Strategies for Dealing with Autocorrelation

For the moment we assume a known autocorrelation matrix $\mathbf{V=KK^T}$. We investigate three approaches to dealing with the autocorrelation in FMRI; these are:

• Colouring (e.g. Friston et al. (1995)), with $\mathbf{S=A}$ where $\mathbf{A}$ is a low-pass filter, giving:

  $$k_{\mbox{\scriptsize {\emph{eff}}}}=\mathbf{c^T(AX)^+AVA^T((AX)^+)^Tc}$$ (6)

  
  

• Variance Correction (where one corrects the statistics for autocorrelation in the data, but neither colours the data nor goes as far as prewhitening), with $\mathbf{S=I}$ giving:

  $$k_{\mbox{\scriptsize {\emph{eff}}}}=\mathbf{c^T(X)^+V((X)^+)^Tc}$$ (7)

  
  

• Prewhitening (e.g. Bullmore et al. (1996)) gives the optimal BLUE, and is obtained by setting $\mathbf{S=K^{-1}}$ giving:

  $$k_{\mbox{\scriptsize {\emph{eff}}}}=\mathbf{c^T(X^TV^{-1}X)^{-1}c}$$ (8)

  
  

For all of these approaches we need an accurate estimate of the autocorrelation matrix $\mathbf{SVS^T}$. Techniques for calculating such an estimate are discussed in the next section.