Spatio-temporal Noise Modelling

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.