r$ for all data sets $k$. We will therefore estimate the initial temporal Eigenbasis from $\mbox{\protect\boldmath$R$}=\frac{1}{K}\sum_k\mbox{\protect\boldmath$R$}_{..k}$, i.e. by the mean data covariance matrix. This corresponds to a PPCA analysis of $\mbox{\protect\boldmath$X$}_{I\times
JK}$, i.e. the original data reshaped into a $(\mbox{\char93  time point})\mbox{\,by\,}
(\mbox{\char93  voxels})\times(\mbox{\char93  subjects/sessions})$ matrix. We use the Laplace approximation to the model order [Minka, 2000] to infer on the number of source processes, $R$. The projection of the data sets onto the matrix $\mbox{\protect\boldmath$U$}_R$ (formed by the first $R$ common Eigenvectors of $\mbox{\protect\boldmath$R$}$) reduces the dimensionality of the data in the temporal domain in order to avoid overfitting. This projection is identical for all $K$ different data sets, given that $\mbox{\protect\boldmath$U$}_R$ is estimated from the mean sample covariance matrix $\mbox{\protect\boldmath$R$}$. Therefore, we can recover the original time-courses by projection onto $\mbox{\protect\boldmath$U$}_R$: when the original data $\mbox{\protect\boldmath$X$}_{IK\times J}$ is transformed into a new set of data $\widetilde{\mbox{\protect\boldmath$X$}}_{RK\times J}$ by projecting each $\mbox{\protect\boldmath$X$}_{..k}$ onto $\mbox{\protect\boldmath$U$}_R$, the original data can be recovered from

\begin{displaymath}
\mbox{\protect\boldmath$X$}_{IK\times
J}=(\mbox{\protect\bol...
...th$U$}_R)\widetilde{\mbox{\protect\boldmath$X$}}_{RK\times J},
\end{displaymath}

where $\mbox{\protect\boldmath$I$}_R$ denotes the identity matrix of rank $R$. If the new data $\widetilde{\mbox{\protect\boldmath$X$}}$ is decomposed such that $\widetilde{\mbox{\protect\boldmath$X$}}_{RK\times J}=\left(\mbox{\protect\boldm...
...protect\boldmath$B$}^{\mbox{\scriptsize\textit{\sffamily {t}}}}+\widetilde{E},
$ then $\mbox{\protect\boldmath$X$}_{IK\times J}=\left(\mbox{\protect\boldmath$C$}\vert...
...protect\boldmath$B$}^{\mbox{\scriptsize\textit{\sffamily {t}}}}+\widetilde{E},
$ where $\mbox{\protect\boldmath$A$}=\mbox{\protect\boldmath$U$}_R\widetilde{\mbox{\protect\boldmath$A$}}$. This approach is different from e.g. [Calhoun et al., 2001], where an individual data set $k$ is projected onto a set of Eigenvectors of the data covariance matrix $\mbox{\protect\boldmath$R$}_{..k}$. As a consequence, each data set in [Calhoun et al., 2001] has a different signal+noise subspace compared to the other data sets.

Similar to the 2-D PICA model, the set of pre-processing steps is iterated in order to obtain estimates for the voxel-wise noise variance $\mbox{\protect\boldmath$V$}$, the PPCA Eigenbasis $\mbox{\protect\boldmath$U$}$ and the model order $R$ before decomposing the reduced data $\widetilde{\mbox{\protect\boldmath$X$}}_{RK\times J}$ into the factor matrices $\widetilde{\mbox{\protect\boldmath$A$}}, \mbox{\protect\boldmath$B$}$ and $\mbox{\protect\boldmath$C$}$ (see [Beckmann and Smith, 2004] for details).


next up previous
Next: Group-level inference Up: Tensor PICA Previous: Relation to mixed-effects GLMs
Christian Beckmann 2004-12-14