Normalisation

Let an unnormalised voxel time series be denoted by $R_t(\ensuremath{\mathbf{x}})$ where $t = 1,\ldots,N_T$ is the time index. Treating this as a time-vector (that is, an $N_T$ by 1 matrix), SPM performs a ``normalisation'':

$$S_t(\ensuremath{\mathbf{x}}) = \frac{R_t(\ensuremath{\mathbf... ...T} R_t(\ensuremath{\mathbf{x}}) R_t(\ensuremath{\mathbf{x}})}}$$ (37)

or, by suppressing indices and using matrix notation, as $S = R / \sqrt{R^\top R}$.

This ``normalisation'' results in the expected sum of the residuals squared being unity. Therefore, the expected value for any particular residual squared is actually:

$$E\{ ( S_t(\ensuremath{\mathbf{x}}) \, )^2 \} = \frac{1}{N_T}.$$ (38)

Consequently, the factor $k$, introduced previously needs to be divided by $\sqrt{N_T}$. This also means that, when taking the sum over the possible time points, it is no longer necessary to normalise the sum. Therefore, the full 4D average for $\lambda$, assuming no temporal correlation, is:

$$\lambda_{11} = \frac{1}{N} \sum_{\ensuremath{\mathbf{x}}} \s... ...{\partial S_t(\ensuremath{\mathbf{x}})}{\partial x} \right)^2.$$ (39)