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The accuracy of estimation, both for PARAFAC and tensor-PICA, depends
on the number of processes, , estimated with each method.
All results presented above have used a value of as estimated via
the Laplace approximation to the model order for the Eigenspectrum of
the data covariance matrix
. 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 and
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 . 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
and . Different colours show the estimation accuracy
for the three spatial maps shown in figure 1.
(A) (B) (C) (D) (E) |
Next: Multi-session FMRI data
Up: Simulated data
Previous: Data set (E)
Christian Beckmann
2004-12-14