Overview of Previous Work

Friston et al. (2000) suggested that current techniques for estimating the autocorrelation (Autoregressive(AR) and $1/f$ models where $f$ is the frequency) are not accurate enough to give prewhitening acceptable bias. Therefore, estimation is made more robust to inaccurate autocorrelation estimates by swamping any intrinsic autocorrelation with the known autocorrelation introduced by band pass filtering, an approach often referred to as ``colouring'' (Friston et al., 1995),  (Worsley and Friston, 1995). For this they use a Gaussian (or similar) low-pass filter matched to the haemodynamic response function (HRF) and a linear high-pass filter which aims to remove the majority of the autocorrelation due to low frequency components. Having shaped the autocorrelation, prewhitening is then not applicable, and the autocorrelation estimate is instead used to correct the variance of univariate linear model parameter estimates and the degrees of freedom used in the GLM.

Although colouring is unbiased, given an accurate autocorrelation estimate, Bullmore et al. (1996) noted the need for serially independent (whitened) residuals to obtain the Best Linear Unbiased Estimates (BLUE) of the GLM parameters. The parameter estimates are ``best'' in the sense that they are the unbiased estimates with the lowest variance. This was achieved using Pseudo-Generalised Least Squares (PGLS) - also known as the Cochrane-Orcutt transformation. The autocorrelation is estimated for the residuals from a first linear model and is then used to ``prewhiten'' the data and the design matrix, for use in a second linear model. The residuals of the second linear model should be close to white noise. Further iterations of this process are possible.

For inference, they ascertain the null distributions using randomisation; that is they randomly reorganise the order of signal intensity values in each observed time series and estimate the test statistic for each. It is important to note that such randomisation of the time series is only valid in the absence of autocorrelations, hence an accurate prewhitening step is necessary. Any randomisation approach on non-white data needs to randomise the data in such a way that the effective structure of the autocorrelation is maintained, for the null distribution to be valid.

To model the autocorrelation, Bullmore proposed an AR model of order 1 (AR(1)), which was shown to model the autocorrelations satisfactorily for the data used in their paper. Purdon and Weisskoff (1998) also suggested using an AR(1) model to do prewhitening, but they also included a white noise component. However, the main focus of their paper was to explore the effect, on the false positive rate, of not taking into account the temporal autocorrelation. For a desired false positive rate of $\alpha=0.05$ they find false positive rates as high as $\alpha=0.16$ in the uncorrected data. At $\alpha=0.02$ the situation worsens further, with uncorrected data giving $\alpha=0.095$. This is because any inaccuracies in the distribution compared with the assumed theoretical distribution are more prominent further down the tail of the distribution.

Locascio et al. (1997) used an Autoregressive Moving Average (ARMA) model (see also Chatfield (1996)) and incorporated it into an overall Contrast Autoregressive and Moving Average (CARMA) model. As well as the ARMA and modelled experimental responses, the CARMA model contains baseline, linear and quadratic terms for the removal of low frequency drift. They fit separate MA and AR models of up to order 3.

Locascio et al. (1997) also suggests that the existence of positive autocorrelation is due to carry over from one time point to the next, stemming from time intervals that are smaller than the actual temporal changes. They describe autocorrelation as the persistence of neuronal activation, cyclical events (presumably they are referring to aliased cardiac and respiratory cycles), or possibly characteristics or artefacts of the measurement process.

Zarahn et al. (1997) and Aguirre et al. (1997) observed $1/f$ noise profiles in FMRI data, and as a result attempted to use a $1/f$ noise model with three parameters to account for temporal autocorrelations. They also carried out a number of water phantom studies to establish how much, if any, of the $1/f$ noise is attributable to physiological processes. They concluded that the same $1/f$ noise was apparent in the phantoms and that therefore the noise was not of physiological origin.

They use their $1/f$ model of the intrinsic autocorrelation together with Worsley and Friston (1995)'s approach of colouring the data. Zarahn proposed to refine this by incorporating the $1/f$ model's representation of the intrinsic autocorrelation into the autocorrelation due to temporal filtering, in order to give a better estimate of the autocorrelation post-temporal filtering. Unfortunately, their attempts fail because they fit the $1/f$ model over the entire brain volume, therefore ignoring the potential for spatial non-stationarity of the noise profile.

Hu et al. (1995) concentrate on the components of the coloured noise in FMRI data that are due to physiological fluctuations. By recording respiration and cardiac cycle data at the same time as the FMRI data is acquired, some of these effects can be removed. This could help to reduce the autocorrelation and potentially improve any autocorrelation estimation subsequently employed. However, these are not the only causes of coloured noise in the data and hence whether or not respiration and cardiac cycle data is available, robust strategies for dealing with autocorrelation are still required.