GLM framework

In the basic GLM, $\mathbf{Y=XB+e}$, $\mathbf{Y}$ is the observed data, $\mathbf{X}$ is the matrix of ``regressors'' (often referred to as the design matrix) and $\mathbf{B}$ are the parameters to be estimated. The errors $\mathbf{e}$ are assumed to have a Normal distribution $N(0,\sigma^2\mathbf{V})$, where $\mathbf{V}$ is the autocorrelation matrix for the time series. There exists (Seber, 1977) a square, nonsingular matrix $\mathbf{K}$ such that $\mathbf{V=KK^T}$, and that $\mathbf{e=K\epsilon}$ where $\mathbf{\epsilon}$ are $N(0,\sigma^2\mathbf{I})$.

Now consider a GLM which incorporates temporal filtering of the data, where $\mathbf{S}$ is the square matrix that performs the temporal filtering via matrix multiplication. $\mathbf{S}$ is a Toeplitz matrix produced from the impulse response; this is directly equivalent to convolving with the impulse response using zero padding. The design matrix is also temporally filtered using $\mathbf{S}$ to reflect the known change in the observed data. We now have:

$$\mathbf{SY=SXB+\eta}$$ (1)

where $\mathbf{\eta}$ is $N(0,\sigma^2\mathbf{SVS^T})$. We use an ordinary least squares (OLS) estimate of $\mathbf{B}$, given by:

$$\mathbf{\hat{B}=(SX)^+SY}$$ (2)

where $\mathbf{(SX)^+}$ is the pseudo-inverse of $\mathbf{(SX)}$ given by $\mathbf{(SX)^+}=((SX)^TSX)^{-1}(SX)^T$. The variance of a contrast $\mathbf{c}$, of these parameter estimates, $\mathbf{\hat{B}}$, is given by:

$$Var\{\mathbf{c^T \hat{B}}\}=k_{\mbox{\scriptsize {\emph{eff}}}} \sigma^2$$

$$k_{\mbox{\scriptsize {\emph{eff}}}}=\mathbf{c^T(SX)^+SVS^T((SX)^+)^Tc}$$ (3)

Note that $k_{\mbox{\scriptsize {\emph{eff}}}}$ is a scalar that scales $\sigma^2$ by an amount that depends upon the design matrix $\mathbf{X}$, the temporal autocorrelation $\mathbf{V}$ and the contrast $\mathbf{c}$ to give the variance of the contrast of parameter estimates. For an estimate of $\sigma^2$ we use (Worsley and Friston, 1995), (Seber, 1977):

$$\hat{\sigma}^2 = \mathbf{\eta^T\eta/trace(RSVS^T)}$$ (4)

where $\mathbf{R=I-SX(SX)^+}$, the residual forming matrix, which can be used to obtain the residuals of the model fit:

$$\mathbf{r} = \mathbf{RSY}$$ (5)