4.1 Simulated data

We acquired whole brain volumes ( $64\times 64\times 21, 4\times 4 \times 6$mm) of FMRI data on a Varian 3T system (TR=3sec; TE=30ms) under resting condition. The data were corrected for subject motion using MCFLIRT [Jenkinson et al., 2002], temporally high-pass filtered (Gaussian-weighted least-squares straight line fitting, with sigma=20.0s) [Marchini and Ripley, 2000] and masked for non-brain voxels using BET [Smith, 2002]. The pre-processed data was used to estimate background noise parameters (voxel-wise means and std. deviations) which were used to generate 3 artificial data sets with Gaussian noise characteristics. Artificial signal was linearly added to the Gaussian background noise data using spatial maps and time courses depicted in figure 1. The time courses correspond to the stimulus trains from a simple block design, a single-event (fixed inter-stimulus interval) design and a single-event (random inter-stimulus interval) convolved with a canonical hæmodynamic response function (Gamma variate with 3s standard-deviation and 6s lag).

Five different data sets ( $196 \mbox{~time points}\times 2800 \mbox{~voxels} \times 3 \mbox{~subjects}$ each) were generated as example FMRI studies with different signal characteristics:

(A)

Each subject's data $\mbox{\protect\boldmath$X$}_{..k}$ contains all three spatial maps shown in figure 1. Each spatial map has a different associated time course: time course 1 modulates spatial map 1, time course 2 modulates spatial map 2 and time course 3 modulates spatial map 3. This defines three spatio-temporal processes which are introduced at different strengths into the individual subjects' data. The 'activation' levels were set to (3,4,5),(2,3,4) and (2,2,3) times the mean noise standard deviation for subjects 1-3. The complete 3-way data conforms to the generative model of equation 1 with $R=3$ source processes in the data.

(B)

Each subject contains spatial map 1 modulated by time-course 1. In addition, subject 2 contains spatial map 2 modulated by time course 2, while subject 3 contains spatial map 3 modulated by time course 3. This data set is a special case of data set (A) with strength set to (3,0,0),(2,3,0) and (2,0,3). The data still conforms to the generative model of equation 1 and is used to demonstrate the performance of PARAFAC and tensor-PICA on data where the matric $\mathbf C$ is sparse, i.e. for data which contains subject-specific source processes in addition to a common source process.

(C)

Like data set (A), but with the individual convolution parameters for the generation of the signal time-courses differing between subjects in mean lag and standard deviation used for the Gamma HRF ($\sigma=3,3.5$ and $4$ seconds, mean lag of $4,5$ and $6$ seconds). This induces small differences in the temporal signal characteristics between subjects. This data set is used to test for robustness against small deviations from the model assumptions in the temporal domain (e.g. small differences between subjects in the BOLD response to the same set of stimuli). Note that this data set still conforms to the tri-linear model, as these different time courses together with the spatial maps can be interpreted as separate source processes (i.e. with $\mbox{\protect\boldmath$A$}$ containing 9 time courses with sets of 3 time courses being close to collinear and with $\mbox{\protect\boldmath$B$}$ containing 9 spatial maps where sets of 3 are identical). The data does not, however, conform to the tensor-PICA model, as the spatial maps are not statistically independent.

(D)

Subject 1 does not contain any 'activation' signal. Subjects 2 and 3 contain 'activation' signal in the area defined by spatial map 2, modulated by the simple block-design (time course 1). Subject 3 also contains extra 'activation' signal in the area defined by spatial map 3, modulated again by time course 1. In addition, all three subjects contain 'nuisance' signals (spatial map 1 modulated by a different time course in each subject). This data simulates cases where FMRI data is confounded by e.g. resting-state networks which are spatially consistent but differ in the temporal characteristics of the resting-state BOLD signal. The data conforms to the tri-linear model when viewed as a set of 5 spatial maps with 5 associated time courses.

(E)

Each data set contains all three spatial maps, but modulated by a different time course, i.e. the association between the spatial maps 1-3 and time courses 1-3 changes between subjects. The data conforms to the tri-linear model when viewed as a set of 9 spatial maps and 9 associated time courses. However, like data set (C) some of the spatial maps are identical and thus not statistically independent.

Figure 1: Artificial spatial maps and time-courses used for the generation of artificial group data.