5.1.6 Accuracy and dimensionality

The accuracy of estimation, both for PARAFAC and tensor-PICA, depends on the number of processes, $R$, estimated with each method. All results presented above have used a value of $R$ as estimated via the Laplace approximation to the model order for the Eigenspectrum of the data covariance matrix $\mbox{\protect\boldmath$R$}_{I\times JK}$. Figure 7 compares the accuracy of estimation for both PARAFAC (P) and tensor-PICA (T) on all 5 data sets (A)-(E) for different values of $R=3,10$ and $40$ in the spatial, temporal and subject domain. Circles denote the source process with highest absolute correlation with one of the three true spatial maps while dots show the correlation of the remaining sources. In almost all cases, the source process with highest spatial correlation also has largest temporal correlation with the associated true time course (or to the best rank-1 approximation)⁷. For data sets (A)-(C), i.e. when signals conform to the generative model of equation 1, the correlations in the spatial and temporal domain between true sources and estimates from tensor-PICA are very high and always clearly identify a single process (i.e. for each 'true' spatial map, one of the estimated spatial maps has high spatial correlations while at the same time all other estimated spatial modes have low spatial correlation). Furthermore, the estimation is relatively robust when estimating a different number of sources. This is of prime importance, since the exact number of source processes is not known a-priori and the Laplace approximation is not expected to always give very accurate results (see [Beckmann and Smith, 2004] for a detailed discussion). The PARAFAC estimates, by comparison, exhibit a stronger dependence on the number of estimated sources $R$. As the number of estimated sources increases, a larger number of source processes show 'spurious' correlations with the true spatial maps. In the case of data sets (D) and (E), the PARAFAC results are significantly worse compared to the tensor-PICA results and do not identify the source processes in any domain. These simulations suggests that tensor-PICA is less sensitive to the model order as well as deviations of the signal content in the data from the generative three-way model.

Figure 7: Accuracy of signal estimation for PARAFAC and tensor-PICA on the artificial FMRI group data (A)-(E). Plots show the correlation between 'true' (or best rank-1) modes and estimated modes in the spatial, temporal and subject domain (top to bottom rows respectively) for PARAFAC (P) and tensor-PICA (T). For each method and data set, the analysis was performed for $R=3,10$ and $40$. Different colours show the estimation accuracy for the three spatial maps shown in figure 1.

        (A)               (B)               (C)               (D)               (E)