In data set (C), different parameters for the HRF convolutions resulted in small differences in the time courses. As a consequence, each of the three spatial maps has a slightly different associated time-course in each subject. As such, there is no longer a single 'true' associated time course across subjects. The induced variation in the temporal domain, however, is small (time courses have temporal correlation) and the different time courses are well approximated by their dominant Eigenvector, i.e. by the best rank-1 approximation. The data permits two different representations: firstly, the signal content can either be approximated as a linear combination of 3 processes (where in the temporal domain the rank-1 approximation to the 3 slightly different time courses is used), or can fully be expressed as a linear combination of 9 processes with large co-linearity in and 3 multiple versions for each of the true spatial sources in .
Figure 4 shows the estimated set of source processes for tensor-PICA and PARAFAC. The tensor-PICA decomposition, due to the independence assumption in the spatial domain, represents the data via a set of 3 source processes. Compared to the tensor-PICA results, the PARAFAC spatial estimates exhibit some cross-talk, e.g. the first spatial map is visibly confounded by map 2 and map 3. Also, in the temporal- and subject domains, PARAFAC finds less accurate estimates of the true source processes. The two approaches differ most significantly in the way in which true spatial maps correlate with each of the estimated maps: while the spatial tensor-PICA decomposition always results in only one source which correlates strongly with the true spatial map, the PARAFAC decomposition shows that, especially for sources 2 and 3, multiple PARAFAC estimates correlate with the true maps. As such, PARAFAC does not represent the signal of interest via 3 different source processes but equally does not find the representation by 9 sources: almost all of the estimated correlated maps show significant amount of cross-talk. Convergence in this case is particularly slow, with 47 times the number of floating point operations compared to tensor-PICA.
(i) tensor-PICA (ii) PARAFAC |