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There has been extensive work on spatially independent temporal
noise modelling. In particular, Friston et al. (12)
and Bullmore et al. (3)
introduced alternative approaches within a
null hypothesis testing general linear model (GLM) framework.
This can also be tackled in an
empirical Bayesian framework to avoid problems with
estimating the temporal autocorrelation from the residuals of the
model fit (9,10).
One of the major limitations of the estimation and
inference of these techniques, is that the standard equations from
the linear modelling statistics literature
(e.g. (40,4)) assume that the
parameters associated with the correlation structure of the noise
are known exactly. The uncertainty associated with these
parameters is not taken into account, resulting in potentially
biased statistics.
One solution to this is to consider a fully Bayesian framework.
This was the approach taken by Genovese (21), in which he took
a full Bayes approach to the modelling of FMRI time series, and
inferred using MCMC. The noise is modelled as deterministic drift
in the data using cubic splines. Gössl et al. (24) also look to
model larger scale drifts over time using a random walk model.
Spatial correlation could arise from several potential sources.
Firstly, there is point spread function due to the fact that we
are performing finite sampling in k-space. Hence, data from an
individual voxel will contain some signal from the tissue around
that voxel, an effect compounded by motion. Secondly, there is
smoothness introduced by motion correction techniques, in
particular due to interpolation. Thirdly, there may be spatially
spread physiological effects, which for our purposes would be
considered as noise -- although these spatial networks of
``activity'' in supposed rest/null data could be of real interest.
Gössl et al. (26) consider
spatial noise in the context of a deterministic spatio-temporal
trend model. However, it is yet to be determined whether it is
adequate to merely model deterministic drift terms, and then
assume that the remaining noise is white.
In this work we will take into account large scale deterministic
temporal fluctuations using
temporal high-pass filtering as a preprocessing step,
and we will model short scale statistical spatial and temporal
noise using autoregressive processes. In future work there is the
possibility to incorporate large scale deterministic fluctuation
modelling into the same model.
Subsections
Next: Scale of Variation
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