5.1.4 Data set (D)

This data set simulates a scenario where subjects differ in spatial extent of signal: subject 1 does not contain any 'activation' signal, while subjects 2 contains 'activation' signal in spatial map 2 and subject 3 contains 'activation' signal in spatial maps 2 and 3. For both subjects 2 and 3, signal is temporally modulated by time course 1. All three subjects contain 'nuisance' signal in spatial area 1, but modulated each time by a different time course (time course $2, 3$ and $2+3$). This simulates spatially consistent but temporally inconsistent effects like resting state networks [Biswal et al., 1996]

Estimated sources are shown in figure 5. The PARAFAC decomposition no longer reflects the spatial or temporal extent of signal well. Similar to data sets (A)-(C), the tensor-PICA decomposition identifies 3 source processes which are strongly correlated with the true spatial maps 1-3. The estimated spatio-temporal decomposition closely matches the way that data was generated. In the case of the main nuisance effect (contained inside spatial map 1), the tensor-PICA decomposition approximates the best rank-1 decomposition of the different time-courses involved as the time-course which best summarises the 3 different temporal signals associated with this single map.

Note that each estimated time course in a tensor-PICA decomposition is calculated from an SVD of a single column of $\mbox{\protect\boldmath$M$}^c$, reshaped into a $I\times K$ matrix. This will not only provide the single time course which best represents the $K$ individual time courses for each column in $\mbox{\protect\boldmath$M$}^c$, but also provides information about the amount of variance that this individual time course explains. For this data, the time courses for estimated source processes 2 and 3 represent $99.27\%$ and $99.57\%$ of the total variance contained in the relevant columns of $\mbox{\protect\boldmath$M$}^c$. By comparison, the time course for source process 1 only represents $54.07\%$ of the variance in the temporal domain. This indicates that the rank-1 approximation of the time-courses associated with spatial map 1 is not very descriptive of the temporal characteristics in each of the subjects.

Figure 5: Tensor-PICA and PARAFAC decomposition results for data set (D) and all three spatial maps.

(i) tensor-PICA (ii) PARAFAC